From 7ddb9a2f897667589b586a33ea7c4883697b8ea0 Mon Sep 17 00:00:00 2001 From: PascalEgn Date: Tue, 16 Jul 2024 12:51:10 +0200 Subject: [PATCH] add hepcrawl parsers --- .github/workflows/test-python-2.yml | 10 +- .gitignore | 2 +- README.rst | 2 +- inspire_utils/__init__.py | 2 +- inspire_utils/date.py | 2 +- inspire_utils/dedupers.py | 2 +- inspire_utils/helpers.py | 2 +- inspire_utils/logging.py | 2 +- inspire_utils/name.py | 2 +- inspire_utils/parsers/__init__.py | 25 + inspire_utils/parsers/arxiv.py | 414 ++++ inspire_utils/parsers/crossref.py | 368 +++ inspire_utils/parsers/elsevier.py | 681 ++++++ inspire_utils/parsers/jats.py | 656 +++++ inspire_utils/query.py | 2 +- inspire_utils/record.py | 2 +- inspire_utils/utils.py | 139 ++ run-tests.sh | 2 +- setup.cfg | 2 +- setup.py | 16 +- tests/data/aps/PhysRevD.102.014505.xml | 3 + .../data/aps/PhysRevD.102.014505_expected.yml | 1962 +++++++++++++++ tests/data/aps/PhysRevD.96.095036.xml | 3 + .../data/aps/PhysRevD.96.095036_expected.yml | 2103 +++++++++++++++++ .../aps/PhysRevD.96.095036_no_date_nodes.xml | 3 + tests/data/aps/PhysRevResearch.5.033093.xml | 311 +++ tests/data/aps/PhysRevX.4.021018.xml | 3 + tests/data/aps/PhysRevX.4.021018_expected.yml | 1322 +++++++++++ tests/data/aps/PhysRevX.7.021021.xml | 562 +++++ tests/data/aps/PhysRevX.7.021021_expected.yml | 1208 ++++++++++ tests/data/aps/PhysRevX.7.021022.xml | 562 +++++ tests/data/aps/PhysRevX.7.021022_expected.yml | 1208 ++++++++++ tests/data/aps/Physics.15.168.xml | 141 ++ tests/data/crossref/2018.3804742.json | 312 +++ tests/data/crossref/2018.3804742_expected.yml | 110 + tests/data/crossref/9781316535783.011.json | 127 + .../crossref/9781316535783.011_expected.yml | 14 + tests/data/crossref/PhysRevB.33.3547.2.json | 159 ++ .../crossref/PhysRevB.33.3547.2_expected.yml | 21 + .../data/crossref/s1463-4988(99)00060-3.json | 1 + .../s1463-4988(99)00060-3_expected.yml | 265 +++ .../data/crossref/sample_crossref_record.json | 728 ++++++ ...ple_crossref_record_with_unknown_type.json | 728 ++++++ tests/data/crossref/tasc.2017.2776938.json | 239 ++ .../crossref/tasc.2017.2776938_expected.yml | 42 + tests/data/elsevier/aphy.2001.6176.xml | 1 + .../data/elsevier/aphy.2001.6176_expected.yml | 329 +++ tests/data/elsevier/j.aim.2021.107831.xml | 1 + .../elsevier/j.aim.2021.107831_expected.yml | 699 ++++++ tests/data/elsevier/j.cpc.2020.107740.xml | 1 + .../elsevier/j.cpc.2020.107740_expected.yml | 302 +++ tests/data/elsevier/j.nima.2019.162728.xml | 1 + .../elsevier/j.nima.2019.162728_expected.yml | 262 ++ tests/data/elsevier/j.nima.2019.162787.xml | 1 + .../elsevier/j.nima.2019.162787_expected.yml | 207 ++ tests/data/elsevier/j.nima.2023.168018.xml | 1 + tests/data/elsevier/j.nimb.2019.04.063.xml | 1 + .../elsevier/j.nimb.2019.04.063_expected.yml | 697 ++++++ .../data/elsevier/j.nuclphysa.2020.121991.xml | 1 + .../j.nuclphysa.2020.121991_expected.yml | 1180 +++++++++ .../data/elsevier/j.nuclphysa.2020.121992.xml | 84 + .../j.nuclphysa.2020.121992_expected.yml | 1180 +++++++++ tests/data/elsevier/j.scib.2020.01.008.xml | 1 + .../elsevier/j.scib.2020.01.008_expected.yml | 1298 ++++++++++ .../record-that-shouldnt-be-harvested.xml | 1 + tests/data/elsevier/sample_consyn_record.xml | 493 ++++ tests/fixtures.py | 38 + tests/test_date.py | 2 +- tests/test_dedupers.py | 2 +- tests/test_helpers.py | 2 +- tests/test_name.py | 2 +- tests/test_parsers_arxiv.py | 85 + tests/test_parsers_crossref.py | 130 + tests/test_parsers_elsevier.py | 176 ++ tests/test_parsers_jats.py | 158 ++ tests/test_record.py | 2 +- 76 files changed, 21780 insertions(+), 28 deletions(-) create mode 100644 inspire_utils/parsers/__init__.py create mode 100644 inspire_utils/parsers/arxiv.py create mode 100644 inspire_utils/parsers/crossref.py create mode 100644 inspire_utils/parsers/elsevier.py create mode 100644 inspire_utils/parsers/jats.py create mode 100644 inspire_utils/utils.py create mode 100644 tests/data/aps/PhysRevD.102.014505.xml create mode 100644 tests/data/aps/PhysRevD.102.014505_expected.yml create mode 100644 tests/data/aps/PhysRevD.96.095036.xml create mode 100644 tests/data/aps/PhysRevD.96.095036_expected.yml create mode 100644 tests/data/aps/PhysRevD.96.095036_no_date_nodes.xml create mode 100644 tests/data/aps/PhysRevResearch.5.033093.xml create mode 100644 tests/data/aps/PhysRevX.4.021018.xml create mode 100644 tests/data/aps/PhysRevX.4.021018_expected.yml create mode 100644 tests/data/aps/PhysRevX.7.021021.xml create mode 100644 tests/data/aps/PhysRevX.7.021021_expected.yml create mode 100644 tests/data/aps/PhysRevX.7.021022.xml create mode 100644 tests/data/aps/PhysRevX.7.021022_expected.yml create mode 100644 tests/data/aps/Physics.15.168.xml create mode 100644 tests/data/crossref/2018.3804742.json create mode 100644 tests/data/crossref/2018.3804742_expected.yml create mode 100644 tests/data/crossref/9781316535783.011.json create mode 100644 tests/data/crossref/9781316535783.011_expected.yml create mode 100644 tests/data/crossref/PhysRevB.33.3547.2.json create mode 100644 tests/data/crossref/PhysRevB.33.3547.2_expected.yml create mode 100644 tests/data/crossref/s1463-4988(99)00060-3.json create mode 100644 tests/data/crossref/s1463-4988(99)00060-3_expected.yml create mode 100644 tests/data/crossref/sample_crossref_record.json create mode 100644 tests/data/crossref/sample_crossref_record_with_unknown_type.json create mode 100644 tests/data/crossref/tasc.2017.2776938.json create mode 100644 tests/data/crossref/tasc.2017.2776938_expected.yml create mode 100644 tests/data/elsevier/aphy.2001.6176.xml create mode 100644 tests/data/elsevier/aphy.2001.6176_expected.yml create mode 100644 tests/data/elsevier/j.aim.2021.107831.xml create mode 100644 tests/data/elsevier/j.aim.2021.107831_expected.yml create mode 100644 tests/data/elsevier/j.cpc.2020.107740.xml create mode 100644 tests/data/elsevier/j.cpc.2020.107740_expected.yml create mode 100644 tests/data/elsevier/j.nima.2019.162728.xml create mode 100644 tests/data/elsevier/j.nima.2019.162728_expected.yml create mode 100644 tests/data/elsevier/j.nima.2019.162787.xml create mode 100644 tests/data/elsevier/j.nima.2019.162787_expected.yml create mode 100644 tests/data/elsevier/j.nima.2023.168018.xml create mode 100644 tests/data/elsevier/j.nimb.2019.04.063.xml create mode 100644 tests/data/elsevier/j.nimb.2019.04.063_expected.yml create mode 100644 tests/data/elsevier/j.nuclphysa.2020.121991.xml create mode 100644 tests/data/elsevier/j.nuclphysa.2020.121991_expected.yml create mode 100644 tests/data/elsevier/j.nuclphysa.2020.121992.xml create mode 100644 tests/data/elsevier/j.nuclphysa.2020.121992_expected.yml create mode 100644 tests/data/elsevier/j.scib.2020.01.008.xml create mode 100644 tests/data/elsevier/j.scib.2020.01.008_expected.yml create mode 100644 tests/data/elsevier/record-that-shouldnt-be-harvested.xml create mode 100644 tests/data/elsevier/sample_consyn_record.xml create mode 100644 tests/fixtures.py create mode 100644 tests/test_parsers_arxiv.py create mode 100644 tests/test_parsers_crossref.py create mode 100644 tests/test_parsers_elsevier.py create mode 100644 tests/test_parsers_jats.py diff --git a/.github/workflows/test-python-2.yml b/.github/workflows/test-python-2.yml index 77a5f1f..b8bc8a5 100644 --- a/.github/workflows/test-python-2.yml +++ b/.github/workflows/test-python-2.yml @@ -10,7 +10,7 @@ jobs: matrix: include: - python: python2 - pip: pip + pip: pip2 steps: - name: Checkout uses: actions/checkout@v2 @@ -20,9 +20,11 @@ jobs: - name: Install python dependencies run: | - ${{ matrix.pip }} install --user --upgrade pip - ${{ matrix.pip }} --no-cache-dir install --user setuptools wheel "urllib3==1.25.11" - ${{ matrix.pip }} --no-cache-dir install --user -e .[tests,elasticsearch5] + wget https://bootstrap.pypa.io/pip/2.7/get-pip.py + ${{ matrix.python }} get-pip.py + ${{ matrix.python }} -m ${{ matrix.pip }} install --user --upgrade pip + ${{ matrix.python }} -m ${{ matrix.pip }} --no-cache-dir install --user setuptools wheel "urllib3==1.25.11" + ${{ matrix.python }} -m ${{ matrix.pip }} --no-cache-dir install --user -e .[tests,elasticsearch5] - name: Show python dependencies run: | diff --git a/.gitignore b/.gitignore index 9118788..381c047 100644 --- a/.gitignore +++ b/.gitignore @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/README.rst b/README.rst index 4acc282..8189cd7 100644 --- a/README.rst +++ b/README.rst @@ -1,6 +1,6 @@ .. This file is part of INSPIRE. - Copyright (C) 2014-2017 CERN. + Copyright (C) 2014-2024 CERN. INSPIRE is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/__init__.py b/inspire_utils/__init__.py index b63c098..3356f1c 100644 --- a/inspire_utils/__init__.py +++ b/inspire_utils/__init__.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/date.py b/inspire_utils/date.py index 0b9bb41..4216579 100644 --- a/inspire_utils/date.py +++ b/inspire_utils/date.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/dedupers.py b/inspire_utils/dedupers.py index e093823..d15edbb 100644 --- a/inspire_utils/dedupers.py +++ b/inspire_utils/dedupers.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/helpers.py b/inspire_utils/helpers.py index 2dd8629..410b837 100644 --- a/inspire_utils/helpers.py +++ b/inspire_utils/helpers.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/logging.py b/inspire_utils/logging.py index 09cd4a7..da252ae 100644 --- a/inspire_utils/logging.py +++ b/inspire_utils/logging.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/name.py b/inspire_utils/name.py index 1a3e61c..6b36f1c 100644 --- a/inspire_utils/name.py +++ b/inspire_utils/name.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/parsers/__init__.py b/inspire_utils/parsers/__init__.py new file mode 100644 index 0000000..855e3d5 --- /dev/null +++ b/inspire_utils/parsers/__init__.py @@ -0,0 +1,25 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +"""Parsers for various metadata formats""" + +from __future__ import absolute_import, division, print_function diff --git a/inspire_utils/parsers/arxiv.py b/inspire_utils/parsers/arxiv.py new file mode 100644 index 0000000..57c033c --- /dev/null +++ b/inspire_utils/parsers/arxiv.py @@ -0,0 +1,414 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. +"""Parser for the arXiv metadata format""" + +from __future__ import absolute_import, division, print_function + +from itertools import chain +import re + +import six +from inspire_schemas.api import LiteratureBuilder +from inspire_schemas.utils import classify_field, normalize_arxiv_category +from ..dedupers import dedupe_list +from ..helpers import maybe_int +from pylatexenc.latex2text import ( + EnvironmentTextSpec, + LatexNodes2Text, + MacroTextSpec, + get_default_latex_context_db, +) + +from ..utils import CONFERENCE_WORDS, THESIS_WORDS, coll_cleanforthe, get_node, split_fullname + +RE_CONFERENCE = re.compile( + r'\b(%s)\b' % '|'.join( + [re.escape(word) for word in CONFERENCE_WORDS] + ), + re.I | re.U, +) +RE_THESIS = re.compile( + r'\b(%s)\b' % '|'.join( + [re.escape(word) for word in THESIS_WORDS] + ), + re.I | re.U, +) +RE_PAGES = re.compile(r'(?i)(\d+)\s*pages?\b') + +RE_DOIS = re.compile(r'[,;\s]+(?=\s*10[.]\d{4,})') + + +def _handle_sqrt(node, l2tobj): + arg = l2tobj.nodelist_to_text(node.nodeargd.argnlist) + format_str = u"\u221a{}" if arg.startswith("(") else u"\u221a({})" + return format_str.format(arg) + + +def get_arxiv_latex_context_db(): + default_db = get_default_latex_context_db() + arxiv_db = default_db.filter_context(keep_categories=["latex-base", "advanced-symbols"]) + arxiv_db.add_context_category( + "overrides", + prepend=True, + macros=[ + MacroTextSpec("sqrt", _handle_sqrt) + ] + ) + + # adapted from https://github.com/phfaist/pylatexenc/issues/32 + arxiv_db.set_unknown_macro_spec(MacroTextSpec("", lambda node: node.latex_verbatim())) + arxiv_db.set_unknown_environment_spec(EnvironmentTextSpec("", lambda node: node.latex_verbatim())) + + return arxiv_db + + +class ArxivParser(object): + """Parser for the arXiv format. + + It can be used directly by invoking the :func:`ArxivParser.parse` method, or be + subclassed to customize its behavior. + + Args: + arxiv_record (Union[str, scrapy.selector.Selector]): the record in arXiv format to parse. + source (Optional[str]): if provided, sets the ``source`` everywhere in + the record. Otherwise, the source is extracted from the arXiv metadata. + """ + _l2t = LatexNodes2Text( + latex_context=get_arxiv_latex_context_db(), + math_mode="verbatim", + strict_latex_spaces="based-on-source", + keep_comments=True, + keep_braced_groups=True, + keep_braced_groups_minlen=2, + ) + + def __init__(self, arxiv_record, source=None): + self.root = self.get_root_node(arxiv_record) + if not source: + source = 'arXiv' + self.builder = LiteratureBuilder(source) + + def parse(self): + """Extract an arXiv record into an Inspire HEP record. + + Returns: + dict: the same record in the Inspire Literature schema. + """ + self.builder.add_abstract(abstract=self.abstract, source=self.source) + self.builder.add_title(title=self.title, source=self.source) + for license in self.licenses: + self.builder.add_license(**license) + for author in self.authors: + self.builder.add_author(author) + self.builder.add_number_of_pages(self.number_of_pages) + self.builder.add_publication_info(**self.publication_info) + for collab in self.collaborations: + self.builder.add_collaboration(collab) + for doi in self.dois: + self.builder.add_doi(**doi) + self.builder.add_preprint_date(self.preprint_date) + if self.public_note: + self.builder.add_public_note(self.public_note, self.source) + for rep_number in self.report_numbers: + self.builder.add_report_number(rep_number, self.source) + self.builder.add_arxiv_eprint(self.arxiv_eprint, self.arxiv_categories) + self.builder.add_private_note(self.private_note) + self.builder.add_document_type(self.document_type) + normalized_categories = [classify_field(arxiv_cat) + for arxiv_cat in self.arxiv_categories] + self.builder.add_inspire_categories(dedupe_list(normalized_categories), 'arxiv') + + return self.builder.record + + def _get_authors_and_collaborations(self, node): + """Parse authors, affiliations and collaborations from the record node. + + Heuristics are used to detect collaborations. In case those are not + reliable, a warning is returned for manual checking. + + Args: + node (Selector): a selector on a record + Returns: + tuple: a tuple of (authors, collaborations, warning) + """ + author_selectors = node.xpath('.//authors//author') + + # take 'for the' out of the general phrases and dont use it in + # affiliations + collab_phrases = [ + 'consortium', ' collab ', 'collaboration', ' team', 'group', + ' on behalf of ', ' representing ', + ] + inst_phrases = ['institute', 'university', 'department', 'center'] + + authors = [] + collaborations = [] + warning_tags = [] + some_affiliation_contains_collaboration = False + + authors_and_affiliations = ( + self._get_author_names_and_affiliations(author) for author in author_selectors + ) + next_author_and_affiliations = ( + self._get_author_names_and_affiliations(author) for author in author_selectors + ) + next(next_author_and_affiliations) + + for (forenames, keyname, affiliations), (next_forenames, next_keyname, _) in six.moves.zip_longest( + authors_and_affiliations, next_author_and_affiliations, + fillvalue=('end of author-list', '', None) + ): + + name_string = " %s %s " % (forenames, keyname) + + # collaborations in affiliation field? Cautious with 'for the' in + # Inst names + affiliations_with_collaborations = [] + affiliations_without_collaborations = [] + for aff in affiliations: + affiliation_contains_collaboration = any( + phrase in aff.lower() for phrase in collab_phrases + ) and not any( + phrase in aff.lower() for phrase in inst_phrases + ) + if affiliation_contains_collaboration: + affiliations_with_collaborations.append(aff) + some_affiliation_contains_collaboration = True + else: + affiliations_without_collaborations.append(aff) + for aff in affiliations_with_collaborations: + coll, author_name = coll_cleanforthe(aff) + if coll and coll not in collaborations: + collaborations.append(coll) + + # Check if name is a collaboration, else append to authors + collaboration_in_name = ' for the ' in name_string.lower() or any( + phrase in name_string.lower() for phrase in collab_phrases + ) + if collaboration_in_name: + coll, author_name = coll_cleanforthe(name_string) + if author_name: + surname, given_names = split_fullname(author_name) + authors.append({ + 'full_name': surname + ', ' + given_names, + 'surname': surname, + 'given_names': given_names, + 'affiliations': [], + }) + if coll and coll not in collaborations: + collaborations.append(coll) + elif name_string.strip() == ':': + # DANGERZONE : this might not be correct - add a warning for the cataloger + warning_tags.append(' %s %s ' % (next_forenames, next_keyname)) + if not some_affiliation_contains_collaboration: + # everything up to now seems to be collaboration info + for author_info in authors: + name_string = " %s %s " % \ + (author_info['given_names'], author_info['surname']) + coll, author_name = coll_cleanforthe(name_string) + if coll and coll not in collaborations: + collaborations.append(coll) + authors = [] + else: + authors.append({ + 'full_name': keyname + ', ' + forenames, + 'surname': keyname, + 'given_names': forenames, + 'affiliations': affiliations_without_collaborations + }) + if warning_tags: + warning = 'WARNING: Colon in authors before %s: Check author list for collaboration names!' % ', '.join(warning_tags) + else: + warning = '' + return authors, collaborations, warning + + @staticmethod + def _get_author_names_and_affiliations(author_node): + forenames = u' '.join( + author_node.xpath('.//forenames//text()').extract() + ) + keyname = u' '.join(author_node.xpath('.//keyname//text()').extract()) + affiliations = author_node.xpath('.//affiliation//text()').extract() + + return forenames, keyname, affiliations + + @property + def preprint_date(self): + preprint_date = self.root.xpath('.//created/text()').extract_first() + + return preprint_date + + @property + def abstract(self): + abstract = self.root.xpath('.//abstract/text()').extract_first() + long_text_fixed = self.fix_long_text(abstract) + return self.latex_to_unicode(long_text_fixed) + + @property + def authors(self): + authors, _, _ = self.authors_and_collaborations + parsed_authors = [self.builder.make_author( + full_name=auth["full_name"], raw_affiliations=auth["affiliations"]) for auth in authors] + + return parsed_authors + + @property + def collaborations(self): + _, collaborations, _ = self.authors_and_collaborations + + return collaborations + + @property + def dois(self): + doi_values = self.root.xpath('.//doi/text()').extract() + doi_values_splitted = chain.from_iterable([re.split(RE_DOIS, doi) for doi in doi_values]) + dois = [ + {'doi': value, 'material': 'publication'} for value in doi_values_splitted + ] + + return dois + + @property + def licenses(self): + licenses = self.root.xpath('.//license/text()').extract() + return [{'url': license, 'material': self.material} for license in licenses] + + @property + def material(self): + return 'preprint' + + @property + def number_of_pages(self): + comments = '; '.join(self.root.xpath('.//comments/text()').extract()) + + found_pages = RE_PAGES.search(comments) + if found_pages: + pages = found_pages.group(1) + return maybe_int(pages) + + return None + + @property + def publication_info(self): + publication_info = { + 'material': 'publication', + 'pubinfo_freetext': self.pubinfo_freetext, + } + + return publication_info + + @property + def pubinfo_freetext(self): + return self.root.xpath('.//journal-ref/text()').extract_first() + + @property + def title(self): + long_text_fixed = self.fix_long_text(self.root.xpath('.//title/text()').extract_first()) + return self.latex_to_unicode(long_text_fixed) + + @staticmethod + def fix_long_text(text): + return re.sub(r'\s+', ' ', text).strip() + + @staticmethod + def get_root_node(arxiv_record): + """Get a selector on the root ``article`` node of the record. + + This can be overridden in case some preprocessing needs to be done on + the XML. + + Args: + arxiv_record(Union[str, scrapy.selector.Selector]): the record in arXiv format. + + Returns: + scrapy.selector.Selector: a selector on the root ``
`` + node. + """ + if isinstance(arxiv_record, six.string_types): + root = get_node(arxiv_record) + else: + root = arxiv_record + root.remove_namespaces() + + return root + + @property + def public_note(self): + comments = '; '.join(self.root.xpath('.//comments/text()').extract()) + + return self.latex_to_unicode(comments) + + @property + def private_note(self): + _, _, warning = self.authors_and_collaborations + + return warning + + @property + def report_numbers(self): + report_numbers = self.root.xpath('.//report-no/text()').extract() + rns = [] + for rn in report_numbers: + rns.extend(rn.split(', ')) + + return rns + + @property + def arxiv_eprint(self): + return self.root.xpath('.//id/text()').extract_first() + + @property + def arxiv_categories(self): + categories = self.root.xpath('.//categories/text()').extract_first(default='[]') + categories = categories.split() + categories_without_old = [normalize_arxiv_category(arxiv_cat) for arxiv_cat in categories] + + return dedupe_list(categories_without_old) + + @property + def document_type(self): + comments = '; '.join(self.root.xpath('.//comments/text()').extract()) + + doctype = 'article' + if RE_THESIS.search(comments): + doctype = 'thesis' + elif RE_CONFERENCE.search(comments): + doctype = 'conference paper' + + return doctype + + @property + def source(self): + return 'arXiv' + + @property + def authors_and_collaborations(self): + if not hasattr(self, '_authors_and_collaborations'): + self._authors_and_collaborations = self._get_authors_and_collaborations(self.root) + return self._authors_and_collaborations + + @classmethod + def latex_to_unicode(cls, latex_string): + try: + return cls._l2t.latex_to_text(latex_string).replace(" ", " ") + except Exception: + return latex_string diff --git a/inspire_utils/parsers/crossref.py b/inspire_utils/parsers/crossref.py new file mode 100644 index 0000000..ca29311 --- /dev/null +++ b/inspire_utils/parsers/crossref.py @@ -0,0 +1,368 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. +"""Parser for the Crossref API metadata format""" +from __future__ import absolute_import, division, print_function + +import itertools +from inspire_schemas.api import LiteratureBuilder, ReferenceBuilder +from ..date import PartialDate +from ..helpers import force_list +from ..record import get_value +from ..dedupers import dedupe_list_of_dicts + +"""Document types for the crossref objects have been extracted +from the following link: https://api.crossref.org/v1/types +""" + +DOC_TYPE_MAP = { + 'book': 'book', + 'book-part': 'book chapter', + 'book-section': 'book chapter', + 'book-series': 'book', + 'book-set': 'book', + 'book-track': 'book chapter', + 'book-chapter': 'book chapter', + 'dissertation': 'thesis', + 'edited-book': 'book', + 'journal-article': 'article', + 'journal-volume': 'article', + 'journal': 'article', + 'monograph': 'book', + 'proceedings': 'proceedings', + 'proceedings-article': 'conference paper', + 'other': 'note', + 'reference-book': 'book', + 'report': 'report', + 'report-series': 'report', +} + + +class CrossrefParser(object): + """Parser for the JSON Crossref format. + + Args: + crossref_record (dict): the record in JSON Crossref API format to parse. + source (Optional[str]): if provided, sets the ``source`` everywhere in + the record. Otherwise, the source is extracted from the Crossref metadata. + """ + def __init__(self, crossref_record, source=None): + self.record = crossref_record.get("message") + if not source: + source = self.material_source + self.builder = LiteratureBuilder(source) + + def parse(self): + """Extract a Crossref record into an Inspire HEP record. + + Returns: + dict: the same record in the Inspire Literature schema. + """ + + self.builder.add_abstract(self.abstract) + for doi in self.dois: + self.builder.add_doi(**doi) + for reference in self.references: + self.builder.add_reference(reference) + self.builder.add_imprint_date(self.imprints) + for author in self.authors: + self.builder.add_author(author) + for license_instance in self.license: + self.builder.add_license(**license_instance) + self.builder.add_publication_info(**self.publication_info) + self.builder.add_title(self.title, subtitle=self.subtitle) + self.builder.add_document_type(self.document_type) + + return self.builder.record + + @property + def document_type(self): + doc_type = self.record.get("type") + return DOC_TYPE_MAP.get(doc_type, "article") + + @property + def title(self): + title = get_value(self.record, "title[0]") + + return title + + @property + def subtitle(self): + subtitle = get_value(self.record, "subtitle[0]") + + return subtitle + + @property + def dois(self): + value = self.record.get("DOI") + dois = [ + {'doi': value, 'material': self.material} + ] + + return dois + + @property + def material_source(self): + return self.record.get("source") + + @property + def material(self): + title = self.title or '' + subtitle = self.subtitle or '' + if title.startswith("Erratum") or subtitle.startswith("Erratum"): + material = 'erratum' + elif title.startswith("Addendum") or subtitle.startswith("Addendum"): + material = 'addendum' + elif title.startswith("Publisher's Note") or subtitle.startswith("Publisher's Note"): + material = 'editorial note' + else: + material = 'publication' + + return material + + @property + def publication_info(self): + publication_info = { + 'artid': self.artid, + 'journal_title': self.journal_title, + 'journal_issue': self.journal_issue, + 'journal_volume': self.journal_volume, + 'page_start': self.page_start, + 'page_end': self.page_end, + 'year': self.year, + 'material': self.material, + 'parent_isbn': self.parent_isbn, + } + + return publication_info + + @property + def parent_isbn(self): + return get_value(self.record, "ISBN[0]") + + @property + def journal_title(self): + if self.document_type == 'book chapter': + return None + + return get_value(self.record, "container-title[0]") + + @property + def artid(self): + return self.record.get("article-number") + + @property + def journal_issue(self): + return self.record.get("issue") + + @property + def journal_volume(self): + return self.record.get("volume") + + @property + def year(self): + date = get_value(self.record, "issued.date-parts[0][0]") + + return date + + @property + def page_start(self): + pages = self.record.get("page") + + if pages: + return pages.split('-')[0] + else: + return None + + @property + def page_end(self): + pages = self.record.get("page") + + if pages and '-' in pages: + return pages.split('-')[1] + else: + return None + + @staticmethod + def get_author_name(author_key): + """Extract an author's name.""" + author_name_list = [author_key.get("family"), author_key.get("given")] + return ', '.join(filter(None, author_name_list)) + + @staticmethod + def get_author_affiliations(author_key): + """Extract an author's affiliations.""" + affiliations = force_list(author_key.get("affiliation")) + + auth_aff = [affiliation.get('name') for affiliation in affiliations] + + return auth_aff + + @staticmethod + def get_author_orcid(author_key): + """Extract an author's orcid.""" + orcid_value = author_key.get('ORCID') + + return [('ORCID', orcid_value)] + + def get_author(self, author_key): + """Extract one author. + + Args: + author_key(dict): a dictionary on a single author. + + Returns: + dict: the parsed author, conforming to the Inspire schema. + """ + author_name = self.get_author_name(author_key) + affiliations = self.get_author_affiliations(author_key) + orcid = self.get_author_orcid(author_key) + + return self.builder.make_author(author_name, raw_affiliations=affiliations, ids=orcid) + + @property + def authors(self): + authors_key = self.record.get("author") + authors = [self.get_author(author) for author in force_list(authors_key)] + + return authors + + @property + def license(self): + license_keys = self.record.get("license") + licenses = [self.get_license(license) for license in force_list(license_keys)] + + return licenses + + def get_license(self, license_key): + """Extract one license. + + Args: + license_key(dict): a dictionary on a single license. + + Returns: + dict: the parsed license, conforming to the Inspire schema. + """ + license = { + 'imposing': self.publisher, + 'material': self.material, + 'url': self.get_license_url(license_key), + } + + return license + + @staticmethod + def get_license_url(license_key): + return license_key.get("URL") + + @property + def publisher(self): + return self.record.get("publisher") + + @property + def abstract(self): + return self.record.get("abstract") + + @property + def imprints(self): + '''issued: Eariest of published-print and published-online + + That is why we use this field to fill the imprints and the publication info. + ''' + + date_parts = get_value(self.record, "issued.date-parts[0]") + + if not date_parts: + return None + + date = PartialDate(*date_parts) + + return date.dumps() + + @property + def references(self): + """Extract a Crossref record into an Inspire HEP references record. + + Returns: + List[dict]: an array of reference schema records, representing + the references in the record + """ + ref_keys = self.record.get("reference") + reference_list = list( + itertools.chain.from_iterable( + self.get_reference(key) for key in force_list(ref_keys) + ) + ) + return dedupe_list_of_dicts(reference_list) + + def get_reference(self, ref_key): + """Extract one reference. + + Args: + ref_key(dict): a dictionary on a single reference. + + Returns: + dict: the parsed reference, as generated by + :class:`inspire_schemas.api.ReferenceBuilder` + """ + builder = ReferenceBuilder() + + journal_title = ref_key.get("journal-title") + if journal_title: + builder.set_journal_title(journal_title) + + journal_volume = ref_key.get("volume") + if journal_volume: + builder.set_journal_volume(journal_volume) + + journal_issue = ref_key.get("issue") + if journal_issue: + builder.set_journal_issue(journal_issue) + + first_page = ref_key.get("first-page") + if first_page: + builder.set_page_artid(page_start=first_page) + + year = ref_key.get("year") + if year: + builder.set_year(year) + + title = ref_key.get("article-title") + if title: + builder.add_title(title) + + isbn = ref_key.get("ISBN") + if isbn: + builder.add_uid(isbn) + + doi = ref_key.get("DOI") + if doi: + builder.add_uid(doi) + + author = ref_key.get("author") + if author: + builder.add_author(author, 'author') + + raw_ref = ref_key.get("unstructured") + if raw_ref: + builder.add_raw_reference(raw_ref, self.material_source) + + yield builder.obj diff --git a/inspire_utils/parsers/elsevier.py b/inspire_utils/parsers/elsevier.py new file mode 100644 index 0000000..209c435 --- /dev/null +++ b/inspire_utils/parsers/elsevier.py @@ -0,0 +1,681 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +from __future__ import absolute_import, division, print_function + +import itertools +import re + +import six +from inspire_schemas.api import LiteratureBuilder, ReferenceBuilder +from ..date import PartialDate +from ..helpers import maybe_int, remove_tags + +from ..utils import get_node + +DOCTYPE_MAPPING = { + "abs": "abstract", + "add": "addendum", + "adv": "advertisement", + "ann": "announcement", + "brv": "book-review", + "cal": "calendar", + "chp": "chapter", + "cnf": "conference", + "con": "contents list", + "cor": "correspondence", + "cop": "copyright", + "crp": "case report", + "dat": "data article", + "dis": "discussion", + "dup": "duplicate", + "edb": "editorial board", + "edi": "editorial", + "err": "erratum", + "exm": "examination", + "fla": "full-length article", + "ind": "index", + "lit": "literature alert", + "lst": "list", + "mic": "micro article", + "mis": "miscellaneous", + "nws": "news", + "ocn": "other contents", + "osp": "original software publication", + "pgl": "practice guideline", + "pnt": "patent report", + "prp": "personal report", + "prv": "product review", + "pub": "publisher's note", + "rem": "removal", + "req": "request for assistance", + "ret": "retraction", + "rev": "review article", + "rpl": "replication studies", + "sco": "short communication", + "ssu": "short survey", + "vid": "video article", +} + +COPYRIGHT_MAPPING = { + "crown": "Crown copyright", + "free-of-copyright": "None", + "full-transfer": "Publisher", + "joint": "Publisher and scientific society", + "limited-transfer": "Authors and publisher", + "other": "Authors", + "society": "Scientific society", + "us-gov": " US government", +} + +DOCTYPES_TO_HARVEST = [ + "full-length article", + "addendum", + "chapter", + "erratum", + "review article", + "short communication", + "short survey", + "publisher's note", + "discussion", +] + + +class ElsevierParser(object): + """Parser for the Elsevier format. + + It can be used directly by invoking the :func:`ElsevierParser.parse` method, or be + subclassed to customize its behavior. + + Args: + elsevier_record (Union[str, scrapy.selector.Selector]): the record in Elsevier format to parse. + source (Optional[str]): if provided, sets the ``source`` everywhere in + the record. Otherwise, the source is extracted from the Elsevier metadata. + """ + + def __init__(self, elsevier_record, source=None): + self.root = self.get_root_node(elsevier_record) + if not source: + source = self.publisher + self.builder = LiteratureBuilder(source) + + def parse(self): + """Extract a Elsevier record into an Inspire HEP record. + + Returns: + dict: the same record in the Inspire Literature schema. + """ + self.builder.add_abstract(self.abstract) + self.builder.add_title(self.title, subtitle=self.subtitle) + self.builder.add_copyright(**self.copyright) + self.builder.add_document_type(self.document_type) + self.builder.add_license(**self.license) + for author in self.authors: + self.builder.add_author(author) + self.builder.add_publication_info(**self.publication_info) + for collab in self.collaborations: + self.builder.add_collaboration(collab) + for doi in self.dois: + self.builder.add_doi(**doi) + for keyword in self.keywords: + self.builder.add_keyword(keyword) + if self.imprints_date: + self.builder.add_imprint_date(self.imprints_date) + elif self.publication_date: + self.builder.add_imprint_date( + self.publication_date.dumps() + ) + for reference in self.references: + self.builder.add_reference(reference) + + return self.builder.record + + @property + def references(self): + """Extract a Elsevier record into an Inspire HEP references record. + + Returns: + List[dict]: an array of reference schema records, representing + the references in the record + """ + ref_nodes = self.root.xpath(".//bib-reference") + return list( + itertools.chain.from_iterable( + self.get_reference_iter(node) for node in ref_nodes + ) + ) + + remove_tags_config_abstract = { + "allowed_tags": ["sup", "sub"], + "allowed_trees": ["math"], + "strip": "self::pub-id|self::issn", + } + + remove_tags_config_title = { + 'allowed_trees': ['math'], + } + + @property + def abstract(self): + abstract_nodes = self.root.xpath(".//head/abstract[not(@graphical)]/abstract-sec/simple-para") + + if not abstract_nodes: + return + + abstract_paragraphs = [remove_tags( + abstract_node, **self.remove_tags_config_abstract + ).strip("/ \n") for abstract_node in abstract_nodes] + abstract = ' '.join(abstract_paragraphs) + return abstract + + @property + def article_type(self): + """Return a article type mapped from abbreviation.""" + abbrv_doctype = self.root.xpath(".//@docsubtype").extract_first() + article_type = DOCTYPE_MAPPING.get(abbrv_doctype) + return article_type + + @property + def artid(self): + artid = self.root.xpath("string(./*/item-info/aid[1])").extract_first() + return artid + + @property + def authors(self): + author_nodes = self.root.xpath("./*/head/author-group") + all_authors = [] + for author_group in author_nodes: + authors = [ + self.get_author(author, author_group) + for author in author_group.xpath("./author") + ] + all_authors.extend(authors) + return all_authors + + @property + def collaborations(self): + collaborations = self.root.xpath( + "./*/head/author-group//collaboration/text/text()" + ).extract() + return collaborations + + @property + def copyright(self): + copyright = { + "holder": self.copyright_holder, + "material": self.material, + "statement": self.copyright_statement, + "year": self.copyright_year, + } + + return copyright + + @property + def copyright_holder(self): + copyright_holder = self.root.xpath( + "string(./*/item-info/copyright[@type][1])" + ).extract_first() + if not copyright_holder: + copyright_type = self.root.xpath( + "./*/item-info/copyright/@type" + ).extract_first() + copyright_holder = COPYRIGHT_MAPPING.get(copyright_type) + + return copyright_holder + + @property + def copyright_statement(self): + copyright_statement = self.root.xpath( + "string(./RDF/Description/copyright[1])" + ).extract_first() + if not copyright_statement: + copyright_statement = self.root.xpath( + "string(./*/item-info/copyright[@type][1])" + ).extract_first() + + return copyright_statement + + @property + def copyright_year(self): + copyright_year = self.root.xpath( + "./*/item-info/copyright[@type]/@year" + ).extract_first() + + return maybe_int(copyright_year) + + @property + def dois(self): + doi = self.root.xpath("string(./RDF/Description/doi[1])").extract_first() + return [{"doi": doi, "material": self.material}] + + @property + def document_type(self): + doctype = None + if self.root.xpath( + "./*[contains(name(),'article') or self::book-review]" + ): + doctype = "article" + elif self.root.xpath("./*[self::book or self::simple-book]"): + doctype = "book" + elif self.root.xpath("./book-chapter"): + doctype = "book chapter" + if self.is_conference_paper: + doctype = "conference paper" + if doctype: + return doctype + + @property + def is_conference_paper(self): + """Decide whether the article is a conference paper.""" + if self.root.xpath("./conference-info"): + return True + journal_issue = self.root.xpath( + "string(./RDF/Description/issueName[1])" + ).extract_first() + if journal_issue: + is_conference = re.findall(r"proceedings|proc.", journal_issue.lower()) + return bool(is_conference) + return False + + @property + def journal_title(self): + jid = self.root.xpath("string(./*/item-info/jid[1])").extract_first(default="") + publication = self.root.xpath( + "string(./RDF/Description/publicationName[1])" + ).extract_first(default=jid) + publication = re.sub(" [S|s]ection", "", publication).replace(",", "").strip() + return publication + + @property + def journal_issue(self): + journal_issue = self.root.xpath( + "string(./serial-issue/issue-info/issue-first[1])" + ).extract_first() + + return journal_issue + + @property + def journal_volume(self): + journal_volume = self.root.xpath( + "string(./RDF/Description/volume[1])" + ).extract_first() + + return journal_volume + + @property + def keywords(self): + keywords = self.root.xpath( + "./*/head/keywords[not(@abr)]/keyword/text/text()" + ).getall() + + return keywords + + @property + def license(self): + license = { + "license": self.license_statement, + "material": self.material, + "url": self.license_url, + } + + return license + + @property + def license_statement(self): + license_statement = self.root.xpath( + "string(./RDF/Description/licenseLine[1])" + ).extract_first() + + return license_statement + + @property + def license_url(self): + license_url = self.root.xpath( + "string(./RDF/Description/openAccessInformation/userLicense[1])" + ).extract_first() + + return license_url + + @property + def material(self): + if self.article_type in ( + "erratum", + "addendum", + "retraction", + "removal", + "duplicate", + ): + material = self.article_type + elif self.article_type in ("editorial", "publisher's note"): + material = "editorial note" + else: + material = "publication" + + return material + + @property + def page_start(self): + page_start = self.root.xpath( + "string(./RDF/Description/startingPage[1])" + ).extract_first() + return page_start + + @property + def page_end(self): + page_end = self.root.xpath( + "string(./RDF/Description/endingPage[1])" + ).extract_first() + return page_end + + @property + def imprints_date(self): + imprints_date = self.root.xpath( + "string(./RDF/Description/availableOnlineInformation/availableOnline)" + ).extract_first() + if imprints_date: + return PartialDate.parse(imprints_date).dumps() + + @property + def publication_date(self): + publication_date = None + publication_date_string = self.root.xpath( + "string(./RDF/Description/coverDisplayDate[1])" + ).extract_first() + if publication_date_string: + try: + publication_date = PartialDate.parse(publication_date_string) + except ValueError: + # in case when date contains month range, eg. July-September 2020 + publication_date = re.sub( + "[A-aZ-z]*-(?=[A-aZ-z])", "", publication_date_string + ) + publication_date = PartialDate.parse(publication_date) + return publication_date + + @property + def publication_info(self): + publication_info = { + "artid": self.artid, + "journal_title": self.journal_title, + "journal_issue": self.journal_issue, + "journal_volume": self.journal_volume, + "material": self.material, + "page_start": self.page_start, + "page_end": self.page_end, + "year": self.year, + } + + return publication_info + + @property + def publisher(self): + publisher = self.root.xpath("string(./RDF/Description/publisher[1])").extract_first( + "Elsevier B.V." + ) + + return publisher + + @property + def subtitle(self): + subtitle = self.root.xpath("string(./*/head/subtitle[1])").extract_first() + + return subtitle + + @property + def title(self): + title = self.root.xpath("./*/head/title[1]").extract_first() + return remove_tags(title, **self.remove_tags_config_title).strip("\n") if title else None + + @property + def year(self): + if self.publication_date: + return self.publication_date.year + + def get_author_affiliations(self, author_node, author_group_node): + """Extract an author's affiliations.""" + ref_ids = author_node.xpath(".//@refid[contains(., 'af')]").extract() + group_affs = author_group_node.xpath("string(./affiliation/textfn[1])").getall() + if ref_ids: + affiliations = self._find_affiliations_by_id(author_group_node, ref_ids) + else: + affiliations = filter(None, group_affs) + return affiliations + + @staticmethod + def _find_affiliations_by_id(author_group, ref_ids): + """Return affiliations with given ids. + + Affiliations should be standardized later. + """ + affiliations_by_id = [] + for aff_id in ref_ids: + affiliation = author_group.xpath( + "string(//affiliation[@id='{}']/textfn[1])".format(aff_id) + ).extract_first() + affiliations_by_id.append(affiliation) + + return affiliations_by_id + + def get_author_emails(self, author_node): + """Extract an author's email addresses.""" + emails = author_node.xpath('string(./e-address[@type="email"][1])').getall() + + return emails + + @staticmethod + def get_author_name(author_node): + """Extract an author's name.""" + surname = author_node.xpath("string(./surname[1])").extract_first() + given_names = author_node.xpath("string(./given-name[1])").extract_first() + suffix = author_node.xpath("string(.//suffix[1])").extract_first() + author_name = ", ".join(el for el in (surname, given_names, suffix) if el) + + return author_name + + @staticmethod + def get_root_node(elsevier_record): + """Get a selector on the root ``article`` node of the record. + + This can be overridden in case some preprocessing needs to be done on + the XML. + + Args: + elsevier_record(Union[str, scrapy.selector.Selector]): the record in Elsevier format. + + Returns: + scrapy.selector.Selector: a selector on the root ``
`` + node. + """ + if isinstance(elsevier_record, six.string_types): + root = get_node(elsevier_record) + else: + root = elsevier_record + root.remove_namespaces() + + return root + + def get_author(self, author_node, author_group_node): + """Extract one author. + + Args: + author_node(scrapy.selector.Selector): a selector on a single + author, e.g. a ````. + + Returns: + dict: the parsed author, conforming to the Inspire schema. + """ + author_name = self.get_author_name(author_node) + emails = self.get_author_emails(author_node) + affiliations = self.get_author_affiliations(author_node, author_group_node) + + return self.builder.make_author( + author_name, raw_affiliations=affiliations, emails=emails + ) + + @staticmethod + def get_reference_authors(ref_node): + """Extract authors from a reference node. + + Args: + ref_node(scrapy.selector.Selector): a selector on a single reference. + + Returns: + List[str]: list of names + """ + authors = ref_node.xpath("./contribution/authors/author") + authors_names = [] + for author in authors: + given_names = author.xpath("string(./given-name[1])").extract_first(default="") + last_names = author.xpath("string(./surname[1])").extract_first(default="") + authors_names.append(" ".join([given_names, last_names]).strip()) + return authors_names + + @staticmethod + def get_reference_editors(ref_node): + """Extract authors of `role` from a reference node. + + Args: + ref_node(scrapy.selector.Selector): a selector on a single reference. + + Returns: + List[str]: list of names + """ + editors = ref_node.xpath(".//editors/authors/author") + editors_names = [] + for editor in editors: + given_names = editor.xpath("string(./given-name[1])").extract_first(default="") + last_names = editor.xpath("string(./surname[1])").extract_first(default="") + editors_names.append(" ".join([given_names, last_names]).strip()) + return editors_names + + @staticmethod + def get_reference_artid(ref_node): + return ref_node.xpath("string(.//article-number[1])").extract_first() + + @staticmethod + def get_reference_pages(ref_node): + first_page = ref_node.xpath("string(.//pages/first-page[1])").extract_first() + last_page = ref_node.xpath("string(.//pages/last-page[1])").extract_first() + return first_page, last_page + + def get_reference_iter(self, ref_node): + """Extract one reference. + + Args: + ref_node(scrapy.selector.Selector): a selector on a single + reference, i.e. ````. + + Yields: + dict: the parsed reference, as generated by + :class:`inspire_schemas.api.ReferenceBuilder` + """ + # handle also unstructured refs + for citation_node in ref_node.xpath("./reference|./other-ref"): + builder = ReferenceBuilder() + + builder.add_raw_reference( + ref_node.extract().strip(), + source=self.builder.source, + ref_format="Elsevier", + ) + + fields = [ + (("string(.//series/title/maintitle[1])"), builder.set_journal_title,), + ( + "string(.//title[parent::edited-book|parent::book]/maintitle[1])", + builder.add_parent_title, + ), + ("string(./publisher/name[1])", builder.set_publisher), + ("string(.//volume-nr[1])", builder.set_journal_volume), + ("string(.//issue-nr[1])", builder.set_journal_issue), + ("string(.//date[1])", builder.set_year), + ("string(.//inter-ref[1])", builder.add_url), + ("string(.//doi[1])", builder.add_uid), + ( + 'string(pub-id[@pub-id-type="other"]' + '[contains(preceding-sibling::text(),"Report No")][1])', + builder.add_report_number, + ), + ("string(./title/maintitle[1])", builder.add_title), + ] + for xpath, field_handler in fields: + value = citation_node.xpath(xpath).extract_first() + citation_node.xpath(xpath) + if value: + field_handler(value) + + label_value = ref_node.xpath("string(./label[1])").extract_first() + builder.set_label(label_value.strip("[]")) + + pages = self.get_reference_pages(citation_node) + artid = self.get_reference_artid(citation_node) + if artid: + builder.set_page_artid(artid=artid) + if any(pages): + builder.set_page_artid(*pages) + + remainder = ( + remove_tags( + citation_node, + strip="self::authors" + "|self::article-number" + "|self::volume-nr" + "|self::issue-nr" + "|self::inter-ref" + "|self::maintitle" + "|self::date" + "|self::label" + "|self::publisher" + "|self::doi" + "|self::pages" + ) + .strip("\"';,. \t\n\r") + .replace("()", "") + ) + if remainder: + builder.add_misc(remainder) + + for editor in self.get_reference_editors(citation_node): + builder.add_author(editor, "editor") + + for author in self.get_reference_authors(citation_node): + builder.add_author(author, "author") + + yield builder.obj + + def attach_fulltext_document(self, file_name, url): + self.builder.add_document(file_name, url, fulltext=True, hidden=True) + + def get_identifier(self): + return self.dois[0]["doi"] + + def should_record_be_harvested(self): + if self.article_type in DOCTYPES_TO_HARVEST and all( + [ + self.title, + self.journal_title, + self.journal_volume, + (self.artid or self.page_start), + ] + ): + return True + return False diff --git a/inspire_utils/parsers/jats.py b/inspire_utils/parsers/jats.py new file mode 100644 index 0000000..5540065 --- /dev/null +++ b/inspire_utils/parsers/jats.py @@ -0,0 +1,656 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +"""Extractrs for various metadata formats""" + +from __future__ import absolute_import, division, print_function + +import itertools + +import six + +from idutils import normalize_orcid +from inspire_schemas.api import LiteratureBuilder, ReferenceBuilder +from inspire_schemas.utils import split_page_artid +from ..date import PartialDate +from ..helpers import maybe_int, remove_tags + +from ..utils import get_node + + +JOURNAL_TITLES_MAPPING = { + "Physics": "APS Physics" +} + + +class JatsParser(object): + """Parser for the JATS format. + + It can be used directly by invoking the :func:`JatsParser.parse` method, or be + subclassed to customize its behavior. + + Args: + jats_record (Union[str, scrapy.selector.Selector]): the record in JATS format to parse. + source (Optional[str]): if provided, sets the ``source`` everywhere in + the record. Otherwise, the source is extracted from the JATS metadata. + """ + def __init__(self, jats_record, source=None): + self.root = self.get_root_node(jats_record) + if not source: + source = self.publisher + self.builder = LiteratureBuilder(source) + + def parse(self): + """Extract a JATS record into an Inspire HEP record. + + Returns: + dict: the same record in the Inspire Literature schema. + """ + self.builder.add_abstract(self.abstract) + self.builder.add_title(self.title, subtitle=self.subtitle) + self.builder.add_copyright(**self.copyright) + self.builder.add_document_type(self.document_type) + self.builder.add_license(**self.license) + for author in self.authors: + self.builder.add_author(author) + self.builder.add_number_of_pages(self.number_of_pages) + self.builder.add_publication_info(**self.publication_info) + for collab in self.collaborations: + self.builder.add_collaboration(collab) + for doi in self.dois: + self.builder.add_doi(**doi) + for keyword in self.keywords: + self.builder.add_keyword(**keyword) + self.builder.add_imprint_date( + self.publication_date.dumps() if self.publication_date else None + ) + for reference in self.references: + self.builder.add_reference(reference) + + return self.builder.record + + @property + def references(self): + """Extract a JATS record into an Inspire HEP references record. + + Returns: + List[dict]: an array of reference schema records, representing + the references in the record + """ + ref_nodes = self.root.xpath('./back/ref-list/ref') + return list( + itertools.chain.from_iterable( + self.get_reference(node) for node in ref_nodes + ) + ) + + remove_tags_config_abstract = { + 'allowed_tags': ['sup', 'sub'], + 'allowed_trees': ['math'], + 'strip': 'self::pub-id|self::issn' + } + + remove_tags_config_title = { + 'allowed_trees': ['math'], + } + + @property + def abstract(self): + abstract_nodes = self.root.xpath('./front//abstract[1]') + + if not abstract_nodes: + return + + abstract = remove_tags(abstract_nodes[0], **self.remove_tags_config_abstract).strip() + return abstract + + @property + def article_type(self): + article_type = self.root.xpath('./@article-type').extract_first() + + return article_type + + @property + def artid(self): + artid = self.root.xpath('./front/article-meta//elocation-id//text()').extract_first() + + return artid + + @property + def authors(self): + author_nodes = self.root.xpath('./front//contrib[@contrib-type="author"]') + authors = [self.get_author(author) for author in author_nodes] + + return authors + + @property + def collaborations(self): + collab_nodes = self.root.xpath( + './front//collab |' + './front//contrib[@contrib-type="collaboration"] |' + './front//on-behalf-of' + ) + collaborations = set( + collab.xpath('string(.)').extract_first() for collab in collab_nodes + ) + + return collaborations + + @property + def copyright(self): + copyright = { + 'holder': self.copyright_holder, + 'material': self.material, + 'statement': self.copyright_statement, + 'year': self.copyright_year, + } + + return copyright + + @property + def copyright_holder(self): + copyright_holder = self.root.xpath('./front//copyright-holder/text()').extract_first() + + return copyright_holder + + @property + def copyright_statement(self): + copyright_statement = self.root.xpath('./front//copyright-statement/text()').extract_first() + + return copyright_statement + + @property + def copyright_year(self): + copyright_year = self.root.xpath('./front//copyright-year/text()').extract_first() + + return maybe_int(copyright_year) + + @property + def dois(self): + doi_values = self.root.xpath('./front/article-meta//article-id[@pub-id-type="doi"]/text()').extract() + dois = [ + {'doi': value, 'material': self.material} for value in doi_values + ] + + if self.material != 'publication': + doi_values = self.root.xpath( + './front/article-meta//related-article[@ext-link-type="doi"]/@href' + ).extract() + related_dois = ({'doi': value} for value in doi_values) + dois.extend(related_dois) + + return dois + + @property + def document_type(self): + if self.is_conference_paper: + document_type = 'conference paper' + else: + document_type = 'article' + + return document_type + + @property + def is_conference_paper(self): + """Decide whether the article is a conference paper.""" + conference_node = self.root.xpath('./front//conference').extract_first() + + return bool(conference_node) + + @property + def journal_title(self): + journal_title = self.root.xpath( + './front/journal-meta//abbrev-journal-title/text() |' + './front/journal-meta//journal-title/text()' + ).extract_first() + + return JOURNAL_TITLES_MAPPING.get(journal_title) or journal_title + + @property + def journal_issue(self): + journal_issue = self.root.xpath('./front/article-meta/issue/text()').extract_first() + + return journal_issue + + @property + def journal_volume(self): + journal_volume = self.root.xpath('./front/article-meta/volume/text()').extract_first() + + return journal_volume + + @property + def keywords(self): + keyword_groups = self.root.xpath('./front//kwd-group') + keywords = itertools.chain.from_iterable(self.get_keywords(group) for group in keyword_groups) + + return keywords + + @property + def license(self): + license = { + 'license': self.license_statement, + 'material': self.material, + 'url': self.license_url, + } + + return license + + @property + def license_statement(self): + license_statement = self.root.xpath('string(./front/article-meta//license)').extract_first().strip() + + return license_statement + + @property + def license_url(self): + url_nodes = ( + './front/article-meta//license_ref/text() |' + './front/article-meta//license/@href |' + './front/article-meta//license//ext-link/@href' + ) + license_url = self.root.xpath(url_nodes).extract_first() + + return license_url + + @property + def material(self): + if self.article_type.startswith('correc'): + material = 'erratum' + elif self.article_type in ('erratum', 'translation', 'addendum', 'reprint'): + material = self.article_type + else: + material = 'publication' + + return material + + @property + def number_of_pages(self): + number_of_pages = maybe_int(self.root.xpath('./front/article-meta//page-count/@count').extract_first()) + + return number_of_pages + + @property + def page_start(self): + page_start = self.root.xpath('./front/article-meta/fpage/text()').extract_first() + + return page_start + + @property + def page_end(self): + page_end = self.root.xpath('./front/article-meta/lpage/text()').extract_first() + + return page_end + + @property + def publication_date(self): + date_nodes = self.root.xpath( + './front//pub-date[@pub-type="ppub"] |' + './front//pub-date[@pub-type="epub"] |' + './front//pub-date[starts-with(@date-type,"pub")] |' + './front//date[starts-with(@date-type,"pub")]' + ) + + if date_nodes: + publication_date = min( + self.get_date(date_node) for date_node in date_nodes + ) + + return publication_date + + @property + def publication_info(self): + publication_info = { + 'artid': self.artid, + 'journal_title': self.journal_title, + 'journal_issue': self.journal_issue, + 'journal_volume': self.journal_volume, + 'material': self.material, + 'page_start': self.page_start, + 'page_end': self.page_end, + 'year': self.year, + } + + return publication_info + + @property + def publisher(self): + publisher = self.root.xpath('./front//publisher-name/text()').extract_first() + + return publisher + + @property + def subtitle(self): + subtitle = self.root.xpath('string(./front//subtitle)').extract_first() + + return subtitle + + @property + def title(self): + title = self.root.xpath('./front//article-title').extract_first() + return remove_tags(title, **self.remove_tags_config_title) + + def get_affiliation(self, id_): + """Get the affiliation with the specified id. + + Args: + id_(str): the value of the ``id`` attribute of the affiliation. + + Returns: + Optional[str]: the affiliation with that id or ``None`` if there is + no match. + """ + affiliation_node = self.root.xpath("//aff[@id=$id_]", id_=id_) + if affiliation_node: + affiliation = remove_tags( + affiliation_node[0], strip="self::label | self::email" + ).strip() + return affiliation + + def get_emails_from_refs(self, id_): + """Get the emails from the node with the specified id. + + Args: + id_(str): the value of the ``id`` attribute of the node. + + Returns: + List[str]: the emails from the node with that id or [] if none found. + """ + email_nodes = self.root.xpath('//aff[@id=$id_]/email/text()', id_=id_) + return email_nodes.extract() + + @property + def year(self): + not_online = ( + 'not(starts-with(@publication-format, "elec"))' + ' and not(starts-with(@publication-format, "online")' + ) + date_nodes = self.root.xpath( + './front//pub-date[@pub-type="ppub"] |' + './front//pub-date[starts-with(@date-type,"pub") and $not_online] |' + './front//date[starts-with(@date-type,"pub") and $not_online]', + not_online=not_online + ) + + if date_nodes: + year = min( + self.get_date(date_node) for date_node in date_nodes + ).year + + return year + + def get_author_affiliations(self, author_node): + """Extract an author's affiliations.""" + raw_referred_ids = author_node.xpath('.//xref[@ref-type="aff"]/@rid').extract() + # Sometimes the rid might have more than one ID (e.g. rid="id0 id1") + referred_ids = set() + for raw_referred_id in raw_referred_ids: + referred_ids.update(set(raw_referred_id.split(' '))) + + affiliations = [ + self.get_affiliation(rid) for rid in referred_ids + if self.get_affiliation(rid) + ] + + return affiliations + + def get_author_emails(self, author_node): + """Extract an author's email addresses.""" + emails = author_node.xpath('.//email/text()').extract() + referred_ids = author_node.xpath('.//xref[@ref-type="aff"]/@rid').extract() + for referred_id in referred_ids: + emails.extend(self.get_emails_from_refs(referred_id)) + + return emails + + @staticmethod + def get_author_name(author_node): + """Extract an author's name.""" + surname = author_node.xpath('.//surname/text()').extract_first() + if not surname: + # the author name is unstructured + author_name = author_node.xpath('string(./string-name)').extract_first() + given_names = author_node.xpath('.//given-names/text()').extract_first() + suffix = author_node.xpath('.//suffix/text()').extract_first() + author_name = ', '.join(el for el in (surname, given_names, suffix) if el) + + return author_name + + @staticmethod + def _get_iso_date(iso_date_string): + try: + iso_date = PartialDate.loads(iso_date_string) + return iso_date + except ValueError: + return + + @staticmethod + def _get_date_from_parts(year, month, day): + possible_dates = [ + [year, month, day], + [year, month], + [year] + ] + # we try different date combinations + # cause even if date part is not None + # it can raise validation error + # (better ask for forgiveness than permission) + for date_parts in possible_dates: + try: + date = PartialDate.from_parts(*date_parts) + return date + except ValueError: + continue + + def get_date(self, date_node): + """Extract a date from a date node. + + Returns: + PartialDate: the parsed date. + """ + iso_string = date_node.xpath('./@iso-8601-date').extract_first(default="") + iso_date = self._get_iso_date(iso_string) + if iso_date: + return iso_date + year = date_node.xpath('string(./year)').extract_first() + month = date_node.xpath('string(./month)').extract_first() + day = date_node.xpath('string(./day)').extract_first() + + date_from_parts = self._get_date_from_parts(year, month, day) + if date_from_parts: + return date_from_parts + + string_date = date_node.xpath('string(./string-date)').extract_first() + try: + parsed_date = PartialDate.parse(string_date) + except ValueError: + parsed_date = None + + return parsed_date + + @staticmethod + def get_keywords(group_node): + """Extract keywords from a keyword group.""" + schema = None + if 'pacs' in group_node.xpath('@kwd-group-type').extract_first(default='').lower(): + schema = 'PACS' + + keywords = (kwd.xpath('string(.)').extract_first() for kwd in group_node.xpath('.//kwd')) + keyword_dicts = ({'keyword': keyword, 'schema': schema} for keyword in keywords) + + return keyword_dicts + + @staticmethod + def get_root_node(jats_record): + """Get a selector on the root ``article`` node of the record. + + This can be overridden in case some preprocessing needs to be done on + the XML. + + Args: + jats_record(Union[str, scrapy.selector.Selector]): the record in JATS format. + + Returns: + scrapy.selector.Selector: a selector on the root ``
`` + node. + """ + if isinstance(jats_record, six.string_types): + root = get_node(jats_record) + else: + root = jats_record + root.remove_namespaces() + + return root + + def get_author(self, author_node): + """Extract one author. + + Args: + author_node(scrapy.selector.Selector): a selector on a single + author, e.g. a ````. + + Returns: + dict: the parsed author, conforming to the Inspire schema. + """ + author_name = self.get_author_name(author_node) + emails = self.get_author_emails(author_node) + affiliations = self.get_author_affiliations(author_node) + orcid = self.get_orcid(author_node) + author_ids = [("ORCID", orcid)] if orcid else [] + return self.builder.make_author( + author_name, + raw_affiliations=affiliations, + emails=emails, + ids=author_ids + ) + + @staticmethod + def get_orcid(author_node): + orcid = author_node.xpath('./contrib-id[@contrib-id-type="orcid"]/text()').extract_first() + if orcid: + return normalize_orcid(orcid) + + @staticmethod + def get_reference_authors(ref_node, role): + """Extract authors of `role` from a reference node. + + Args: + ref_node(scrapy.selector.Selector): a selector on a single reference. + role(str): author role + + Returns: + List[str]: list of names + """ + return ref_node.xpath( + './person-group[@person-group-type=$role]/string-name/text()', + role=role + ).extract() + + def get_reference(self, ref_node): + """Extract one reference. + + Args: + ref_node(scrapy.selector.Selector): a selector on a single + reference, i.e. ````. + + Returns: + dict: the parsed reference, as generated by + :class:`inspire_schemas.api.ReferenceBuilder` + """ + for citation_node in ref_node.xpath('./mixed-citation'): + builder = ReferenceBuilder() + + builder.add_raw_reference( + ref_node.extract().strip(), + source=self.builder.source, + ref_format='JATS' + ) + + fields = [ + ( + ( + 'self::node()[@publication-type="journal" ' + 'or @publication-type="eprint"]/source/text()' + ), + builder.set_journal_title, + ), + ( + 'self::node()[@publication-type="book"]/source/text()', + builder.add_parent_title, + ), + ('./publisher-name/text()', builder.set_publisher), + ('./volume/text()', builder.set_journal_volume), + ('./issue/text()', builder.set_journal_issue), + ('./year/text()', builder.set_year), + ('./pub-id[@pub-id-type="arxiv"]/text()', builder.add_uid), + ('./pub-id[@pub-id-type="doi"]/text()', builder.add_uid), + ( + 'pub-id[@pub-id-type="other"]' + '[contains(preceding-sibling::text(),"Report No")]/text()', + builder.add_report_number + ), + ('./article-title/text()', builder.add_title), + ('../label/text()', lambda x: builder.set_label(x.strip('[].'))) + ] + + for xpath, field_handler in fields: + value = citation_node.xpath(xpath).extract_first() + citation_node.xpath(xpath) + if value: + field_handler(value) + + remainder = ( + remove_tags( + citation_node, + strip="self::person-group" + "|self::pub-id" + "|self::article-title" + "|self::volume" + "|self::issue" + "|self::year" + "|self::label" + "|self::publisher-name" + '|self::source[../@publication-type!="proc"]' + "|self::object-id" + "|self::page-range" + "|self::issn", + ) + .strip("\"';,. \t\n\r") + .replace("()", "") + ) + if remainder: + builder.add_misc(remainder) + + for editor in self.get_reference_authors(citation_node, 'editor'): + builder.add_author(editor, 'editor') + + for author in self.get_reference_authors(citation_node, 'author'): + builder.add_author(author, 'author') + + page_range = citation_node.xpath('./page-range/text()').extract_first() + if page_range: + page_artid = split_page_artid(page_range) + builder.set_page_artid(*page_artid) + + yield builder.obj + + def attach_fulltext_document(self, file_name, url): + self.builder.add_document(file_name, url, fulltext=True, hidden=True) diff --git a/inspire_utils/query.py b/inspire_utils/query.py index 374491e..75c7996 100644 --- a/inspire_utils/query.py +++ b/inspire_utils/query.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/record.py b/inspire_utils/record.py index e43f18a..ec7e7b7 100644 --- a/inspire_utils/record.py +++ b/inspire_utils/record.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/inspire_utils/utils.py b/inspire_utils/utils.py new file mode 100644 index 0000000..511d1fc --- /dev/null +++ b/inspire_utils/utils.py @@ -0,0 +1,139 @@ +from __future__ import ( + absolute_import, + division, + print_function, +) +import re +from scrapy.selector import Selector + + +RE_FOR_THE = re.compile( + r'\b(?:for|on behalf of|representing)\b', + re.IGNORECASE, +) +INST_PHRASES = ['for the development', ] + + +def get_node(text, namespaces=None): + """Get a scrapy selector for the given text node.""" + node = Selector(text=text, type="xml") + if namespaces: + for ns in namespaces: + node.register_namespace(ns[0], ns[1]) + return node + + +def coll_cleanforthe(coll): + """ Cleanup collaboration, try to find author """ + author = None + + if any(phrase for phrase in INST_PHRASES if phrase in coll.lower()): + # don't touch it, doesn't look like a collaboration + return coll, author + + coll = coll.strip('.; ') + + if RE_FOR_THE.search(coll): + # get strings leading and trailing 'for the' + (lead, trail) = RE_FOR_THE.split(coll, maxsplit=1) + if re.search(r'\w', lead): + author = lead.strip() + if re.search(r'\w', trail): + coll = trail + + coll = re.sub('(?i)^ *the ', '', coll) + coll = re.sub('(?i) *collaborations? *', '', coll) + coll = coll.strip() + + return coll, author + + +def split_fullname(author, switch_name_order=False): + """Split an author name to surname and given names. + + It accepts author strings with and without comma separation. + As default surname is first in case of comma separation, otherwise last. + Multi-part surnames are incorrectly detected in strings without comma + separation. + """ + if not author: + return "", "" + + if "," in author: + fullname = [n.strip() for n in author.split(',')] + surname_first = True + else: + fullname = [n.strip() for n in author.split()] + surname_first = False + + if switch_name_order: + surname_first = not surname_first + + if surname_first: + surname = fullname[0] + given_names = " ".join(fullname[1:]) + else: + surname = fullname[-1] + given_names = " ".join(fullname[:-1]) + + return surname, given_names + + +CONFERENCE_WORDS = [ + 'colloquium', + 'colloquiums', + 'conf', + 'conference', + 'conferences', + 'contrib', + 'contributed', + 'contribution', + 'contributions', + 'forum', + 'lecture', + 'lectures', + 'meeting', + 'meetings', + 'pres', + 'presented', + 'proc', + 'proceeding', + 'proceedings', + 'rencontre', + 'rencontres', + 'school', + 'schools', + 'seminar', + 'seminars', + 'symp', + 'symposium', + 'symposiums', + 'talk', + 'talks', + 'workshop', + 'workshops' +] + +THESIS_WORDS = [ + 'diploma', + 'diplomarbeit', + 'diplome', + 'dissertation', + 'doctoraal', + 'doctoral', + 'doctorat', + 'doctorate', + 'doktorarbeit', + 'dottorato', + 'habilitationsschrift', + 'hochschule', + 'inauguraldissertation', + 'memoire', + 'phd', + 'proefschrift', + 'schlussbericht', + 'staatsexamensarbeit', + 'tesi', + 'thesis', + 'travail' +] diff --git a/run-tests.sh b/run-tests.sh index f988587..8caf506 100755 --- a/run-tests.sh +++ b/run-tests.sh @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/setup.cfg b/setup.cfg index d99a59d..e846e1b 100644 --- a/setup.cfg +++ b/setup.cfg @@ -20,4 +20,4 @@ include = inspire_utils/*.py addopts = --cov=inspire_utils --cov-report=term-missing:skip-covered [flake8] -ignore = E501 FI12 FI14 FI15 FI16 FI17 FI50 FI51 FI53 W504 FI18 +ignore = E501 FI12 FI14 FI15 FI16 FI17 FI50 FI51 FI53 W504 W743 FI18 diff --git a/setup.py b/setup.py index 6f4a2b1..943f953 100644 --- a/setup.py +++ b/setup.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by @@ -39,7 +39,10 @@ 'nameparser~=1.1,>=1.1.3;python_version >= "3.6"', 'python-dateutil~=2.0,>=2.6.1', 'six~=1.0,>=1.10.0', - 'urllib3~=1.0,<=1.26.12' + 'urllib3~=1.0,<=1.26.12', + 'inspire_schemas', + 'scrapy', + 'pylatexenc', ] docs_require = [] @@ -48,16 +51,15 @@ 'flake8-future-import~=0.0,>=0.4.3', 'mock~=2.0,>=2.0.0', 'pytest-cov~=2.0,>=2.5.1', - 'pytest~=8.0,>=8.0.2', + 'pytest>=4.6.0;python_version <= "2.7"', + 'pytest>=8.0.2;python_version >= "3.6"', + 'unicode-string-literal>=1.1;python_version <= "2.7"', + 'deepdiff' ] extras_require = { 'docs': docs_require, 'tests': tests_require, - 'tests:python_version=="2.7"': [ - 'unicode-string-literal~=1.0,>=1.1', - 'pytest~=4.0,>=4.6.0', - ], } extras_require['all'] = [] diff --git a/tests/data/aps/PhysRevD.102.014505.xml b/tests/data/aps/PhysRevD.102.014505.xml new file mode 100644 index 0000000..7d60996 --- /dev/null +++ b/tests/data/aps/PhysRevD.102.014505.xml @@ -0,0 +1,3 @@ + + +
PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.102.014505ARTICLESLattice field theories, lattice QCDCharting the scaling region of the Ising universality class in two and three dimensionsCHARTING THE SCALING REGION OF THE ISING …MICHELE CASELLE AND MARIANNA SORBAhttps://orcid.org/0000-0001-5488-142XCaselleMichele1https://orcid.org/0000-0002-7582-0121SorbaMarianna1,2,*Department of Physics, University of Turin and INFN, Turin, Via Pietro Giuria 1, I-10125 Turin, ItalySISSA and INFN, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy

msorba@sissa.it

10July20201July202010210145055June202018June2020Published by the American Physical Society2020authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

We study the behavior of a universal combination of susceptibility and correlation length in the Ising model in two and three dimensions, in presence of both magnetic and thermal perturbations, in the neighborhood of the critical point. In three dimensions we address the problem using a parametric representation of the equation of state. In two dimensions we make use of the exact integrability of the model along the thermal and the magnetic axes. Our results can be used as a sort of “reference frame” to chart the critical region of the model. While our results can be applied in principle to any possible realization of the Ising universality class, we address in particular, as specific examples, three instances of Ising behavior in finite temperature QCD related in various ways to the deconfinement transition. In particular, in the last of these examples, we study the critical ending point in the finite density, finite temperature phase diagram of QCD. In this finite density framework, due to the well-known sign problem, Monte Carlo simulations are not possible and thus a direct comparison of experimental results with quantum field theory & statistical mechanics predictions like the one we discuss in this paper may be important. Moreover in this example it is particularly difficult to disentangle “magnetic-like” from “thermal-like” observables and thus an explicit charting of the neighborhood of the critical point can be particularly useful.

INTRODUCTION

Despite its apparent simplicity the Ising model is one of the cornerstones of modern statistical mechanics. Over the years it has become a theoretical laboratory to test new ideas, ranging from symmetry breaking to conformal field theories. Moreover, thanks to its exact solvability in two dimensions [1,2] and the ease with which it can be simulated in three dimensions, it has been widely used as a benchmark to test new numerical approaches and innovative approximations.

The corresponding universality class, in the renormalization group sense [3], is of central importance in theoretical physics due to its many experimental realizations in different physical contexts, ranging from condensed matter to high energy physics. At the same time it describes the critical behavior of a lot of different spin models and, in the limit of high temperatures, also of gauge theories.

From a statistical mechanics point of view it represents the simplest way to describe systems with short-range interactions and a scalar order parameter (density or uniaxial magnetization) which undergo a symmetry breaking phase transition. From a quantum field theory (QFT) point of view it is the simplest example of a unitary conformal field theory (CFT) [4] perturbed by only two relevant operators: the “spin” operator (which is Z2 odd) and the “energy” operator (Z2 even) [5].

Thanks to integrability, conformal perturbation, and bootstrap [6,7] lots of results are known, both in two and in three dimensions, on the behavior of the model at the critical point, or when only one of the two perturbing operators is present. However, typically, the interesting regime for most of the experimental realizations of the model is when both the perturbing operators are present and much less is known in this situation.

The aim of this paper is to partially fill this gap by studying a suitable universal combination of thermodynamic quantities (see below for the precise definition) in the presence of both perturbing operators. In three dimensions we shall address the problem using a parametric representation of the equation of state [8], while in two dimensions we shall make use of the exact integrability of the model in presence of a single perturbation [5]. Using these tools we shall be able to predict the value of this quantity in the whole phase space of the model in the neighborhood of the critical point. These values can be used as a sort of “reference frame” to chart the critical region of the model.

The universal combination that we shall study involves the magnetic susceptibility and thus our proposal is particularly effective when the model is characterized by an explicit Z2 symmetry. When this is not the case, like for the liquid-vapor transition or for the finite density QCD example that we shall discuss below, the explicit knowledge of our universal combination may help to identify the exact directions in the phase space of the model with respect to which the magnetic susceptibility must be evaluated.

Thanks to universality, our results hold not only for the standard nearest neighbor Ising model, but also for any possible realization of the Ising universality class and in fact we shall use the high precision Monte Carlo estimates obtained from an improved version of the Ising model to benchmark and test our results [9–15].

In particular, we shall concentrate in the second part of the paper on realizations in the context of high energy physics, suggested by the lattice regularization of QCD. We shall discuss three instances of Ising behavior in finite temperature QCD related in various ways to the deconfinement transition. In the last of these examples, we shall address the critical ending point of finite density QCD. In this case, due to the well-known sign problem, Monte Carlo simulations are not possible and thus a direct comparison of experimental results with QFT/statistical mechanics predictions like the one we discuss in this paper may be important.

This paper is organized as follows. Section II is devoted to a general introduction to the model and to the universal combination of thermodynamic quantities which is the main subject of the paper. In Sec. III we shall address the problem in three dimensions using a suitable parametric representation of the equation of state of the model. We shall also show that the same approach cannot be used in two dimensions. In Sec. IV we shall then address the two-dimensional case using appropriate expansions around the exact solutions of the model. Finally Sec. V will be devoted to the discussion of a set of examples in high temperature QCD. We collected in the Appendices some additional material which may be useful to reproduce our numerical analysis.

GENERAL INFORMATION ON THE ISING UNIVERSALITY CLASS

The Ising model has a global Z2 symmetry and is characterized by two relevant operators which encode the Z2 odd (σ) and Z2 even (ε) perturbations of the critical point.

From a QFT point of view, the model in the vicinity of the critical point can be written as a perturbed conformal field theory S=SCFT+t∫ddxε(x)+H∫ddxσ(x),where ε(x) and σ(x) are the energy and spin operators and represent the continuum limits of the lattice operators ∑⟨ij⟩σiσj and ∑iσi respectively. These operators are conjugated to the reduced temperature t=1Tc(T-Tc) and magnetic field H, which measure the deviation from the critical point. The action SCFT is the conformal-invariant action of the model at the critical point. In two dimensions this is the action of a free massless Majorana fermion with central charge c=1/2.

Thanks to the exact integrability of the model for H=0 (pure thermal perturbation) and for t=0 (pure magnetic perturbation) much is known of this QFT in two dimensions. In particular all the critical exponents and the universal ratios are known exactly and, as we shall see below, reliable expansions around the integrable lines can be constructed for several observables.

In three dimensions there are not exact results, but from the recent progress of the bootstrap approach and the improvement of Monte Carlo methods several universal quantities can be evaluated with very high precision.

The most important realization of this QFT is the spin Ising model on a cubic (in d=3) or square (in d=2) lattice, which we shall use in the following to fix notations. As it is well known, the model is defined by the following energy function, E({σi})=-J∑⟨ij⟩σiσj-H^∑i=1Nσi,where the spins σi can take the values σi=±1, the index i labels the sites of the lattice, the symbol ⟨ij⟩ means that the sum is performed over pairs of nearest neighbor sites, J is the coupling strength between spins (we assume a positive isotropic interaction so that for all pairs of nearest neighbor spins Jij=J>0), and H^ is the external magnetic field.

The partition function of the model is Z=∑{σi}e-1kBTE({σi}).Let us define β=J/kBT and H=H^/kBT. For H=0 the model is explicitly Z2 symmetric and is characterized by two phases, a low temperature phase in which the Z2 symmetry is spontaneously broken and a spontaneous magnetization is present and a high temperature phase in which the Z2 symmetry is restored. The two phases are separated by a critical point. If one switches on the magnetic field it becomes apparent that the low T phase is actually a line of first order phase transitions which ends with the critical point. In the following we shall be interested in the scaling region in the vicinity of this critical point. Standard renormalization group arguments tell us that in this limit the irrelevant operators of the model can be neglected and the behavior is completely described only by the two relevant operators ε, σ and one can perform a continuum limit of the model which leads exactly to the QFT described by Eq. (1).

From the partition function defined above it is easy to obtain all the thermodynamic observables. In particular, following the standard notation we have for the magnetization and the magnetic susceptibility M=-∂log(Z)∂H,χ=-∂2log(Z)∂H2=∂M∂H.The exponential correlation length can be extracted from the large distance decay of the spin-spin connected correlator as ⟨σ(x)σ(0)⟩c∼e-|x|/ξ|x|→+∞,where ⟨σ(x)σ(0)⟩c≡⟨σ(x)σ(0)⟩-⟨σ(0)⟩2.

In several practical applications also useful is the so-called second-moment correlation length which is defined through the second moment of the spin-spin correlation function as ξ2≡[12d∫ddx|x|2⟨σ(x)σ(0)⟩c∫ddx⟨σ(x)σ(0)⟩c]1/2,and it is simpler to evaluate than ξ both in numerical simulations and in experiments.

In the scaling limit the critical behavior of all thermodynamic quantities is controlled by the two “scaling exponents” xε and xσ which are universal and are shared by all physical realizations of the Ising universality class. The corresponding amplitudes are not universal, but one can construct suitable combinations in which the nonuniversal features of the model cancel out (see Appendix A) and represent testable predictions of the Ising QFT to be compared with any possible realization of the Ising universality class.

While this is a well studied subject when only a single perturbation is present, its extension to the whole scaling region of the model, where both the H and t perturbations are present, is not straightforward.

The main goal of this paper is to show that such an extension can be easily obtained making use of a parametric representation of the model and that the resulting universal quantities can be used as a natural reference frame to chart the scaling region of the Ising universality class.

While the parametric approach is completely general and could be applied in principle to any universal combination of thermodynamic quantities, in this paper we shall study in particular the following ratio, Ω=(χ(t,H)Γ-)(ξ-ξ(t,H))γ/ν,and its natural extension to the second moment correlation length Ω2=(χ(t,H)Γ-)(ξ2,-ξ2(t,H))γ/ν,where Γ-, ξ-, and ξ2,- denote the amplitudes of χ, ξ, and ξ2 along the t<0, H=0 axis (see Appendix A for detailed definitions and normalizations).

This choice is motivated by the fact that the two observables which appear in the ratio are rather easy to evaluate, both in numerical simulations and in experiments, since they only involve derivatives or correlations of the order parameter and are normalized with respect to the values they have along the critical line of first order phase transitions, which is easy to identify (again, both numerically and experimentally).

The main drawback of this choice is that it assumes an explicit realization of the Z2 symmetry. While in many interesting applications, like for the liquid-vapor transition in which the role of the perturbing parameter is played by the density, this is not the case and the Z2 symmetry is just an “emergent” symmetry. The typical approach in these cases, following Rehr and Mermin [16], is to realize the t, H perturbations as suitable linear combinations of the actual variables of the model.

In the scaling region, when both the relevant perturbations are present, all the thermodynamic observables depend on the scaling combination

Notice that our definition of η differs from that of Ref. [5] by a factor 2π.

η≡t|H|d-xεd-xσ=t|H|1βδ.

The three limits in which only one of the two perturbations is present (H=0, t<0), (H≠0, t=0), and (H=0,t>0) correspond respectively to η=-∞, η=0, and η=+∞. In these limits Ω can be written in terms of the standard universal amplitude ratios Q2, Γ+/Γ-, and ξ-/ξ+ (see Appendix A) as follows: Ω(η)=1,η=-∞,Ω(η)=1Q2(Γ+Γ-)(ξ-ξ+)γ/ν,η=0,Ω(η)=(Γ+Γ-)(ξ-ξ+)γ/ν,η=+∞.

These values can be used as benchmarks to test the reliability of our estimates and as “anchors” of the reference frame we are constructing.

PARAMETRIC REPRESENTATION

It is useful to introduce a parametric representation of the critical equation of state that not only satisfies the scaling hypothesis but additionally allows a simpler implementation of the analytic properties of the equation of state itself. Following [17], we express the thermodynamic variables t, H in terms of a couple of parameters R, θ both positive. Intuitively, the first measures the distance form the critical point in the (t,H) plane while the latter corresponds to the angular displacement along lines of constant R around the critical point. The parametrization of t and H results in a parametric expression of M as well, explicitly {M=m0Rβθ,t=R(1-θ2),H=h0Rβδh(θ).

Calling θ0>1 the smallest positive zero of the function h(θ), we see from the system [Eq. (11)] that the domain of interest in the (R,θ) plane is 0≤θ≤θ0 for every R≥0 and that θ=θ0 corresponds to the H=0, t<0 axis, θ=1 to the H≠0, t=0 axis, and θ=0 to the H=0, t>0 one.

The key point of the whole analysis is the determination of the function h(θ). There are some general properties which h(θ) must satisfy. It must be analytic in this physical domain in order to satisfy the regularity properties of the critical equation of state, i.e., the so-called Griffiths analyticity [18]. Moreover, it must be an odd function of θ because of the Z2 symmetry of the system. The most general choice is thus a polynomial of the type h(θ)=θ+∑n=1kh2n+1θ2n+1.Using standard QFT methods [8,19–21] one can extract the coefficients h2n+1 from a high temperature expansion of the free energy of the model. Using a variational method it is possible to obtain reliable and stable estimates of the coefficients up to h7 in three dimensions and to h11 in two dimensions. We can use the known amplitude ratios as benchmarks to evaluate the reliability of these parametric representations. As we will see h7 will be enough to obtain estimates which in three dimensions agree within the errors with the amplitude ratios. The situation is worse in two dimensions, and this will prompt us to address the 2D case with a different approach.

A similar parametric representation can be introduced also for the correlation length. Following [22] we parametrize the square mass of the underlying QFT, i.e., ξ-2 as follows, ξ-2=R2νa0(1+cθ2),and similarly, for the second moment correlation length ξ2-2=R2ν(a0)2nd(1+c2θ2).

The constants c and c2 can be fixed using the universal ratios ξ+/ξ- and ξ2,+/ξ2,- respectively and we use then the Q2 and (Q2)2nd ratios to test the parametric representation.

Truncating at the quadratic order Eqs. (13) and (14) could seem a too drastic approximation, but we will see below that in d=3 it gives quite good results. The same is not true in d=2 where, however, as we anticipated above, we shall use a different approach to evaluate Ω(η).

Using the above results we may construct a parametric representation of Ω as a function of θ Ω(θ)=Ω0(1-θ2+2βθ2)(1+cθ2)γ2ν2βδθh(θ)+(1-θ2)h′(θ)with Ω0=(1-θ02)h′(θ0)(1-θ02+2βθ02)(1+cθ02)γ2νand a similar expression for Ω2(θ) with c→c2.

There are a few universal features of Ω that we may deduce from this expression and hold for any possible realization of the Ising universality class:

Ω is a monotonic decreasing function of θ [see Fig. 1 for a plot of Ω(θ) in d=3], and therefore it can be inverted. From any given experimental estimate of Ω a precise value of θ may be extracted.

Using Ω we may identify the critical isothermal line, which corresponds to Ωiso≡Ω(θ=1) (Ωiso=1.08376… in d=3).

There is a maximum value of Ω which corresponds to Ωmax=Ω(θ=0)=Ω0 (Ωmax=1.35502… in d=3).

110.1103/PhysRevD.102.014505.f1

Result for Ω(θ) according to Eq. (15) with 0≤θ≤θ0 in d=3.

In practical applications one may be interested in the expression of Ω as a function of η=t/|H|1βδ. It is useful to introduce a universal version of the scaling variable defined as η˜=(h0)1βδη.

Using the parametric representation of Eq. (11) it is possible to write the expansion of θ as a function of η˜ in the neighborhood of the three singular points η˜=(±∞,0) and then Ω as a function of η˜, which we plot in Fig. 2 in the d=3 case (see Appendix B for more details on this expansion).

210.1103/PhysRevD.102.014505.f2

Result for Ω(η˜) in the three limits of η˜ given in Eqs. (B8), (B9), and (B10) in d=3.

Results in <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>

In three dimensions we have [8,20] h(θ)=θ+h3θ3+h5θ5+h7θ7+O(θ9)=θ-0.736743θ3+0.008904θ5-0.000472θ7+O(θ9),and the resulting value for the smallest positive root is θ02=1.37861.

Inserting these values into Eq. (B4) we find Γ+/Γ-∼4.76, which is in good agreement (with a difference of the order of 1%) with the Monte Carlo estimate Γ+/Γ-=4.714(4) reported in [11,13].

Using the known values of ξ+/ξ- and ξ2,+/ξ2,- we then obtain c=0.0416 and c2=0.0776. To check the reliability of this parametric representation we may use them to predict Q2 and (Q2)2nd: we end up with Q2∼1.250 and (Q2)2nd∼1.209, which, again, are in agreement (with a difference of the order of 3%) with the best numerical estimates reported in Eqs. (A8) and (A9).

This tells us that we can trust the parametric representation of Ω(θ) discussed above. We plot the result in Fig. 1.

We find in particular Ωiso=1.08376… and Ωmax=1.35502…. In Fig. 2 we plot the expression of Ω as a function of η˜.

Results in <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>

In two dimensions the situation is not as good as in d=3. We have [21] h(θ)=θ+h3θ3+h5θ5+h7θ7+h9θ9+h11θ11+O(θ13)=θ-1.07745θ3+0.146609θ5+0.0224263θ7+0.00549457θ9+0.00612906θ11+O(θ13),and the smallest positive zero is θ02=1.16951.

From this we find Γ+/Γ-∼39.63, which is 5% away from the exact value: Γ+/Γ-=37.6936520… [23]. For the correlation length we get, from the exact values of ξ+/ξ- and ξ2,+/ξ2,-, the following estimates: c=-0.75678… and c2=-0.60933…. Plugging these values into the expression for Q2 and (Q2)2nd we find for instance Q2=5.3342… and (Q2)2nd=3.52360, which are rather far from the exact values Q2=3.23513834… and (Q2)2nd=2.8355305… reported in [23]. This prompted us to address the study of the behavior of Ω(η) in d=2 with a different approach.

EXACT EXPRESSION FOR <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> IN THE <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula> CASE

In d=2, one can obtain much more precise results performing a perturbative expansion around the exact solutions along the two axes H=0 and t=0. A powerful tool to study the free energy of a perturbed CFT is the well-known truncated conformal space approach (TCSA) [24,25]. In our case, thanks to the exact mapping of the H=0 model to the QFT of a free Majorana fermion it is possible to construct a more effective version of the TCSA which uses the free fermions as a basis, the “truncated free-fermion space approach” [5]. With this approach it is possible to evaluate the free energy for almost all values of η [5,26–28] and from that of the susceptibility. With similar methods it is also possible to study the perturbed mass spectrum of the theory [26–28] and hence the correlation length ξ which is the inverse of the lowest mass of the spectrum M1. The resulting expansions for χ and M1, in the three singular limits, are reported in Appendix C.

Combining these quantities we obtain a precise estimate for Ω, which we plot in Fig. 3. The resulting polynomial expansions in the three regions of interest are Ω(η)=∑nΩn-(-η)58n,η→-∞,Ω(η)=∑nΩn0ηn,η∼0,Ω(η)=∑nΩn+η58n,η→+∞.

310.1103/PhysRevD.102.014505.f3

Result for Ω(η) in the three limits of η given in Eq. (20) in d=2.

The coefficients are reported in Table I.

I10.1103/PhysRevD.102.014505.t1

Expansion coefficients of Ω(η) in the three regimes of interest, according to Eq. (20).

nΩn-Ωn0Ωn+013.463 9611.20641⋯9.078⋯20.400 3121.23⋯3-0.075 963⋯⋯40.025 182⋯⋯5-0.030 408 9⋯⋯60.006 086⋯-0.094 337-0.001 91⋯⋯80.004 10⋯⋯9-0.001 035⋯⋯

As expected, the three limiting cases Ω(±∞),Ω(0) (which correspond to the n=0 values in the table) agree with the universal values obtained plugging into Eq. (10) the universal ratios quoted in Eq. (A5).

THREE EXAMPLES OF ISING BEHAVIOR IN FINITE TEMPERATURE QCD

Among the many physical realizations of the Ising universality class in this paper we decided to focus on three examples taken from high energy physics and in particular from the lattice regularization of QCD at finite temperature.

The deconfinement transition in the <inline-formula><mml:math display="inline"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> pure gauge theory

The most direct realization of the Ising universality class in lattice gauge theories (LGTs) is given by the deconfinement transition of pure gauge theories with a symmetry group G which has Z2 as center. This result is based on the Svetitsky-Yaffe approach [29] to the study of finite temperature (d+1)-dimensional pure gauge theories. The main idea of [29] is to construct a d-dimensional effective theory from the original one by integrating out the spacelike links and keeping as only remaining degrees of freedom the Polyakov loops. These loops are then treated as spins of an effective d-dimensional model whose global symmetry must coincide with the center of the original gauge group. If both the phase transitions of the original gauge theory and that of the effective spin model are continuous, they must belong to the same universality class, and one can use the effective model (which is usually much simpler than the original gauge theory) to extract informations on the deconfinement transition of the original model. These results are very general: if in particular we focus on gauge theories with a gauge group G whose center is Z2 [like for instance, Z2 itself, SU(2), or Sp(2N)], the deconfinement transition will belong to the Ising universality class. A well studied example of this correspondence is the SU(2) model [30–32]. Even if the gauge group is not SU(3) and the model only contains gluonic degrees of freedom this simplified model shares a lot of properties with QCD: the presence of a confining flux tube at low temperatures, a deconfined phase at high temperature, a rich glueball spectrum, asymptotic freedom. For these reasons it has been studied a lot in the past both in (2+1) [33–35] and in (3+1) [36] dimensions. Thanks to the Svetitsky-Yaffe construction, several gauge-invariant observables of the SU(2) model can be mapped to equivalent Ising observables:

The Polyakov loop is mapped to the spin (Z2 odd) operator and thus the Polyakov loop susceptibility is mapped to the magnetic susceptibility χ of the Ising model.

The deconfining transition of the gauge model corresponds to the magnetization transition of the Ising model. In particular, the confining phase (low temperature of the gauge model) is mapped to the Z2 symmetric phase (high temperature phase) of the Ising model, while the deconfined phase is mapped to the broken symmetry phase of the Ising model.

The Wilson action (i.e., the trace of the ordered product of the gauge field along the links of a plaquette) is mapped to the energy operator

Actually it is mapped to the most general Z2 even Ising observable, which is a mixture of the identity and energy operators, but the identity operator plays no role in this context.

of the Ising model.

The screening mass of the gauge model in the deconfined phase is mapped to the mass of the Ising model in the low temperature phase, while Ntσ, where σ is the string tension and Nt is the inverse temperature of the gauge model (i.e., the size Nt of the lattice in the compactified direction which defines the finite temperature setting in LGTs) is mapped to the inverse of the high temperature correlation length of the Ising model.

This mapping has been widely used in the past years to predict the behavior of various physical observables of the gauge model near the deconfinement transition, like for instance the short distance behavior of the Polyakov loop correlator [37], the width of the flux tube [38], the Hagedorn-like behavior of the glueball spectrum [39] or the behavior of the universal ξ/ξ2 ratio [40]. More generally the mapping allows one to relate all the thermodynamic observables of the two models and in particular also Ω(η), which can thus be evaluated in the SU(2) LGT and then compared with the QFT prediction discussed in the previous section.

QCD with dynamical quarks and the Columbia plot

The situation is different if we study full QCD, i.e., if we include dynamical quarks in the model. In this case the center symmetry is explicitly broken by the Dirac operator and all the above considerations do not hold anymore. For physical values of the quark masses there is no phase transition between the high temperature quark-gluon plasma phase and the low temperature confined phase which are only separated by a smooth crossover [41,42]. However, in the phase diagram of the model one finds a rich structure of phase transitions as the values of the masses of the quarks are varied [43]. This pattern of phase transitions is summarized in the well-known Columbia plot, which we report here in Fig. 4. The plot describes the nature of the phase transitions as a function of the quark masses. On the two axes are respectively the masses of up-down quarks (x axis) and the mass of the strange quark (y axis). We see in the central part of the plot a wide region, where the physical point lies, in which there is no phase transition but only a crossover between the two phases. In the top-right and in the bottom left corners there are two regions where the transition is of the first order. These regions end with two lines of second order transitions which are expected to both be of the Ising type.

410.1103/PhysRevD.102.014505.f4

Columbia plot.

The phase diagram reported in the Columbia plot can be studied with standard Monte Carlo simulations, and in the vicinity of the critical lines the results of these simulations could be mapped using our tools to the Ising phase diagram and then compared with the Ising predictions as we discussed above for the simpler case of the pure gauge SU(2) model.

It is important to notice that the two critical regions are of a different nature. The one in the top-right corner is a deconfinement transition similar to the one discussed in the previous section. In fact in the limit of infinite mass quarks the model becomes a pure gauge theory. In this limit the SU(3) gauge model has a first order deconfinement transition [differently from the SU(2) one discussed above which is of second order]. As the mass of the quarks decreases the gap in the order parameter decreases and the first order region ends into a critical line of Ising-like phase transitions.

The critical region in the bottom left portion of the Columbia plot has a completely different origin. It describes the restoration of the chiral symmetry in QCD at finite temperature and small quark masses. In QCD with three massless quark flavors the chiral phase transition is expected to be first order and to remain of first order even for small but nonzero values of the quark masses. As the quark masses increase, the gap in the order parameter decreases and the first order region terminates in a critical line of the Ising type. Above this line chiral symmetry is restored through a smooth crossover. In this case, the Z2 symmetry is an emergent symmetry and the identification of the H and t axes of the equivalent 3D Ising model is not trivial (see a discussion on this issue in [44,45] and in Sec. V C) and thus a universal charting of the scaling region could indeed be useful.

Even if the precise location of the critical line is still debated, it seems that the physical point is not too far from this bottom left Ising line. If this is the case, then our analysis could be applied, maybe with some degree of approximation, also to the physical point.

The critical ending point of the QCD phase diagram at finite chemical potential

The most interesting application of our results is for QCD at finite baryon density, which is realized by adding a finite chemical potential μ to the QCD Lagrangian. This regime is particularly interesting since it can be explored experimentally in heavy-ion collisions [46–53], and at the same time it cannot be studied using Monte Carlo simulations due to the well-known sign problem (see for instance [54,55] for a discussion of the sign problem in this context).

In this regime the QCD phase diagram is expected to reveal interesting novel phases [56,57]. In particular it is widely expected that the hadronic phase (low T, low μ) should be separated from the quark-gluon plasma phase (high T, high μ) by a line of first order transitions with a critical end point at finite critical values of T and μ (see Fig. 5) which should again belong to the Ising universality class [58–65].

510.1103/PhysRevD.102.014505.f5

QCD phase diagram at finite chemical potential.

Also for this model, as for the liquid-vapor transition (or the chiral transition discussed above), the Z2 symmetry is not realized explicitly but it is just an emergent symmetry. This class of models is typically addressed with the “mixing-of-coordinates" scheme proposed in [16]. The approach was pursued for the finite density QCD case in [61,63], leading to very interesting results. In both cases the mapping between Ising and QCD coordinates was performed via the parametric representation of the equation of state. In this respect, the explicit expression of Ω in terms of θ that we discussed in this paper could be used as a shortcut in the process and could facilitate the identification of Ising-like behaviors in the experimental data.

As more and more experimental results are obtained, it will become possible to directly test them with universal predictions from the Ising model and it will be important to have a precise charting of the Ising phase diagram to organize results and drive our understanding of strongly coupled QCD in this regime. Our paper is a first step in this direction. We proposed and studied one particular combination, chosen for its simplicity from a theoretical point of view and its accessibility from an experimental and numerical point of view, but other combinations are possible and could be studied using, as we suggest here, parametric representations in d=3 and/or expansion around the exactly integrable solutions in d=2.

ACKNOWLEDGMENTS

We thank C. Bonati, M. Hasenbusch, and M. Panero for useful comments and suggestions.

CRITICAL AMPLITUDES AND UNIVERSAL AMPLITUDE RATIOS

We list below the scaling behavior of the observables used in the main text: ξ≈ξ+t-ν,ξ2≈ξ2,+t-ν,t>0,H=0,ξ≈ξ-(-t)-ν,ξ2≈ξ2,-(-t)-ν,t<0,H=0,ξ≈ξc|H|-νc,ξ2≈ξ2,c|H|-νc,t=0,H≠0,χ≈Γ+t-γ,t>0,H=0,χ≈Γ-(-t)-γ,M≈B(-t)β,t<0,H=0,χ≈Γc|H|-γc,M≈Bc|H|1δ,t=0,H≠0,where the critical indices are defined in terms of the scaling exponents as follows: β=xσ(d-xε),δ=(d-xσ)xσ,γ=(d-2xσ)d-xε,ν=1d-xε,γc=(d-2xσ)(d-xσ),νc=1(d-xσ).

From these definitions it is possible to construct the following universal amplitude ratios: Γ+Γ-,ξ+ξ-,Q2=(Γ+Γc)(ξcξ+)γ/ν,ξ2,+ξ2,-,(Q2)2nd=(Γ+Γc)(ξ2,cξ2,+)γ/ν.

Exact results for the amplitude ratios in <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>

In d=2, thanks to the exact integrability of the two relevant perturbations all the above universal quantities are known exactly [23]: xε=1,xσ=18,Γ+Γ-=37.6936520…,ξ+ξ-=2,Q2=3.23513834…,ξ2,+ξ2,-=3.16249504…,(Q2)2nd=2.8355305….

Numerical estimates of the amplitude ratios in <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math></inline-formula>

In three dimensions there are no exact results but, thanks to the recent improvement of the bootstrap approach [6,7,66] and to the remarkable precision of recent Monte Carlo simulations [11,13,15], reliable numerical estimates for all these quantities exist: xε=1.412625(10),xσ=0.5181489(10),Γ+Γ-=4.714(4),ξ+ξ-=1.896(3),Q2=1.207(2),ξ2,+ξ2,-=1.940(2),(Q2)2nd=1.179(2).

It is also possible to choose realizations of the model in which the amplitude of the first irrelevant operator is tuned toward zero, thus allowing a more efficient approach to the fixed point. This is for instance the idea followed in [9–15] to improve the numerical estimates of universal quantities using Monte Carlo simulations.

USEFUL RESULTS IN THE PARAMETRIC REPRESENTATIONFixing the nonuniversal constant <inline-formula><mml:math display="inline"><mml:msub><mml:mi>h</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math></inline-formula>

The scaling parameter η is defined modulo a nonuniversal constant h0 which sets its scale and depends on the specific model at hand, i.e., on the specific realization of the Ising universality class in which one is interested. However, given such a realization, it is rather easy to fix h0. The simplest option is to measure (numerically or experimentally) the magnetization M and the susceptibility χ along two directions (or in the same direction) in the (t,H) plane, for instance along the critical line of first order phase transitions. Then from the ratio of the two amplitudes one can easily extract h0. We report here for completeness the corresponding expressions in the case in which one measures, besides the amplitude B of the magnetization, the value of Γ+, or that of Γ-, h0=BΓ+(θ02-1)βθ0or h0=-BΓ-(θ02-1)γ+β-1(1-θ02+2βθ02)θ0h′(θ0).

For instance, in the case of the 3D Ising model defined by Eqs. (2) and (3) we have [67] B=1.6920(5) and Γ-=0.2394(5) to which corresponds h0∼0.923, while in the case of the model tuned so as to eliminate the first irrelevant operator discussed above we find [13] B=1.9875(3) and Γ+=0.14300(5), to which corresponds h0∼8.62.

Parametric representation of the magnetic susceptibility

From the parametric representation of the critical equation of state [Eq. (11)] we may obtain the magnetic susceptibility as follows: χ(R,θ)=(m0h0)R-γ1-θ2+2βθ22βδθh(θ)+(1-θ2)h′(θ),and from this expression it is easy to extract the amplitude ratio Γ+Γ-=-(θ02-1)1-γh′(θ0)(1-θ02+2βθ02).

Expansion of <inline-formula><mml:math display="inline"><mml:mi>θ</mml:mi></mml:math></inline-formula> as a function of <inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>η</mml:mi><mml:mo accent="true" stretchy="false">˜</mml:mo></mml:mover></mml:math></inline-formula> in the neighborhood of the three singular points <inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>η</mml:mi><mml:mo accent="true" stretchy="false">˜</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mo>±</mml:mo><mml:mi>∞</mml:mi><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

We report here the first few terms: in the θ→θ0 limit θ(η˜)=θ0+(θ02-1)βδh′(θ0)(1-η˜)βδ+[2βδθ0(θ02-1)2βδ(θ02-1)(h′(θ0))2-(θ02-1)2βδh′′(θ0)2(h′(θ0))3(1-η˜)2βδ+O[(1-η˜)3βδ],in the θ→1 limit θ(η˜)=1-12(h(1))1βδη˜+2h′(1)-βδh(1)8βδh(1)(h(1))2βδη˜2+O(η˜3),and finally in the θ→0 limit θ(η˜)=0+(1η˜)βδ-h′′(0)2(1η˜)2βδ+O[(1η˜)3βδ].

Expansions of <inline-formula><mml:math display="inline"><mml:mi mathvariant="normal">Ω</mml:mi></mml:math></inline-formula> as a function of <inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>η</mml:mi><mml:mo accent="true" stretchy="false">˜</mml:mo></mml:mover></mml:math></inline-formula>

Plugging the above expansions into the expression for Ω(θ) we find

We report here only the first term for the three expansions to avoid too complex formulas; it is straightforward to obtain the next orders.

for Ω(η˜): in the η˜→-∞ limit Ω(η˜)=1+θ0(θ02-1)βδh′(θ0)⁢[cγν(1+cθ02)-4β[1+θ02(2β-1)](θ02-1)+2βδ(θ02-1)-h′′(θ0)θ0h′(θ0)](1-η˜)βδ+O[(1-η˜)2βδ],in the η˜→0 limit Ω(η˜)=Ω0(1+c)γ2νδh(1)+Ω0(1+c)γ2ν2δh(1)⁢[h′(1)(βδ-1)-δh(1)(β-1)βδh(1)-cγν(1+c)](h(1))1βδη˜+O(η˜2),and finally in the η˜→+∞ limit Ω(η˜)=Ω0-Ω0h′′(0)(1η˜)βδ+O[(1η˜)2βδ].

Upon substitution c→c2 the same results are valid for Ω2(η˜).

EXACT RESULTS IN <inline-formula><mml:math display="inline"><mml:mi>d</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula>

We report here the expansions, in the three regimes of interest, for χ and for M1≡1/ξ, obtained using the results reported in [5,26–28]: χ(η)=(-t)-74∑nχn-(-η)158n,η→-∞,χ(η)=|H|-1415∑nχn0ηn,η∼0,χ(η)=t-74∑nχn+η158n,η→+∞,1ξ(η)=(-t)∑nmn-(-η)54n,η→-∞,1ξ(η)=|H|815∑nmn0ηn,η∼0,1ξ(η)=t∑nmn+η54n,η→+∞,where we have already substituted the known values of the critical indices of the model. The coefficients are reported in Tables II and III.

II10.1103/PhysRevD.102.014505.t2

Expansion coefficients of the magnetic susceptibility in the three regimes of interest, according to Eq. (C1).

nχn-χn0χn+00.003 926 870.085 172 10.148 001 231-2.982 96×10-40.498 574⋯23.349 04×10-51.675 52-0.004 074 283-4.7920×10-63.331 85⋯48.2208×10-70.907 03 1.1818×10-45-1.635×10-7-20.939⋯63.694×10-8-85.959-3.4330×10-67-9.335×10-9-169.4⋯82.610×10-9-10.329.951×10-89-7.995×10-101162⋯102.67×10-104288-2.87×10-911-9.62×10-11⋯⋯123.74×10-11⋯8.34×10-11III10.1103/PhysRevD.102.014505.t3

Expansion coefficients of the correlation length in the three regimes of interest, according to Eq. (C2).

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diff --git a/tests/data/aps/PhysRevD.102.014505_expected.yml b/tests/data/aps/PhysRevD.102.014505_expected.yml new file mode 100644 index 0000000..c720c62 --- /dev/null +++ b/tests/data/aps/PhysRevD.102.014505_expected.yml @@ -0,0 +1,1962 @@ +abstract: 'We study the behavior of a universal combination of susceptibility and correlation length in the Ising model in two and three dimensions, in presence of both magnetic and thermal perturbations, in the neighborhood of the critical point. In three dimensions we address the problem using a parametric representation of the equation of state. In two dimensions we make use of the exact integrability of the model along the thermal and the magnetic axes. Our results can be used as a sort of “reference frame” to chart the critical region of the model. While our results can be applied in principle to any possible realization of the Ising universality class, we address in particular, as specific examples, three instances of Ising behavior in finite temperature QCD related in various ways to the deconfinement transition. In particular, in the last of these examples, we study the critical ending point in the finite density, finite temperature phase diagram of QCD. In this finite density framework, due to the well-known sign problem, Monte Carlo simulations are not possible and thus a direct comparison of experimental results with quantum field theory & statistical mechanics predictions like the one we discuss in this paper may be important. Moreover in this example it is particularly difficult to disentangle “magnetic-like” from “thermal-like” observables and thus an explicit charting of the neighborhood of the critical point can be particularly useful.' +copyright_holder: authors +copyright_statement: Published by the American Physical Society +copyright_year: 2020 +document_type: article +license_url: https://creativecommons.org/licenses/by/4.0/ +license_statement: Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. +article_type: research-article +journal_title: Physical Review D +material: publication +publisher: American Physical Society +year: 2020 +authors: +- full_name: Caselle, Michele + raw_affiliations: + - value: Department of Physics, University of Turin and INFN, Turin, Via Pietro + Giuria 1, I-10125 Turin, Italy + source: American Physical Society + ids: + - schema: ORCID + value: 0000-0001-5488-142X +- full_name: Sorba, Marianna + ids: + - schema: ORCID + value: 0000-0002-7582-0121 + raw_affiliations: + - value: Department of Physics, University of Turin and INFN, Turin, Via Pietro + Giuria 1, I-10125 Turin, Italy + source: American Physical Society + - value: SISSA and INFN, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy + source: American Physical Society +artid: '014505' +title: Charting the scaling region of the Ising universality class in two and three dimensions +number_of_pages: 12 +dois: +- material: publication + doi: 10.1103/PhysRevD.102.014505 +journal_volume: '102' +journal_issue: '1' +is_conference_paper: false +publication_date: 2020-07-01 +documents: +- key: PhysRevD.102.014505.xml + url: http://example.org/PhysRevD.102.014505.xml + source: American Physical Society + fulltext: true + hidden: true +references: + - raw_refs: + - schema: JATS + value: 1L. 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PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.96.095036ARTICLESBeyond the standard modelPrecision tests and fine tuning in twin Higgs modelsPRECISION TESTS AND FINE TUNING IN TWIN HIGGS …CONTINO et al.https://orcid.org/0000-0003-1813-2645ContinoRoberto1,*GrecoDavide2,†MahbubaniRakhi3,‡RattazziRiccardo2,§TorreRiccardo2,∥Scuola Normale Superiore and INFN, Pisa IT-56125, ItalyTheoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne CH-1015, SwitzerlandTheoretical Physics Department, CERN, Geneva CH-1211, Switzerland

roberto.contino@sns.it

davide.greco@epfl.ch

rakhi.mahbubani@cern.ch

riccardo.rattazzi@epfl.ch

riccardo.torre@epfl.ch

29November20171November201796909503627February20177November2017Published by the American Physical Society2017authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

We analyze the parametric structure of twin Higgs (TH) theories and assess the gain in fine tuning which they enable compared to extensions of the standard model with colored top partners. Estimates show that, at least in the simplest realizations of the TH idea, the separation between the mass of new colored particles and the electroweak scale is controlled by the coupling strength of the underlying UV theory, and that a parametric gain is achieved only for strongly-coupled dynamics. Motivated by this consideration we focus on one of these simple realizations, namely composite TH theories, and study how well such constructions can reproduce electroweak precision data. The most important effect of the twin states is found to be the infrared contribution to the Higgs quartic coupling, while direct corrections to electroweak observables are subleading and negligible. We perform a careful fit to the electroweak data including the leading-logarithmic corrections to the Higgs quartic up to three loops. Our analysis shows that agreement with electroweak precision tests can be achieved with only a moderate amount of tuning, in the range 5%–10%, in theories where colored states have mass of order 3–5 TeV and are thus out of reach of the LHC. For these levels of tuning, larger masses are excluded by a perturbativity bound, which makes these theories possibly discoverable, hence falsifiable, at a future 100 TeV collider.

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschunghttp://dx.doi.org/10.13039/501100001711Swiss National Science FoundationFonds National Suisse de la Recherche ScientifiqueFondo Nazionale Svizzero per la Ricerca ScientificaFonds National SuisseFondo Nazionale SvizzeroSchweizerischer NationalfondsSNFSNSFFNShttp://sws.geonames.org/2658434/NSF-CH200020-150060200020-169696200021-160190H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663H2020 Excellent Science - European Research CouncilEuropean Research CouncilERChttp://sws.geonames.org/6695072/267985614577INTRODUCTION

The principle of naturalness offers arguably the main motivation for exploring physics at around the weak scale. According to naturalness, the plausibility of specific parameter choices in quantum field theory must be assessed using symmetries and selection rules. When viewing the standard model (SM) as an effective field theory valid below a physical cutoff scale and considering only the known interactions of the Higgs boson, we expect the following corrections to its mass.

We take mh2=2mH2=λhv2/2 with ⟨H⟩=v/2=174  GeV, which corresponds to a potential V=-mH2|H|2+λh4|H|4.

δmh2=3yt24π2Λt2-9g2232π2Λg22-3g1232π2Λg12-3λh8π2Λh2+⋯,where each Λ represents the physical cutoff scale in a different sector of the theory. The above equation is simply dictated by symmetry: dilatations (dimensional analysis) determine the scale dependence and the broken shift symmetry of the Higgs field sets the coupling dependence. Unsurprisingly, these contributions arise in any explicit UV completion of the SM, although in some cases there may be other larger ones. According to Eq. (1.1), any given (large) value of the scale of new physics can be associated with a (small) number ε, which characterizes the accuracy at which the different contributions to the mass must cancel among themselves, in order to reproduce the observed value mh≃125  GeV. As the largest loop factor is due to the top Yukawa coupling, according to Eq. (1.1) the scale ΛNP where new states must first appear is related to mh2 and ε via ΛNP2∼4π23yt2×mh2ε⇒ΛNP∼0.451εTeV.The dimensionless quantity ε measures how finely-tuned mh2 is, given ΛNP, and can therefore be regarded as a measure of the tuning. Notice that the contributions from g22 and λh in Eq. (1.1) correspond to ΛNP=1.1  TeV/ε and ΛNP=1.3  TeV/ε, respectively. Although not significantly different from the relation in the top sector, these scales would still be large enough to push new states out of direct reach of the LHC for ε∼0.1.

Indeed, for a given ε, Eq. (1.2) only provides an upper bound for ΛNP; in the more fundamental UV theory there can in principle exist larger corrections to mh2 which are not captured by Eq. (1.1). In particular, in the minimal supersymmetric SM (MSSM) with high-scale mediation of the soft terms, δmh2 in Eq. (1.1) is logarithmically enhanced by RG evolution above the weak scale. In that case, Eq. (1.2) is modified as follows: ΛNP2∼2π23yt2×1lnΛUV/ΛNP×mh2ε,where ΛNP corresponds to the overall mass of the stops and ΛUV≫ΛNP is the scale of mediation of the soft terms. However, for generic composite Higgs (CH) models, as well as for supersymmetric models with low-scale mediation, Eq. (1.2) provides a fair estimate of the relation between the scale of new physics and the amount of tuning, the Higgs mass being fully generated by quantum corrections at around the weak scale. If the origin of mh is normally termed soft in the MSMM with large ΛUV [Eq. (1.3)], it should then be termed supersoft in models respecting Eq. (1.2). As is well known, and shown by Eq. (1.3), the natural expectation in the MSSM for ΛUV≳100  TeV is ΛNP∼mZ∼mh. In view of this, soft scenarios were already somewhat constrained by direct searches at LEP and Tevatron, whereas the natural range of the scale of supersoft models is only now being probed at the LHC.

Equation (1.2) sets an absolute upper bound on ΛNP for a given fine tuning ε, but does not give any information on its nature. In particular it does not specify the quantum numbers of the new states that enter the theory at or below this scale. Indeed, the most relevant states associated with the top sector, the so-called top partners, are bosonic in standard supersymmetric models and fermionic in CH models. Nonetheless, one common feature of these standard scenarios is that the top partners carry SM quantum numbers, color in particular. They are thus copiously produced in hadronic collisions, making the LHC a good probe of these scenarios. Yet there remains the logical possibility that the states that are primarily responsible for the origin of the Higgs mass at or below ΛNP are not charged under the SM, and thus much harder to produce and detect at the LHC. The twin Higgs (TH) is probably the most interesting of the (few) ideas that take this approach [1–22]. This is primarily because the TH mechanism can, at least in principle, be implemented in a SM extension valid up to ultrahigh scales. The structure of TH models is such that the states at the threshold ΛNP in Eq. (1.2) carry quantum numbers under the gauge group of a copy, a twin, of the SM, but are neutral under the SM gauge group. These twin states, of which the twin tops are particularly relevant, are thus poorly produced at the LHC. The theory must also contain states with SM quantum numbers, but their mass m* is boosted with respect to ΛNP roughly by a factor g*/gSM, where g* describes the coupling strength of the new dynamics, while gSM represents a generic SM coupling. As discussed in the next section, depending on the structure of the model, gSM can be either the top Yukawa or the square root of the Higgs quartic. As a result, given the tuning ε, the squared mass of the new colored and charged states is roughly given by m*2∼4π23yt2×mh2ε×(g*gSM)2.For g*>gSM, we could define these model as effectively hypersoft, in that, for fixed fine tuning, the gap between the SM-charged states and the weak scale is even larger than that in supersoft models. In practice the above equation implies that, for strong g*, the new states are out of reach of the LHC even for mild tuning (see Sec. II A for a more precise statement). Equation (1.4) synthesizes the potential relevance of the TH mechanism, and makes it clear that the new dynamics must be rather strong for the mechanism to work. Given the hierarchy problem, it then seems almost inevitable to make the TH a composite TH (although it could also be a supersymmetric composite TH). Realizations of the TH mechanism within the paradigm of CH models with fermion partial compositeness [23] have already been proposed, both in the holographic and effective theory setups [6,9,13,14].

It is important to recognize that the factor that boosts the mass of the states with SM gauge quantum numbers in Eq. (1.4) is the coupling g* itself. Because of this, strong-dynamics effects in the Higgs sector, which are described in the low-energy theory by nonrenormalizable operators with coefficients proportional to powers of g*/m*, do not “decouple” when these states are made heavier, at fixed fine tuning ε. In the standard parametrics of the CH, m*/g* is of the order of f, the decay constant of the σ-model within which the Higgs doublet emerges as a pseudo-Nambu-Goldstone boson (pNGB). Then ξ≡v2/f2, as well as being a measure of the fine tuning through ε=2ξ, also measures the relative deviation of the Higgs couplings from the SM ones, in the TH like in any CH model.

The factor of two difference between the fine tuning ε and ξ is due to the Z2 symmetry of the Higgs potential in the TH models, as shown in Sec. II A [7].

Recent Higgs coupling measurements roughly constrain ξ≲10–20% [24], and a sensitivity of order 5% is expected in the high-luminosity phase of the LHC [25]. However Higgs loop effects in precision Z-pole observables measured at LEP already limit ξ≲5% [26,27]. Having to live with this few-percent tuning would somewhat undermine the motivation for the clever TH construction. In ordinary CH models this strong constraint on ξ can in principle be relaxed thanks to compensating corrections to the T^ parameter coming from the top partners. In the most natural models, these are proportional to yt4v2/m*2 and thus, unlike the Higgs-sector contribution, decouple when m* is increased. This makes it hard to realize such a compensatory effect in the most distinctive range of parameters for TH models, where m*∼5–10  TeV. Alternatively one could consider including custodial-breaking couplings larger than yt in the top-partner sector. Unfortunately these give rise to equally-enhanced contributions to the Higgs potential, which would in turn require further ad-hoc cancellations.

As already observed in the literature [13,14] another important aspect of TH models is that calculable IR-dominated contributions to the Higgs quartic coupling almost saturate its observed value. Though a welcome property in principle, this sets even stronger constraints on additional UV contributions, such as those induced by extra sources of custodial breaking. In this paper we study the correlation between these effects, in order to better assess the relevance of the TH construction as a valid alternative to more standard ideas about EW-scale physics. Several such studies already exist for standard composite Higgs scenarios [28–30]. In extending these to the TH we shall encounter an additional obstacle to gaining full benefit from the TH boost in Eq. (1.4): the model structure requires rather “big” multiplets, implying a large number of degrees of freedom. This results in an upper bound for the coupling that is parametrically smaller than 4π by naive dimensional analysis (NDA); hence the boost factor is similarly depressed. We shall discuss in detail how serious and unavoidable a limitation this is.

The paper is organized as follows: in Sec. II we discuss the general structure and parametrics of TH models, followed by Sec. III where we discuss the more specific class of composite TH models we focus on for the purpose of our study. In Secs. IV and V we present our computations of the basic physical quantities: the Higgs potential and precision electroweak parameters (S^,T^,δgLb). Section VI is devoted to a discussion of the resulting constraints on the model and an appraisal of the whole TH scenario. Our conclusions are presented in Sec. VII.

THE TWIN HIGGS SCENARIOStructure and parametrics

In this section we outline the essential aspects of the TH mechanism. Up to details and variants which are not crucial for the present discussion, the TH scenario involves an exact duplicate, SM˜, of the SM fields and interactions, underpinned by a Z2 symmetry. In practice this Z2 must be explicitly broken in order to obtain a realistic phenomenology, and perhaps more importantly, a realistic cosmology [19–21]. However the sources of Z2 breaking can have a structure and size that makes them irrelevant in the discussion of naturalness in electroweak symmetry breaking, which is the main goal of this section.

Our basic assumption is that the SM and its twin emerge from a more fundamental Z2-symmetric theory at the scale m*, at which new states with SM quantum numbers, color in particular, first appear. In order to get a feel for the mechanism and its parametrics, it is sufficient to focus on the most general potential for two Higgs doublets H and H˜, invariant under the gauge group GSM×G˜SM, with GSM=SU(3)c×SU(2)L×U(1)Y, as well as a Z2: V(H,H˜)=-mH2(|H|2+|H˜|2)+λH4(|H|2+|H˜|2)2+λ^h8(|H|4+|H˜|4).Strictly speaking, the above potential does not have minima with realistic “tunable” ⟨H⟩. This goal can be achieved by the simple addition of a naturally small Z2-breaking mass term which, while changing the vacuum expectation value, does not affect the estimates of fine tuning, and hence will be neglected for the purposes of this discussion. Like for the SM Higgs, the most general potential is accidentally invariant under a custodial SO(4)×SO˜(4). Notice however that in the limit λ^h→0, the additional Z2 enhances the custodial symmetry to SO(8), where H≡H⊕H˜≡8. In this exact limit, if H˜ acquired an expectation value ⟨H˜⟩≡f/2, all 4 components of the ordinary Higgs H would remain exactly massless NGBs. Of course the SM and SM˜ gauge and Yukawa couplings, along with λ^h, explicitly break the SO(8) symmetry that protects the Higgs. Consider however the scenario where these other couplings, which are known to be weak, can be treated as small SO(8)-breaking perturbations of a stronger SO(8)-preserving underlying dynamics, of which the quartic coupling λH is a manifestation. In this situation we can reassess the relation between the SM Higgs mass, the amount of tuning and the scale m* where new states charged under the SM are first encountered, treating λ^h as a small perturbation of λH. At zeroth order, i.e., neglecting λ^h, we can expand around the vacuum ⟨H˜⟩2=2mH2/λH≡f2/2, ⟨H⟩=0. The spectrum consists of a heavy scalar σ, with mass mσ=2mH=λHf/2, corresponding to the radial mode, 3 NGBs eaten by the twin gauge bosons, which get masses ∼gf/2 and the massless H. When turning on λ^h, SO(8) is broken explicitly and H acquires a potential. At leading order in a λ^h/λH expansion the result is simply given by substituting |H˜|2=f2/2-|H|2 in Eq. (2.1).

Notice that the effective Higgs quartic receives approximately equal contributions from |H|4 and |H˜|4. This is a well-known and interesting property of the TH, see for instance Ref. [7].

The quartic coupling and the correction to the squared mass are then given by λh≃λ^hδmH2∼λ^hf2/8≃(λh/2λH)mH2.As mentioned above, we assume that mH2 also receives an independent contribution from a Z2-breaking mass term, which can be ignored in the estimates of tuning. Note that in terms of the physical masses of the Higgs, mh, and of its heavy twin, mσ, we have precisely the same numerical relation δmh2=(λh/2λH)mσ2. The amount of tuning ε, defined as mh2/δmh2, is given by ε=2ξ=2v2/f2.

Our estimate of δmH2 in Eq. (2.2) is based on a simplifying approximation where the SO(8)-breaking quartic is taken Z2-symmetric. In general we could allow different couplings λ^h and λ^h˜ for |H|4 and |H˜|4 respectively, constrained by the requirement λ^h+λ^h˜≃2λh. As the estimate of δmH2 in Eq. (2.2) is determined by the |H˜|4 term, it is clear that a reduction of λ^h˜, with λh fixed, would improve the tuning, as emphasized in Ref. [22]. As discussed in Sec. IV, however, a significant fraction of the contribution to λ^h and λ^h˜ is coming from RG evolution due to the top and twin top. According to our analysis, λ^h/λ^h˜ varies between 1.5 in the simplest models to 3 in models where λ^h˜ is purely IR-dominated as in Ref. [22]. Though interesting, this gain does not change our parametric estimates.

The ratio λh/λH is the crucial parameter in the game. Indeed it is through Eq. (2.2) that mH is sensitive to quantum corrections to the Lagrangian mass parameter mH, or, equivalently, that the physical Higgs mass mh is sensitive to the physical mass of the radial mode mσ. In particular, what matters is the correlation of mσ with, and its sensitivity to, m*, where new states with SM quantum numbers appear. One can think of three basic scenarios for that relation, which we now illustrate, ordering them by increasing level of model-building ingenuity. Beyond these scenarios there is the option of tadpole-dominated electroweak symmetry breaking, which we shall briefly discuss at the end.

Subhypersoft scenario

The simplest option is given by models with mσ∼m*. Supersymmetric TH models with medium- to high-scale soft-term mediation belong to this class [7], with m* representing the soft mass of the squarks. Like in the MSSM, mH, and therefore mσ, is generated via RG evolution: two decades of running are sufficient to obtain mσ∼m*. Another example is composite TH models [13,14]. In their simplest incarnation they are characterized by one overall mass scale m* and coupling g* [31], so that by construction one has mσ∼m* and λH∼g*2. As discussed below Eq. (2.2), in both these scenarios one then expects δmh2∼(λh/2λH)m*2. It is interesting to compare this result to the leading top-sector contribution in Eq. (1.1). For that purpose it is worth noticing that, as discussed in Sec. IV, in TH models the RG-induced contribution to the Higgs quartic coupling Δλh|RG∼(3yt4/π2)lnm*/mt (more than) saturates its experimental value λh∼0.5 for m*∼3–10  TeV.

For this naive estimate we have taken the twin-top contribution equal to the top one, so that the result is just twice the SM one. For a more precise statement see Sec. IV.

We can thus write δmh2∼(λh/2λH)m*2∼3yt42π21λHln(m*/mt)m*2≡3yt22π2×yt2g*2×ln(m*/mt)×m*2which should be compared to the first term on the right-hand side of Eq. (1.1). Accounting for the possibility of tuning we then have m*∼0.45×g*2yt×1ln(m*/mt)×1ε  TeV.Compared to Eq. (1.2), the mass of colored states is on one hand parametrically boosted by the ratio g*/(2yt), and on the other it is mildly decreased by the logarithm. The motivation for, and gain in, the ongoing work on the simplest realization of the TH idea pivot upon the above g*/yt. The basic question is how high g* can be pushed without leading to a breakdown of the effective description. One goal of this paper is to investigate to what extent one can realistically gain from this parameter in more explicit CH realizations. Applying naive dimensional analysis (NDA) one would be tempted to say that g* as big as ∼4π makes sense, in which case m*∼10  TeV would only cost a mild ε∼0.1 tuning. However such an estimate seems quantitatively too naive. For instance, by focusing on the simple toy model whose potential is given by Eq. (2.1), we can associate the upper bound on λH≡g*2, to the point where perturbation theory breaks down. One possible way to proceed is to consider the one loop beta function μdλHdμ=N+832π2λH2,and to estimate the maximum value of the coupling λH as that for which ΔλH/λH∼O(1) through one e-folding of RG evolution. We find λH=2mσ2f2≲32π2N+8⇒mσf≲π,for  N=8,which also gives g*∼λH≲2π, corresponding to a significantly smaller maximal gain in Eq. (2.4) with respect to the NDA estimate. In Sec. III B we shall perform alternative estimates in more specific CH constructions, obtaining similar results.

It is perhaps too narrow-minded to stick rigidly to such estimates to determine the boost that g*/(2yt) can give to m*. Although it is parametrically true that the stronger the coupling g*, the heavier the colored partners can be at fixed tuning, the above debate over factors of a few make it difficult to be more specific in our estimates. In any case the gain permitted by Eq. (2.4) is probably less than one might naively have hoped, making it fair to question the motivation for the TH, at least in its “subhypersoft” realization. With this reservation in mind, we continue our exploration of the TH in the belief that the connection between naturalness and LHC signatures is so crucial that it must be analyzed in all its possible guises.

Concerning in particular composite TH scenarios one last important model building issue concerns the origin of the Higgs quartic λh. In generic CH it is known that the contribution to λh that arises at O(yt2) is too large when g* is strong. Given that the TH mechanism demands g* as strong as possible then composite TH models must ensure that the leading O(yt2) contribution is absent so that λh arises at O(yt4). As discussed in Ref. [13], this property is not guaranteed but it can be easily ensured provided the couplings that give rise to yt via partial compositeness respect specific selection rules.

Hypersoft scenario

The second option corresponds to the structurally robust situation where mσ2 is one loop factor smaller than m*2. This is for instance achieved if H is a PNG-boson octet multiplet associated to the spontaneous breaking SO(9)→SO(8) in a model with fundamental scale m*. Another option would be to have a supersymmetric model where supersymmetric masses of order m* are mediated to the stops at the very scale m* at which H is massless. Of course in both cases a precise computation of mσ2 would require the full theory. However a parametrically correct estimate can be given by considering the quadratically divergent 1-loop corrections in the low energy theory, in the same spirit of Eq. (1.1). As yt and λH are expected to be the dominant couplings the analogue of Eqs. (1.1) and (2.2) imply δmh2∼λh2λH(3yt24π2+5λH16π2)m*2=(yt2λH+512)3λh8π2m*2.Very roughly, for λH≳yt2, top effects become subdominant and the natural value for mh becomes controlled by λh, like the term induced by the Higgs quartic in Eq. (1.1). In the absence of tuning this roughly corresponds to the technicolor limit m*∼4πv, while allowing for fine tuning we have m*∼1.4×1ε  TeV.It should be said that in this scenario there is no extra boost of m* at fixed tuning by taking λH>yt2. Indeed the choice λH∼yt2 is preferable as concerns electroweak precision tests (EWPT). It is well known that RG evolution in the effective theory below mσ gives rise to corrections to the S^ and T^ parameters as discussed in Sec. V [32]. In view of the relation ε=2v2/f2 this gives a direct connection between fine-tuning electroweak precision data, and the mass of the twin Higgs mσ. At fixed v2/f2, EWPT then favor the smallest possible mσ=λHf/2, that is the smallest λH∼yt2. The most plausible spectrum in this class of models is roughly the following: the twin scalar σ and the twin tops appear around the same scale ∼ytf/2, below the colored partners who live at m*. The presence of the somewhat light scalar σ is one of interesting features of this class of models.

Superhypersoft scenario

This option is a clever variant of the previous one, where below the scale m* approximate supersymmetry survives in the Higgs sector in such a way that the leading contribution to δmH2 proportional to λH is purely due to the top sector [7]. In that way Eq. (2.7) reduces to δmh2∼λh2λH(3yt24π2)m*2=yt2λH×3λh8π2m*2.so that by choosing g*>yt one can push the scale m* further up with fixed fine tuning ε m*∼1.4×g*2yt×1ε  TeV.In principle even under the conservative assumption that g*∼2π is the maximal allowed value, this scenario seemingly allows m*∼14  TeV with a mild ε∼0.1 tuning.

It should be said that in order to realize this scenario one would need to complete H into a pair of chiral superfield octets Hu and Hd, along the lines of Ref. [7], as well as add a singlet superfield S in order to generate the Higgs quartic via the superpotential trilinear g*SHuHd. Obviously this is a very far-fetched scenario combining all possible ideas to explain the smallness of the weak scale: supersymmetry, compositeness, and the twin Higgs mechanism.

Alternative vacuum dynamics: Tadpole induced EWSB

In all the scenarios discussed so far the tuning of the Higgs vacuum expectation value (vev) and that of the Higgs mass coincided: ε, which controls the tuning of mh2 according to Eqs. (2.4), (2.8), and (2.10), is equal to 2v2/f2, which measures the tuning of the vev. This was because the only tuning in the Higgs potential was associated with the small quadratic term, while the quartic was assumed to be of the right size without the need for further cancellations (see, e.g., the discussion in Ref. [33]). Experimentally however, one can distinguish between the need for tuning that originates from measurements of Higgs and electroweak observables, which are controlled by v2/f2, and that coming from direct searches for top partners. Currently, with bounds on colored top partners at just around 1 TeV [34,35], but with Higgs couplings already bounded to lie within 10%–20% of their SM value [24], the only reason for tuning in all TH scenarios is to achieve a small v2/f2. It is then fair to consider options that reduce or eliminate only the tuning of v2/f2. As argued in Ref. [18], this can be achieved by modifying the H scalar vacuum dynamics, and having its vev induced instead by a tadpole mixing with an additional electroweak-breaking technicolor (TC) sector [36–38]. In order to preserve the Z2 symmetry one adds two twin TC sectors, both characterized by a mass scale mTC and a decay constant fTC∼mTC/4π (i.e., it is parametrically convenient to assume gTC∼4π). Below the TC scale the dynamics in the visible and twin sectors is complemented by Goldstone triplets πa and π˜a which can be embedded into doublet fields according to Σ=fTCeiπaσa(01),Σ˜=fTCeiπ˜aσa(01),and are assumed to mix with H and H˜ via the effective potential terms Vtadpole=M2(H†Σ+H˜†Σ˜)+H.c.Assuming mTC≪mH the H˜ vacuum dynamics is not significantly modified, but, for mTC>mh, Vtadpole acts like a rigid tadpole term for H. The expectation value ⟨H⟩ is thus determined by balancing such a tadpole against the gauge-invariant |H|2 mass term; the latter will then roughly coincide with mh2. In order for this to work, by Eq. (2.2) the SO(8)-breaking quartic λ^h should be negative, resulting in v∼(M2/mh2)fTC. It is easy to convince oneself that the corrections to Higgs couplings are O(fTC2/v2): present bounds can then be satisfied for fTC∼v/10≃80  GeV. In turn, the value of v/f is controlled by f and can thus be naturally small. The TC scale is roughly mTC∼4πfTC∼600–800  GeV, while the noneaten pNGB π in Eq. (2.11) have a mass mπ2∼M2v/fTC∼mh2(v/fTC)2∼400  GeV. The latter value, although rather low, is probably large enough to satisfy constraints from direct searches. In our opinion, what may be more problematic are EWPT, in view of the effects from the TC sector, which shares some of the vices of ordinary TC. The IR contributions to S^ and T^, associated with the splitting mπa<mTC, are here smaller than the analogues of ordinary technicolor (there associated with the splitting mW≪mTC). However the UV contribution to S^ is parametrically the same as in ordinary TC, in particular it is enhanced at large NTC. Even at NTC=2, staying within the allowed (S^,T^) ellipse still requires a correlated contribution from ΔT^, which in principle should also be counted as tuning. In spite of this, models with tadpole-induced EWSB represent a clever variant where, technically, the dynamics of EWSB does not currently appear tuned. A thorough analysis of the constraints is certainly warranted.

THE COMPOSITE TWIN HIGGS

In this section and in the remainder of the paper, we will focus on the CH realization of the TH, which belongs to the subhypersoft class of models. In this simple and well-motivated context we shall discuss EWPT, fine tuning, and structural consistency of the model.

Our basic structural assumption is that at a generic UV scale ΛUV≫m*, our theory can be decomposed into two sectors: a strongly-interacting composite sector and a weakly-interacting elementary sector. The composite sector is assumed to be endowed with the global symmetry G=SO(8)×U(1)X×Z2 and to be approximately scale- (conformal) invariant down to the scale m*, at which it develops a mass gap. We assume the overall interaction strength at the resonance mass scale m* to be roughly described by one parameter g* [31]. The large separation of mass scales ΛUV≫m* is assumed to arise naturally, in that the occurrence of the mass gap m* is controlled by either a marginally relevant deformation, or by a relevant deformation whose smallness is controlled by some global symmetry. At the scale m*, SO(8)×U(1)X×Z2 is spontaneously broken to the subgroup H=SO(7)×U(1)X, giving rise to seven NGBs in the 7 of SO(7) with decay constant f∼m*/g*. The subgroup U(1)X does not participate to the spontaneous breaking, but its presence is needed to reproduce the hypercharges of the SM fermions, similarly to CH models. The elementary sector consists in turn of two separate weakly interacting sectors: one containing the visible SM fermions and gauge bosons, corresponding to the SM gauge group GSM=SU(3)c×SU(2)L×U(1)Y; the other containing the twin SM with the same fermion content and a SM˜ gauge group G˜SM=SU˜(3)c×SU˜(2)L. The external Z2 symmetry, or twin parity, interchanges these two copies. For simplicity, and following [13], we choose not to introduce a mirror hypercharge field. This is our only source of explicit twin-parity breaking, and affects neither our discussion of fine tuning, nor that of precision electroweak measurements.

The elementary and composite sectors are coupled according to the paradigm of partial compositeness [23]. The elementary EW gauge bosons couple to the strong dynamics as a result of the weak gauging of the SU(2)L×U(1)Y×SU˜(2)L subgroup of the global SO(8)×U(1)X. A linear mixing with the global conserved currents is thus induced: LmixV⊃g2WμαJαμ+g1BμJBμ+g˜2W˜μαJ˜αμ,where g1,2 and g˜2 denote the SM and twin weak gauge couplings, JBμ≡J3Rμ+JXμ and Jμ, J˜μ and JXμ are the currents associated respectively to the SU(2)L, SU˜(2)L and U(1)X generators. The elementary fermions mix analogously with various operators transforming as linear representations of SO(8) that are generated in the far UV by the strongly interacting dynamics. The mixing Lagrangian takes the schematic form: LmixF⊃q¯LαΔαAORA+t¯RΘAOLA+q˜¯LαΔ˜αAO˜RA+t¯RΘ˜AO˜LA+H.c.,where, following, e.g., Ref. [39], we introduced spurions ΔαA, Δ˜αA, ΘA, and Θ˜A in order to uplift the elementary fields to linear representations of SO(8), and match the quantum numbers of the composite operators. The left-handed mixings ΔαA, Δ˜αA necessarily break SO(8) since qL only partially fills a multiplet of SO(8). The right-handed mixings, instead, may or may not break SO(8). The breaking of SO(8) gives rise to a potential for the NGBs at one loop and the physical Higgs is turned into a pNGB. We conclude by noticing that g1,2 and g˜2 correspond to quasimarginal couplings which start off weak in the UV, and remain weak down to m*. The fermion mixings could be either relevant or marginal, and it is possible that some may correspond to interactions that grow as strong as g* at the IR scale m* [40]. In particular, as is well known, there is some advantage as regards tuning in considering the right mixings ΘA and Θ˜A to be strong. In that case one may even imagine the IR scale to be precisely generated by the corresponding deformation of the fixed point. While this latter option may be interesting from a top-down perspective, it would play no appreciable role in our low-energy phenomenological discussion.

A simplified model

In order to proceed we now consider a specific realization of the composite TH and introduce a concrete simplified effective Lagrangian description of its dynamics. Our model captures the most important features of this class of theories, like the pNGB nature of the Higgs field, and provides at the same time a simple framework for the interactions between the elementary fields and the composite states, vectors and fermions. We make use of this effective model as an example of a specific scenario in which we can compute EW observables, and study the feasibility of the TH idea as a new paradigm for physics at the EW scale.

We write down an effective Lagrangian for the composite TH model using the Callan-Coleman-Wess-Zumino (CCWZ) construction [41,42], and generalizing the simpler case of a two-site model developed in Ref. [13]. According to the CCWZ technique, a Lagrangian invariant under the global SO(8) group can be written following the rules of a local SO(7) symmetry. The basic building blocks are the Goldstone matrix Σ(Π), which encodes the seven NGBs, Π, present in the theory, and the operators dμ(Π) and Eμ(Π) resulting from the Maurer-Cartan form constructed with the Goldstone matrix. An external U(1)X group is also added to the global invariance in order to reproduce the correct fermion hypercharges [13]. The CCWZ approach is reviewed and applied to the SO(8)/SO(7) coset in Appendix A.

Before proceeding, we would like to recall the simplified model philosophy of Ref. [43], which we essentially employ. In a generic composite theory, the mass scale m* would control both the cutoff of the low energy σ-model and the mass of the resonances. In that case no effective Lagrangian method is expected to be applicable to describe the resonances. So, in order to produce a manageable effective Lagrangian we thus consider a Lagrangian for resonances that can, at least in principle, be made lighter that m*. One more structured way to proceed could be to consider a deconstructed extra-dimension where the mass of the lightest resonances, corresponding to the inverse compactification length, is parametrically separated from the 5D cut-off, interpreted as m*. Here we do not go that far and simply consider a set of resonances that happen to be a bit lighter than m*. We do so to give a structural dignity to our effective Lagrangian, though at the end, for our numerical analysis, we just take the resonances a factor of 2 below m*. We believe that is a fair procedure given our purpose of estimating the parametric consistency of the general TH scenario.

We start our analysis of the effective Lagrangian with the bosonic sector. Together with the elementary SM gauge bosons, the W’s and B, we introduce the twin partners W˜ to gauge the SU˜(2)L group. As representative of the composite dynamics, we restrict our interest to the heavy spin-1 resonances transforming under the adjoint of SO(7) and to a vector singlet. We therefore introduce a set of vectors ρμa which form a 21 of SO(7) and the gauge vector associated with the external U(1)X, which we call ρμX. The Lagrangian for the bosonic sector can be written as Lbosonic=Lπ+LcompV+LelemV+LmixV.The first term describes the elementary gauge bosons masses and the NGBs dynamics and is given by Lπ=f24Tr[dμdμ].The second term, LcompV, is a purely composite term, generated at the scale m* after confinement; it reduces to the kinetic terms for the ρ vectors, namely: LcompV=-14gρ2ρμνaρμνa-14gρX2ρμνXρXμν,where ρμνa=∂μρνa-∂νρμa-fabcρμbρνc, ρμνX=∂μρνX-∂νρμX, and gρ and gρX are the coupling strengths for the composite spin-1 bosons. The third term in Eq. (3.3), LelemV, is a purely elementary interaction, produced at the scale ΛUV where the elementary fields are formally introduced. Also this Lagrangian can contain only the kinetic terms for the elementary fields: LelemV=-14g12BμνBμν-14g22WμνaWaμν-14g˜22W˜μνaW˜aμν,where g1, g2, and g˜2 denote the weak gauge couplings. The last term in the Lagrangian (3.3), LmixV, is a mixing term between the elementary and composite sectors originating from partial compositeness. We have:

Notice that in the Lagrangian (3.7), the parameters f, Mρ, MρX, gρ, and gρX are all independent. It is common to define the parameters aρ=Mρ/(gρf) and aρX=MρX/(gρXf), which are expected to be O(1). In our analysis we set aρ=1/2 corresponding to the two-site model value (see the last paragraph of this section) and aρX=1.

LmixV=Mρ22gρ2(Tr[ρμaTa21-Eμ])2+MρX22gρX2(ρμX-Bμ)2,where Ta21 are the SO(8) generators in the adjoint of SO(7) (see Appendix A).

We now introduce the Lagrangian for the fermionic sector. This depends on the choice of quantum numbers for the composite operators in Eq. (3.2). The minimal option is to choose OR and O˜R to be in the fundamental representation of SO(8), whereas the operators OL and O˜L are singlets of the global group. Therefore, the elementary SM doublet and its twin must be embedded into fundamental representations of SO(8), whereas the tR and the t˜R are complete singlets under the global SO(8) invariance. This choice is particularly useful to generalize our discussion to the case of a fully-composite right-handed top. From the low-energy perspective, the linear mixing between composite operators and elementary fields translates into a linear coupling between the latter and a layer of fermionic resonances excited from the vacuum by the operators in the fundamental and singlet representations of the global group. Decomposing the 8 of SO(8) as 8=7+1 under SO(7), we introduce a set of fermionic resonances filling a complete fundamental representation of SO(7) and another set consisting of just one singlet.

Notice that in general we should introduce two different singlets in our Lagrangian. One corresponds to a full SO(8) singlet, while the other is the SO(7) singlet appearing in the decomposition 8=7+1 of the fundamental of SO(8) under the SO(7) subgroup. We will further simplify our study identifying the two singlets with just one composite particle.

We denote with Ψ7 the fermionic resonances in the septuplet and with Ψ1 the singlet, both charged under SU(3)c. Together with them, we must introduce analogous composite states charged under SU˜(3)c; we use the corresponding notation Ψ˜7 and Ψ˜1. We refer to Ref. [13] for the complete expression of Ψ7 and Ψ˜7 in terms of the constituent fermions.

The fermionic effective Lagrangian is split into three parts, which have the same meaning as the analogous distinctions we made for the bosonic sector of the theory: Lfermionic=LcompF+LelemF+LmixF.The fully composite term is given by: LcompF=Ψ¯7(iD7-MΨ)Ψ7+Ψ¯1(iD1-MS)Ψ1+Ψ˜¯7(i∇-M˜Ψ)Ψ˜7+Ψ˜¯1(i∂-M˜S)Ψ˜1+(icLΨ¯7LidiΨ1L+icRΨ¯7RidiΨ1R+ic˜LΨ˜¯7LidiΨ˜1L+ic˜RΨ˜¯7RidiΨ˜1R+H.c.),where D7μ=∇μ+iXBμ, D1μ=∂μ+iXBμ, and ∇μ=∂μ+iEμ. We have introduced two sets of O(1) coefficients, cL and cR and their twins, for the interactions mediated by the dμ operator. Considering the elementary part of the Lagrangian, it comprises just the kinetic terms for the doublets and right-handed tops: LelemF=q¯LiDqL+t¯RiDtR+q˜¯LiDq˜L+t˜¯Ri∂t˜R.The final term in our classification is the elementary/composite mixing that we write again following the prescription of partial compositeness. With our choice of quantum numbers for the composite operators, the spurions in Eq. (3.2) can be matched to dimensionless couplings according to ΔαA=(00iyL-yL0×4iyLyL000×4),ΘA=yR,and Δ˜αA=(0×400iy˜L-y˜L0×4iy˜Ly˜L00),Θ˜A=y˜R,where we have introduced the elementary/composite mixing parameters yL, yR and their twin counterparts. These dimensionless y’s control the strength of the interaction between the elementary and composite resonance fields, according to the Lagrangian: LmixF=f(q¯LαΔαAΣAiΨ7i+q¯LαΔαAΣA8Ψ1+yRt¯RΨ1+H.c.)+f(q˜¯LαΔ˜αAΣAiΨ˜7i+q˜¯LαΔ˜αAΣA8Ψ˜1+y˜Rt˜¯RΨ˜1+H.c.).Depending on the UV boundary condition and the relevance or marginality of the operators appearing in Eq. (3.2), the y’s can vary from weak to O(g*). Correspondingly the light fermions vary from being completely elementary (for y weak) to effectively fully composite (for y∼g*). For reasons that will become clear, given yt∼yLyR/g*, it is convenient to take yL≃y˜L∼yt, i.e., weak left mixing, and yR≃y˜R∼g*. For such strong right-handed mixing the right-handed tops can be practically considered part of the strong sector.

The last term that we need to introduce in the effective Lagrangian describes the interactions between the vector and fermion resonances and originates completely in the composite sector. We have: LcompVF=∑i=L,R[αiΨ¯7i(ρ-E)Ψ7i+α7iΨ¯7i(ρX-B)Ψ7i+α1iΨ¯1i(ρX-B)Ψ1i+α˜iΨ˜¯7i(ρ-E)Ψ˜7i+α˜7iΨ˜¯7i(ρX-B)Ψ˜7i+α˜1iΨ˜¯1i(ρX-B)Ψ˜1i],where all the coefficients αi appearing in the Lagrangian are O(1) parameters.

We conclude the discussion of our effective Lagrangian by clarifying its two-site model limit [44] (see also Ref. [45]). This is obtained by combining the singlet and the septuplet into a complete representation of SO(8), so that the model enjoys an enhanced SO(8)L×SO(8)R global symmetry. This is achieved by setting cL=cR=c˜L=c˜R=0 and all the αi equal to 1. Moreover, we have to impose Mρ=gρf/2, so that the heavy vector resonances can be reinterpreted as gauge fields of SO(7). As shown in Ref. [44], with this choice of the free parameters the Higgs potential becomes calculable up to only a logarithmic divergence, that one can regulate by imposing just one renormalization condition. In the subsequent sections, we will extensively analyze the EW precision constraints in the general case, as well as in the two-site limit.

Perturbativity of the simplified model

In Sec. II A 1 it was noted that a TH construction typically involves large multiplicities of states and, as a consequence, the dynamics responsible for its UV completion cannot be maximally strongly coupled. This in turn limits the improvement in fine tuning that can be achieved compared to standard scenarios of EWSB. In our naive estimates of Eqs. (2.3), (2.7), and (2.9) the interaction strength of the UV theory was controlled by the σ-model quartic coupling λH or, equivalently, by mσ/f. By considering the λH one-loop β-function [Eq. (2.5)] we estimated the maximal value of λH as the one corresponding to an O(1) relative change through one e-folding of RG evolution. For an SO(8)/SO(7) σ-model this led to λH≲2π, or, equivalently, mσ/f≲π.

Alternatively, the limit set by perturbativity on the UV interaction strength may also be estimated in the effective theory described by the nonlinear σ-model by determining the energy scale at which tree-level scattering amplitudes become nonperturbative. For concreteness, we considered the following two types of scattering processes: ππ→ππ and ππ→ψ¯ψ, where π are the NGBs and ψ={Ψ7,Ψ˜7} denotes a composite fermion transforming in the fundamental of SO(7). Other processes can (and should) be considered, with the actual bound being given by the strongest of the constraints obtained in this way.

Requiring that the process ππ→ππ stay perturbative up the cutoff scale m* gives the bound Mρf∼MΨf≲m*f<4πN-2≃5.1,where the second inequality is valid in a generic SO(N)/SO(N-1) nonlinear σ-model, and we have set N=8 in the last step. More details on how this result was obtained can be found in Appendix G. Equation (3.15) in fact corresponds to a limit on the interaction strength of the UV theory, given that the couplings among fermion and vector resonances are of order MΨ/f and Mρ/f, respectively. Perturbativity of the scattering amplitude for ππ→ψ¯ψ instead gives (see Appendix G for details) Mρf∼MΨf≲m*f<122πNf≃4πNf,where Nf is the multiplicity of composite fermions (including the number of colors and families). Our simplified model with one family of composite fermions has Nf=6, which gives a limit similar to Eq. (3.15): MΨ/f≲5.3. A model with three families of composite quarks and leptons has instead Nf=24, from which follows the stronger bound MΨ/f≲2.6.

As a third alternative, one could analyze when 1-loop corrections to a given observable become of the same order as its tree-level value. We applied this criterion to our simplified model by considering the S^ parameter, the new physics contribution to which includes a tree-level correction from heavy vectors given by Eq. (5.6), and a one-loop correction due to heavy fermions, which can be found in Appendix E. By requiring that the one-loop term be smaller than the tree-level correction, we obtain a bound on the strong coupling constant gρ. As an illustration, we consider the two-site model limit cL=cR=0 and Mρ=gρf/2 and keep the dominant UV contribution to S^ in Eq. (E7) which is logarithmically sensitive to the cutoff. By setting m*=2MΨ, we find: ΔS^1-loopΔS^tree<1⇒Mρf=gρ2<π2log2≃2.7.

The perturbative limits obtained from Eqs. (3.15), (3.16), and (3.17) are comparable to that on λH derived in Sec. II A 1. As already discussed there, one could take any of these results as indicative of the maximal interaction strength in the underlying UV dynamics, though none of them should be considered as a sharp exclusion condition. In our analysis of EW observables we will make use of Eq. (3.15) with N=8 and of Eq. (3.16) with Nf=24 to highlight the regions of parameter space where our perturbative calculation is less reliable. We use both limits as a measure of the intrinsic uncertainty which is inevitably associated with this type of estimation.

HIGGS EFFECTIVE POTENTIAL

As anticipated in the general discussion of Sec. II A, a potential for the Higgs boson is generated at the scale m* by loops of heavy states through the SO(8)-breaking couplings of the elementary fields to the strong sector. Once written in terms of the Higgs boson h (where H†H=f2sin2(h/f)/2, H˜†H˜=f2cos2(h/f)/2), at 1-loop this UV threshold contribution has the form [13]: V(m*)f4=332π2[116g12gρ2L1+(yL2-y˜L2)gΨ2L2]sin2hf+3yL464π2F1(sin4hf+cos4hf),where gΨ≡MΨ/f, L1, L2, F1 are O(1) dimensionless functions of the masses and couplings of the theory and the explicit expression of the function F1 is reported in Eq. (C1) of Appendix C. The first term in the above equation originates from Z2-breaking effects.

Subleading Z2-breaking terms have been neglected for simplicity. The complete expressions are given in Ref. [13].

The second term, generated by loops of fermions, is Z2 symmetric and explicitly violates the SO(8) invariance; it thus corresponds to the (UV part of the) last term of Eq. (2.1). Upon electroweak symmetry breaking, Eq. (4.1) contributes to the physical Higgs mass an amount equal to δmh2|UV=3yL44π2F1f2ξ(1-ξ),where ξ controls the degree of vacuum misalignment: ξ≡sin2⟨h⟩f=v2f2.

Below the scale m* an important contribution to the potential arises from loops of light states, in particular from the top quark and from its twin. The bulk of this IR contribution is captured by the RG evolution of the Higgs potential from the scale m* down to the electroweak scale. As noted in previous studies (see, e.g., Ref. [13]), for sufficiently large m* this IR effect dominates over the UV threshold correction and can reproduce the experimental Higgs mass almost entirely. An analogous IR correction to the Higgs quartic arises in SUSY theories with large stop masses, from loops of top quarks. The distinctive feature of any TH scenario, including our model, is the additional twin top contribution.

The Higgs effective action, including the leading O(ξ) corrections associated with operators of dimension 6, was computed at 1-loop in Ref. [13]; the resulting IR contribution to mh2 was found to be δmh2|IR1-loop=3yt48π2f2ξ(1-ξ)(logm*2mt2+logm*2m˜t2),where yt denotes the top Yukawa coupling. The two single-log terms in parentheses correspond to the IR contributions to the effective Higgs quartic λh from the top quark and twin top respectively. Leading-logarithmic corrections of the form (αlog)n, arising at higher loops have however an important numerical impact.

Here α=gSM2/4π, with gSM being any large SM coupling, i.e., gS and yt.

For example, (αlog)2 corrections generated by 2-loop diagrams (mostly due to the running of the top and twin top Yukawa couplings, that are induced by respectively QCD and twin QCD) are expected to give a ∼30% reduction in the Higgs mass for m*≃5  TeV.

We have computed the IR contribution to the Higgs mass in a combined expansion in ξ and (αlog). We have included the LO electroweak contribution, all terms up to NLO in αt and αS, and some contributions, expected to be leading for not too large m*, at NNLO in αS. We report the details in Appendix C.

The value of (δmh2|IR)1/2 is shown in Fig. 1 as a function of m* for ξ=0.1. The upper and lower curves represent respectively the LO and NLO calculation in our combined (ξ,αlog) expansion. Numerically, the naive expectation is confirmed, as the NLO correction decreases (δmh2|IR)1/2 by ∼35% for m*=5  TeV. The dotted curve, indicated as NNLO*, includes NNLO contributions of order αtξ2log, αtαSξlog2, and αtαS2log3 (see Appendix C). Additional contributions of order αt2ξlog2, αt2αSlog3 and αt3log3 are not included. An attempt to include these contributions has been made in Ref. [46]. However, the calculation presented there misses some additional contributions from the twin GB and does not represent the full NNLO calculation. The picture that we get from our NNLO* calculation and the incomplete result of Ref. [46], is that the NNLO* gives an overestimate of the IR contribution to the Higgs mass as the aforementioned neglected effects are expected to give a reduction. Given the lack of a complete NNLO calculation, we estimate our uncertainty on the IR Higgs mass as the green shaded region lying between the LO and NLO results in Fig. 1. In order to choose, within this uncertainty, a value that is as close as possible to the full NNLO result, in the rest of the paper we take as input value for the IR correction to the Higgs mass the average of the LO and NLO results, corresponding to the solid black line in the figure.

110.1103/PhysRevD.96.095036.f1

IR contribution to the Higgs mass as a function of the scale m* for ξ=0.1. The dot-dashed (upper) and dashed (lower) curves denote the LO and NLO result in a combined perturbative expansion in (αlog) and ξ. The dotted curve contains the pure QCD part of the NNLO correction (see Appendix C for details). The shaded region represents the uncertainty in our estimate. The central value of this band, given by the solid black line and computed as the average of the LO and NLO results, is the value that we use as our numerical estimate throughout the paper.

The plot of Fig. 1 illustrates one of the characteristic features of TH models: the IR contribution to the Higgs mass largely accounts for its experimental value and is completely predicted by the theory in terms of the low-energy particle content (SM plus twin states). In particular, considering the aforementioned input value and choosing as a benchmark values m*=5  TeV and ξ=0.1, we find that the contributions of the SM and twin light degrees of freedom account for around 50% and 40% of the Higgs mass squared, respectively, so that almost the entire experimental value of the Higgs mass can be due only to the IR degrees of freedom. Threshold effects arising at the UV matching scale, on the other hand, are model dependent but give a subleading correction. An accurate prediction of the Higgs mass and an assessment of the plausibility of the model thus requires a precise determination of its IR contribution. Indeed the difference between the IR contribution of Fig. 1 and the measured value mh=125  GeV must be accounted for by the UV threshold contribution in Eq. (4.2); for our previous benchmark choice of m*, about 12% of the Higgs mass should be generated by the UV physics. This translates into a generic constraint on the size of yL, a parameter upon which electroweak precision observables (EWPO) crucially depend, thus creating a nontrivial correlation between the Higgs mass, EWPO and naturalness.

ELECTROWEAK PRECISION OBSERVABLES

In this section we compute the contribution of the new states described by our simplified model to the EWPO. Although it neglects the effects of the heavier resonances, our calculation is expected to give a fair assessment of the size of the corrections due to the full strong dynamics, and in particular to reproduce the correlations among different observables.

It is well known that, under the assumption of quark and lepton universality, short-distance corrections to the electroweak observables due to heavy new physics can be expressed in terms of four parameters, S^, T^, W, Y, defined in Ref. [47] (see also Ref. [48] for an equivalent analysis) as a generalization of the parametrization introduced by Peskin and Takeuchi in Refs. [49,50]. Two additional parameters, δgLb and δgRb, can be added to account for the modified couplings of the Z boson to left- and right-handed bottom quarks respectively.

We define δgLb and δgRb in terms of the following effective Lagrangian in the unitary gauge: Leff⊃g22cWZμb¯γμ[(gLbSM+δgLb)(1-γ5)+(gRbSM+δgRb)(1+γ5)]b+⋯where the dots stand for higher-derivative terms and gLbSM=-1/2+sW2/3, gRbSM=sW2/3.

A naive estimate shows that in CH theories, including our TH model, W and Y are subdominant in an expansion in the weak couplings [31] and can thus be neglected. The small coupling of the right-handed bottom quark to the strong dynamics makes also δgRb small and negligible in our model. We thus focus on S^, T^, and δgLb, and compute them by including effects from the exchange of vector and fermion resonances, and from Higgs compositeness.

We work at the 1-loop level and at leading order in the electroweak couplings and perform an expansion in inverse powers of the new physics scale. In this limit, the twin states do not affect the EWPO as a consequence of their being neutral under the SM gauge group. Deviations from the SM predictions arise only from heavy states with SM quantum numbers and are parametrically the same as in ordinary CH models with singlet tR. This can be easily shown by means of naive dimensional analysis and symmetries as follows. Twin tops interact with the SM fields only through higher-dimensional operators. The operators relevant for the EWPO are those involving either a SM current or a derivative of the hypercharge field strength: OBt˜=g′mW2∂μBμνt˜¯γνt˜,Oqt˜=1v2q¯LγμqLt˜¯γμt˜,OHt˜=iv2H†Dμ↔Ht˜¯γμt˜,where t˜ indicates either a right- or left-handed twin top.

Notice that OHt˜ can be rewritten in terms of the other two operators by using the equations of motion, but it is still useful to consider it in our discussion.

The first two operators of Eq. (5.2) are generated at the scale m* by the tree-level exchange of the ρX. Their coefficients (in a basis with canonical kinetic terms) are respectively of order (mW2/m*2)(y˜/g*)2 and (yL2v2/m*2)⁢(y˜/g*)2, where y˜ equals either y˜L or y˜R depending on the chirality of t˜. The third operator breaks custodial isospin and the only way it can be generated is via the exchange of weakly coupled elementary fields at loop level. Given that the contribution to EWPO is further suppressed by t˜ loops, the third operator can affect EWPO only at, at least, two loops and is thus clearly negligible. By closing the t˜ loops the first two operators can give rise to effects that are schematically of the form BB, Bq¯q, or (q¯q)2. The formally quadratically divergent piece of the loop integral renormalizes the corresponding dimension-6 operators. For instance the second structure gives Cg′16π2yL2m*2(y˜g*)4∂νBμνq¯LγμqLwith C an O(1) coefficient which depends on the details of the physics at the scale m*. Using the equations of motion for B, the above operator gives rise to a correction to the Zbb¯ vertex of relative size δgLbgLb∼g′216π2yL2v2m*2(y˜g*)4which, even assuming y˜∼g*, is O(g′/yt)2 suppressed with respect to the leading visible sector effect we discuss below. Aside the quadratically divergent piece there is also a logarithmic divergent piece whose overall coefficient is calculable. The result is further suppressed with respect to the above contribution by a factor (mt˜2/m*2)ln(mt˜2/m*2).

An additional contribution could in principle come from loops of the extra three “twin” NGBs contained in the coset SO(8)/SO(7). Simple inspection however shows that there is no corresponding 1-loop diagram contributing to the EWPO. In the end we conclude that the effect of twin loops is negligible.

Since the effects from the twin sector can be neglected, the corrections to S^, T^, and δgLb are parametrically the same as in ordinary CH models. We now give a concise review of the contributions to each of these quantities, distinguishing between the threshold correction generated at the scale m* and the contribution arising from the RG evolution down to the electroweak scale. For recent analyses of the EWPO in the context of SO(5)/SO(4) CH models see for example Refs. [29,30,45].

<inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>S</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math></inline-formula> parameter

The leading contribution to the S^ parameter arises at tree level from the exchange of spin-1 resonances. Since only the (3,1) and (1,3) components of the spin-1 multiplet contribute, its expression is the same as in SO(5)/SO(4) composite-Higgs theories

We neglect for simplicity a contribution from the operator Eμνρμν, which also arises at tree level. See for example the discussion in Refs. [45,51].

: ΔS^ρ=g222gρ2ξ.In our numerical analysis presented in Sec. VI we use the two-site model relation Mρ=gρf/2 to rewrite ΔS^ρ=mW2Mρ2.The 1-loop contribution from loops of spin-1 and fermion resonances is subdominant (by a factor g*2/16π2) and will be neglected for simplicity in the following. Nevertheless, we explicitly computed the fermionic contribution (see Appendix E) to monitor the validity of the perturbative expansion and estimate the limit of strong coupling in our model (a discussion on this aspect was given in Sec. III B). An additional threshold correction to S^, naively of the same order as Eq. (5.6), arises from the exchange of cutoff modes at m*. As already anticipated, we neglect this correction in the following. In this respect our calculation is subject to an O(1) uncertainty and should rather be considered as an estimate, possibly more refined than a naive one, which takes the correlations among different observables into account.

Besides the UV threshold effects described above, S^ gets an IR contribution from RG evolution down to the electroweak scale. The leading effect of this type comes from the compositeness of the Higgs boson, and is the same as in SO(5)/SO(4) CH models [32]: ΔS^h=g22192π2ξlogm*2mh2.In the effective theory below m* this corresponds to the evolution of the dimension-6 operators OW=ig2mW2H†σiDμ↔HDνWμνi,OB=ig′2mW2H†Dμ↔H∂νBμνinduced by a 1-loop insertion of OH=12v2∂μ(H†H)∂μ(H†H).Denoting with c¯i the coefficients of the effective operators and working at leading order in the SM couplings, the RG evolution can be expressed as c¯i(μ)=(δij+γijlogμM)c¯j(M),where γij is the anomalous dimension matrix (computed at leading order in the SM couplings). The S^ parameter gets a correction ΔS^=(c¯W(mZ)+c¯B(mZ))ξ, and one has γW,H+γB,H=-g22/(96π2). An additional contribution to the running arises from insertions of the current-current operators OHq=iv2q¯LγμqLH†Dμ↔H,OHq′=iv2q¯LγμσiqLH†σiDμ↔H,OHt=iv2t¯RγμtRH†Dμ↔Hin a loop of top quarks. This is however suppressed by a factor yL2/g*2 compared to Eq. (5.7) and will be neglected. The suppression arises because the current-current operators are generated at the matching scale with coefficients proportional to yL2.

The total correction to the S^ parameter in our model is ΔS^=ΔS^ρ+ΔS^h, with the two contributions given by Eqs. (5.6) and (5.7).

<inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math></inline-formula> parameter

Tree-level contributions to the T^ parameter are forbidden in our model by the SO(3) custodial symmetry preserved by the strong dynamics, and can only arise via loops involving the elementary states. A non-vanishing effect arises at the 1-loop level corresponding to a violation of custodial isospin by two units. The leading contribution comes from loops of fermions and is proportional to yL4, given that the spurionic transformation rule of yL is that of a doublet, while yR is a singlet. We find: ΔT^Ψ=aUVNcyL216π2yL2v2MΨ2+aIRNcyt216π2yL2v2MΨ2logM12mt2,where aUV,IR are O(1) coefficients whose values are reported in Appendix E and we have defined M1≡MS2+yR2f2. The result is finite and does not depend on the cutoff scale m*. The first term corresponds to the UV threshold correction generated at the scale μ=M1∼MΨ. The second term instead encodes the IR running from the threshold scale down to low energy, due to loops of top quarks. In the effective theory below M1 it corresponds to the RG evolution of the dimension-6 operator OT=12v2(H†Dμ↔H)2due to insertions of the current-current operators of Eq. (5.11). In particular, ΔT^=c^T(mZ)ξ and one has γT,Ht=-γT,Hq=3yt2/4π2, γT,Hq′=0. Notice that the size of the second contribution with respect to the first is O[(yt/yL)2log(M12/mt2)]: for yt∼yL, that is for fully composite tR, the IR dominated contribution is formally logarithmically enhanced and dominant.

Further contributions to T^ come from loops of spin-1 resonances, the exchange of cutoff modes and Higgs compositeness. The latter is due to the modified couplings of the composite Higgs to vector bosons and reads [32]: ΔT^h=-3g1264π2ξlogm*2mh2.In the effective theory it corresponds to the running of OT due to the insertion of the operator OH in a loop with hypercharge. The contribution is of the form of Eq. (5.10) with γT,H=3g12/32π2. The exchange of spin-1 resonances gives a UV threshold correction which is also proportional to g12 (as a spurion, the hypercharge coupling transforms as an isospin triplet), but without any log enhancement. It is thus subleading compared to Eq. (5.14) and we will neglect it for simplicity (see Ref. [45] for the corresponding computation in the context of SO(5)/SO(4) models). Finally, we also omit the effect of the cutoff modes because it is incalculable, although naively this is of the same order as the contribution from states included in our simplified model. Our result is thus subject to an O(1) uncertainty.

The total contribution to the T^ parameter in our model is therefore ΔT^=ΔT^h+ΔT^Ψ with the two contributions given by Eqs. (5.12) and (5.14).

<inline-formula><mml:math display="inline"><mml:mi>δ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In the limit of vanishing transferred momentum, tree-level corrections to δgLb are forbidden by the PLR parity of the strong dynamics that exchanges SU(2)L with SU(2)R in the visible SO(4) and SU˜(2)L with SU˜(2)R in the twin SO˜(4) (see Appendix A for details). This is a simple extension of the PLR symmetry of CH models which protects the Zbb¯ coupling from large corrections [52]. In our case PLR is an element of the unbroken SO(7) and keeps the vacuum unchanged. It is thus an exact invariance of the strong dynamics, differently from SO(5)/SO(4) models where it is accidental at O(p2). The gauge couplings g1,2 and yL explicitly break it, while yR preserves it. At finite external momentum δgLb gets a non-vanishing tree-level contribution: (δgLb)tree=f2ξ8Mρ2[g12(αL+α7L)-g22αL]yL2f2MΨ2+yL2f2.In the effective theory below M1, this correction arises from the dimension-6 operators OBq=g′mW2∂μBμνq¯LγνqL,OWq=gmW2DμWμνaq¯LγνσaqL,.It is of order (yL2/g*2)(g2/g*2)ξ, hence a factor g2/g*2 smaller than the naive expectation in absence of the PLR protection.

At the 1-loop level, corrections to δgLb arise from the virtual exchange of heavy fermion and vector states. The leading effect comes at O(yL4) from loops of heavy fermions (the corresponding diagrams are those of Figs. 4 and 5) and reads (δgLb)Ψ=yL216π2NcyL2v2MΨ2(bUV+cUVlogm*2MΨ2)+bIRyt216π2NcyL2v2MΨ2logM12mt2.The expressions of the O(1) coefficients bUV,IR and cUV are reported in Appendix E. The first term is logarithmically divergent and encodes the UV threshold correction at the matching scale. The divergence comes, in particular, from diagrams where the fermion loop is connected to the b-quark current through the exchange of a spin-1 resonance [29]. A simple operator analysis shows that the threshold contribution from the vector resonances in the adjoint of SO(7) identically vanishes in our model (see Appendix D for details). An additional UV threshold contribution to δgLb arises from diagrams where the spin-1 resonances circulate in the loop. For simplicity we will not include such effect in our analysis (see Ref. [30] for the corresponding computation in the context of SO(5)/SO(4) models). It is however easy to show that there is no possible diagram with ρX circulating in the loop as a consequence of its quantum numbers, while the corresponding contribution from vector resonances in the adjoint of SO(7) is nonvanishing in this case.

The second term in Eq. (5.17) accounts for the IR running down to the electroweak scale. In the effective theory below M1 one has δgLb=-(c¯Hq(mZ)+c¯Hq′(mZ))/2, hence the IR correction arises from the evolution of the operators OHq and OHq′ due to loops of top quarks. In this case the operators that contribute to the running via their 1-loop insertion are those of Eq. (5.11) as well as the following four-quark operators [53]: OLR=(q¯LγμqL)(t¯RγμtR),OLL=(q¯LγμqL)(q¯LγμqL),OLL′=(q¯LσaγμqL)(q¯LσaγμqL).In fact, the operators contributing at O(yL2yt2) to Eq. (5.10) are only those generated at O(yL2) at the matching scale; these are OHt, the linear combination OHq-OHq′ (even under PLR), and OLR (generated via the exchange of ρX).

The operators OLL and OLL′ are generated at O(yL4) by the tree-level exchange of both ρX and ρ.

Notice finally that the relative size of the IR and UV contributions to δgLb is O[(yt/yL)2log(M12/mt2)] precisely like in the case of ΔT^Ψ.

It is interesting that in our model the fermionic corrections to δgLb and T^ are parametrically of the same order and their signs tend to be correlated. It is for example well known that a heavy fermion with the quantum numbers of tR gives a positive correction to both quantities [28,54–56]. We have verified that this is also the case in our model for MS≪MΨ∼Mρ (light singlet).

In this limit one has ΔT^Ψ≃3(δgLb)Ψ.

Conversely, a light septuplet (MΨ≪MS∼Mρ) gives a negative contribution to both δgLb and T^.

The existence of a similar sign correlation in the limit of a light (2,2) has been pointed out in the context of SO(5)/SO(4) CH models, see Ref. [29].

 Although in general the expressions for ΔT^Ψ and (δgLb)Ψ are uncorrelated, their signs tend to be the same whenever the contribution from ρX to Eq. (5.17) is subleading. The sign correlation can instead be broken if ρX contributes significantly to δgLb [in particular, (δgLb)Ψ can be negative for αiL=-αiR]. The importance of the above considerations lies in the fact that EW precision data prefer a positive T^ and a negative δgLb. Situations when both quantities have the same sign are thus experimentally disfavored.

Considering that no additional correction to δgLb arises from Higgs compositeness, and that we neglect as before the incalculable effect due to cutoff states, the total contribution in our model is δgLb=(δgLb)tree+(δgLb)Ψ, with the two contributions given by eqs. (5.15) and (5.17).

RESULTS AND DISCUSSION

We are now ready to translate the prediction for the Higgs mass and the EWPO into bounds on the parameter space of our simplified model and for the composite TH in general. We are interested in quantifying the degree of fine tuning that our construction suffers when requiring the mass scale of the heavy fermions to lie above the ultimate experimental reach of the LHC. As discussed in Sec. II, this scale receives the largest boost from the TH mechanism (without a corresponding increase in the fine tuning of the Higgs mass) in the regime of parameters corresponding to a fully strongly coupled theory, where no quantitatively precise EFT description is allowed. Our computations of physical quantities in this most relevant regime should then be interpreted as an educated naive dimensional analysis (eNDA) estimate, where one hopes to capture the generic size of effects beyond the naivest 4π counting, and including factors of a few related to multiplet size, to spin, and to numerical accidents. In the limit where Mρ/f and MΨ/f are significantly below their perturbative upper bounds our computations are well defined. eNDA then corresponds to assuming that the results do not change by more than O(1) (i.e. less than O(5) to be more explicit) when extrapolating to a scenario where the resonance mass scale sits at strong coupling. In practice we shall consider the resonant masses up to their perturbativity bound and vary the αi and ci parameters within an O(1) range.

Notice indeed that (α=1,c=0) and (α=0,c=1/2) correspond to specific limits at weak coupling, namely the two-site model and the linear sigma model respectively. This suggests that their natural range is O(1).

In view of the generous parameter space that we shall explore our analysis should be viewed as conservative, in the sense that a realistic TH model will never do better.

Let us now describe the various pieces of our analysis. Consider first the Higgs potential, where the dependence on physics at the resonance mass scale is encapsulated in the function F1 [Eq. (4.2)] which controls the UV threshold correction to the Higgs quartic. It is calculable in our simplified model and the result is O(1) [its expression is reported in Eq. (C1)], but it can easily be made a bit smaller at the price of some mild tuning by varying the field content or the representations of the heavy fermions. In order to account for these options and thus broaden the scope of our analysis we will treat F1 as a free O(1) parameter. The value of F1 has a direct impact on the size of the left-handed top mixing yL, since δmh2|UV∼yL4F1, and hence controls the interplay between the Higgs potential and EWPO. Specifically, as we already stressed, a smaller F1 implies a larger value of yL, which in turn gives a larger ΔT^∝yL4v2/MΨ2. This could help improve the compatibility with EWPT even for large MΨ, at the cost of a small additional tuning due to the need for a clever maneuver in the S^,T^ plane to get back into the ellipse, as well as the fact that F1 is generically expected to be O(1). In the following we will thus treat F1 as an input parameter and use Eqs. (4.2) and (B4) to fix yL and yR in terms of the Higgs and top quark experimental masses. Our final results will be shown for two different choices of F1, namely F1=1 and F1=0.3, in order to illustrate how the bounds are affected by changing the size of the UV threshold correction to the Higgs potential.

The EWPO and the Higgs mass computed in the previous sections depend on several parameters, in particular on the mass spectrum of resonances (see Appendix B), the parameters ci, αi of Eqs. (3.9), (3.14), and the parameter F1 discussed above. In order to focus on the situation where resonances can escape detection at the LHC, we will assume that their masses are all comparable and that they lie at or just below the cutoff scale. In order to simplify the numerical analysis we thus set MΨ=MS=M˜Ψ=M˜S=Mρ=MρX=m*/2. The factor of two difference between MΨ and m* is chosen to avoid setting all UV logarithms of the form log(m*/MΨ) to zero, while not making them artificially large. As a further simplification we set cL=cR≡c, α7L=α1L, and α7R=α1R. The parameter αL appears only in the tree-level contribution to δgLb, see Eq. (5.15), and we fix it equal to 1 for simplicity. Even though the above choices represent a significant reduction of the whole available parameter space, for the purpose of our analysis they represent a sufficiently rich set where EWPT can be successfully passed.

Let us now discuss the numerical bounds on the parameter space of our simplified model. They have been obtained by fixing the top and Higgs masses to their experimental value and performing the numerical fit described in Appendix F. As experimental inputs, we use the PDG values of the top quark pole mass mt=173.21±0.51±0.71 (see last paragraph of Appendix C), and of the Higgs mass, mh=125.09±0.24  GeV [57]. Figure 2 shows the results of the fit in the (MΨ,ξ) plane for F1=0.3 (left panel) and F1=1 (right panel). In both panels we have set c=0, which corresponds to the two-site model limit of our simplified Lagrangian. The yellow regions correspond to the points that pass the χ2 test at 95% confidence level (CL), see Appendix F for details. Solid black contours denote the regions for which α1L=-α1R=1, while dashed contours surround the regions obtained with α1L=α1R=1. The areas in blue are theoretically inaccessible. The lower left region in dark blue, in particular, corresponds to MΨ/f≡gΨ<yL. The upper dark- and light-blue regions correspond instead to points violating the perturbative limits on gΨ given by Eq. (3.15) with N=8, and Eq. (3.16) with Nf=24, respectively (see Sec. III B for a discussion). The difference between these two regions can be taken as an indication of the uncertainty associated with the perturbative bound.

Notice that because of our choice m*=2MΨ, the scale m* lies a factor of 2 above the pertubative cutoff. This is compatible with the semiquantitative nature of our estimates. As we stated previously we insisted in keeping m*=2MΨ because it implies a more generic contribution to electroweak precision observables.

210.1103/PhysRevD.96.095036.f2

Allowed regions in the (MΨ,ξ) plane for F1=0.3 (left panel) and F1=1 (right panel). See the text for an explanation of the different regions and of the choice of parameters.

In the left panel of Fig. 2 the allowed (lighter yellow) region extends up to ξ≃0.2 for masses MΨ in the 2–3 TeV range. Such large values of ξ are possible in this case because the fermionic contribution to ΔT^Ψ turns out to be sufficiently large and positive to compensate for both the negative ΔT^h in Eq. (5.14) and the positive ΔS^ρ and ΔS^h in Eqs. (5.6), (5.7). For larger MΨ the fermionic contribution ΔT^Ψ becomes too small and this compensation no longer occurs. In this case, however, the strongest bound comes from the perturbativity limit (blue region), which makes points with large MΨ at fixed ξ theoretically inaccessible. Notice that large values of ξ become excluded if one considers the choice α1L=α1R=1 leading to the dashed contour. The large difference between the solid and dashed curves (i.e. lighter and darker yellow regions) depends on the sign correlation between ΔT^Ψ and δgLb. In the case of the solid line, the signs are anticorrelated (e.g., positive ΔT^Ψ and negative δgLb), allowing for the compensation effect by ΔT^Ψ. In the case of the dashed line, instead, the signs of the two parameters are correlated (both positive), so that when ΔT^Ψ is large, δgLb is also large and positive. This makes it more difficult to pass the χ2 test, since data prefer a negative δgLb.

In the right panel of Fig. 2, obtained with F1=1, the allowed yellow region shrinks because the larger value of F1 implies a smaller yL hence a smaller ΔT^Ψ. In this case the χ2 test is passed only for ξ<0.06, and the difference between the solid and dashed lines is small since the large and positive ΔS^ always dominates the fit. Masses MΨ larger than ∼5  TeV are excluded by the perturbative bound, unless one considers smaller values of ξ.

The results of Fig. 2 can change significantly if the parameter c is allowed to be different from zero. In particular, as one can verify from our formulas in Appendix E, positive values of c increase ΔT^ and as a result the allowed regions in Fig. 2 shift to the right toward larger values of MΨ. In this case the perturbative bound excludes a large portion of the region passing the χ2 test. The effect of varying c is illustrated in Fig. 3, which shows the 95% CL allowed regions in the plane (c,α) for F1=0.3 (left panel) and F1=1 (right panel). In both panels we have set α≡α1L=-α1R (ensuring positive ΔT^Ψ and negative δgLb). The yellow, orange, and red regions correspond, respectively, to ξ=0.05, ξ=0.1, and ξ=0.15, with the masses of the resonances fixed at their perturbative upper bound Mρ=MΨ=4πf/N-2 (which for N=8 gives ∼6/4/3.2  TeV). Note that increasing F1 (reducing yL) shifts the allowed region toward positive values of c, since as mentioned above, smaller yL requires a larger positive c to get a large enough ΔT^Ψ. Obviously, larger values of ξ correspond to smaller allowed regions, as is clear from Fig. 2. Finally, notice that the vertically symmetric structure of the allowed regions is due to the quadratic dependence of δgLb on α. From these plots, one can see that for resonances conceivably out of direct reach of the LHC and for ξ∼0.1, corresponding to about 20% tuning of the Higgs mass, both α and c are allowed to span a good fraction of their expected O(1) range. No dramatic extra tuning in these parameters seems therefore necessary to meet the constraints of EWPT. In particular, considering the plot for F1=1 (right panel), one notices that the bulk of the allowed region is at positive c. For instance by choosing c∼0.2–0.5 the plot in the (ξ,MΨ) plane for F1=1 becomes quite similar to the one at the left of Fig. 3 valid for F1=0.3: there exists a “peak” centered at MΨ∼2–4  TeV and extending up to ξ∼0.2. The specific choice c=0 is thus particularly restrictive for F1=1 (right panel of Fig. 2), but this restriction is lifted for positive c. Overall we conclude that for ξ∼0.1 and for resonances just beyond the LHC reach, the correct value of the Higgs quartic can be obtained and EWPT passed with only a mild additional tuning associated with a sign correlation α1L=-α1R, and a correlation between c and F1 (e.g. c>0 for F1=1). These correlations allow the various contributions to T^ and δgLb to compensate for each other, achieving agreement with EWPT. If forced to quantify the tuning inherent in these effects, we could estimate it to be around 1/4=(1/2)×(1/2), given about 1/2 of the plausible choices for both αi and ci are allowed.

310.1103/PhysRevD.96.095036.f3

Allowed regions in the (c,α) plane, with c=cL=cR, for F1=0.3 (left panel) and F1=1 (right panel). The yellow, orange, and red regions correspond to ξ=0.05, 0.1 and 0.15 respectively. See the text for an explanation of the choice of the other parameters.

SUMMARY

In this paper we tried to assess how plausible a scenario yielding no new particles at the LHC can be obtained using the TH construction. We distinguished three possible classes of models: the subhypersoft, the hypersoft and the superhypersoft, with increasing degree of technical complexity and decreasing (technical) fine tuning. We then focused on the CH incarnation of the simplest option, the subhypersoft scenario, where the boost factor for the mass of colored partners [Eq. (2.4)] at fixed tuning is roughly given by g*2yt×1ln(m*/mt).Here the gain derives entirely from the relative coupling strength g*/2yt, making the marriage of twinning and compositeness practically obligatory. We attempted a more precise estimate of the upper limit on g*/2yt, as compared with previous studies (e.g. Ref. [13]). We found by independent but consistent estimates, that the bound ranges from ∼3 in a toy sigma model [Eq. (2.6)] to ∼5 in a simplified CH model [Eq. (3.15)], with both limits somewhat below the NDA estimate of 4π∼12. Consequently for a mild tuning ε∼0.1 the upper bound on the mass of the resonances with SM quantum numbers is closer to the 3–5 TeV range than it is to 10 TeV. This gain, despite being less spectacular than naively expected, is still sufficient to push these states out of direct reach of the LHC, provided we resort to full strong coupling. In practice this implies no real computational advantage from considering holographic realizations of composite TH constructions: since the boost factor is controlled by the KK coupling, the 5D description breaks down in precisely the most interesting regime, where the KK coupling is strong. In this situation computations based on an explicit 5D construction, such as the ones studied in Refs. [9,16] for instance, are no better than numerical estimates made in our simplified model. Indeed we have checked that EWPT can be satisfied in a sizable portion of the parameter space, given some interplay among the various contributions. In particular the IR corrections to T^ and S^ are enhanced by ln(m*/mh), and for ξ>0.1 the compensating contribution to T^, which decreases like 1/m*2, is necessary. Given that perturbativity limits m* to be below 5 TeV for ξ>0.1 (see the upper blue exclusion region in Fig. 2) this compensation in EWPT can still take place at the price of a moderate extra tuning. For ξ of order a few percent on the other hand, EWPT would be passed without any additional tuning, while the masses of SM-charged resonances would be pushed up to the 10 TeV range, where nothing less than a 100 TeV collider would be required to discover them, and that barely so [58,59].

Although EWPT work similarly in the CH and composite TH frameworks, the two are crucially different when it comes to contributions to the Higgs quartic. In the CH these are enhanced when g*, i.e. m*/f, is strong and, as discussed for instance in Ref. [33], in order to avoid additional tuning of the Higgs quartic g* cannot be too large. According to the study in Refs. [60–63] the corresponding upper bound on the mass of the colored top partners in CH reads roughly m*/f≲1.5; this should be compared to the upper bound m*/f≲5 from strong coupling we found in Eq. (3.15). The Higgs quartic protection afforded by the TH mechanism allows us to take m*/f as large as possible, allowing the colored partners to be heavier at fixed f, hence at fixed fine tuning ξ. In the end the gain is about a factor of 5/1.5∼3, not impressive, but sufficient to place the colored partners outside of LHC reach for a mild tuning ξ∼0.1.

Finally, we comment on the classes of models not covered in this paper: the hypersoft and superhypersoft scenarios. The latter requires combining supersymmetry and compositeness with the TH mechanism, which, while logically possible, does not correspond to any existing construction. Such a construction would need to be rather ingenious, and we currently do not feel compelled to provide it, given the already rather epicyclic nature of the TH scenario. The simpler hypersoft scenario, though also clever, can by contrast be implemented in a straightforward manner, via, e.g., a tumbling SO(9)→SO(8)→SO(7) pattern of symmetry breaking. The advantage of this approach is that it allows us to remain within the weakly-coupled domain, due to the presence of a relatively light twin Higgs scalar mode σ, whose mass can be parametrically close to that of the twin tops, ∼ytf (around 1 TeV for ξ∼0.1). As well as giving rise to distinctive experimental signatures due to mixing with the SM Higgs [64], the mass of the light σ acts as a UV cutoff for the IR contributions to S^ and T^ in Eqs. (5.7) and (5.14) [12]. For sufficiently light σ then, less or no interplay between the various contributions is required in order to pass EWPT. Together with calculability, this property may well single out the hypersoft scenario as the most plausible TH construction.

ACKNOWLEDGMENTS

We would like to thank Andrey Katz, Alberto Mariotti, Kin Mimouni, Giuliano Panico, Diego Redigolo, Matteo Salvarezza, and Andrea Wulzer for useful discussions. The Swiss National Science Foundation partially supported the work of D. G. and R. R. under Contracts No. 200020-150060 and No. 200020-169696, the work of R. C. under Contract No. 200021-160190, and the work of R. T. under the Sinergia network CRSII2-160814. The work of R. C. was partly supported by the ERC Advanced Grant No. 267985 Electroweak Symmetry Breaking, Flavour and Dark Matter: One Solution for Three Mysteries (DaMeSyFla). R. M. is supported by ERC Grant No. 614577 HICCUP High Impact Cross Section Calculations for Ultimate Precision.

<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> GENERATORS AND CCWZ VARIABLES

In this Appendix we define the generators of the SO(8) algebra and describe the SO(8)/SO(7) symmetry-breaking pattern, introducing the CCWZ variables for our model. We refer the reader to Refs. [41,42] for a detailed analysis of this procedure and we closely follow the notation of Ref. [13].

We start by listing the twenty-eight generators of SO(8) and decomposing them into irreducible representations of the unbroken subgroup SO(7): 28=7⊕21. They can be compactly written as: (Tij)kl=i2(δikδjl-δilδjk),with i,j,k,l=1,…,8. We choose to align the vacuum expectation value responsible for the spontaneous breaking of SO(8) to SO(7) along the 8th component: ϕ0=f(0,0,0,0,0,0,0,1)t. With this choice, the broken and unbroken generators, transforming, respectively, in the 7 and 21 of SO(7), are (Tβ7)γρ=i2(δ8γδβρ-δ8ρδβγ),(Tαβ21)γρ=i2(δαγδβρ-δαρδβγ),where α,β=1,…,7 and γ,ρ=1,…,8. It is useful to identify the subgroups SO(4)∼SU(2)L×SU(2)R and SO˜(4)∼SU˜(2)L×SU˜(2)R of SO(8); they are generated by: (TL)α=(tLα000),(TR)α=(tRα000),(T˜L)α=(000tLα),(T˜R)α=(000tRα),where tLα and tRα are 4×4 matrices defined as (tL,Rα)ij=-i2[12εαβγ(δiβδjγ-δjβδiγ)±(δiαδj4-δjαδi4)]with α=1, 2, 3 and i,j=1,…,4. The elementary SM and twin vector bosons gauge, respectively, a subgroup SU(2)L×U(1)Y of SO(4), with U(1)Y≡U(1)R3, and the subgroup SU˜(2)L inside SO˜(4). This choice corresponds to having zero vacuum misalignment at tree level.

The spontaneous breaking of SO(8) to SO(7) delivers seven NGBs, that we collect in the vector Π=(π2,π1,-π3,π4,π˜2,π˜1,-π˜3)t. The first four transform as a fundamental of SO(4) and form the Higgs doublet; the remaining three are singlets of SO(4) and are thus neutral under the SM gauge group. All together they can be arranged in the Goldstone matrix Σ(Π)=ei2fΠ·T7=[I7-ΠΠtΠt·Π(1-cos(Πt·Πf))ΠΠt·Πsin(Πt·Πf)-ΠΠt·Πsin(Πt·Πf)cos(Πt·Πf)].The latter transforms nonlinearly under the action of an SO(8) group element g, according to the standard relation: Σ(Π)→g·Σ(Π)·h†(Π,g),where h(Π,g)∈SO(7) depends on g and Π(x). We identify the Higgs boson with the NGB along the generator T47; in the unitary gauge, all the remaining NGBs are nonpropagating fields and the Π vector becomes Π|unitary gauge=(0,…,π4=⟨h⟩+h(x),…,0),where ξ=sin2(⟨h⟩/f)=v2/f2. In this case the Σ matrix simplifies to: Σ(Π)|unitary gauge=ei2fπ4T47=[I30000cosπ4f0sinπ4f00I300-sinπ4f0cosπ4f].

Given the above symmetry breaking pattern, it is possible to define a LR parity, PLR=diag(-1,-1,-1,+1,-1,-1,-1,+1),which exchanges SU(2)L with SU(2)R inside SO(4) and SU˜(2)L with SU˜(2)R inside SO˜(4). The corresponding action on the fields is such that π4 is even, while all the other NGBs are odd. As already noticed in Sec. V C, PLR is an element of both SO(8) and SO(7), which means that it is an exact symmetry of the strong dynamics and acts linearly on the physical spectrum of fields.

The CCWZ variables dμ and Eμ are defined as usual through the Maurer-Cartan form, Σ†(Π)DμΣ(Π)≡idμi(Π)Ti7+iEμa(Π)Ta21,as the components along the broken and unbroken generators of SO(8) respectively. The derivative Dμ is covariant with respect to the external SM and twin gauge fields: Dμ=∂μ-iAμATA,withAμATA=g2Wμα(TL)α+g1Bμ(TR)3+g˜2W˜μα(T˜L)α.Under the action of a global element g∈SO(8), dμ and Eμ transform with the rules of a local SO(7) transformation: dμ≡dμiTi7→h(Π,g)dμh†(Π,g),Eμ≡EμaTa21→h(Π,g)(Eμ-i∂μ)h†(Π,g).It is straightforward to derive the explicit expressions of dμ and Eμ from the Maurer-Cartan relation in Eq. (A10). They are however lengthy and not very illuminating, so we do not report them here. We can easily obtain the mass spectrum of the gauge sector of our theory from the NGB kinetic term; in the unitary gauge and after rotating to the mass eigenstate basis, we find: Lmass=f24(dμi)2⊃g224f2ξWμ+Wμ-+(g12+g22)8f2ξZμZμ+g˜228f2(1-ξ)∑i=13(W˜μi)2.

MASS MATRICES AND SPECTRUM

In this Appendix, we briefly discuss the mass matrices of the different charged sectors in the composite TH model and the related particle spectrum. We refer to Ref. [13] for the expressions of the Ψ7 and Ψ˜7 multiplets in terms of their component heavy fermions.

We start by considering the fields that do not have the right quantum numbers to mix with the elementary SM and twin quarks and whose mass is therefore independent of the mixing parameters yL,R and y˜L,R. These are the composite fermions X5/3, D˜1 and D˜-1, with charges 5/3, 1 and -1 respectively; their mass is exactly given by the Lagrangian parameters MΨ (for the first one), and M˜Ψ (for the last two).

The remaining sectors have charge -1/3, 0, and 2/3 and because of the elementary/composite mixing the associated mass matrices are in general nondiagonal and must be diagonalized by a proper field rotation. The simplest case is the (-1/3)-charged sector, containing the bottom quark and the heavy B field; the mass matrix in the {b,B} basis is M-1/3=(0fyL0-MΨ).After rotation, we find a massless bottom quark (it has no mass since we are not including the bR in the model), and a massive B particle with mB2=MΨ2+yL2f2.

As regards the sector of charge 2/3, it contains seven different particles: the top quark, the toplike heavy states T and X2/3 and four composite fermions that do not participate in the SM weak interactions, S2/31,⋯S2/34. In the {t,T,X2/3,S2/31,…,S2/34} basis, the mass matrix is given by: M2/3=(012fyL(1-ξ+1)-12fyL(1-ξ-1)0-fyLξ20-MΨ00000-MΨ00000-MΨ×I30fyR000-MS).The states S2/31,S2/32,S2/33 completely decouple from the elementary sector and do not mix with the top quark. Their mass is therefore exactly given by the Lagrangian parameter MΨ. The remaining 4×4 matrix is in general too complicated to be analytically diagonalized, but one can easily find the spectrum in perturbation theory by expanding M2/3 for ξ≪1, which is in general a phenomenologically viable limit. The leading order expression for the masses is then: mt2≃f42yL2yR2MS2+yR2f2ξ+O(ξ2),mX2/32=MΨ2,mT2≃MΨ2+yL2f2(1-ξ2)+O(ξ2),mS2/342≃MS2+yR2f2+yL2f2MS22(MS2+yR2f2)ξ+O(ξ2).The X2/3 fermion can be also decoupled and its mass is exactly equal to MΨ. On the contrary, the other three particles mix with each other and their mass gets corrected after EWSB (as expected, the top mass is generated for nonzero values of ξ).

Finally, we analyze the neutral sector of our model. It comprises eight fields, the twin top and bottom quarks, and six of the composite fermions contained in the Ψ˜7 multiplet. Working in the field basis {t˜,b˜,D˜01,D˜02,U˜01,…,U˜04}, the mass matrix reads: M0=(00-12fξy˜L12fξy˜L00-ify˜L2-f1-ξy˜L20000-ify˜L2fy˜L20000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ0fy˜R000000-M˜S).After diagonalization, we find one massless eigenvalue corresponding to the twin bottom quark (it does not acquire mass since we are not introducing the b˜R field). Four of the neutral heavy fermions completely decouple and acquire the following masses mU˜01=mU˜03=mD˜02=M˜Ψ,mU˜022=M˜Ψ2+y˜L2f2.The elementary/composite mixing induces instead corrections to the masses of the remaining neutral particles; at leading order in ξ we find: mt˜2≃f42y˜L2y˜R2M˜S2+y˜R2f2(1-ξ)+O(ξ2),mD˜012≃M˜Ψ2+12y˜L2f2(1+ξ)+O(ξ2),mU˜042≃M˜S2+y˜R2f2+y˜L2f2M˜S22(M˜S2+y˜R2f2)(1-ξ)+O(ξ2).

We conclude by noticing that the masses of the particles in different charged sectors are not unrelated to each other, but must be connected according to the action of the twin symmetry. In particular, it is obvious that the two singlets S2/34 and U˜04 form an exact twin pair, as it is the case for each SM quark and the corresponding twin partner. The remaining pairs can be easily found from the spectrum and correspond to the implementation of the twin symmetry in the composite sector as defined in [13].

RG IMPROVEMENT OF THE HIGGS EFFECTIVE POTENTIAL

In this Appendix we describe the computation of the Higgs effective potential and its RG improvement. First of all, the UV threshold correction can be computed with a standard Coleman-Weinberg (CW) procedure, from which we can easily derive the function F1 of Eq. (4.2). We find: F1=-14[1+MS4(MS2+f2yR2)2-MS2+MΨ2-f2yR2MS2-MΨ2+f2yR2logm*2MΨ2+MS2(MS2(MΨ2+f2yR2)+2f2yR2MΨ2+MS4)(MS2+f2yR2)2(MS2-MΨ2+f2yR2)logm*2MS2+f2yR2].

The IR contribution to the Higgs mass can be organized using a joint expansion in ξ and (αilog), where αi=gi2/(4π) for couplings gi. Schematically: δmh2|IR=mt2αt4πt[a1+b1ξ+b2αt4πt+b3αS4πt+c1ξ2+c2ξαS4πt+c3αS216π2t2+c4ξαt4πt+c5αt216π2t2+c6αt4παS4πt2]-a2mW2αEW4πt+twin,where t=logm*2/mt2 and all the couplings are evaluated at the scale m*. Here ai, bi, ci are the O(1) coefficients of the LO, NLO, and NNLO terms respectively. The twin contribution is obtained by substituting all αi, ai, bi, ci with the corresponding tilded quantities and t with t˜=logm*2/mt˜2. The calculation of the LO and NLO terms is straightforward and all these contributions are included in our calculation. The NNLO terms indicated in blue are also simple to evaluate, and are included in our final result, indicated as NNLO*. The calculation of the remaining NNLO terms is more complicated, since it involves the running of several higher dimensional operators involving both the SM and twin fields. An attempt to compute the full potential at NNLO has been presented in Ref. [46]. However, while that result includes the contribution of the SM Goldstone bosons, additional contributions induced by the twin Goldstones, which are expected to arise at NNLO, are not included. Since the full calculation at NNLO is beyond the scope of this paper, we limit ourselves to only include the colored contributions in Eq. (C2). Comparing with Ref. [46] suggests that our NNLO* result gives an overestimate of the IR Higgs mass; that this is the case at large m* is also evident from Fig. 1. For this reason we consider the NNLO* curve only as an indication of the importance of the NNLO correction, and use as final input for our numerical analysis the average of the LO and NLO results as described in Sec. IV.

We present now a procedure for computing the aforementioned contributions to the Higgs mass based on the approach of Ref. [46]. The RG improvement of the Higgs effective potential can be obtained by solving the one-loop β-function of the vacuum energy in the background of the Higgs field. The fermion contribution to the vacuum energy is given by dVf(hc,t)imprdt=Nc16π2[Mt(hc,t)4+Mt˜(hc,t)4],whereas the leading gauge contribution is given by Vg(hc,t)=316π2[2MW(hc)2+MZ(hc)2+3MW˜(hc)2]t.

In order to compute the improved potential to NNLO* accuracy, we must input the gauge boson masses at tree-level and the renormalized top and twin top masses with leading-log accuracy in the corresponding Yukawa couplings and with next-to-leading-log accuracy in the strong gauge couplings. After solving Eq. (C3) for the effective potential, one must properly account for the Higgs wave function renormalization before expanding around the minimum of the potential in order to compute the physical Higgs mass. We fill in the salient details below.

On integrating out the heavy degrees of freedom, the composite twin Higgs model at the scale m* can be represented by an effective Lagrangian that is invariant under a global SU(3)c×SU(2)L×U(1)Y symmetry, identified with the gauge group of the SM, as well as an additional SU(3)˜c×SU(2)˜L+R. SU(3)c˜ is the gauged twin color and SU(2)˜L+R is a global twin custodial group, broken only by fermion interactions, under which the twin gauge bosons (W˜) and Goldstone bosons (GBs) transform as triplets. We work in the gaugeless limit g=g′=g˜=0, since we are only interesting in capturing the leading contributions to the Higgs potential that parametrically depend on yt, y˜t, gS, and g˜S.

The relevant light degrees of freedom are the SM fermion doublet, QL, and singlet, tR, and their twin counterparts t˜L and t˜R, the gluons and their twins; the Higgs doublet (including the massless GBs) and the twin GBs. For economy of notation we define a twin Higgs doublet in analogy with the SM one, H˜=12(2π˜+h˜+iπ˜0),withh˜=f1-2H†H+π˜2f2,making explicit the dependence on the twin Higgs which is integrated out at tree-level, thus realizing the SO(8) symmetry nonlinearly.

Since the background field calculation at leading-log order requires corrections to the fermion masses and Higgs wave function at only one loop, we can neglect in the effective Lagrangian at the scale m*, any operator that has vanishing coefficients at tree level. We also omit all operators that cannot be predicted solely in terms of the low-energy parameters of the theory (yt and y˜t) and f, including products of Higgs and fermion currents that arise from integrating out composite fermions.

In any case, these do not contribute to the effective potential at this order.

We make the following field redefinition: fsin(|Π|f)|Π|Πi→Πi,for  |Π|=2H†H+π˜2which allows the effective Lagrangian, containing a pure scalar sector LS and a fermionic sector LF, to be expressed in an especially compact form. The relevant operators in each sector are given below: LS=DμH†DμH+12∂μπ˜∂μπ˜+cHOH+cHπ˜OHπ˜+cπ˜Oπ˜2f2+2dHH†Hf2OH,and LF=ib¯L∂bL+ib¯R∂bR+it¯L∂tL+it¯R∂tR-yt[Q¯LHctR+H.c.]+ib˜¯L∂b˜L+ib˜¯R∂b˜R+it˜¯L∂t˜L+it˜¯R∂t˜R-y˜t[Q˜¯LH˜ct˜R+H.c.].where OH=∂μ(H†H)∂μ(H†H),Oπ˜=∂μπ˜2∂μπ˜2,OHπ˜=∂μ(H†H)∂μπ˜2.We have defined a twin fermion doublet Q˜L=(t˜L,b˜L)T, transforming under the twin-sector symmetries, and the Yukawa couplings are defined at tree-level as yt(m*)=yLyRfMS2+yR2f2andy˜t(m*)=yL˜yR˜fM˜S2+yR˜2f2.

The Wilson coefficients at the scale m* have the boundary values: cH(m*)=cHπ˜(m*)=cπ˜(m*)=dH(m*)=1,while the Z2 symmetry enforces yt(m*)=y˜t(m*)andαs(m*)=αs˜(m*)

We now expand the effective Lagrangian in the background of the Higgs field h=hc+h^ to obtain: LS=Zπ(hc)(12∂μπ0∂μπ0+∂μπ+∂μπ-)+12Zh^(hc)∂μh^∂μh^+12Zπ˜(hc)∂μπ˜∂μπ˜,where we defined Zπ(hc)=1,Zh^(hc)=1+cHhc2f2+dHhc4f4,Zπ˜(hc)=1,and LF=∑ψiZψ(hc)ψ¯∂ψ-mt(hc)t¯t-mb(hc)b¯b-mt˜(hc)t˜¯t˜-mb˜(hc)b˜¯b˜-yt(hc)2h^t¯t-iyt(hc)2π0t¯γ5t-iyt(hc)2π-b¯(1+γ5)t+iyt(hc)2π+t¯(1-γ5)b+y˜t(hc)2h^t˜¯t˜-iyπ˜t˜t˜(hc)2π˜0t˜¯γ5t˜-iyπ˜t˜b˜(hc)2π˜-b˜¯(1+γ5)t˜+iyπ˜t˜b˜(hc)2π˜+t˜¯(1-γ5)b˜.Here ψ runs over all the Weyl fermions {tL,tR,bL,bR,t˜L,t˜R,b˜L,b˜R} and we defined Zψ(hc)=1,yt(hc)=yt,mt(hc)=ythc2,mb(hc)=0,yπ˜t˜b˜(hc)=y˜t,y˜t(hc)=y˜thcf11-hc2f2,mt˜(hc)=y˜tf21-hc2f2mb˜(hc)=0.

We can now compute the running masses of the top and twin top in the background: Mt(hc,t)≈mt(hc)[1+(gS24π2-3yt(hc)24(4π)2Zh^(hc))t+22gS4(4π)4t2],Mt˜(hc,t)≈mt˜(hc)[1+(g˜S24π2-3y˜t(hc)24(4π)2Zh^(hc))t+22g˜S4(4π)4t2],with t=log(m*2/μ2). The values of the parameters in the Higgs background are simply read off from the effective Lagrangian at the scale m*. Notice that in the background, only yt and y˜t are scale-dependent, while hc is “frozen”; hence the running of the quark mass is identical to that of the corresponding Yukawa coupling.

We can now substitute Eqs. (C17) into the RG equation (C3), integrate the result and expand at the required order in hc/f to get Vf(hc,t)impr=Nc64π2{[yt4hc4+y˜t4f4(1-hc2f2)2]t+yt4hc42(gS2π2-3yt2(4π)2)t2+2396π4[gS4yt4hc4+g˜S4y˜t4f4(1-hc2f2)2]t3+y˜t4f432π2(1-hc2f2)2[16g˜S2-3y˜t2hc2f2(1-(1+cH)hc2f2)]t2}.This is the improved effective potential written in terms of parameters computed at the scale m*. We can now treat hc as a quantum field, fluctuating around its minimum hc=h+⟨h⟩. We then have to take into account that the potential (C18) has been computed in a noncanonical basis for hc, where its kinetic term coefficient, including only the leading logarithmic running, is Zhc(t)=1+3yt2(4π)2t.Notice that only the top contributes to the one-loop running of the hc wave function, since the twin top only couples quadratically to hc, giving rise to a one-loop contribution that is not logarithmically divergent. After normalizing hc using Eq. (C19), including a Z2 breaking mass term necessary to achieve a viable minimum and minimizing the potential, hc acquires a vacuum expectation value hc=h+⟨h⟩. This makes the wave function of h (the Higgs field in the minimum) noncanonical again, due to the presence of cH: Zh(⟨h⟩)=1+cH⟨h⟩2f2.After normalizing h taking into account also this last effect, and including the LO gauge contribution, the desired contribution to the Higgs mass reads: δmh2|IR=3v28π2{(yt4t+yt˜4t˜)(1-cHξ+(cH2-dH)ξ2)-116[(3g24+2g12g22+g14)t+3g˜24t˜]+132π2(16yt4gS2t2+16yt˜4g˜S2t˜2)(1-cHξ)+132π2(-15yt6t2-12yt2y˜t4t˜2+3yt˜6t˜2(1+cH))+2396π4(yt4gS4t3+yt˜4g˜S4t˜3)},where we have highlighted the various contributions present in Eq. (C2) with the corresponding colors and set t=logm*2/mt2, t˜=logm*2/mt˜2. The Higgs vev has been defined by fixing the W mass according to mW2=g2⟨h⟩24,⇒⟨h⟩2=v2=12GF=246  GeV.

A numerical determination of the IR contribution δmh2|IR can be obtained by making use of the experimental value of the top quark mass to fix yt(m*) and y˜t(m*). In fact, the 1-loop RG equation for yt is decoupled from the twin sector at the order we are interested in and can be easily solved. Including only the orders required to match our calculation of the Higgs mass squared in Eq. (C21) we get: yt(m*)=yt˜(m*)=yt(mt)[1-(gS(mt)24π2-9yt(mt)264π2)logm*2mt2+22gS(mt)4(4π)4log2m*2mt2],gS(m*)=g˜S(m*)=gS(mt)[1-7gS(mt)232π2logm*2mt2],which fixes yt(m*) in terms of yt(mt). As an input to our numerical analysis we use the PDG combination for the top quark pole mass mt=173.21±0.51±0.71  GeV [57]. This is converted into the top Yukawa coupling in the MS¯ scheme ytMS¯(mt)=0.936±0.005 by making use of Eq. (62) of Ref. [65]. We then use yt(mt)=ytMS¯(mt). For the strong interaction, we run the parameter measured at the scale of the Z boson mass, gS(mZ)∼1.22, to the scale mt to obtain gS(mt).

OPERATOR ANALYSIS OF THE HEAVY-VECTOR CONTRIBUTION TO <inline-formula><mml:math display="inline"><mml:mi>δ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this Appendix we discuss the UV threshold contribution to δgLb generated by the tree-level exchange of the composite vectors ρ [adjoint of SO(7)] and ρX [singlet of SO(7)] at zero transferred momentum. This effect arises at leading order from diagrams with a loop of heavy fermions, as in Fig. 5. Our simple effective operator analysis will show that the contribution of the ρ identically vanishes, in agreement with the explicit calculation in the simplified model.

An adjoint of SO(7) decomposes under the custodial SU(2)L×SU(2)R as: 21=(3,1)+(1,3)+3×(2,2)+3×(1,1).The first two representations contain the vector resonances that are typically predicted by ordinary CH models, namely ρL and ρR. They mix at tree-level with the Z boson and in general contribute to δgLb. The remaining resonances do not have the right quantum numbers to both mix with the Z boson and couple to the left-handed bottom quark due to isospin conservation. As a result, only the components ρL and ρR inside the 21 can give a contribution to δgLb at the 1-loop level.

In order to analyze such effect, we make use of an operator approach. We classify the operators that can be generated at the scale m* by integrating out the composite states, focusing on those which can modify the Zbb¯ vertex at zero transferred momentum. In general, since an exact PLR invariance implies vanishing correction to gLb at zero transferred momentum, any δgLb must be generated proportional to some spurionic coupling breaking this symmetry. In our model, the only coupling breaking PLR in the fermion sector is yL, and a nonvanishing δgLb arises at order yL4. The effective operators can be constructed using the CCWZ formalism in terms of the covariant spurion χL=Σ†Δ†ΔΣ,where Δ is defined in Eq. (3.11). By construction χL is a Hermitian complex matrix. Under the action of an element g∈SO(8), it transforms as a 21a+27s+7+1+1 of SO(7) (where the 7 is complex), and its formal transformation rule is χL→h(Π,g)χLh†(Π,g),h∈SO(7).As a second ingredient to build the effective operators, we uplift the elementary doublet qL into a 7+1 representation of SO(7) by dressing it with NGBs: QL=(Σ†Δ†qL).We will denote with QL(7) and QL(1) respectively the septuplet and singlet components of QL. Since QL(1) does not contain bL (it depends only on tL), only QL(7) is of interest for the present analysis. Under a SO(8) transformation QL(7)→h(Π,g)QL(7).

The effective operators contributing to δgLb can be thus constructed in terms of χL, dμ, and QL(7). We find that the exchange of ρμ in the diagram of Fig. 5 can generate two independent operators, O21=Q¯L(7)γμTaQL(7)Tr(dμχLTa),O21′=Q¯L(7)γμTaQL(7)(dμTaχL)88,where Ta is an SO(8) generator in the adjoint of SO(7); the exchange of ρμX gives rise to other two

Additional structures constructed in terms of dμ and χL can be rewritten in terms of those appearing in Eqs. (D6) and (D7), hence they do not generate new linearly independent operators. Notice that Tr(dμχLTa)∝faa^b^dμa^(χL(7))b^, (dμTaχL)88∝faa^b^dμa^(χL(7)*)b^, (dμχL)88=-dμa^(χL(7))a^, Tr(dμχL)=-dμa^(χL(7))a^+dμa^(χL(7)*)a^, where χL(7) denotes the component of χL transforming as a (complex) fundamental of SO(7). A similar classification in the context of SO(5)/SO(4) models in Ref. [29] found only one operator, corresponding to the linear combination O1′-O1.

: O1=Q¯L(7)γμQL(7)Tr(dμχL),O1′=Q¯L(7)γμQL(7)(dμχL)88.Simple inspection reveals that only the septuplet component of χL contributes in the above equations. One can easily check that the operators of Eq. (D6) give a vanishing contribution to δgLb. In particular, the terms generated by the exchange of the (2,2) and (1,1) components of the ρ give (as expected) an identically vanishing contribution. Those arising from ρL and ρR [obtained by setting Ta in Eq. (D6) equal to respectively one of the (3,1) and (1,3) generators] give instead an equal and opposite correction to gLb. This is in agreement with the results of a direct calculation in the simplified model, from which one finds that the contributions from ρL and ρR cancel each other. Finally, a nonvanishing δgLb arises from the operators of Eq. (D7) generated by the exchange of ρX. Upon expanding in powers of the Higgs doublet, O1 and O1′ both match the dimension-6 operator OHq of Eq. (5.11) and differ only by higher-order terms.

EXPLICIT FORMULAS FOR THE EWPO

In this Appendix we report the results of our calculation of the electroweak precision observables, in particular we collect here the explicit expression of the coefficients aUV, aIR of Eq. (5.12) and bUV, cUV, bIR of Eq. (5.17).

Let us start considering the T^ parameter. For convenience, we split the UV contribution into two parts, redefining aUV as: aUV=aUVFin+aUVLoglog(MΨ2MS2+f2yR2).The coefficients aUVFin and aUVLog are obtained through a straightforward calculation, but their expressions are complicated functions of the Lagrangian parameters. We thus show them only in the limit cL=cR≡c and MΨ=MS≡M, for simplicity. For aIR we give instead the complete expression. We find: aUVFin=112(-12M4f4yR4+6f2M2yR2(f2yR2+M2)2+9M6(f2yR2+M2)3-8)+c(-5f8yR8-2f6M2yR6+7f4M4yR4+12f2M6yR2+4M8)2f4yR4(f2yR2+M2)2+c2(f2yR2+2M2)2(3f2yR2-5M2)2f4yR4(f2yR2+M2),aUVLog=f6yR6-f4M2yR4-f2M4yR2-2M62f6yR6+2c(2f6yR6-3f4M2yR4+3f2M4yR2+2M6)f6yR6+c2(f2M4yR2-10M6)f6yR6,aIR=12+MS2MΨ22(MS2+f2yR2)2+2cLMSMΨ+2cRf2yR2MS2+f2yR2.

The derivation of δgbL at 1-loop level is more involved and requires the computation of a series of diagrams. As explained in the text, we focus on those featuring a loop of fermions and NGBs (see Fig. 4), and that one with a loop of fermion and the tree-level exchange of a heavy vector (see Fig. 5). The coefficients cUV is generated only by the latter diagram; we find: cUV=α7L(α1R+α7R)(1+2cR)gρX2f2MρX2MΨ22(MΨ2+yL2f2).We remind the reader that in our numerical analysis we use MρX/(gρXf)=1, see footnote 5. We redefine the other two coefficients as bIR=δIR+δ¯IRbUV=(δUVFin+δ¯UVFin)+(δUVLog+δ¯UVLog)log(MΨ2MS2+f2yR2),where δIR, δUVFin, and δUVLog are generated by the diagrams in Fig. 4 only, whereas δ¯IR, δ¯UVFin, and δ¯UVLog parametrize the correction due to the tree-level exchange of a heavy spin-1 singlet in Fig. 5. As before, we report the expression of the UV parameters in the limit cL=cR≡c, MΨ=MS≡M, for simplicity; in the case of the coefficients with a bar, generated by the diagram of Fig. 5, we further set α7L=α1L and α7R=α1R. We find: δUVFin=-2f6yR6-4f4M2yR4-4f2M4yR2+M612(f2yR2+M2)3-c(f6yR6+4f4M2yR4-2f2M4yR2+M6)62f2yR2(f2yR2+M2)2+16c2M2(3f2yR2+M2-2f2yR2)-c3M232f2yR2,δUVLog=16-M26f2yR2-c3M432f4yR4-c2M43f4yR4-c(-f4yR4+2f2M2yR2+M4)62f4yR4,δ¯UVFin=f2M2α1LgρX2(f4yR4(α1L+2α1R)+f2M2yR2(3α1L+8α1R)+2M4(α1L+α1R))4MρX2(f2yL2+M2)(f2yR2+M2)2+cf2M2α1LgρX2(f2yR2(2α1L+α1R)+2M2α1L)2MρX2(f2yL2+M2)(f2yR2+M2),δ¯UVLog=cM4α1LgρX2(α1L-α1R)2yR2MρX2(f2yL2+M2)+f2M2α1Lα1RgρX22MρX2(f2yL2+M2).For the IR coefficients we give instead the full expressions. We find: δIR=16+MS2MΨ26(MS2+f2yR2)2+2cLMSMΨ3(MS2+f2yR2)+2cRf2yR212(MS2+f2yR2),δ¯IR=α7Lα1RgρX2MρX2f4MΨ2yR22(f2yL2+MΨ2)(f2yR2+MS2).Notice that the IR corrections aIR and bIR are related to each other and parametrize the running of the effective coefficients c¯Hq, c¯Hq′, and c¯Ht, as explained in the main text.

410.1103/PhysRevD.96.095036.f4

Diagrams with a loop of fermions and NGBs contributing to the ZbLb¯L vertex. Here fi indicates any fermion (both heavy and light) in our simplified model; π± are the charged NGBs.

510.1103/PhysRevD.96.095036.f5

The diagram contributing to ZbLb¯L with a loop of fermions and tree-level exchange of a heavy vector. Here fi and fj denote generic fermions in the theory, both heavy and light, whereas ρν indicates the heavy vector.

Finally, we report the contribution to S^ generated in our simplified model by loops of heavy fermions. We do not include this correction in our electroweak fit, because in the perturbative region of the parameter space it is subdominant with respect to the tree-level shift of Eq. (5.6). Rather, we use this computation as an additional way to estimate the perturbativity bound, as discussed in Sec. III B. Analogously to what we did for T^ and δgLb, we parametrize the fermionic contribution to S^ as: ΔS^Ψ=g228π2ξ[(1-cL2-cR2)logm*2MΨ2+(1-c˜L2-c˜R2)logm*2M˜Ψ2]+g2216π2ξ[sUVFin+sUVLoglog(MΨ2MS2+f2yR2)+s˜UVFin+s˜UVLoglog(M˜Ψ2M˜S2+f2y˜R2)]+g2216π2ξsIRyL2f2MΨ2logM12mt2.Terms in the first line are logarithmically sensitive to the UV cutoff, the second line contains the UV threshold corrections, while the IR running appears in the third line. The UV thresholds include a contribution from the twin composites Ψ˜7 and Ψ˜1, parametrized by s˜UVFin and s˜UVLog. At leading order in yL, by virtue of the twin parity invariance of the strong sector, such a contribution can be obtained from that of Ψ7 and Ψ1 (i.e. from sUVFin and sUVLog) by simply interchanging the tilded quantities with the untilded ones. Higher orders in yL break this symmetry and generate different corrections in the two sectors. We performed the computation of the UV coefficients for yL=0, whereas sIR is derived up to order yL2. We find: sUVFin=12-6cLcRMSMΨ(f2yR2+MS2+MΨ2)(f2yR2+MS2-MΨ2)2+(cR2+cL2)6(24MS2MΨ2(f2yR2+MS2-MΨ2)2-7),sUVLog=-2(MS2+f2yR2)(MS2-MΨ2+f2yR2)3[6cLcRMSMΨ3+cR2MS2(f2yR2+MS2-3MΨ2)+cL2(f2yR2+MS2)(f2yR2+MS2-3MΨ2)],sIR=MS2MΨ4-f2yL2((f2yR2+MS2)(MS2-f2yR2)+MS2MΨ2)+MΨ2(f2yR2+MS2)26MΨ2(f2yR2+MS2)2-cR22f2yR2(MΨ2-f2yL2)3MΨ2(f2yR2+MS2)-cL2MS(MΨ2-f2yL2)3MΨ(f2yR2+MS2).

THE EW FIT

For our analysis of the electroweak observables we make use of the fit to the parameters ε1,2,3,b [66–68] performed in Ref. [69] (see also Ref. [26]). The central values there obtained for the shifts Δεi≡εi-εiSM and the corresponding correlation matrix are Δε1=0.0007±0.0010Δε2=-0.0001±0.0009Δε3=0.0006±0.0009Δεb=0.0003±0.0013ρ=(10.80.86−0.330.810.51−0.320.860.511−0.22−0.33−0.32−0.221).We can directly relate Δε1 to ΔT^ and Δε3 to ΔS^ by using the results of Refs. [45,47], and furthermore Δεb=-2δgbL. We set Δε2=0 in our study, since its effect is subdominant in our model as well as in CH models [47]. We thus make use of Eq. (F1) to perform a χ2 test of the compatibility of our predictions with the experimental constraints. The χ2 function is defined as usual: χ2=∑ij(Δεi-μi)(σ2)ij-1(Δεj-μj),(σ)ij2=σiρijσj,where μi and σi denote respectively the mean values and the standard deviations of Eq. (F1), while Δεi indicates the theoretical prediction for each EW observable computed in terms of the Lagrangian parameters. After deriving the χ2, we perform a fit by scanning over the points in our parameter space keeping only those for which Δχ2≡χ2-χmin2<7.82, the latter condition corresponding to the 95% confidence level with 3 degrees of freedom. Using this procedure, we convert the experimental constraints into bounds over the plane (MΨ,ξ).

ESTIMATES OF THE PERTURBATIVITY BOUND

This Appendix contains details on the derivation of the perturbative limits discussed in Sec. III B. As explained there, we consider the processes πaπb→πcπd and πaπb→ψ¯cψd, where ψ={Ψ7,Ψ˜7} and all indices transform under the fundamental representation of the unbroken SO(7). In order to better monitor how the results depend on the multiplicity of NGBs and fermions, we perform the calculation for a generic SO(N)/SO(N-1) coset with Nf composite fermions ψ in the fundamental of SO(N-1). Taking N=8 and Nf=2×3=6 thus reproduces the simplified model of Sec. III A.

The perturbative limits are obtained by first expressing the scattering amplitudes in terms of components with definite SO(N-1) quantum numbers. In the case of SO(7) the product of two fundamentals decomposes as 7⊗7=1⊕21a⊕27s, where the indices a and s label respectively the antisymmetric and symmetric two-index representations. A completely analogous decomposition holds in the general case of SO(N)/SO(N-1),

One has N⊗N=1⊕[N(N-1)/2]a⊕[N(N+1)/2-1]s.

but for simplicity we will use the SO(7) notation in the following to label the various components. The leading tree-level contributions to the scattering amplitudes arise from the contact interaction generated by the expansion of the NGB kinetic term of Eq. (3.4) and from the NGB-fermion interactions of Eq. (3.9). The structure of the corresponding vertices implies that the four-NGB amplitude has components in all three irreducible representations of SO(N-1) and contains all partial waves. The amplitude with two NGBs and two fermions, instead, has only the antisymmetric component of SO(N-1) and starts with the p-wave. At energies much larger than all masses the amplitudes read M(πaπb→πcπd)=sf2δabδcd+tf2δacδbd+uf2δadδbc,M(πaπb→Ψ¯7LcΨ7Ld)=s2f2sinθ(δacδbd-δadδbc),M(πaπb→Ψ¯7RcΨ7Rd)=s2f2sinθ(δacδbd-δadδbc).They decompose into irreducible representations of SO(N-1) as follows: M(1)(πaπb→πcπd)=(N-2)sf2,M(21)(πaπb→Ψ¯7LcΨ7Ld)=s2f2sinθ,M(21)(πaπb→πcπd)=sf2cosθ,M(21)(πaπb→Ψ¯7RcΨ7Rd)=s2f2sinθ.M(27)(πaπb→πcπd)=-sf2,Performing a partial wave decomposition we get M(r)=∑λi,λfMλi,λf(r)=16πk(i)k(f)∑j=0∞aj(r)(2j+1)⁢∑λi,λfDλi,λfj(θ),where λi, λf are the initial and final state total helicities, and k(i)(k(f)) is equal to either 1 or 2 depending on whether the two particles in the initial (final) state are distinguishable or identical respectively. In the above equation M(r) should be considered as a matrix acting on the space of different channels. The coefficients aj(r) are given by aj(r)=132πk(i)k(f)∫0πdθ∑λi,λfDλi,λfj(θ)Mλi,λf(r).and act as matrices on the space of (elastic and inelastic) channels with total angular momentum j and SO(N-1) irreducible representations r. They can be rewritten as a function of the scattering phase as aj(r)=e2iδj(r)-12i∼δj(r).Our NDA estimate of the perturbativity bound is derived by requiring this phase to be smaller than maximal: |δj(r)|<π2⇒|aj(r)|<π2

Let us consider first the case r=1, corresponding to the amplitude singlet of SO(N-1). The only contribution comes from the four-NGB channel. Since the helicities of the initial and final states are all zeros, in this particular case the Wigner functions Dλi,λfj(θ) reduce to the Legendre polynomials: aj(1)=164π∫0πdθPj(cosθ)M(1).The first and strongest perturbativity constraint comes from the s-wave amplitude, which corresponds to j=0. We find: a0(1)=N-232πsf2,where N=8 in our case. From Eqs. (G6) and (G8), one obtains the constraint of Eq. (3.15).

We analyze now the constraint from the scattering in the antisymmetric representation, r=21. In this case, both the NGB and the fermion channels contribute; the process ππ→ππ is however independent of the fermion and Goldstone multiplicities and can be neglected in the limit of N and Nf. The process involving fermions is a function of Nf and generates a perturbative limit which is comparable and complementary to the previous one. We have: aj(21)=∑λf=±1132π∫0πdθD0,λfj(θ)M0,λf(21).As anticipated, this equation vanishes for j=0, so that the strongest constraint is now derived for p-wave scattering, with j=1. We have a1(21)=Nf242πsf2.From Eqs. (G6) and (G10) follows the constraint of Eq. (3.16).

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diff --git a/tests/data/aps/PhysRevD.96.095036_expected.yml b/tests/data/aps/PhysRevD.96.095036_expected.yml new file mode 100644 index 0000000..eaac575 --- /dev/null +++ b/tests/data/aps/PhysRevD.96.095036_expected.yml @@ -0,0 +1,2103 @@ +abstract: We analyze the parametric structure of twin Higgs (TH) theories and assess the gain in fine tuning which they enable compared to extensions of the standard model with colored top partners. Estimates show that, at least in the simplest realizations of the TH idea, the separation between the mass of new colored particles and the electroweak scale is controlled by the coupling strength of the underlying UV theory, and that a parametric gain is achieved only for strongly-coupled dynamics. Motivated by this consideration we focus on one of these simple realizations, namely composite TH theories, and study how well such constructions can reproduce electroweak precision data. The most important effect of the twin states is found to be the infrared contribution to the Higgs quartic coupling, while direct corrections to electroweak observables are subleading and negligible. We perform a careful fit to the electroweak data including the leading-logarithmic corrections to the Higgs quartic up to three loops. Our analysis shows that agreement with electroweak precision tests can be achieved with only a moderate amount of tuning, in the range 5%–10%, in theories where colored states have mass of order 3–5 TeV and are thus out of reach of the LHC. For these levels of tuning, larger masses are excluded by a perturbativity bound, which makes these theories possibly discoverable, hence falsifiable, at a future 100 TeV collider. +copyright_holder: authors +copyright_statement: Published by the American Physical Society +copyright_year: 2017 +document_type: article +license_url: https://creativecommons.org/licenses/by/4.0/ +license_statement: Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +article_type: research-article +journal_title: Physical Review D +material: publication +publisher: American Physical Society +year: 2017 +authors: +- raw_affiliations: + - source: American Physical Society + value: Scuola Normale Superiore and INFN, Pisa IT-56125, Italy + full_name: Contino, Roberto + ids: + - schema: ORCID + value: 0000-0003-1813-2645 +- raw_affiliations: + - source: American Physical Society + value: Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne + CH-1015, Switzerland + full_name: Greco, Davide +- raw_affiliations: + - source: American Physical Society + value: Theoretical Physics Department, CERN, Geneva CH-1211, Switzerland + full_name: Mahbubani, Rakhi +- raw_affiliations: + - source: American Physical Society + value: Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne + CH-1015, Switzerland + full_name: Rattazzi, Riccardo +- raw_affiliations: + - source: American Physical Society + value: Theoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne + CH-1015, Switzerland + full_name: Torre, Riccardo +artid: '095036' +title: Precision tests and fine tuning in twin Higgs models +number_of_pages: 31 +dois: +- material: publication + doi: 10.1103/PhysRevD.96.095036 +journal_volume: '96' +journal_issue: '9' +is_conference_paper: false +publication_date: 2017-11-01 +documents: +- key: PhysRevD.96.095036.xml + url: http://example.org/PhysRevD.96.095036.xml + source: American Physical Society + fulltext: true + hidden: true +references: +- reference: + publication_info: + journal_volume: '96' + year: 2006 + artid: '231802' + journal_title: Phys. Rev. Lett. + authors: + - inspire_role: author + full_name: Chacko, Z. + - inspire_role: author + full_name: Goh, H.-S. + - inspire_role: author + full_name: Harnik, R. + title: + title: 'The Twin Higgs: Natural Electroweak Breaking from Mirror Symmetry' + label: '1' + dois: + - 10.1103/PhysRevLett.96.231802 + raw_refs: + - source: American Physical Society + value: '1Z. Chacko, H.-S. + Goh, and R. Harnik, + The Twin Higgs: Natural Electroweak Breaking from Mirror Symmetry, + Phys. Rev. Lett. 96, 231802 + (2006).PRLTAO0031-900710.1103/PhysRevLett.96.231802' + schema: JATS +- reference: + authors: + - inspire_role: author + full_name: Barbieri, R. + - inspire_role: author + full_name: Gregoire, T. + - inspire_role: author + full_name: Hall, L.J. + title: + title: Mirror world at the large hadron collider + label: '2' + arxiv_eprint: hep-ph/0509242 + raw_refs: + - source: American Physical Society + value: 2R. Barbieri, T. + Gregoire, and L. J. Hall, + Mirror world at the large hadron collider, arXiv:hep-ph/0509242. + schema: JATS +- reference: + publication_info: + journal_volume: '2006' + page_start: '126' + journal_issue: '01' + artid: '126' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Chacko, Z. + - inspire_role: author + full_name: Nomura, Y. + - inspire_role: author + full_name: Papucci, M. + - inspire_role: author + full_name: Perez, G. + title: + title: Natural little hierarchy from a partially Goldstone twin Higgs + label: '3' + dois: + - 10.1088/1126-6708/2006/01/126 + raw_refs: + - source: American Physical Society + value: 3Z. Chacko, Y. + Nomura, M. Papucci, and G. + Perez, Natural little hierarchy + from a partially Goldstone twin Higgs, J. High Energy + Phys. 01 (2006) 126.JHEPFG1029-847910.1088/1126-6708/2006/01/126 + schema: JATS +- reference: + publication_info: + journal_volume: '2006' + page_start: '108' + journal_issue: '01' + artid: '108' + journal_title: J. 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Procedia + 273–275, 2219 (2016).PPHRCK1875-389210.1016/j.nuclphysbps.2015.09.361 + schema: JATS diff --git a/tests/data/aps/PhysRevD.96.095036_no_date_nodes.xml b/tests/data/aps/PhysRevD.96.095036_no_date_nodes.xml new file mode 100644 index 0000000..d638505 --- /dev/null +++ b/tests/data/aps/PhysRevD.96.095036_no_date_nodes.xml @@ -0,0 +1,3 @@ + + +
PRDPRVDAQPhysical Review DPhys. Rev. D2470-00102470-0029American Physical Society10.1103/PhysRevD.96.095036ARTICLESBeyond the standard modelPrecision tests and fine tuning in twin Higgs modelsPRECISION TESTS AND FINE TUNING IN TWIN HIGGS …CONTINO et al.ContinoRoberto1,*GrecoDavide2,†MahbubaniRakhi3,‡RattazziRiccardo2,§TorreRiccardo2,∥Scuola Normale Superiore and INFN, Pisa IT-56125, ItalyTheoretical Particle Physics Laboratory, Institute of Physics, EPFL, Lausanne CH-1015, SwitzerlandTheoretical Physics Department, CERN, Geneva CH-1211, Switzerland

roberto.contino@sns.it

davide.greco@epfl.ch

rakhi.mahbubani@cern.ch

riccardo.rattazzi@epfl.ch

riccardo.torre@epfl.ch

969095036Published by the American Physical Society2017authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

We analyze the parametric structure of twin Higgs (TH) theories and assess the gain in fine tuning which they enable compared to extensions of the standard model with colored top partners. Estimates show that, at least in the simplest realizations of the TH idea, the separation between the mass of new colored particles and the electroweak scale is controlled by the coupling strength of the underlying UV theory, and that a parametric gain is achieved only for strongly-coupled dynamics. Motivated by this consideration we focus on one of these simple realizations, namely composite TH theories, and study how well such constructions can reproduce electroweak precision data. The most important effect of the twin states is found to be the infrared contribution to the Higgs quartic coupling, while direct corrections to electroweak observables are subleading and negligible. We perform a careful fit to the electroweak data including the leading-logarithmic corrections to the Higgs quartic up to three loops. Our analysis shows that agreement with electroweak precision tests can be achieved with only a moderate amount of tuning, in the range 5%–10%, in theories where colored states have mass of order 3–5 TeV and are thus out of reach of the LHC. For these levels of tuning, larger masses are excluded by a perturbativity bound, which makes these theories possibly discoverable, hence falsifiable, at a future 100 TeV collider.

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschunghttp://dx.doi.org/10.13039/501100001711Swiss National Science FoundationFonds National Suisse de la Recherche ScientifiqueFondo Nazionale Svizzero per la Ricerca ScientificaFonds National SuisseFondo Nazionale SvizzeroSchweizerischer NationalfondsSNFSNSFFNShttp://sws.geonames.org/2658434/NSF-CH200020-150060200020-169696200021-160190H2020 European Research Councilhttp://dx.doi.org/10.13039/100010663H2020 Excellent Science - European Research CouncilEuropean Research CouncilERChttp://sws.geonames.org/6695072/267985614577INTRODUCTION

The principle of naturalness offers arguably the main motivation for exploring physics at around the weak scale. According to naturalness, the plausibility of specific parameter choices in quantum field theory must be assessed using symmetries and selection rules. When viewing the standard model (SM) as an effective field theory valid below a physical cutoff scale and considering only the known interactions of the Higgs boson, we expect the following corrections to its mass.

We take mh2=2mH2=λhv2/2 with ⟨H⟩=v/2=174  GeV, which corresponds to a potential V=-mH2|H|2+λh4|H|4.

δmh2=3yt24π2Λt2-9g2232π2Λg22-3g1232π2Λg12-3λh8π2Λh2+⋯,where each Λ represents the physical cutoff scale in a different sector of the theory. The above equation is simply dictated by symmetry: dilatations (dimensional analysis) determine the scale dependence and the broken shift symmetry of the Higgs field sets the coupling dependence. Unsurprisingly, these contributions arise in any explicit UV completion of the SM, although in some cases there may be other larger ones. According to Eq. (1.1), any given (large) value of the scale of new physics can be associated with a (small) number ε, which characterizes the accuracy at which the different contributions to the mass must cancel among themselves, in order to reproduce the observed value mh≃125  GeV. As the largest loop factor is due to the top Yukawa coupling, according to Eq. (1.1) the scale ΛNP where new states must first appear is related to mh2 and ε via ΛNP2∼4π23yt2×mh2ε⇒ΛNP∼0.451εTeV.The dimensionless quantity ε measures how finely-tuned mh2 is, given ΛNP, and can therefore be regarded as a measure of the tuning. Notice that the contributions from g22 and λh in Eq. (1.1) correspond to ΛNP=1.1  TeV/ε and ΛNP=1.3  TeV/ε, respectively. Although not significantly different from the relation in the top sector, these scales would still be large enough to push new states out of direct reach of the LHC for ε∼0.1.

Indeed, for a given ε, Eq. (1.2) only provides an upper bound for ΛNP; in the more fundamental UV theory there can in principle exist larger corrections to mh2 which are not captured by Eq. (1.1). In particular, in the minimal supersymmetric SM (MSSM) with high-scale mediation of the soft terms, δmh2 in Eq. (1.1) is logarithmically enhanced by RG evolution above the weak scale. In that case, Eq. (1.2) is modified as follows: ΛNP2∼2π23yt2×1lnΛUV/ΛNP×mh2ε,where ΛNP corresponds to the overall mass of the stops and ΛUV≫ΛNP is the scale of mediation of the soft terms. However, for generic composite Higgs (CH) models, as well as for supersymmetric models with low-scale mediation, Eq. (1.2) provides a fair estimate of the relation between the scale of new physics and the amount of tuning, the Higgs mass being fully generated by quantum corrections at around the weak scale. If the origin of mh is normally termed soft in the MSMM with large ΛUV [Eq. (1.3)], it should then be termed supersoft in models respecting Eq. (1.2). As is well known, and shown by Eq. (1.3), the natural expectation in the MSSM for ΛUV≳100  TeV is ΛNP∼mZ∼mh. In view of this, soft scenarios were already somewhat constrained by direct searches at LEP and Tevatron, whereas the natural range of the scale of supersoft models is only now being probed at the LHC.

Equation (1.2) sets an absolute upper bound on ΛNP for a given fine tuning ε, but does not give any information on its nature. In particular it does not specify the quantum numbers of the new states that enter the theory at or below this scale. Indeed, the most relevant states associated with the top sector, the so-called top partners, are bosonic in standard supersymmetric models and fermionic in CH models. Nonetheless, one common feature of these standard scenarios is that the top partners carry SM quantum numbers, color in particular. They are thus copiously produced in hadronic collisions, making the LHC a good probe of these scenarios. Yet there remains the logical possibility that the states that are primarily responsible for the origin of the Higgs mass at or below ΛNP are not charged under the SM, and thus much harder to produce and detect at the LHC. The twin Higgs (TH) is probably the most interesting of the (few) ideas that take this approach [1–22]. This is primarily because the TH mechanism can, at least in principle, be implemented in a SM extension valid up to ultrahigh scales. The structure of TH models is such that the states at the threshold ΛNP in Eq. (1.2) carry quantum numbers under the gauge group of a copy, a twin, of the SM, but are neutral under the SM gauge group. These twin states, of which the twin tops are particularly relevant, are thus poorly produced at the LHC. The theory must also contain states with SM quantum numbers, but their mass m* is boosted with respect to ΛNP roughly by a factor g*/gSM, where g* describes the coupling strength of the new dynamics, while gSM represents a generic SM coupling. As discussed in the next section, depending on the structure of the model, gSM can be either the top Yukawa or the square root of the Higgs quartic. As a result, given the tuning ε, the squared mass of the new colored and charged states is roughly given by m*2∼4π23yt2×mh2ε×(g*gSM)2.For g*>gSM, we could define these model as effectively hypersoft, in that, for fixed fine tuning, the gap between the SM-charged states and the weak scale is even larger than that in supersoft models. In practice the above equation implies that, for strong g*, the new states are out of reach of the LHC even for mild tuning (see Sec. II A for a more precise statement). Equation (1.4) synthesizes the potential relevance of the TH mechanism, and makes it clear that the new dynamics must be rather strong for the mechanism to work. Given the hierarchy problem, it then seems almost inevitable to make the TH a composite TH (although it could also be a supersymmetric composite TH). Realizations of the TH mechanism within the paradigm of CH models with fermion partial compositeness [23] have already been proposed, both in the holographic and effective theory setups [6,9,13,14].

It is important to recognize that the factor that boosts the mass of the states with SM gauge quantum numbers in Eq. (1.4) is the coupling g* itself. Because of this, strong-dynamics effects in the Higgs sector, which are described in the low-energy theory by nonrenormalizable operators with coefficients proportional to powers of g*/m*, do not “decouple” when these states are made heavier, at fixed fine tuning ε. In the standard parametrics of the CH, m*/g* is of the order of f, the decay constant of the σ-model within which the Higgs doublet emerges as a pseudo-Nambu-Goldstone boson (pNGB). Then ξ≡v2/f2, as well as being a measure of the fine tuning through ε=2ξ, also measures the relative deviation of the Higgs couplings from the SM ones, in the TH like in any CH model.

The factor of two difference between the fine tuning ε and ξ is due to the Z2 symmetry of the Higgs potential in the TH models, as shown in Sec. II A [7].

Recent Higgs coupling measurements roughly constrain ξ≲10–20% [24], and a sensitivity of order 5% is expected in the high-luminosity phase of the LHC [25]. However Higgs loop effects in precision Z-pole observables measured at LEP already limit ξ≲5% [26,27]. Having to live with this few-percent tuning would somewhat undermine the motivation for the clever TH construction. In ordinary CH models this strong constraint on ξ can in principle be relaxed thanks to compensating corrections to the T^ parameter coming from the top partners. In the most natural models, these are proportional to yt4v2/m*2 and thus, unlike the Higgs-sector contribution, decouple when m* is increased. This makes it hard to realize such a compensatory effect in the most distinctive range of parameters for TH models, where m*∼5–10  TeV. Alternatively one could consider including custodial-breaking couplings larger than yt in the top-partner sector. Unfortunately these give rise to equally-enhanced contributions to the Higgs potential, which would in turn require further ad-hoc cancellations.

As already observed in the literature [13,14] another important aspect of TH models is that calculable IR-dominated contributions to the Higgs quartic coupling almost saturate its observed value. Though a welcome property in principle, this sets even stronger constraints on additional UV contributions, such as those induced by extra sources of custodial breaking. In this paper we study the correlation between these effects, in order to better assess the relevance of the TH construction as a valid alternative to more standard ideas about EW-scale physics. Several such studies already exist for standard composite Higgs scenarios [28–30]. In extending these to the TH we shall encounter an additional obstacle to gaining full benefit from the TH boost in Eq. (1.4): the model structure requires rather “big” multiplets, implying a large number of degrees of freedom. This results in an upper bound for the coupling that is parametrically smaller than 4π by naive dimensional analysis (NDA); hence the boost factor is similarly depressed. We shall discuss in detail how serious and unavoidable a limitation this is.

The paper is organized as follows: in Sec. II we discuss the general structure and parametrics of TH models, followed by Sec. III where we discuss the more specific class of composite TH models we focus on for the purpose of our study. In Secs. IV and V we present our computations of the basic physical quantities: the Higgs potential and precision electroweak parameters (S^,T^,δgLb). Section VI is devoted to a discussion of the resulting constraints on the model and an appraisal of the whole TH scenario. Our conclusions are presented in Sec. VII.

THE TWIN HIGGS SCENARIOStructure and parametrics

In this section we outline the essential aspects of the TH mechanism. Up to details and variants which are not crucial for the present discussion, the TH scenario involves an exact duplicate, SM˜, of the SM fields and interactions, underpinned by a Z2 symmetry. In practice this Z2 must be explicitly broken in order to obtain a realistic phenomenology, and perhaps more importantly, a realistic cosmology [19–21]. However the sources of Z2 breaking can have a structure and size that makes them irrelevant in the discussion of naturalness in electroweak symmetry breaking, which is the main goal of this section.

Our basic assumption is that the SM and its twin emerge from a more fundamental Z2-symmetric theory at the scale m*, at which new states with SM quantum numbers, color in particular, first appear. In order to get a feel for the mechanism and its parametrics, it is sufficient to focus on the most general potential for two Higgs doublets H and H˜, invariant under the gauge group GSM×G˜SM, with GSM=SU(3)c×SU(2)L×U(1)Y, as well as a Z2: V(H,H˜)=-mH2(|H|2+|H˜|2)+λH4(|H|2+|H˜|2)2+λ^h8(|H|4+|H˜|4).Strictly speaking, the above potential does not have minima with realistic “tunable” ⟨H⟩. This goal can be achieved by the simple addition of a naturally small Z2-breaking mass term which, while changing the vacuum expectation value, does not affect the estimates of fine tuning, and hence will be neglected for the purposes of this discussion. Like for the SM Higgs, the most general potential is accidentally invariant under a custodial SO(4)×SO˜(4). Notice however that in the limit λ^h→0, the additional Z2 enhances the custodial symmetry to SO(8), where H≡H⊕H˜≡8. In this exact limit, if H˜ acquired an expectation value ⟨H˜⟩≡f/2, all 4 components of the ordinary Higgs H would remain exactly massless NGBs. Of course the SM and SM˜ gauge and Yukawa couplings, along with λ^h, explicitly break the SO(8) symmetry that protects the Higgs. Consider however the scenario where these other couplings, which are known to be weak, can be treated as small SO(8)-breaking perturbations of a stronger SO(8)-preserving underlying dynamics, of which the quartic coupling λH is a manifestation. In this situation we can reassess the relation between the SM Higgs mass, the amount of tuning and the scale m* where new states charged under the SM are first encountered, treating λ^h as a small perturbation of λH. At zeroth order, i.e., neglecting λ^h, we can expand around the vacuum ⟨H˜⟩2=2mH2/λH≡f2/2, ⟨H⟩=0. The spectrum consists of a heavy scalar σ, with mass mσ=2mH=λHf/2, corresponding to the radial mode, 3 NGBs eaten by the twin gauge bosons, which get masses ∼gf/2 and the massless H. When turning on λ^h, SO(8) is broken explicitly and H acquires a potential. At leading order in a λ^h/λH expansion the result is simply given by substituting |H˜|2=f2/2-|H|2 in Eq. (2.1).

Notice that the effective Higgs quartic receives approximately equal contributions from |H|4 and |H˜|4. This is a well-known and interesting property of the TH, see for instance Ref. [7].

The quartic coupling and the correction to the squared mass are then given by λh≃λ^hδmH2∼λ^hf2/8≃(λh/2λH)mH2.As mentioned above, we assume that mH2 also receives an independent contribution from a Z2-breaking mass term, which can be ignored in the estimates of tuning. Note that in terms of the physical masses of the Higgs, mh, and of its heavy twin, mσ, we have precisely the same numerical relation δmh2=(λh/2λH)mσ2. The amount of tuning ε, defined as mh2/δmh2, is given by ε=2ξ=2v2/f2.

Our estimate of δmH2 in Eq. (2.2) is based on a simplifying approximation where the SO(8)-breaking quartic is taken Z2-symmetric. In general we could allow different couplings λ^h and λ^h˜ for |H|4 and |H˜|4 respectively, constrained by the requirement λ^h+λ^h˜≃2λh. As the estimate of δmH2 in Eq. (2.2) is determined by the |H˜|4 term, it is clear that a reduction of λ^h˜, with λh fixed, would improve the tuning, as emphasized in Ref. [22]. As discussed in Sec. IV, however, a significant fraction of the contribution to λ^h and λ^h˜ is coming from RG evolution due to the top and twin top. According to our analysis, λ^h/λ^h˜ varies between 1.5 in the simplest models to 3 in models where λ^h˜ is purely IR-dominated as in Ref. [22]. Though interesting, this gain does not change our parametric estimates.

The ratio λh/λH is the crucial parameter in the game. Indeed it is through Eq. (2.2) that mH is sensitive to quantum corrections to the Lagrangian mass parameter mH, or, equivalently, that the physical Higgs mass mh is sensitive to the physical mass of the radial mode mσ. In particular, what matters is the correlation of mσ with, and its sensitivity to, m*, where new states with SM quantum numbers appear. One can think of three basic scenarios for that relation, which we now illustrate, ordering them by increasing level of model-building ingenuity. Beyond these scenarios there is the option of tadpole-dominated electroweak symmetry breaking, which we shall briefly discuss at the end.

Subhypersoft scenario

The simplest option is given by models with mσ∼m*. Supersymmetric TH models with medium- to high-scale soft-term mediation belong to this class [7], with m* representing the soft mass of the squarks. Like in the MSSM, mH, and therefore mσ, is generated via RG evolution: two decades of running are sufficient to obtain mσ∼m*. Another example is composite TH models [13,14]. In their simplest incarnation they are characterized by one overall mass scale m* and coupling g* [31], so that by construction one has mσ∼m* and λH∼g*2. As discussed below Eq. (2.2), in both these scenarios one then expects δmh2∼(λh/2λH)m*2. It is interesting to compare this result to the leading top-sector contribution in Eq. (1.1). For that purpose it is worth noticing that, as discussed in Sec. IV, in TH models the RG-induced contribution to the Higgs quartic coupling Δλh|RG∼(3yt4/π2)lnm*/mt (more than) saturates its experimental value λh∼0.5 for m*∼3–10  TeV.

For this naive estimate we have taken the twin-top contribution equal to the top one, so that the result is just twice the SM one. For a more precise statement see Sec. IV.

We can thus write δmh2∼(λh/2λH)m*2∼3yt42π21λHln(m*/mt)m*2≡3yt22π2×yt2g*2×ln(m*/mt)×m*2which should be compared to the first term on the right-hand side of Eq. (1.1). Accounting for the possibility of tuning we then have m*∼0.45×g*2yt×1ln(m*/mt)×1ε  TeV.Compared to Eq. (1.2), the mass of colored states is on one hand parametrically boosted by the ratio g*/(2yt), and on the other it is mildly decreased by the logarithm. The motivation for, and gain in, the ongoing work on the simplest realization of the TH idea pivot upon the above g*/yt. The basic question is how high g* can be pushed without leading to a breakdown of the effective description. One goal of this paper is to investigate to what extent one can realistically gain from this parameter in more explicit CH realizations. Applying naive dimensional analysis (NDA) one would be tempted to say that g* as big as ∼4π makes sense, in which case m*∼10  TeV would only cost a mild ε∼0.1 tuning. However such an estimate seems quantitatively too naive. For instance, by focusing on the simple toy model whose potential is given by Eq. (2.1), we can associate the upper bound on λH≡g*2, to the point where perturbation theory breaks down. One possible way to proceed is to consider the one loop beta function μdλHdμ=N+832π2λH2,and to estimate the maximum value of the coupling λH as that for which ΔλH/λH∼O(1) through one e-folding of RG evolution. We find λH=2mσ2f2≲32π2N+8⇒mσf≲π,for  N=8,which also gives g*∼λH≲2π, corresponding to a significantly smaller maximal gain in Eq. (2.4) with respect to the NDA estimate. In Sec. III B we shall perform alternative estimates in more specific CH constructions, obtaining similar results.

It is perhaps too narrow-minded to stick rigidly to such estimates to determine the boost that g*/(2yt) can give to m*. Although it is parametrically true that the stronger the coupling g*, the heavier the colored partners can be at fixed tuning, the above debate over factors of a few make it difficult to be more specific in our estimates. In any case the gain permitted by Eq. (2.4) is probably less than one might naively have hoped, making it fair to question the motivation for the TH, at least in its “subhypersoft” realization. With this reservation in mind, we continue our exploration of the TH in the belief that the connection between naturalness and LHC signatures is so crucial that it must be analyzed in all its possible guises.

Concerning in particular composite TH scenarios one last important model building issue concerns the origin of the Higgs quartic λh. In generic CH it is known that the contribution to λh that arises at O(yt2) is too large when g* is strong. Given that the TH mechanism demands g* as strong as possible then composite TH models must ensure that the leading O(yt2) contribution is absent so that λh arises at O(yt4). As discussed in Ref. [13], this property is not guaranteed but it can be easily ensured provided the couplings that give rise to yt via partial compositeness respect specific selection rules.

Hypersoft scenario

The second option corresponds to the structurally robust situation where mσ2 is one loop factor smaller than m*2. This is for instance achieved if H is a PNG-boson octet multiplet associated to the spontaneous breaking SO(9)→SO(8) in a model with fundamental scale m*. Another option would be to have a supersymmetric model where supersymmetric masses of order m* are mediated to the stops at the very scale m* at which H is massless. Of course in both cases a precise computation of mσ2 would require the full theory. However a parametrically correct estimate can be given by considering the quadratically divergent 1-loop corrections in the low energy theory, in the same spirit of Eq. (1.1). As yt and λH are expected to be the dominant couplings the analogue of Eqs. (1.1) and (2.2) imply δmh2∼λh2λH(3yt24π2+5λH16π2)m*2=(yt2λH+512)3λh8π2m*2.Very roughly, for λH≳yt2, top effects become subdominant and the natural value for mh becomes controlled by λh, like the term induced by the Higgs quartic in Eq. (1.1). In the absence of tuning this roughly corresponds to the technicolor limit m*∼4πv, while allowing for fine tuning we have m*∼1.4×1ε  TeV.It should be said that in this scenario there is no extra boost of m* at fixed tuning by taking λH>yt2. Indeed the choice λH∼yt2 is preferable as concerns electroweak precision tests (EWPT). It is well known that RG evolution in the effective theory below mσ gives rise to corrections to the S^ and T^ parameters as discussed in Sec. V [32]. In view of the relation ε=2v2/f2 this gives a direct connection between fine-tuning electroweak precision data, and the mass of the twin Higgs mσ. At fixed v2/f2, EWPT then favor the smallest possible mσ=λHf/2, that is the smallest λH∼yt2. The most plausible spectrum in this class of models is roughly the following: the twin scalar σ and the twin tops appear around the same scale ∼ytf/2, below the colored partners who live at m*. The presence of the somewhat light scalar σ is one of interesting features of this class of models.

Superhypersoft scenario

This option is a clever variant of the previous one, where below the scale m* approximate supersymmetry survives in the Higgs sector in such a way that the leading contribution to δmH2 proportional to λH is purely due to the top sector [7]. In that way Eq. (2.7) reduces to δmh2∼λh2λH(3yt24π2)m*2=yt2λH×3λh8π2m*2.so that by choosing g*>yt one can push the scale m* further up with fixed fine tuning ε m*∼1.4×g*2yt×1ε  TeV.In principle even under the conservative assumption that g*∼2π is the maximal allowed value, this scenario seemingly allows m*∼14  TeV with a mild ε∼0.1 tuning.

It should be said that in order to realize this scenario one would need to complete H into a pair of chiral superfield octets Hu and Hd, along the lines of Ref. [7], as well as add a singlet superfield S in order to generate the Higgs quartic via the superpotential trilinear g*SHuHd. Obviously this is a very far-fetched scenario combining all possible ideas to explain the smallness of the weak scale: supersymmetry, compositeness, and the twin Higgs mechanism.

Alternative vacuum dynamics: Tadpole induced EWSB

In all the scenarios discussed so far the tuning of the Higgs vacuum expectation value (vev) and that of the Higgs mass coincided: ε, which controls the tuning of mh2 according to Eqs. (2.4), (2.8), and (2.10), is equal to 2v2/f2, which measures the tuning of the vev. This was because the only tuning in the Higgs potential was associated with the small quadratic term, while the quartic was assumed to be of the right size without the need for further cancellations (see, e.g., the discussion in Ref. [33]). Experimentally however, one can distinguish between the need for tuning that originates from measurements of Higgs and electroweak observables, which are controlled by v2/f2, and that coming from direct searches for top partners. Currently, with bounds on colored top partners at just around 1 TeV [34,35], but with Higgs couplings already bounded to lie within 10%–20% of their SM value [24], the only reason for tuning in all TH scenarios is to achieve a small v2/f2. It is then fair to consider options that reduce or eliminate only the tuning of v2/f2. As argued in Ref. [18], this can be achieved by modifying the H scalar vacuum dynamics, and having its vev induced instead by a tadpole mixing with an additional electroweak-breaking technicolor (TC) sector [36–38]. In order to preserve the Z2 symmetry one adds two twin TC sectors, both characterized by a mass scale mTC and a decay constant fTC∼mTC/4π (i.e., it is parametrically convenient to assume gTC∼4π). Below the TC scale the dynamics in the visible and twin sectors is complemented by Goldstone triplets πa and π˜a which can be embedded into doublet fields according to Σ=fTCeiπaσa(01),Σ˜=fTCeiπ˜aσa(01),and are assumed to mix with H and H˜ via the effective potential terms Vtadpole=M2(H†Σ+H˜†Σ˜)+H.c.Assuming mTC≪mH the H˜ vacuum dynamics is not significantly modified, but, for mTC>mh, Vtadpole acts like a rigid tadpole term for H. The expectation value ⟨H⟩ is thus determined by balancing such a tadpole against the gauge-invariant |H|2 mass term; the latter will then roughly coincide with mh2. In order for this to work, by Eq. (2.2) the SO(8)-breaking quartic λ^h should be negative, resulting in v∼(M2/mh2)fTC. It is easy to convince oneself that the corrections to Higgs couplings are O(fTC2/v2): present bounds can then be satisfied for fTC∼v/10≃80  GeV. In turn, the value of v/f is controlled by f and can thus be naturally small. The TC scale is roughly mTC∼4πfTC∼600–800  GeV, while the noneaten pNGB π in Eq. (2.11) have a mass mπ2∼M2v/fTC∼mh2(v/fTC)2∼400  GeV. The latter value, although rather low, is probably large enough to satisfy constraints from direct searches. In our opinion, what may be more problematic are EWPT, in view of the effects from the TC sector, which shares some of the vices of ordinary TC. The IR contributions to S^ and T^, associated with the splitting mπa<mTC, are here smaller than the analogues of ordinary technicolor (there associated with the splitting mW≪mTC). However the UV contribution to S^ is parametrically the same as in ordinary TC, in particular it is enhanced at large NTC. Even at NTC=2, staying within the allowed (S^,T^) ellipse still requires a correlated contribution from ΔT^, which in principle should also be counted as tuning. In spite of this, models with tadpole-induced EWSB represent a clever variant where, technically, the dynamics of EWSB does not currently appear tuned. A thorough analysis of the constraints is certainly warranted.

THE COMPOSITE TWIN HIGGS

In this section and in the remainder of the paper, we will focus on the CH realization of the TH, which belongs to the subhypersoft class of models. In this simple and well-motivated context we shall discuss EWPT, fine tuning, and structural consistency of the model.

Our basic structural assumption is that at a generic UV scale ΛUV≫m*, our theory can be decomposed into two sectors: a strongly-interacting composite sector and a weakly-interacting elementary sector. The composite sector is assumed to be endowed with the global symmetry G=SO(8)×U(1)X×Z2 and to be approximately scale- (conformal) invariant down to the scale m*, at which it develops a mass gap. We assume the overall interaction strength at the resonance mass scale m* to be roughly described by one parameter g* [31]. The large separation of mass scales ΛUV≫m* is assumed to arise naturally, in that the occurrence of the mass gap m* is controlled by either a marginally relevant deformation, or by a relevant deformation whose smallness is controlled by some global symmetry. At the scale m*, SO(8)×U(1)X×Z2 is spontaneously broken to the subgroup H=SO(7)×U(1)X, giving rise to seven NGBs in the 7 of SO(7) with decay constant f∼m*/g*. The subgroup U(1)X does not participate to the spontaneous breaking, but its presence is needed to reproduce the hypercharges of the SM fermions, similarly to CH models. The elementary sector consists in turn of two separate weakly interacting sectors: one containing the visible SM fermions and gauge bosons, corresponding to the SM gauge group GSM=SU(3)c×SU(2)L×U(1)Y; the other containing the twin SM with the same fermion content and a SM˜ gauge group G˜SM=SU˜(3)c×SU˜(2)L. The external Z2 symmetry, or twin parity, interchanges these two copies. For simplicity, and following [13], we choose not to introduce a mirror hypercharge field. This is our only source of explicit twin-parity breaking, and affects neither our discussion of fine tuning, nor that of precision electroweak measurements.

The elementary and composite sectors are coupled according to the paradigm of partial compositeness [23]. The elementary EW gauge bosons couple to the strong dynamics as a result of the weak gauging of the SU(2)L×U(1)Y×SU˜(2)L subgroup of the global SO(8)×U(1)X. A linear mixing with the global conserved currents is thus induced: LmixV⊃g2WμαJαμ+g1BμJBμ+g˜2W˜μαJ˜αμ,where g1,2 and g˜2 denote the SM and twin weak gauge couplings, JBμ≡J3Rμ+JXμ and Jμ, J˜μ and JXμ are the currents associated respectively to the SU(2)L, SU˜(2)L and U(1)X generators. The elementary fermions mix analogously with various operators transforming as linear representations of SO(8) that are generated in the far UV by the strongly interacting dynamics. The mixing Lagrangian takes the schematic form: LmixF⊃q¯LαΔαAORA+t¯RΘAOLA+q˜¯LαΔ˜αAO˜RA+t¯RΘ˜AO˜LA+H.c.,where, following, e.g., Ref. [39], we introduced spurions ΔαA, Δ˜αA, ΘA, and Θ˜A in order to uplift the elementary fields to linear representations of SO(8), and match the quantum numbers of the composite operators. The left-handed mixings ΔαA, Δ˜αA necessarily break SO(8) since qL only partially fills a multiplet of SO(8). The right-handed mixings, instead, may or may not break SO(8). The breaking of SO(8) gives rise to a potential for the NGBs at one loop and the physical Higgs is turned into a pNGB. We conclude by noticing that g1,2 and g˜2 correspond to quasimarginal couplings which start off weak in the UV, and remain weak down to m*. The fermion mixings could be either relevant or marginal, and it is possible that some may correspond to interactions that grow as strong as g* at the IR scale m* [40]. In particular, as is well known, there is some advantage as regards tuning in considering the right mixings ΘA and Θ˜A to be strong. In that case one may even imagine the IR scale to be precisely generated by the corresponding deformation of the fixed point. While this latter option may be interesting from a top-down perspective, it would play no appreciable role in our low-energy phenomenological discussion.

A simplified model

In order to proceed we now consider a specific realization of the composite TH and introduce a concrete simplified effective Lagrangian description of its dynamics. Our model captures the most important features of this class of theories, like the pNGB nature of the Higgs field, and provides at the same time a simple framework for the interactions between the elementary fields and the composite states, vectors and fermions. We make use of this effective model as an example of a specific scenario in which we can compute EW observables, and study the feasibility of the TH idea as a new paradigm for physics at the EW scale.

We write down an effective Lagrangian for the composite TH model using the Callan-Coleman-Wess-Zumino (CCWZ) construction [41,42], and generalizing the simpler case of a two-site model developed in Ref. [13]. According to the CCWZ technique, a Lagrangian invariant under the global SO(8) group can be written following the rules of a local SO(7) symmetry. The basic building blocks are the Goldstone matrix Σ(Π), which encodes the seven NGBs, Π, present in the theory, and the operators dμ(Π) and Eμ(Π) resulting from the Maurer-Cartan form constructed with the Goldstone matrix. An external U(1)X group is also added to the global invariance in order to reproduce the correct fermion hypercharges [13]. The CCWZ approach is reviewed and applied to the SO(8)/SO(7) coset in Appendix A.

Before proceeding, we would like to recall the simplified model philosophy of Ref. [43], which we essentially employ. In a generic composite theory, the mass scale m* would control both the cutoff of the low energy σ-model and the mass of the resonances. In that case no effective Lagrangian method is expected to be applicable to describe the resonances. So, in order to produce a manageable effective Lagrangian we thus consider a Lagrangian for resonances that can, at least in principle, be made lighter that m*. One more structured way to proceed could be to consider a deconstructed extra-dimension where the mass of the lightest resonances, corresponding to the inverse compactification length, is parametrically separated from the 5D cut-off, interpreted as m*. Here we do not go that far and simply consider a set of resonances that happen to be a bit lighter than m*. We do so to give a structural dignity to our effective Lagrangian, though at the end, for our numerical analysis, we just take the resonances a factor of 2 below m*. We believe that is a fair procedure given our purpose of estimating the parametric consistency of the general TH scenario.

We start our analysis of the effective Lagrangian with the bosonic sector. Together with the elementary SM gauge bosons, the W’s and B, we introduce the twin partners W˜ to gauge the SU˜(2)L group. As representative of the composite dynamics, we restrict our interest to the heavy spin-1 resonances transforming under the adjoint of SO(7) and to a vector singlet. We therefore introduce a set of vectors ρμa which form a 21 of SO(7) and the gauge vector associated with the external U(1)X, which we call ρμX. The Lagrangian for the bosonic sector can be written as Lbosonic=Lπ+LcompV+LelemV+LmixV.The first term describes the elementary gauge bosons masses and the NGBs dynamics and is given by Lπ=f24Tr[dμdμ].The second term, LcompV, is a purely composite term, generated at the scale m* after confinement; it reduces to the kinetic terms for the ρ vectors, namely: LcompV=-14gρ2ρμνaρμνa-14gρX2ρμνXρXμν,where ρμνa=∂μρνa-∂νρμa-fabcρμbρνc, ρμνX=∂μρνX-∂νρμX, and gρ and gρX are the coupling strengths for the composite spin-1 bosons. The third term in Eq. (3.3), LelemV, is a purely elementary interaction, produced at the scale ΛUV where the elementary fields are formally introduced. Also this Lagrangian can contain only the kinetic terms for the elementary fields: LelemV=-14g12BμνBμν-14g22WμνaWaμν-14g˜22W˜μνaW˜aμν,where g1, g2, and g˜2 denote the weak gauge couplings. The last term in the Lagrangian (3.3), LmixV, is a mixing term between the elementary and composite sectors originating from partial compositeness. We have:

Notice that in the Lagrangian (3.7), the parameters f, Mρ, MρX, gρ, and gρX are all independent. It is common to define the parameters aρ=Mρ/(gρf) and aρX=MρX/(gρXf), which are expected to be O(1). In our analysis we set aρ=1/2 corresponding to the two-site model value (see the last paragraph of this section) and aρX=1.

LmixV=Mρ22gρ2(Tr[ρμaTa21-Eμ])2+MρX22gρX2(ρμX-Bμ)2,where Ta21 are the SO(8) generators in the adjoint of SO(7) (see Appendix A).

We now introduce the Lagrangian for the fermionic sector. This depends on the choice of quantum numbers for the composite operators in Eq. (3.2). The minimal option is to choose OR and O˜R to be in the fundamental representation of SO(8), whereas the operators OL and O˜L are singlets of the global group. Therefore, the elementary SM doublet and its twin must be embedded into fundamental representations of SO(8), whereas the tR and the t˜R are complete singlets under the global SO(8) invariance. This choice is particularly useful to generalize our discussion to the case of a fully-composite right-handed top. From the low-energy perspective, the linear mixing between composite operators and elementary fields translates into a linear coupling between the latter and a layer of fermionic resonances excited from the vacuum by the operators in the fundamental and singlet representations of the global group. Decomposing the 8 of SO(8) as 8=7+1 under SO(7), we introduce a set of fermionic resonances filling a complete fundamental representation of SO(7) and another set consisting of just one singlet.

Notice that in general we should introduce two different singlets in our Lagrangian. One corresponds to a full SO(8) singlet, while the other is the SO(7) singlet appearing in the decomposition 8=7+1 of the fundamental of SO(8) under the SO(7) subgroup. We will further simplify our study identifying the two singlets with just one composite particle.

We denote with Ψ7 the fermionic resonances in the septuplet and with Ψ1 the singlet, both charged under SU(3)c. Together with them, we must introduce analogous composite states charged under SU˜(3)c; we use the corresponding notation Ψ˜7 and Ψ˜1. We refer to Ref. [13] for the complete expression of Ψ7 and Ψ˜7 in terms of the constituent fermions.

The fermionic effective Lagrangian is split into three parts, which have the same meaning as the analogous distinctions we made for the bosonic sector of the theory: Lfermionic=LcompF+LelemF+LmixF.The fully composite term is given by: LcompF=Ψ¯7(iD7-MΨ)Ψ7+Ψ¯1(iD1-MS)Ψ1+Ψ˜¯7(i∇-M˜Ψ)Ψ˜7+Ψ˜¯1(i∂-M˜S)Ψ˜1+(icLΨ¯7LidiΨ1L+icRΨ¯7RidiΨ1R+ic˜LΨ˜¯7LidiΨ˜1L+ic˜RΨ˜¯7RidiΨ˜1R+H.c.),where D7μ=∇μ+iXBμ, D1μ=∂μ+iXBμ, and ∇μ=∂μ+iEμ. We have introduced two sets of O(1) coefficients, cL and cR and their twins, for the interactions mediated by the dμ operator. Considering the elementary part of the Lagrangian, it comprises just the kinetic terms for the doublets and right-handed tops: LelemF=q¯LiDqL+t¯RiDtR+q˜¯LiDq˜L+t˜¯Ri∂t˜R.The final term in our classification is the elementary/composite mixing that we write again following the prescription of partial compositeness. With our choice of quantum numbers for the composite operators, the spurions in Eq. (3.2) can be matched to dimensionless couplings according to ΔαA=(00iyL-yL0×4iyLyL000×4),ΘA=yR,and Δ˜αA=(0×400iy˜L-y˜L0×4iy˜Ly˜L00),Θ˜A=y˜R,where we have introduced the elementary/composite mixing parameters yL, yR and their twin counterparts. These dimensionless y’s control the strength of the interaction between the elementary and composite resonance fields, according to the Lagrangian: LmixF=f(q¯LαΔαAΣAiΨ7i+q¯LαΔαAΣA8Ψ1+yRt¯RΨ1+H.c.)+f(q˜¯LαΔ˜αAΣAiΨ˜7i+q˜¯LαΔ˜αAΣA8Ψ˜1+y˜Rt˜¯RΨ˜1+H.c.).Depending on the UV boundary condition and the relevance or marginality of the operators appearing in Eq. (3.2), the y’s can vary from weak to O(g*). Correspondingly the light fermions vary from being completely elementary (for y weak) to effectively fully composite (for y∼g*). For reasons that will become clear, given yt∼yLyR/g*, it is convenient to take yL≃y˜L∼yt, i.e., weak left mixing, and yR≃y˜R∼g*. For such strong right-handed mixing the right-handed tops can be practically considered part of the strong sector.

The last term that we need to introduce in the effective Lagrangian describes the interactions between the vector and fermion resonances and originates completely in the composite sector. We have: LcompVF=∑i=L,R[αiΨ¯7i(ρ-E)Ψ7i+α7iΨ¯7i(ρX-B)Ψ7i+α1iΨ¯1i(ρX-B)Ψ1i+α˜iΨ˜¯7i(ρ-E)Ψ˜7i+α˜7iΨ˜¯7i(ρX-B)Ψ˜7i+α˜1iΨ˜¯1i(ρX-B)Ψ˜1i],where all the coefficients αi appearing in the Lagrangian are O(1) parameters.

We conclude the discussion of our effective Lagrangian by clarifying its two-site model limit [44] (see also Ref. [45]). This is obtained by combining the singlet and the septuplet into a complete representation of SO(8), so that the model enjoys an enhanced SO(8)L×SO(8)R global symmetry. This is achieved by setting cL=cR=c˜L=c˜R=0 and all the αi equal to 1. Moreover, we have to impose Mρ=gρf/2, so that the heavy vector resonances can be reinterpreted as gauge fields of SO(7). As shown in Ref. [44], with this choice of the free parameters the Higgs potential becomes calculable up to only a logarithmic divergence, that one can regulate by imposing just one renormalization condition. In the subsequent sections, we will extensively analyze the EW precision constraints in the general case, as well as in the two-site limit.

Perturbativity of the simplified model

In Sec. II A 1 it was noted that a TH construction typically involves large multiplicities of states and, as a consequence, the dynamics responsible for its UV completion cannot be maximally strongly coupled. This in turn limits the improvement in fine tuning that can be achieved compared to standard scenarios of EWSB. In our naive estimates of Eqs. (2.3), (2.7), and (2.9) the interaction strength of the UV theory was controlled by the σ-model quartic coupling λH or, equivalently, by mσ/f. By considering the λH one-loop β-function [Eq. (2.5)] we estimated the maximal value of λH as the one corresponding to an O(1) relative change through one e-folding of RG evolution. For an SO(8)/SO(7) σ-model this led to λH≲2π, or, equivalently, mσ/f≲π.

Alternatively, the limit set by perturbativity on the UV interaction strength may also be estimated in the effective theory described by the nonlinear σ-model by determining the energy scale at which tree-level scattering amplitudes become nonperturbative. For concreteness, we considered the following two types of scattering processes: ππ→ππ and ππ→ψ¯ψ, where π are the NGBs and ψ={Ψ7,Ψ˜7} denotes a composite fermion transforming in the fundamental of SO(7). Other processes can (and should) be considered, with the actual bound being given by the strongest of the constraints obtained in this way.

Requiring that the process ππ→ππ stay perturbative up the cutoff scale m* gives the bound Mρf∼MΨf≲m*f<4πN-2≃5.1,where the second inequality is valid in a generic SO(N)/SO(N-1) nonlinear σ-model, and we have set N=8 in the last step. More details on how this result was obtained can be found in Appendix G. Equation (3.15) in fact corresponds to a limit on the interaction strength of the UV theory, given that the couplings among fermion and vector resonances are of order MΨ/f and Mρ/f, respectively. Perturbativity of the scattering amplitude for ππ→ψ¯ψ instead gives (see Appendix G for details) Mρf∼MΨf≲m*f<122πNf≃4πNf,where Nf is the multiplicity of composite fermions (including the number of colors and families). Our simplified model with one family of composite fermions has Nf=6, which gives a limit similar to Eq. (3.15): MΨ/f≲5.3. A model with three families of composite quarks and leptons has instead Nf=24, from which follows the stronger bound MΨ/f≲2.6.

As a third alternative, one could analyze when 1-loop corrections to a given observable become of the same order as its tree-level value. We applied this criterion to our simplified model by considering the S^ parameter, the new physics contribution to which includes a tree-level correction from heavy vectors given by Eq. (5.6), and a one-loop correction due to heavy fermions, which can be found in Appendix E. By requiring that the one-loop term be smaller than the tree-level correction, we obtain a bound on the strong coupling constant gρ. As an illustration, we consider the two-site model limit cL=cR=0 and Mρ=gρf/2 and keep the dominant UV contribution to S^ in Eq. (E7) which is logarithmically sensitive to the cutoff. By setting m*=2MΨ, we find: ΔS^1-loopΔS^tree<1⇒Mρf=gρ2<π2log2≃2.7.

The perturbative limits obtained from Eqs. (3.15), (3.16), and (3.17) are comparable to that on λH derived in Sec. II A 1. As already discussed there, one could take any of these results as indicative of the maximal interaction strength in the underlying UV dynamics, though none of them should be considered as a sharp exclusion condition. In our analysis of EW observables we will make use of Eq. (3.15) with N=8 and of Eq. (3.16) with Nf=24 to highlight the regions of parameter space where our perturbative calculation is less reliable. We use both limits as a measure of the intrinsic uncertainty which is inevitably associated with this type of estimation.

HIGGS EFFECTIVE POTENTIAL

As anticipated in the general discussion of Sec. II A, a potential for the Higgs boson is generated at the scale m* by loops of heavy states through the SO(8)-breaking couplings of the elementary fields to the strong sector. Once written in terms of the Higgs boson h (where H†H=f2sin2(h/f)/2, H˜†H˜=f2cos2(h/f)/2), at 1-loop this UV threshold contribution has the form [13]: V(m*)f4=332π2[116g12gρ2L1+(yL2-y˜L2)gΨ2L2]sin2hf+3yL464π2F1(sin4hf+cos4hf),where gΨ≡MΨ/f, L1, L2, F1 are O(1) dimensionless functions of the masses and couplings of the theory and the explicit expression of the function F1 is reported in Eq. (C1) of Appendix C. The first term in the above equation originates from Z2-breaking effects.

Subleading Z2-breaking terms have been neglected for simplicity. The complete expressions are given in Ref. [13].

The second term, generated by loops of fermions, is Z2 symmetric and explicitly violates the SO(8) invariance; it thus corresponds to the (UV part of the) last term of Eq. (2.1). Upon electroweak symmetry breaking, Eq. (4.1) contributes to the physical Higgs mass an amount equal to δmh2|UV=3yL44π2F1f2ξ(1-ξ),where ξ controls the degree of vacuum misalignment: ξ≡sin2⟨h⟩f=v2f2.

Below the scale m* an important contribution to the potential arises from loops of light states, in particular from the top quark and from its twin. The bulk of this IR contribution is captured by the RG evolution of the Higgs potential from the scale m* down to the electroweak scale. As noted in previous studies (see, e.g., Ref. [13]), for sufficiently large m* this IR effect dominates over the UV threshold correction and can reproduce the experimental Higgs mass almost entirely. An analogous IR correction to the Higgs quartic arises in SUSY theories with large stop masses, from loops of top quarks. The distinctive feature of any TH scenario, including our model, is the additional twin top contribution.

The Higgs effective action, including the leading O(ξ) corrections associated with operators of dimension 6, was computed at 1-loop in Ref. [13]; the resulting IR contribution to mh2 was found to be δmh2|IR1-loop=3yt48π2f2ξ(1-ξ)(logm*2mt2+logm*2m˜t2),where yt denotes the top Yukawa coupling. The two single-log terms in parentheses correspond to the IR contributions to the effective Higgs quartic λh from the top quark and twin top respectively. Leading-logarithmic corrections of the form (αlog)n, arising at higher loops have however an important numerical impact.

Here α=gSM2/4π, with gSM being any large SM coupling, i.e., gS and yt.

For example, (αlog)2 corrections generated by 2-loop diagrams (mostly due to the running of the top and twin top Yukawa couplings, that are induced by respectively QCD and twin QCD) are expected to give a ∼30% reduction in the Higgs mass for m*≃5  TeV.

We have computed the IR contribution to the Higgs mass in a combined expansion in ξ and (αlog). We have included the LO electroweak contribution, all terms up to NLO in αt and αS, and some contributions, expected to be leading for not too large m*, at NNLO in αS. We report the details in Appendix C.

The value of (δmh2|IR)1/2 is shown in Fig. 1 as a function of m* for ξ=0.1. The upper and lower curves represent respectively the LO and NLO calculation in our combined (ξ,αlog) expansion. Numerically, the naive expectation is confirmed, as the NLO correction decreases (δmh2|IR)1/2 by ∼35% for m*=5  TeV. The dotted curve, indicated as NNLO*, includes NNLO contributions of order αtξ2log, αtαSξlog2, and αtαS2log3 (see Appendix C). Additional contributions of order αt2ξlog2, αt2αSlog3 and αt3log3 are not included. An attempt to include these contributions has been made in Ref. [46]. However, the calculation presented there misses some additional contributions from the twin GB and does not represent the full NNLO calculation. The picture that we get from our NNLO* calculation and the incomplete result of Ref. [46], is that the NNLO* gives an overestimate of the IR contribution to the Higgs mass as the aforementioned neglected effects are expected to give a reduction. Given the lack of a complete NNLO calculation, we estimate our uncertainty on the IR Higgs mass as the green shaded region lying between the LO and NLO results in Fig. 1. In order to choose, within this uncertainty, a value that is as close as possible to the full NNLO result, in the rest of the paper we take as input value for the IR correction to the Higgs mass the average of the LO and NLO results, corresponding to the solid black line in the figure.

110.1103/PhysRevD.96.095036.f1

IR contribution to the Higgs mass as a function of the scale m* for ξ=0.1. The dot-dashed (upper) and dashed (lower) curves denote the LO and NLO result in a combined perturbative expansion in (αlog) and ξ. The dotted curve contains the pure QCD part of the NNLO correction (see Appendix C for details). The shaded region represents the uncertainty in our estimate. The central value of this band, given by the solid black line and computed as the average of the LO and NLO results, is the value that we use as our numerical estimate throughout the paper.

The plot of Fig. 1 illustrates one of the characteristic features of TH models: the IR contribution to the Higgs mass largely accounts for its experimental value and is completely predicted by the theory in terms of the low-energy particle content (SM plus twin states). In particular, considering the aforementioned input value and choosing as a benchmark values m*=5  TeV and ξ=0.1, we find that the contributions of the SM and twin light degrees of freedom account for around 50% and 40% of the Higgs mass squared, respectively, so that almost the entire experimental value of the Higgs mass can be due only to the IR degrees of freedom. Threshold effects arising at the UV matching scale, on the other hand, are model dependent but give a subleading correction. An accurate prediction of the Higgs mass and an assessment of the plausibility of the model thus requires a precise determination of its IR contribution. Indeed the difference between the IR contribution of Fig. 1 and the measured value mh=125  GeV must be accounted for by the UV threshold contribution in Eq. (4.2); for our previous benchmark choice of m*, about 12% of the Higgs mass should be generated by the UV physics. This translates into a generic constraint on the size of yL, a parameter upon which electroweak precision observables (EWPO) crucially depend, thus creating a nontrivial correlation between the Higgs mass, EWPO and naturalness.

ELECTROWEAK PRECISION OBSERVABLES

In this section we compute the contribution of the new states described by our simplified model to the EWPO. Although it neglects the effects of the heavier resonances, our calculation is expected to give a fair assessment of the size of the corrections due to the full strong dynamics, and in particular to reproduce the correlations among different observables.

It is well known that, under the assumption of quark and lepton universality, short-distance corrections to the electroweak observables due to heavy new physics can be expressed in terms of four parameters, S^, T^, W, Y, defined in Ref. [47] (see also Ref. [48] for an equivalent analysis) as a generalization of the parametrization introduced by Peskin and Takeuchi in Refs. [49,50]. Two additional parameters, δgLb and δgRb, can be added to account for the modified couplings of the Z boson to left- and right-handed bottom quarks respectively.

We define δgLb and δgRb in terms of the following effective Lagrangian in the unitary gauge: Leff⊃g22cWZμb¯γμ[(gLbSM+δgLb)(1-γ5)+(gRbSM+δgRb)(1+γ5)]b+⋯where the dots stand for higher-derivative terms and gLbSM=-1/2+sW2/3, gRbSM=sW2/3.

A naive estimate shows that in CH theories, including our TH model, W and Y are subdominant in an expansion in the weak couplings [31] and can thus be neglected. The small coupling of the right-handed bottom quark to the strong dynamics makes also δgRb small and negligible in our model. We thus focus on S^, T^, and δgLb, and compute them by including effects from the exchange of vector and fermion resonances, and from Higgs compositeness.

We work at the 1-loop level and at leading order in the electroweak couplings and perform an expansion in inverse powers of the new physics scale. In this limit, the twin states do not affect the EWPO as a consequence of their being neutral under the SM gauge group. Deviations from the SM predictions arise only from heavy states with SM quantum numbers and are parametrically the same as in ordinary CH models with singlet tR. This can be easily shown by means of naive dimensional analysis and symmetries as follows. Twin tops interact with the SM fields only through higher-dimensional operators. The operators relevant for the EWPO are those involving either a SM current or a derivative of the hypercharge field strength: OBt˜=g′mW2∂μBμνt˜¯γνt˜,Oqt˜=1v2q¯LγμqLt˜¯γμt˜,OHt˜=iv2H†Dμ↔Ht˜¯γμt˜,where t˜ indicates either a right- or left-handed twin top.

Notice that OHt˜ can be rewritten in terms of the other two operators by using the equations of motion, but it is still useful to consider it in our discussion.

The first two operators of Eq. (5.2) are generated at the scale m* by the tree-level exchange of the ρX. Their coefficients (in a basis with canonical kinetic terms) are respectively of order (mW2/m*2)(y˜/g*)2 and (yL2v2/m*2)⁢(y˜/g*)2, where y˜ equals either y˜L or y˜R depending on the chirality of t˜. The third operator breaks custodial isospin and the only way it can be generated is via the exchange of weakly coupled elementary fields at loop level. Given that the contribution to EWPO is further suppressed by t˜ loops, the third operator can affect EWPO only at, at least, two loops and is thus clearly negligible. By closing the t˜ loops the first two operators can give rise to effects that are schematically of the form BB, Bq¯q, or (q¯q)2. The formally quadratically divergent piece of the loop integral renormalizes the corresponding dimension-6 operators. For instance the second structure gives Cg′16π2yL2m*2(y˜g*)4∂νBμνq¯LγμqLwith C an O(1) coefficient which depends on the details of the physics at the scale m*. Using the equations of motion for B, the above operator gives rise to a correction to the Zbb¯ vertex of relative size δgLbgLb∼g′216π2yL2v2m*2(y˜g*)4which, even assuming y˜∼g*, is O(g′/yt)2 suppressed with respect to the leading visible sector effect we discuss below. Aside the quadratically divergent piece there is also a logarithmic divergent piece whose overall coefficient is calculable. The result is further suppressed with respect to the above contribution by a factor (mt˜2/m*2)ln(mt˜2/m*2).

An additional contribution could in principle come from loops of the extra three “twin” NGBs contained in the coset SO(8)/SO(7). Simple inspection however shows that there is no corresponding 1-loop diagram contributing to the EWPO. In the end we conclude that the effect of twin loops is negligible.

Since the effects from the twin sector can be neglected, the corrections to S^, T^, and δgLb are parametrically the same as in ordinary CH models. We now give a concise review of the contributions to each of these quantities, distinguishing between the threshold correction generated at the scale m* and the contribution arising from the RG evolution down to the electroweak scale. For recent analyses of the EWPO in the context of SO(5)/SO(4) CH models see for example Refs. [29,30,45].

<inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>S</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math></inline-formula> parameter

The leading contribution to the S^ parameter arises at tree level from the exchange of spin-1 resonances. Since only the (3,1) and (1,3) components of the spin-1 multiplet contribute, its expression is the same as in SO(5)/SO(4) composite-Higgs theories

We neglect for simplicity a contribution from the operator Eμνρμν, which also arises at tree level. See for example the discussion in Refs. [45,51].

: ΔS^ρ=g222gρ2ξ.In our numerical analysis presented in Sec. VI we use the two-site model relation Mρ=gρf/2 to rewrite ΔS^ρ=mW2Mρ2.The 1-loop contribution from loops of spin-1 and fermion resonances is subdominant (by a factor g*2/16π2) and will be neglected for simplicity in the following. Nevertheless, we explicitly computed the fermionic contribution (see Appendix E) to monitor the validity of the perturbative expansion and estimate the limit of strong coupling in our model (a discussion on this aspect was given in Sec. III B). An additional threshold correction to S^, naively of the same order as Eq. (5.6), arises from the exchange of cutoff modes at m*. As already anticipated, we neglect this correction in the following. In this respect our calculation is subject to an O(1) uncertainty and should rather be considered as an estimate, possibly more refined than a naive one, which takes the correlations among different observables into account.

Besides the UV threshold effects described above, S^ gets an IR contribution from RG evolution down to the electroweak scale. The leading effect of this type comes from the compositeness of the Higgs boson, and is the same as in SO(5)/SO(4) CH models [32]: ΔS^h=g22192π2ξlogm*2mh2.In the effective theory below m* this corresponds to the evolution of the dimension-6 operators OW=ig2mW2H†σiDμ↔HDνWμνi,OB=ig′2mW2H†Dμ↔H∂νBμνinduced by a 1-loop insertion of OH=12v2∂μ(H†H)∂μ(H†H).Denoting with c¯i the coefficients of the effective operators and working at leading order in the SM couplings, the RG evolution can be expressed as c¯i(μ)=(δij+γijlogμM)c¯j(M),where γij is the anomalous dimension matrix (computed at leading order in the SM couplings). The S^ parameter gets a correction ΔS^=(c¯W(mZ)+c¯B(mZ))ξ, and one has γW,H+γB,H=-g22/(96π2). An additional contribution to the running arises from insertions of the current-current operators OHq=iv2q¯LγμqLH†Dμ↔H,OHq′=iv2q¯LγμσiqLH†σiDμ↔H,OHt=iv2t¯RγμtRH†Dμ↔Hin a loop of top quarks. This is however suppressed by a factor yL2/g*2 compared to Eq. (5.7) and will be neglected. The suppression arises because the current-current operators are generated at the matching scale with coefficients proportional to yL2.

The total correction to the S^ parameter in our model is ΔS^=ΔS^ρ+ΔS^h, with the two contributions given by Eqs. (5.6) and (5.7).

<inline-formula><mml:math display="inline"><mml:mover accent="true"><mml:mi>T</mml:mi><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:math></inline-formula> parameter

Tree-level contributions to the T^ parameter are forbidden in our model by the SO(3) custodial symmetry preserved by the strong dynamics, and can only arise via loops involving the elementary states. A non-vanishing effect arises at the 1-loop level corresponding to a violation of custodial isospin by two units. The leading contribution comes from loops of fermions and is proportional to yL4, given that the spurionic transformation rule of yL is that of a doublet, while yR is a singlet. We find: ΔT^Ψ=aUVNcyL216π2yL2v2MΨ2+aIRNcyt216π2yL2v2MΨ2logM12mt2,where aUV,IR are O(1) coefficients whose values are reported in Appendix E and we have defined M1≡MS2+yR2f2. The result is finite and does not depend on the cutoff scale m*. The first term corresponds to the UV threshold correction generated at the scale μ=M1∼MΨ. The second term instead encodes the IR running from the threshold scale down to low energy, due to loops of top quarks. In the effective theory below M1 it corresponds to the RG evolution of the dimension-6 operator OT=12v2(H†Dμ↔H)2due to insertions of the current-current operators of Eq. (5.11). In particular, ΔT^=c^T(mZ)ξ and one has γT,Ht=-γT,Hq=3yt2/4π2, γT,Hq′=0. Notice that the size of the second contribution with respect to the first is O[(yt/yL)2log(M12/mt2)]: for yt∼yL, that is for fully composite tR, the IR dominated contribution is formally logarithmically enhanced and dominant.

Further contributions to T^ come from loops of spin-1 resonances, the exchange of cutoff modes and Higgs compositeness. The latter is due to the modified couplings of the composite Higgs to vector bosons and reads [32]: ΔT^h=-3g1264π2ξlogm*2mh2.In the effective theory it corresponds to the running of OT due to the insertion of the operator OH in a loop with hypercharge. The contribution is of the form of Eq. (5.10) with γT,H=3g12/32π2. The exchange of spin-1 resonances gives a UV threshold correction which is also proportional to g12 (as a spurion, the hypercharge coupling transforms as an isospin triplet), but without any log enhancement. It is thus subleading compared to Eq. (5.14) and we will neglect it for simplicity (see Ref. [45] for the corresponding computation in the context of SO(5)/SO(4) models). Finally, we also omit the effect of the cutoff modes because it is incalculable, although naively this is of the same order as the contribution from states included in our simplified model. Our result is thus subject to an O(1) uncertainty.

The total contribution to the T^ parameter in our model is therefore ΔT^=ΔT^h+ΔT^Ψ with the two contributions given by Eqs. (5.12) and (5.14).

<inline-formula><mml:math display="inline"><mml:mi>δ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In the limit of vanishing transferred momentum, tree-level corrections to δgLb are forbidden by the PLR parity of the strong dynamics that exchanges SU(2)L with SU(2)R in the visible SO(4) and SU˜(2)L with SU˜(2)R in the twin SO˜(4) (see Appendix A for details). This is a simple extension of the PLR symmetry of CH models which protects the Zbb¯ coupling from large corrections [52]. In our case PLR is an element of the unbroken SO(7) and keeps the vacuum unchanged. It is thus an exact invariance of the strong dynamics, differently from SO(5)/SO(4) models where it is accidental at O(p2). The gauge couplings g1,2 and yL explicitly break it, while yR preserves it. At finite external momentum δgLb gets a non-vanishing tree-level contribution: (δgLb)tree=f2ξ8Mρ2[g12(αL+α7L)-g22αL]yL2f2MΨ2+yL2f2.In the effective theory below M1, this correction arises from the dimension-6 operators OBq=g′mW2∂μBμνq¯LγνqL,OWq=gmW2DμWμνaq¯LγνσaqL,.It is of order (yL2/g*2)(g2/g*2)ξ, hence a factor g2/g*2 smaller than the naive expectation in absence of the PLR protection.

At the 1-loop level, corrections to δgLb arise from the virtual exchange of heavy fermion and vector states. The leading effect comes at O(yL4) from loops of heavy fermions (the corresponding diagrams are those of Figs. 4 and 5) and reads (δgLb)Ψ=yL216π2NcyL2v2MΨ2(bUV+cUVlogm*2MΨ2)+bIRyt216π2NcyL2v2MΨ2logM12mt2.The expressions of the O(1) coefficients bUV,IR and cUV are reported in Appendix E. The first term is logarithmically divergent and encodes the UV threshold correction at the matching scale. The divergence comes, in particular, from diagrams where the fermion loop is connected to the b-quark current through the exchange of a spin-1 resonance [29]. A simple operator analysis shows that the threshold contribution from the vector resonances in the adjoint of SO(7) identically vanishes in our model (see Appendix D for details). An additional UV threshold contribution to δgLb arises from diagrams where the spin-1 resonances circulate in the loop. For simplicity we will not include such effect in our analysis (see Ref. [30] for the corresponding computation in the context of SO(5)/SO(4) models). It is however easy to show that there is no possible diagram with ρX circulating in the loop as a consequence of its quantum numbers, while the corresponding contribution from vector resonances in the adjoint of SO(7) is nonvanishing in this case.

The second term in Eq. (5.17) accounts for the IR running down to the electroweak scale. In the effective theory below M1 one has δgLb=-(c¯Hq(mZ)+c¯Hq′(mZ))/2, hence the IR correction arises from the evolution of the operators OHq and OHq′ due to loops of top quarks. In this case the operators that contribute to the running via their 1-loop insertion are those of Eq. (5.11) as well as the following four-quark operators [53]: OLR=(q¯LγμqL)(t¯RγμtR),OLL=(q¯LγμqL)(q¯LγμqL),OLL′=(q¯LσaγμqL)(q¯LσaγμqL).In fact, the operators contributing at O(yL2yt2) to Eq. (5.10) are only those generated at O(yL2) at the matching scale; these are OHt, the linear combination OHq-OHq′ (even under PLR), and OLR (generated via the exchange of ρX).

The operators OLL and OLL′ are generated at O(yL4) by the tree-level exchange of both ρX and ρ.

Notice finally that the relative size of the IR and UV contributions to δgLb is O[(yt/yL)2log(M12/mt2)] precisely like in the case of ΔT^Ψ.

It is interesting that in our model the fermionic corrections to δgLb and T^ are parametrically of the same order and their signs tend to be correlated. It is for example well known that a heavy fermion with the quantum numbers of tR gives a positive correction to both quantities [28,54–56]. We have verified that this is also the case in our model for MS≪MΨ∼Mρ (light singlet).

In this limit one has ΔT^Ψ≃3(δgLb)Ψ.

Conversely, a light septuplet (MΨ≪MS∼Mρ) gives a negative contribution to both δgLb and T^.

The existence of a similar sign correlation in the limit of a light (2,2) has been pointed out in the context of SO(5)/SO(4) CH models, see Ref. [29].

 Although in general the expressions for ΔT^Ψ and (δgLb)Ψ are uncorrelated, their signs tend to be the same whenever the contribution from ρX to Eq. (5.17) is subleading. The sign correlation can instead be broken if ρX contributes significantly to δgLb [in particular, (δgLb)Ψ can be negative for αiL=-αiR]. The importance of the above considerations lies in the fact that EW precision data prefer a positive T^ and a negative δgLb. Situations when both quantities have the same sign are thus experimentally disfavored.

Considering that no additional correction to δgLb arises from Higgs compositeness, and that we neglect as before the incalculable effect due to cutoff states, the total contribution in our model is δgLb=(δgLb)tree+(δgLb)Ψ, with the two contributions given by eqs. (5.15) and (5.17).

RESULTS AND DISCUSSION

We are now ready to translate the prediction for the Higgs mass and the EWPO into bounds on the parameter space of our simplified model and for the composite TH in general. We are interested in quantifying the degree of fine tuning that our construction suffers when requiring the mass scale of the heavy fermions to lie above the ultimate experimental reach of the LHC. As discussed in Sec. II, this scale receives the largest boost from the TH mechanism (without a corresponding increase in the fine tuning of the Higgs mass) in the regime of parameters corresponding to a fully strongly coupled theory, where no quantitatively precise EFT description is allowed. Our computations of physical quantities in this most relevant regime should then be interpreted as an educated naive dimensional analysis (eNDA) estimate, where one hopes to capture the generic size of effects beyond the naivest 4π counting, and including factors of a few related to multiplet size, to spin, and to numerical accidents. In the limit where Mρ/f and MΨ/f are significantly below their perturbative upper bounds our computations are well defined. eNDA then corresponds to assuming that the results do not change by more than O(1) (i.e. less than O(5) to be more explicit) when extrapolating to a scenario where the resonance mass scale sits at strong coupling. In practice we shall consider the resonant masses up to their perturbativity bound and vary the αi and ci parameters within an O(1) range.

Notice indeed that (α=1,c=0) and (α=0,c=1/2) correspond to specific limits at weak coupling, namely the two-site model and the linear sigma model respectively. This suggests that their natural range is O(1).

In view of the generous parameter space that we shall explore our analysis should be viewed as conservative, in the sense that a realistic TH model will never do better.

Let us now describe the various pieces of our analysis. Consider first the Higgs potential, where the dependence on physics at the resonance mass scale is encapsulated in the function F1 [Eq. (4.2)] which controls the UV threshold correction to the Higgs quartic. It is calculable in our simplified model and the result is O(1) [its expression is reported in Eq. (C1)], but it can easily be made a bit smaller at the price of some mild tuning by varying the field content or the representations of the heavy fermions. In order to account for these options and thus broaden the scope of our analysis we will treat F1 as a free O(1) parameter. The value of F1 has a direct impact on the size of the left-handed top mixing yL, since δmh2|UV∼yL4F1, and hence controls the interplay between the Higgs potential and EWPO. Specifically, as we already stressed, a smaller F1 implies a larger value of yL, which in turn gives a larger ΔT^∝yL4v2/MΨ2. This could help improve the compatibility with EWPT even for large MΨ, at the cost of a small additional tuning due to the need for a clever maneuver in the S^,T^ plane to get back into the ellipse, as well as the fact that F1 is generically expected to be O(1). In the following we will thus treat F1 as an input parameter and use Eqs. (4.2) and (B4) to fix yL and yR in terms of the Higgs and top quark experimental masses. Our final results will be shown for two different choices of F1, namely F1=1 and F1=0.3, in order to illustrate how the bounds are affected by changing the size of the UV threshold correction to the Higgs potential.

The EWPO and the Higgs mass computed in the previous sections depend on several parameters, in particular on the mass spectrum of resonances (see Appendix B), the parameters ci, αi of Eqs. (3.9), (3.14), and the parameter F1 discussed above. In order to focus on the situation where resonances can escape detection at the LHC, we will assume that their masses are all comparable and that they lie at or just below the cutoff scale. In order to simplify the numerical analysis we thus set MΨ=MS=M˜Ψ=M˜S=Mρ=MρX=m*/2. The factor of two difference between MΨ and m* is chosen to avoid setting all UV logarithms of the form log(m*/MΨ) to zero, while not making them artificially large. As a further simplification we set cL=cR≡c, α7L=α1L, and α7R=α1R. The parameter αL appears only in the tree-level contribution to δgLb, see Eq. (5.15), and we fix it equal to 1 for simplicity. Even though the above choices represent a significant reduction of the whole available parameter space, for the purpose of our analysis they represent a sufficiently rich set where EWPT can be successfully passed.

Let us now discuss the numerical bounds on the parameter space of our simplified model. They have been obtained by fixing the top and Higgs masses to their experimental value and performing the numerical fit described in Appendix F. As experimental inputs, we use the PDG values of the top quark pole mass mt=173.21±0.51±0.71 (see last paragraph of Appendix C), and of the Higgs mass, mh=125.09±0.24  GeV [57]. Figure 2 shows the results of the fit in the (MΨ,ξ) plane for F1=0.3 (left panel) and F1=1 (right panel). In both panels we have set c=0, which corresponds to the two-site model limit of our simplified Lagrangian. The yellow regions correspond to the points that pass the χ2 test at 95% confidence level (CL), see Appendix F for details. Solid black contours denote the regions for which α1L=-α1R=1, while dashed contours surround the regions obtained with α1L=α1R=1. The areas in blue are theoretically inaccessible. The lower left region in dark blue, in particular, corresponds to MΨ/f≡gΨ<yL. The upper dark- and light-blue regions correspond instead to points violating the perturbative limits on gΨ given by Eq. (3.15) with N=8, and Eq. (3.16) with Nf=24, respectively (see Sec. III B for a discussion). The difference between these two regions can be taken as an indication of the uncertainty associated with the perturbative bound.

Notice that because of our choice m*=2MΨ, the scale m* lies a factor of 2 above the pertubative cutoff. This is compatible with the semiquantitative nature of our estimates. As we stated previously we insisted in keeping m*=2MΨ because it implies a more generic contribution to electroweak precision observables.

210.1103/PhysRevD.96.095036.f2

Allowed regions in the (MΨ,ξ) plane for F1=0.3 (left panel) and F1=1 (right panel). See the text for an explanation of the different regions and of the choice of parameters.

In the left panel of Fig. 2 the allowed (lighter yellow) region extends up to ξ≃0.2 for masses MΨ in the 2–3 TeV range. Such large values of ξ are possible in this case because the fermionic contribution to ΔT^Ψ turns out to be sufficiently large and positive to compensate for both the negative ΔT^h in Eq. (5.14) and the positive ΔS^ρ and ΔS^h in Eqs. (5.6), (5.7). For larger MΨ the fermionic contribution ΔT^Ψ becomes too small and this compensation no longer occurs. In this case, however, the strongest bound comes from the perturbativity limit (blue region), which makes points with large MΨ at fixed ξ theoretically inaccessible. Notice that large values of ξ become excluded if one considers the choice α1L=α1R=1 leading to the dashed contour. The large difference between the solid and dashed curves (i.e. lighter and darker yellow regions) depends on the sign correlation between ΔT^Ψ and δgLb. In the case of the solid line, the signs are anticorrelated (e.g., positive ΔT^Ψ and negative δgLb), allowing for the compensation effect by ΔT^Ψ. In the case of the dashed line, instead, the signs of the two parameters are correlated (both positive), so that when ΔT^Ψ is large, δgLb is also large and positive. This makes it more difficult to pass the χ2 test, since data prefer a negative δgLb.

In the right panel of Fig. 2, obtained with F1=1, the allowed yellow region shrinks because the larger value of F1 implies a smaller yL hence a smaller ΔT^Ψ. In this case the χ2 test is passed only for ξ<0.06, and the difference between the solid and dashed lines is small since the large and positive ΔS^ always dominates the fit. Masses MΨ larger than ∼5  TeV are excluded by the perturbative bound, unless one considers smaller values of ξ.

The results of Fig. 2 can change significantly if the parameter c is allowed to be different from zero. In particular, as one can verify from our formulas in Appendix E, positive values of c increase ΔT^ and as a result the allowed regions in Fig. 2 shift to the right toward larger values of MΨ. In this case the perturbative bound excludes a large portion of the region passing the χ2 test. The effect of varying c is illustrated in Fig. 3, which shows the 95% CL allowed regions in the plane (c,α) for F1=0.3 (left panel) and F1=1 (right panel). In both panels we have set α≡α1L=-α1R (ensuring positive ΔT^Ψ and negative δgLb). The yellow, orange, and red regions correspond, respectively, to ξ=0.05, ξ=0.1, and ξ=0.15, with the masses of the resonances fixed at their perturbative upper bound Mρ=MΨ=4πf/N-2 (which for N=8 gives ∼6/4/3.2  TeV). Note that increasing F1 (reducing yL) shifts the allowed region toward positive values of c, since as mentioned above, smaller yL requires a larger positive c to get a large enough ΔT^Ψ. Obviously, larger values of ξ correspond to smaller allowed regions, as is clear from Fig. 2. Finally, notice that the vertically symmetric structure of the allowed regions is due to the quadratic dependence of δgLb on α. From these plots, one can see that for resonances conceivably out of direct reach of the LHC and for ξ∼0.1, corresponding to about 20% tuning of the Higgs mass, both α and c are allowed to span a good fraction of their expected O(1) range. No dramatic extra tuning in these parameters seems therefore necessary to meet the constraints of EWPT. In particular, considering the plot for F1=1 (right panel), one notices that the bulk of the allowed region is at positive c. For instance by choosing c∼0.2–0.5 the plot in the (ξ,MΨ) plane for F1=1 becomes quite similar to the one at the left of Fig. 3 valid for F1=0.3: there exists a “peak” centered at MΨ∼2–4  TeV and extending up to ξ∼0.2. The specific choice c=0 is thus particularly restrictive for F1=1 (right panel of Fig. 2), but this restriction is lifted for positive c. Overall we conclude that for ξ∼0.1 and for resonances just beyond the LHC reach, the correct value of the Higgs quartic can be obtained and EWPT passed with only a mild additional tuning associated with a sign correlation α1L=-α1R, and a correlation between c and F1 (e.g. c>0 for F1=1). These correlations allow the various contributions to T^ and δgLb to compensate for each other, achieving agreement with EWPT. If forced to quantify the tuning inherent in these effects, we could estimate it to be around 1/4=(1/2)×(1/2), given about 1/2 of the plausible choices for both αi and ci are allowed.

310.1103/PhysRevD.96.095036.f3

Allowed regions in the (c,α) plane, with c=cL=cR, for F1=0.3 (left panel) and F1=1 (right panel). The yellow, orange, and red regions correspond to ξ=0.05, 0.1 and 0.15 respectively. See the text for an explanation of the choice of the other parameters.

SUMMARY

In this paper we tried to assess how plausible a scenario yielding no new particles at the LHC can be obtained using the TH construction. We distinguished three possible classes of models: the subhypersoft, the hypersoft and the superhypersoft, with increasing degree of technical complexity and decreasing (technical) fine tuning. We then focused on the CH incarnation of the simplest option, the subhypersoft scenario, where the boost factor for the mass of colored partners [Eq. (2.4)] at fixed tuning is roughly given by g*2yt×1ln(m*/mt).Here the gain derives entirely from the relative coupling strength g*/2yt, making the marriage of twinning and compositeness practically obligatory. We attempted a more precise estimate of the upper limit on g*/2yt, as compared with previous studies (e.g. Ref. [13]). We found by independent but consistent estimates, that the bound ranges from ∼3 in a toy sigma model [Eq. (2.6)] to ∼5 in a simplified CH model [Eq. (3.15)], with both limits somewhat below the NDA estimate of 4π∼12. Consequently for a mild tuning ε∼0.1 the upper bound on the mass of the resonances with SM quantum numbers is closer to the 3–5 TeV range than it is to 10 TeV. This gain, despite being less spectacular than naively expected, is still sufficient to push these states out of direct reach of the LHC, provided we resort to full strong coupling. In practice this implies no real computational advantage from considering holographic realizations of composite TH constructions: since the boost factor is controlled by the KK coupling, the 5D description breaks down in precisely the most interesting regime, where the KK coupling is strong. In this situation computations based on an explicit 5D construction, such as the ones studied in Refs. [9,16] for instance, are no better than numerical estimates made in our simplified model. Indeed we have checked that EWPT can be satisfied in a sizable portion of the parameter space, given some interplay among the various contributions. In particular the IR corrections to T^ and S^ are enhanced by ln(m*/mh), and for ξ>0.1 the compensating contribution to T^, which decreases like 1/m*2, is necessary. Given that perturbativity limits m* to be below 5 TeV for ξ>0.1 (see the upper blue exclusion region in Fig. 2) this compensation in EWPT can still take place at the price of a moderate extra tuning. For ξ of order a few percent on the other hand, EWPT would be passed without any additional tuning, while the masses of SM-charged resonances would be pushed up to the 10 TeV range, where nothing less than a 100 TeV collider would be required to discover them, and that barely so [58,59].

Although EWPT work similarly in the CH and composite TH frameworks, the two are crucially different when it comes to contributions to the Higgs quartic. In the CH these are enhanced when g*, i.e. m*/f, is strong and, as discussed for instance in Ref. [33], in order to avoid additional tuning of the Higgs quartic g* cannot be too large. According to the study in Refs. [60–63] the corresponding upper bound on the mass of the colored top partners in CH reads roughly m*/f≲1.5; this should be compared to the upper bound m*/f≲5 from strong coupling we found in Eq. (3.15). The Higgs quartic protection afforded by the TH mechanism allows us to take m*/f as large as possible, allowing the colored partners to be heavier at fixed f, hence at fixed fine tuning ξ. In the end the gain is about a factor of 5/1.5∼3, not impressive, but sufficient to place the colored partners outside of LHC reach for a mild tuning ξ∼0.1.

Finally, we comment on the classes of models not covered in this paper: the hypersoft and superhypersoft scenarios. The latter requires combining supersymmetry and compositeness with the TH mechanism, which, while logically possible, does not correspond to any existing construction. Such a construction would need to be rather ingenious, and we currently do not feel compelled to provide it, given the already rather epicyclic nature of the TH scenario. The simpler hypersoft scenario, though also clever, can by contrast be implemented in a straightforward manner, via, e.g., a tumbling SO(9)→SO(8)→SO(7) pattern of symmetry breaking. The advantage of this approach is that it allows us to remain within the weakly-coupled domain, due to the presence of a relatively light twin Higgs scalar mode σ, whose mass can be parametrically close to that of the twin tops, ∼ytf (around 1 TeV for ξ∼0.1). As well as giving rise to distinctive experimental signatures due to mixing with the SM Higgs [64], the mass of the light σ acts as a UV cutoff for the IR contributions to S^ and T^ in Eqs. (5.7) and (5.14) [12]. For sufficiently light σ then, less or no interplay between the various contributions is required in order to pass EWPT. Together with calculability, this property may well single out the hypersoft scenario as the most plausible TH construction.

ACKNOWLEDGMENTS

We would like to thank Andrey Katz, Alberto Mariotti, Kin Mimouni, Giuliano Panico, Diego Redigolo, Matteo Salvarezza, and Andrea Wulzer for useful discussions. The Swiss National Science Foundation partially supported the work of D. G. and R. R. under Contracts No. 200020-150060 and No. 200020-169696, the work of R. C. under Contract No. 200021-160190, and the work of R. T. under the Sinergia network CRSII2-160814. The work of R. C. was partly supported by the ERC Advanced Grant No. 267985 Electroweak Symmetry Breaking, Flavour and Dark Matter: One Solution for Three Mysteries (DaMeSyFla). R. M. is supported by ERC Grant No. 614577 HICCUP High Impact Cross Section Calculations for Ultimate Precision.

<inline-formula><mml:math display="inline"><mml:mi>S</mml:mi><mml:mi>O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>8</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> GENERATORS AND CCWZ VARIABLES

In this Appendix we define the generators of the SO(8) algebra and describe the SO(8)/SO(7) symmetry-breaking pattern, introducing the CCWZ variables for our model. We refer the reader to Refs. [41,42] for a detailed analysis of this procedure and we closely follow the notation of Ref. [13].

We start by listing the twenty-eight generators of SO(8) and decomposing them into irreducible representations of the unbroken subgroup SO(7): 28=7⊕21. They can be compactly written as: (Tij)kl=i2(δikδjl-δilδjk),with i,j,k,l=1,…,8. We choose to align the vacuum expectation value responsible for the spontaneous breaking of SO(8) to SO(7) along the 8th component: ϕ0=f(0,0,0,0,0,0,0,1)t. With this choice, the broken and unbroken generators, transforming, respectively, in the 7 and 21 of SO(7), are (Tβ7)γρ=i2(δ8γδβρ-δ8ρδβγ),(Tαβ21)γρ=i2(δαγδβρ-δαρδβγ),where α,β=1,…,7 and γ,ρ=1,…,8. It is useful to identify the subgroups SO(4)∼SU(2)L×SU(2)R and SO˜(4)∼SU˜(2)L×SU˜(2)R of SO(8); they are generated by: (TL)α=(tLα000),(TR)α=(tRα000),(T˜L)α=(000tLα),(T˜R)α=(000tRα),where tLα and tRα are 4×4 matrices defined as (tL,Rα)ij=-i2[12εαβγ(δiβδjγ-δjβδiγ)±(δiαδj4-δjαδi4)]with α=1, 2, 3 and i,j=1,…,4. The elementary SM and twin vector bosons gauge, respectively, a subgroup SU(2)L×U(1)Y of SO(4), with U(1)Y≡U(1)R3, and the subgroup SU˜(2)L inside SO˜(4). This choice corresponds to having zero vacuum misalignment at tree level.

The spontaneous breaking of SO(8) to SO(7) delivers seven NGBs, that we collect in the vector Π=(π2,π1,-π3,π4,π˜2,π˜1,-π˜3)t. The first four transform as a fundamental of SO(4) and form the Higgs doublet; the remaining three are singlets of SO(4) and are thus neutral under the SM gauge group. All together they can be arranged in the Goldstone matrix Σ(Π)=ei2fΠ·T7=[I7-ΠΠtΠt·Π(1-cos(Πt·Πf))ΠΠt·Πsin(Πt·Πf)-ΠΠt·Πsin(Πt·Πf)cos(Πt·Πf)].The latter transforms nonlinearly under the action of an SO(8) group element g, according to the standard relation: Σ(Π)→g·Σ(Π)·h†(Π,g),where h(Π,g)∈SO(7) depends on g and Π(x). We identify the Higgs boson with the NGB along the generator T47; in the unitary gauge, all the remaining NGBs are nonpropagating fields and the Π vector becomes Π|unitary gauge=(0,…,π4=⟨h⟩+h(x),…,0),where ξ=sin2(⟨h⟩/f)=v2/f2. In this case the Σ matrix simplifies to: Σ(Π)|unitary gauge=ei2fπ4T47=[I30000cosπ4f0sinπ4f00I300-sinπ4f0cosπ4f].

Given the above symmetry breaking pattern, it is possible to define a LR parity, PLR=diag(-1,-1,-1,+1,-1,-1,-1,+1),which exchanges SU(2)L with SU(2)R inside SO(4) and SU˜(2)L with SU˜(2)R inside SO˜(4). The corresponding action on the fields is such that π4 is even, while all the other NGBs are odd. As already noticed in Sec. V C, PLR is an element of both SO(8) and SO(7), which means that it is an exact symmetry of the strong dynamics and acts linearly on the physical spectrum of fields.

The CCWZ variables dμ and Eμ are defined as usual through the Maurer-Cartan form, Σ†(Π)DμΣ(Π)≡idμi(Π)Ti7+iEμa(Π)Ta21,as the components along the broken and unbroken generators of SO(8) respectively. The derivative Dμ is covariant with respect to the external SM and twin gauge fields: Dμ=∂μ-iAμATA,withAμATA=g2Wμα(TL)α+g1Bμ(TR)3+g˜2W˜μα(T˜L)α.Under the action of a global element g∈SO(8), dμ and Eμ transform with the rules of a local SO(7) transformation: dμ≡dμiTi7→h(Π,g)dμh†(Π,g),Eμ≡EμaTa21→h(Π,g)(Eμ-i∂μ)h†(Π,g).It is straightforward to derive the explicit expressions of dμ and Eμ from the Maurer-Cartan relation in Eq. (A10). They are however lengthy and not very illuminating, so we do not report them here. We can easily obtain the mass spectrum of the gauge sector of our theory from the NGB kinetic term; in the unitary gauge and after rotating to the mass eigenstate basis, we find: Lmass=f24(dμi)2⊃g224f2ξWμ+Wμ-+(g12+g22)8f2ξZμZμ+g˜228f2(1-ξ)∑i=13(W˜μi)2.

MASS MATRICES AND SPECTRUM

In this Appendix, we briefly discuss the mass matrices of the different charged sectors in the composite TH model and the related particle spectrum. We refer to Ref. [13] for the expressions of the Ψ7 and Ψ˜7 multiplets in terms of their component heavy fermions.

We start by considering the fields that do not have the right quantum numbers to mix with the elementary SM and twin quarks and whose mass is therefore independent of the mixing parameters yL,R and y˜L,R. These are the composite fermions X5/3, D˜1 and D˜-1, with charges 5/3, 1 and -1 respectively; their mass is exactly given by the Lagrangian parameters MΨ (for the first one), and M˜Ψ (for the last two).

The remaining sectors have charge -1/3, 0, and 2/3 and because of the elementary/composite mixing the associated mass matrices are in general nondiagonal and must be diagonalized by a proper field rotation. The simplest case is the (-1/3)-charged sector, containing the bottom quark and the heavy B field; the mass matrix in the {b,B} basis is M-1/3=(0fyL0-MΨ).After rotation, we find a massless bottom quark (it has no mass since we are not including the bR in the model), and a massive B particle with mB2=MΨ2+yL2f2.

As regards the sector of charge 2/3, it contains seven different particles: the top quark, the toplike heavy states T and X2/3 and four composite fermions that do not participate in the SM weak interactions, S2/31,⋯S2/34. In the {t,T,X2/3,S2/31,…,S2/34} basis, the mass matrix is given by: M2/3=(012fyL(1-ξ+1)-12fyL(1-ξ-1)0-fyLξ20-MΨ00000-MΨ00000-MΨ×I30fyR000-MS).The states S2/31,S2/32,S2/33 completely decouple from the elementary sector and do not mix with the top quark. Their mass is therefore exactly given by the Lagrangian parameter MΨ. The remaining 4×4 matrix is in general too complicated to be analytically diagonalized, but one can easily find the spectrum in perturbation theory by expanding M2/3 for ξ≪1, which is in general a phenomenologically viable limit. The leading order expression for the masses is then: mt2≃f42yL2yR2MS2+yR2f2ξ+O(ξ2),mX2/32=MΨ2,mT2≃MΨ2+yL2f2(1-ξ2)+O(ξ2),mS2/342≃MS2+yR2f2+yL2f2MS22(MS2+yR2f2)ξ+O(ξ2).The X2/3 fermion can be also decoupled and its mass is exactly equal to MΨ. On the contrary, the other three particles mix with each other and their mass gets corrected after EWSB (as expected, the top mass is generated for nonzero values of ξ).

Finally, we analyze the neutral sector of our model. It comprises eight fields, the twin top and bottom quarks, and six of the composite fermions contained in the Ψ˜7 multiplet. Working in the field basis {t˜,b˜,D˜01,D˜02,U˜01,…,U˜04}, the mass matrix reads: M0=(00-12fξy˜L12fξy˜L00-ify˜L2-f1-ξy˜L20000-ify˜L2fy˜L20000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ00000000-M˜Ψ0fy˜R000000-M˜S).After diagonalization, we find one massless eigenvalue corresponding to the twin bottom quark (it does not acquire mass since we are not introducing the b˜R field). Four of the neutral heavy fermions completely decouple and acquire the following masses mU˜01=mU˜03=mD˜02=M˜Ψ,mU˜022=M˜Ψ2+y˜L2f2.The elementary/composite mixing induces instead corrections to the masses of the remaining neutral particles; at leading order in ξ we find: mt˜2≃f42y˜L2y˜R2M˜S2+y˜R2f2(1-ξ)+O(ξ2),mD˜012≃M˜Ψ2+12y˜L2f2(1+ξ)+O(ξ2),mU˜042≃M˜S2+y˜R2f2+y˜L2f2M˜S22(M˜S2+y˜R2f2)(1-ξ)+O(ξ2).

We conclude by noticing that the masses of the particles in different charged sectors are not unrelated to each other, but must be connected according to the action of the twin symmetry. In particular, it is obvious that the two singlets S2/34 and U˜04 form an exact twin pair, as it is the case for each SM quark and the corresponding twin partner. The remaining pairs can be easily found from the spectrum and correspond to the implementation of the twin symmetry in the composite sector as defined in [13].

RG IMPROVEMENT OF THE HIGGS EFFECTIVE POTENTIAL

In this Appendix we describe the computation of the Higgs effective potential and its RG improvement. First of all, the UV threshold correction can be computed with a standard Coleman-Weinberg (CW) procedure, from which we can easily derive the function F1 of Eq. (4.2). We find: F1=-14[1+MS4(MS2+f2yR2)2-MS2+MΨ2-f2yR2MS2-MΨ2+f2yR2logm*2MΨ2+MS2(MS2(MΨ2+f2yR2)+2f2yR2MΨ2+MS4)(MS2+f2yR2)2(MS2-MΨ2+f2yR2)logm*2MS2+f2yR2].

The IR contribution to the Higgs mass can be organized using a joint expansion in ξ and (αilog), where αi=gi2/(4π) for couplings gi. Schematically: δmh2|IR=mt2αt4πt[a1+b1ξ+b2αt4πt+b3αS4πt+c1ξ2+c2ξαS4πt+c3αS216π2t2+c4ξαt4πt+c5αt216π2t2+c6αt4παS4πt2]-a2mW2αEW4πt+twin,where t=logm*2/mt2 and all the couplings are evaluated at the scale m*. Here ai, bi, ci are the O(1) coefficients of the LO, NLO, and NNLO terms respectively. The twin contribution is obtained by substituting all αi, ai, bi, ci with the corresponding tilded quantities and t with t˜=logm*2/mt˜2. The calculation of the LO and NLO terms is straightforward and all these contributions are included in our calculation. The NNLO terms indicated in blue are also simple to evaluate, and are included in our final result, indicated as NNLO*. The calculation of the remaining NNLO terms is more complicated, since it involves the running of several higher dimensional operators involving both the SM and twin fields. An attempt to compute the full potential at NNLO has been presented in Ref. [46]. However, while that result includes the contribution of the SM Goldstone bosons, additional contributions induced by the twin Goldstones, which are expected to arise at NNLO, are not included. Since the full calculation at NNLO is beyond the scope of this paper, we limit ourselves to only include the colored contributions in Eq. (C2). Comparing with Ref. [46] suggests that our NNLO* result gives an overestimate of the IR Higgs mass; that this is the case at large m* is also evident from Fig. 1. For this reason we consider the NNLO* curve only as an indication of the importance of the NNLO correction, and use as final input for our numerical analysis the average of the LO and NLO results as described in Sec. IV.

We present now a procedure for computing the aforementioned contributions to the Higgs mass based on the approach of Ref. [46]. The RG improvement of the Higgs effective potential can be obtained by solving the one-loop β-function of the vacuum energy in the background of the Higgs field. The fermion contribution to the vacuum energy is given by dVf(hc,t)imprdt=Nc16π2[Mt(hc,t)4+Mt˜(hc,t)4],whereas the leading gauge contribution is given by Vg(hc,t)=316π2[2MW(hc)2+MZ(hc)2+3MW˜(hc)2]t.

In order to compute the improved potential to NNLO* accuracy, we must input the gauge boson masses at tree-level and the renormalized top and twin top masses with leading-log accuracy in the corresponding Yukawa couplings and with next-to-leading-log accuracy in the strong gauge couplings. After solving Eq. (C3) for the effective potential, one must properly account for the Higgs wave function renormalization before expanding around the minimum of the potential in order to compute the physical Higgs mass. We fill in the salient details below.

On integrating out the heavy degrees of freedom, the composite twin Higgs model at the scale m* can be represented by an effective Lagrangian that is invariant under a global SU(3)c×SU(2)L×U(1)Y symmetry, identified with the gauge group of the SM, as well as an additional SU(3)˜c×SU(2)˜L+R. SU(3)c˜ is the gauged twin color and SU(2)˜L+R is a global twin custodial group, broken only by fermion interactions, under which the twin gauge bosons (W˜) and Goldstone bosons (GBs) transform as triplets. We work in the gaugeless limit g=g′=g˜=0, since we are only interesting in capturing the leading contributions to the Higgs potential that parametrically depend on yt, y˜t, gS, and g˜S.

The relevant light degrees of freedom are the SM fermion doublet, QL, and singlet, tR, and their twin counterparts t˜L and t˜R, the gluons and their twins; the Higgs doublet (including the massless GBs) and the twin GBs. For economy of notation we define a twin Higgs doublet in analogy with the SM one, H˜=12(2π˜+h˜+iπ˜0),withh˜=f1-2H†H+π˜2f2,making explicit the dependence on the twin Higgs which is integrated out at tree-level, thus realizing the SO(8) symmetry nonlinearly.

Since the background field calculation at leading-log order requires corrections to the fermion masses and Higgs wave function at only one loop, we can neglect in the effective Lagrangian at the scale m*, any operator that has vanishing coefficients at tree level. We also omit all operators that cannot be predicted solely in terms of the low-energy parameters of the theory (yt and y˜t) and f, including products of Higgs and fermion currents that arise from integrating out composite fermions.

In any case, these do not contribute to the effective potential at this order.

We make the following field redefinition: fsin(|Π|f)|Π|Πi→Πi,for  |Π|=2H†H+π˜2which allows the effective Lagrangian, containing a pure scalar sector LS and a fermionic sector LF, to be expressed in an especially compact form. The relevant operators in each sector are given below: LS=DμH†DμH+12∂μπ˜∂μπ˜+cHOH+cHπ˜OHπ˜+cπ˜Oπ˜2f2+2dHH†Hf2OH,and LF=ib¯L∂bL+ib¯R∂bR+it¯L∂tL+it¯R∂tR-yt[Q¯LHctR+H.c.]+ib˜¯L∂b˜L+ib˜¯R∂b˜R+it˜¯L∂t˜L+it˜¯R∂t˜R-y˜t[Q˜¯LH˜ct˜R+H.c.].where OH=∂μ(H†H)∂μ(H†H),Oπ˜=∂μπ˜2∂μπ˜2,OHπ˜=∂μ(H†H)∂μπ˜2.We have defined a twin fermion doublet Q˜L=(t˜L,b˜L)T, transforming under the twin-sector symmetries, and the Yukawa couplings are defined at tree-level as yt(m*)=yLyRfMS2+yR2f2andy˜t(m*)=yL˜yR˜fM˜S2+yR˜2f2.

The Wilson coefficients at the scale m* have the boundary values: cH(m*)=cHπ˜(m*)=cπ˜(m*)=dH(m*)=1,while the Z2 symmetry enforces yt(m*)=y˜t(m*)andαs(m*)=αs˜(m*)

We now expand the effective Lagrangian in the background of the Higgs field h=hc+h^ to obtain: LS=Zπ(hc)(12∂μπ0∂μπ0+∂μπ+∂μπ-)+12Zh^(hc)∂μh^∂μh^+12Zπ˜(hc)∂μπ˜∂μπ˜,where we defined Zπ(hc)=1,Zh^(hc)=1+cHhc2f2+dHhc4f4,Zπ˜(hc)=1,and LF=∑ψiZψ(hc)ψ¯∂ψ-mt(hc)t¯t-mb(hc)b¯b-mt˜(hc)t˜¯t˜-mb˜(hc)b˜¯b˜-yt(hc)2h^t¯t-iyt(hc)2π0t¯γ5t-iyt(hc)2π-b¯(1+γ5)t+iyt(hc)2π+t¯(1-γ5)b+y˜t(hc)2h^t˜¯t˜-iyπ˜t˜t˜(hc)2π˜0t˜¯γ5t˜-iyπ˜t˜b˜(hc)2π˜-b˜¯(1+γ5)t˜+iyπ˜t˜b˜(hc)2π˜+t˜¯(1-γ5)b˜.Here ψ runs over all the Weyl fermions {tL,tR,bL,bR,t˜L,t˜R,b˜L,b˜R} and we defined Zψ(hc)=1,yt(hc)=yt,mt(hc)=ythc2,mb(hc)=0,yπ˜t˜b˜(hc)=y˜t,y˜t(hc)=y˜thcf11-hc2f2,mt˜(hc)=y˜tf21-hc2f2mb˜(hc)=0.

We can now compute the running masses of the top and twin top in the background: Mt(hc,t)≈mt(hc)[1+(gS24π2-3yt(hc)24(4π)2Zh^(hc))t+22gS4(4π)4t2],Mt˜(hc,t)≈mt˜(hc)[1+(g˜S24π2-3y˜t(hc)24(4π)2Zh^(hc))t+22g˜S4(4π)4t2],with t=log(m*2/μ2). The values of the parameters in the Higgs background are simply read off from the effective Lagrangian at the scale m*. Notice that in the background, only yt and y˜t are scale-dependent, while hc is “frozen”; hence the running of the quark mass is identical to that of the corresponding Yukawa coupling.

We can now substitute Eqs. (C17) into the RG equation (C3), integrate the result and expand at the required order in hc/f to get Vf(hc,t)impr=Nc64π2{[yt4hc4+y˜t4f4(1-hc2f2)2]t+yt4hc42(gS2π2-3yt2(4π)2)t2+2396π4[gS4yt4hc4+g˜S4y˜t4f4(1-hc2f2)2]t3+y˜t4f432π2(1-hc2f2)2[16g˜S2-3y˜t2hc2f2(1-(1+cH)hc2f2)]t2}.This is the improved effective potential written in terms of parameters computed at the scale m*. We can now treat hc as a quantum field, fluctuating around its minimum hc=h+⟨h⟩. We then have to take into account that the potential (C18) has been computed in a noncanonical basis for hc, where its kinetic term coefficient, including only the leading logarithmic running, is Zhc(t)=1+3yt2(4π)2t.Notice that only the top contributes to the one-loop running of the hc wave function, since the twin top only couples quadratically to hc, giving rise to a one-loop contribution that is not logarithmically divergent. After normalizing hc using Eq. (C19), including a Z2 breaking mass term necessary to achieve a viable minimum and minimizing the potential, hc acquires a vacuum expectation value hc=h+⟨h⟩. This makes the wave function of h (the Higgs field in the minimum) noncanonical again, due to the presence of cH: Zh(⟨h⟩)=1+cH⟨h⟩2f2.After normalizing h taking into account also this last effect, and including the LO gauge contribution, the desired contribution to the Higgs mass reads: δmh2|IR=3v28π2{(yt4t+yt˜4t˜)(1-cHξ+(cH2-dH)ξ2)-116[(3g24+2g12g22+g14)t+3g˜24t˜]+132π2(16yt4gS2t2+16yt˜4g˜S2t˜2)(1-cHξ)+132π2(-15yt6t2-12yt2y˜t4t˜2+3yt˜6t˜2(1+cH))+2396π4(yt4gS4t3+yt˜4g˜S4t˜3)},where we have highlighted the various contributions present in Eq. (C2) with the corresponding colors and set t=logm*2/mt2, t˜=logm*2/mt˜2. The Higgs vev has been defined by fixing the W mass according to mW2=g2⟨h⟩24,⇒⟨h⟩2=v2=12GF=246  GeV.

A numerical determination of the IR contribution δmh2|IR can be obtained by making use of the experimental value of the top quark mass to fix yt(m*) and y˜t(m*). In fact, the 1-loop RG equation for yt is decoupled from the twin sector at the order we are interested in and can be easily solved. Including only the orders required to match our calculation of the Higgs mass squared in Eq. (C21) we get: yt(m*)=yt˜(m*)=yt(mt)[1-(gS(mt)24π2-9yt(mt)264π2)logm*2mt2+22gS(mt)4(4π)4log2m*2mt2],gS(m*)=g˜S(m*)=gS(mt)[1-7gS(mt)232π2logm*2mt2],which fixes yt(m*) in terms of yt(mt). As an input to our numerical analysis we use the PDG combination for the top quark pole mass mt=173.21±0.51±0.71  GeV [57]. This is converted into the top Yukawa coupling in the MS¯ scheme ytMS¯(mt)=0.936±0.005 by making use of Eq. (62) of Ref. [65]. We then use yt(mt)=ytMS¯(mt). For the strong interaction, we run the parameter measured at the scale of the Z boson mass, gS(mZ)∼1.22, to the scale mt to obtain gS(mt).

OPERATOR ANALYSIS OF THE HEAVY-VECTOR CONTRIBUTION TO <inline-formula><mml:math display="inline"><mml:mi>δ</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>L</mml:mi><mml:mi>b</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

In this Appendix we discuss the UV threshold contribution to δgLb generated by the tree-level exchange of the composite vectors ρ [adjoint of SO(7)] and ρX [singlet of SO(7)] at zero transferred momentum. This effect arises at leading order from diagrams with a loop of heavy fermions, as in Fig. 5. Our simple effective operator analysis will show that the contribution of the ρ identically vanishes, in agreement with the explicit calculation in the simplified model.

An adjoint of SO(7) decomposes under the custodial SU(2)L×SU(2)R as: 21=(3,1)+(1,3)+3×(2,2)+3×(1,1).The first two representations contain the vector resonances that are typically predicted by ordinary CH models, namely ρL and ρR. They mix at tree-level with the Z boson and in general contribute to δgLb. The remaining resonances do not have the right quantum numbers to both mix with the Z boson and couple to the left-handed bottom quark due to isospin conservation. As a result, only the components ρL and ρR inside the 21 can give a contribution to δgLb at the 1-loop level.

In order to analyze such effect, we make use of an operator approach. We classify the operators that can be generated at the scale m* by integrating out the composite states, focusing on those which can modify the Zbb¯ vertex at zero transferred momentum. In general, since an exact PLR invariance implies vanishing correction to gLb at zero transferred momentum, any δgLb must be generated proportional to some spurionic coupling breaking this symmetry. In our model, the only coupling breaking PLR in the fermion sector is yL, and a nonvanishing δgLb arises at order yL4. The effective operators can be constructed using the CCWZ formalism in terms of the covariant spurion χL=Σ†Δ†ΔΣ,where Δ is defined in Eq. (3.11). By construction χL is a Hermitian complex matrix. Under the action of an element g∈SO(8), it transforms as a 21a+27s+7+1+1 of SO(7) (where the 7 is complex), and its formal transformation rule is χL→h(Π,g)χLh†(Π,g),h∈SO(7).As a second ingredient to build the effective operators, we uplift the elementary doublet qL into a 7+1 representation of SO(7) by dressing it with NGBs: QL=(Σ†Δ†qL).We will denote with QL(7) and QL(1) respectively the septuplet and singlet components of QL. Since QL(1) does not contain bL (it depends only on tL), only QL(7) is of interest for the present analysis. Under a SO(8) transformation QL(7)→h(Π,g)QL(7).

The effective operators contributing to δgLb can be thus constructed in terms of χL, dμ, and QL(7). We find that the exchange of ρμ in the diagram of Fig. 5 can generate two independent operators, O21=Q¯L(7)γμTaQL(7)Tr(dμχLTa),O21′=Q¯L(7)γμTaQL(7)(dμTaχL)88,where Ta is an SO(8) generator in the adjoint of SO(7); the exchange of ρμX gives rise to other two

Additional structures constructed in terms of dμ and χL can be rewritten in terms of those appearing in Eqs. (D6) and (D7), hence they do not generate new linearly independent operators. Notice that Tr(dμχLTa)∝faa^b^dμa^(χL(7))b^, (dμTaχL)88∝faa^b^dμa^(χL(7)*)b^, (dμχL)88=-dμa^(χL(7))a^, Tr(dμχL)=-dμa^(χL(7))a^+dμa^(χL(7)*)a^, where χL(7) denotes the component of χL transforming as a (complex) fundamental of SO(7). A similar classification in the context of SO(5)/SO(4) models in Ref. [29] found only one operator, corresponding to the linear combination O1′-O1.

: O1=Q¯L(7)γμQL(7)Tr(dμχL),O1′=Q¯L(7)γμQL(7)(dμχL)88.Simple inspection reveals that only the septuplet component of χL contributes in the above equations. One can easily check that the operators of Eq. (D6) give a vanishing contribution to δgLb. In particular, the terms generated by the exchange of the (2,2) and (1,1) components of the ρ give (as expected) an identically vanishing contribution. Those arising from ρL and ρR [obtained by setting Ta in Eq. (D6) equal to respectively one of the (3,1) and (1,3) generators] give instead an equal and opposite correction to gLb. This is in agreement with the results of a direct calculation in the simplified model, from which one finds that the contributions from ρL and ρR cancel each other. Finally, a nonvanishing δgLb arises from the operators of Eq. (D7) generated by the exchange of ρX. Upon expanding in powers of the Higgs doublet, O1 and O1′ both match the dimension-6 operator OHq of Eq. (5.11) and differ only by higher-order terms.

EXPLICIT FORMULAS FOR THE EWPO

In this Appendix we report the results of our calculation of the electroweak precision observables, in particular we collect here the explicit expression of the coefficients aUV, aIR of Eq. (5.12) and bUV, cUV, bIR of Eq. (5.17).

Let us start considering the T^ parameter. For convenience, we split the UV contribution into two parts, redefining aUV as: aUV=aUVFin+aUVLoglog(MΨ2MS2+f2yR2).The coefficients aUVFin and aUVLog are obtained through a straightforward calculation, but their expressions are complicated functions of the Lagrangian parameters. We thus show them only in the limit cL=cR≡c and MΨ=MS≡M, for simplicity. For aIR we give instead the complete expression. We find: aUVFin=112(-12M4f4yR4+6f2M2yR2(f2yR2+M2)2+9M6(f2yR2+M2)3-8)+c(-5f8yR8-2f6M2yR6+7f4M4yR4+12f2M6yR2+4M8)2f4yR4(f2yR2+M2)2+c2(f2yR2+2M2)2(3f2yR2-5M2)2f4yR4(f2yR2+M2),aUVLog=f6yR6-f4M2yR4-f2M4yR2-2M62f6yR6+2c(2f6yR6-3f4M2yR4+3f2M4yR2+2M6)f6yR6+c2(f2M4yR2-10M6)f6yR6,aIR=12+MS2MΨ22(MS2+f2yR2)2+2cLMSMΨ+2cRf2yR2MS2+f2yR2.

The derivation of δgbL at 1-loop level is more involved and requires the computation of a series of diagrams. As explained in the text, we focus on those featuring a loop of fermions and NGBs (see Fig. 4), and that one with a loop of fermion and the tree-level exchange of a heavy vector (see Fig. 5). The coefficients cUV is generated only by the latter diagram; we find: cUV=α7L(α1R+α7R)(1+2cR)gρX2f2MρX2MΨ22(MΨ2+yL2f2).We remind the reader that in our numerical analysis we use MρX/(gρXf)=1, see footnote 5. We redefine the other two coefficients as bIR=δIR+δ¯IRbUV=(δUVFin+δ¯UVFin)+(δUVLog+δ¯UVLog)log(MΨ2MS2+f2yR2),where δIR, δUVFin, and δUVLog are generated by the diagrams in Fig. 4 only, whereas δ¯IR, δ¯UVFin, and δ¯UVLog parametrize the correction due to the tree-level exchange of a heavy spin-1 singlet in Fig. 5. As before, we report the expression of the UV parameters in the limit cL=cR≡c, MΨ=MS≡M, for simplicity; in the case of the coefficients with a bar, generated by the diagram of Fig. 5, we further set α7L=α1L and α7R=α1R. We find: δUVFin=-2f6yR6-4f4M2yR4-4f2M4yR2+M612(f2yR2+M2)3-c(f6yR6+4f4M2yR4-2f2M4yR2+M6)62f2yR2(f2yR2+M2)2+16c2M2(3f2yR2+M2-2f2yR2)-c3M232f2yR2,δUVLog=16-M26f2yR2-c3M432f4yR4-c2M43f4yR4-c(-f4yR4+2f2M2yR2+M4)62f4yR4,δ¯UVFin=f2M2α1LgρX2(f4yR4(α1L+2α1R)+f2M2yR2(3α1L+8α1R)+2M4(α1L+α1R))4MρX2(f2yL2+M2)(f2yR2+M2)2+cf2M2α1LgρX2(f2yR2(2α1L+α1R)+2M2α1L)2MρX2(f2yL2+M2)(f2yR2+M2),δ¯UVLog=cM4α1LgρX2(α1L-α1R)2yR2MρX2(f2yL2+M2)+f2M2α1Lα1RgρX22MρX2(f2yL2+M2).For the IR coefficients we give instead the full expressions. We find: δIR=16+MS2MΨ26(MS2+f2yR2)2+2cLMSMΨ3(MS2+f2yR2)+2cRf2yR212(MS2+f2yR2),δ¯IR=α7Lα1RgρX2MρX2f4MΨ2yR22(f2yL2+MΨ2)(f2yR2+MS2).Notice that the IR corrections aIR and bIR are related to each other and parametrize the running of the effective coefficients c¯Hq, c¯Hq′, and c¯Ht, as explained in the main text.

410.1103/PhysRevD.96.095036.f4

Diagrams with a loop of fermions and NGBs contributing to the ZbLb¯L vertex. Here fi indicates any fermion (both heavy and light) in our simplified model; π± are the charged NGBs.

510.1103/PhysRevD.96.095036.f5

The diagram contributing to ZbLb¯L with a loop of fermions and tree-level exchange of a heavy vector. Here fi and fj denote generic fermions in the theory, both heavy and light, whereas ρν indicates the heavy vector.

Finally, we report the contribution to S^ generated in our simplified model by loops of heavy fermions. We do not include this correction in our electroweak fit, because in the perturbative region of the parameter space it is subdominant with respect to the tree-level shift of Eq. (5.6). Rather, we use this computation as an additional way to estimate the perturbativity bound, as discussed in Sec. III B. Analogously to what we did for T^ and δgLb, we parametrize the fermionic contribution to S^ as: ΔS^Ψ=g228π2ξ[(1-cL2-cR2)logm*2MΨ2+(1-c˜L2-c˜R2)logm*2M˜Ψ2]+g2216π2ξ[sUVFin+sUVLoglog(MΨ2MS2+f2yR2)+s˜UVFin+s˜UVLoglog(M˜Ψ2M˜S2+f2y˜R2)]+g2216π2ξsIRyL2f2MΨ2logM12mt2.Terms in the first line are logarithmically sensitive to the UV cutoff, the second line contains the UV threshold corrections, while the IR running appears in the third line. The UV thresholds include a contribution from the twin composites Ψ˜7 and Ψ˜1, parametrized by s˜UVFin and s˜UVLog. At leading order in yL, by virtue of the twin parity invariance of the strong sector, such a contribution can be obtained from that of Ψ7 and Ψ1 (i.e. from sUVFin and sUVLog) by simply interchanging the tilded quantities with the untilded ones. Higher orders in yL break this symmetry and generate different corrections in the two sectors. We performed the computation of the UV coefficients for yL=0, whereas sIR is derived up to order yL2. We find: sUVFin=12-6cLcRMSMΨ(f2yR2+MS2+MΨ2)(f2yR2+MS2-MΨ2)2+(cR2+cL2)6(24MS2MΨ2(f2yR2+MS2-MΨ2)2-7),sUVLog=-2(MS2+f2yR2)(MS2-MΨ2+f2yR2)3[6cLcRMSMΨ3+cR2MS2(f2yR2+MS2-3MΨ2)+cL2(f2yR2+MS2)(f2yR2+MS2-3MΨ2)],sIR=MS2MΨ4-f2yL2((f2yR2+MS2)(MS2-f2yR2)+MS2MΨ2)+MΨ2(f2yR2+MS2)26MΨ2(f2yR2+MS2)2-cR22f2yR2(MΨ2-f2yL2)3MΨ2(f2yR2+MS2)-cL2MS(MΨ2-f2yL2)3MΨ(f2yR2+MS2).

THE EW FIT

For our analysis of the electroweak observables we make use of the fit to the parameters ε1,2,3,b [66–68] performed in Ref. [69] (see also Ref. [26]). The central values there obtained for the shifts Δεi≡εi-εiSM and the corresponding correlation matrix are Δε1=0.0007±0.0010Δε2=-0.0001±0.0009Δε3=0.0006±0.0009Δεb=0.0003±0.0013ρ=(10.80.86−0.330.810.51−0.320.860.511−0.22−0.33−0.32−0.221).We can directly relate Δε1 to ΔT^ and Δε3 to ΔS^ by using the results of Refs. [45,47], and furthermore Δεb=-2δgbL. We set Δε2=0 in our study, since its effect is subdominant in our model as well as in CH models [47]. We thus make use of Eq. (F1) to perform a χ2 test of the compatibility of our predictions with the experimental constraints. The χ2 function is defined as usual: χ2=∑ij(Δεi-μi)(σ2)ij-1(Δεj-μj),(σ)ij2=σiρijσj,where μi and σi denote respectively the mean values and the standard deviations of Eq. (F1), while Δεi indicates the theoretical prediction for each EW observable computed in terms of the Lagrangian parameters. After deriving the χ2, we perform a fit by scanning over the points in our parameter space keeping only those for which Δχ2≡χ2-χmin2<7.82, the latter condition corresponding to the 95% confidence level with 3 degrees of freedom. Using this procedure, we convert the experimental constraints into bounds over the plane (MΨ,ξ).

ESTIMATES OF THE PERTURBATIVITY BOUND

This Appendix contains details on the derivation of the perturbative limits discussed in Sec. III B. As explained there, we consider the processes πaπb→πcπd and πaπb→ψ¯cψd, where ψ={Ψ7,Ψ˜7} and all indices transform under the fundamental representation of the unbroken SO(7). In order to better monitor how the results depend on the multiplicity of NGBs and fermions, we perform the calculation for a generic SO(N)/SO(N-1) coset with Nf composite fermions ψ in the fundamental of SO(N-1). Taking N=8 and Nf=2×3=6 thus reproduces the simplified model of Sec. III A.

The perturbative limits are obtained by first expressing the scattering amplitudes in terms of components with definite SO(N-1) quantum numbers. In the case of SO(7) the product of two fundamentals decomposes as 7⊗7=1⊕21a⊕27s, where the indices a and s label respectively the antisymmetric and symmetric two-index representations. A completely analogous decomposition holds in the general case of SO(N)/SO(N-1),

One has N⊗N=1⊕[N(N-1)/2]a⊕[N(N+1)/2-1]s.

but for simplicity we will use the SO(7) notation in the following to label the various components. The leading tree-level contributions to the scattering amplitudes arise from the contact interaction generated by the expansion of the NGB kinetic term of Eq. (3.4) and from the NGB-fermion interactions of Eq. (3.9). The structure of the corresponding vertices implies that the four-NGB amplitude has components in all three irreducible representations of SO(N-1) and contains all partial waves. The amplitude with two NGBs and two fermions, instead, has only the antisymmetric component of SO(N-1) and starts with the p-wave. At energies much larger than all masses the amplitudes read M(πaπb→πcπd)=sf2δabδcd+tf2δacδbd+uf2δadδbc,M(πaπb→Ψ¯7LcΨ7Ld)=s2f2sinθ(δacδbd-δadδbc),M(πaπb→Ψ¯7RcΨ7Rd)=s2f2sinθ(δacδbd-δadδbc).They decompose into irreducible representations of SO(N-1) as follows: M(1)(πaπb→πcπd)=(N-2)sf2,M(21)(πaπb→Ψ¯7LcΨ7Ld)=s2f2sinθ,M(21)(πaπb→πcπd)=sf2cosθ,M(21)(πaπb→Ψ¯7RcΨ7Rd)=s2f2sinθ.M(27)(πaπb→πcπd)=-sf2,Performing a partial wave decomposition we get M(r)=∑λi,λfMλi,λf(r)=16πk(i)k(f)∑j=0∞aj(r)(2j+1)⁢∑λi,λfDλi,λfj(θ),where λi, λf are the initial and final state total helicities, and k(i)(k(f)) is equal to either 1 or 2 depending on whether the two particles in the initial (final) state are distinguishable or identical respectively. In the above equation M(r) should be considered as a matrix acting on the space of different channels. The coefficients aj(r) are given by aj(r)=132πk(i)k(f)∫0πdθ∑λi,λfDλi,λfj(θ)Mλi,λf(r).and act as matrices on the space of (elastic and inelastic) channels with total angular momentum j and SO(N-1) irreducible representations r. They can be rewritten as a function of the scattering phase as aj(r)=e2iδj(r)-12i∼δj(r).Our NDA estimate of the perturbativity bound is derived by requiring this phase to be smaller than maximal: |δj(r)|<π2⇒|aj(r)|<π2

Let us consider first the case r=1, corresponding to the amplitude singlet of SO(N-1). The only contribution comes from the four-NGB channel. Since the helicities of the initial and final states are all zeros, in this particular case the Wigner functions Dλi,λfj(θ) reduce to the Legendre polynomials: aj(1)=164π∫0πdθPj(cosθ)M(1).The first and strongest perturbativity constraint comes from the s-wave amplitude, which corresponds to j=0. We find: a0(1)=N-232πsf2,where N=8 in our case. From Eqs. (G6) and (G8), one obtains the constraint of Eq. (3.15).

We analyze now the constraint from the scattering in the antisymmetric representation, r=21. In this case, both the NGB and the fermion channels contribute; the process ππ→ππ is however independent of the fermion and Goldstone multiplicities and can be neglected in the limit of N and Nf. The process involving fermions is a function of Nf and generates a perturbative limit which is comparable and complementary to the previous one. We have: aj(21)=∑λf=±1132π∫0πdθD0,λfj(θ)M0,λf(21).As anticipated, this equation vanishes for j=0, so that the strongest constraint is now derived for p-wave scattering, with j=1. We have a1(21)=Nf242πsf2.From Eqs. (G6) and (G10) follows the constraint of Eq. (3.16).

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diff --git a/tests/data/aps/PhysRevResearch.5.033093.xml b/tests/data/aps/PhysRevResearch.5.033093.xml new file mode 100644 index 0000000..2a3b59d --- /dev/null +++ b/tests/data/aps/PhysRevResearch.5.033093.xml @@ -0,0 +1,311 @@ + + +
+ + +PRRESEARCH +PRRHAI + +Physical Review Research +Phys. Rev. Res. + +2643-1564 + +American Physical Society + + + +10.1103/PhysRevResearch.5.033093 + + +ARTICLES + + + +Motional ground-state cooling of single atoms in state-dependent optical tweezers +MOTIONAL GROUND-STATE COOLING OF SINGLE ATOMS … +C. HÖLZL et al. + + +https://orcid.org/0000-0002-2176-1031 +HölzlC. +1 + +https://orcid.org/0000-0001-5527-5878 +GötzelmannA. +1 + +https://orcid.org/0009-0007-6940-7916 +WirthM. +1 + +https://orcid.org/0000-0002-1305-4011 +SafronovaM. S. +2,3 + +https://orcid.org/0000-0001-9763-9131 +WeberS. +4 + +https://orcid.org/0000-0002-9106-3001 +MeinertF. +1 +* + +5. Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany +Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA +Joint Quantum Institute, National Institute of Standards and Technology and the University of Maryland, College Park, Maryland 20742, USA +Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany + + +

f.meinert@physik.uni-stuttgart.de

 + +9August2023 +August2023 +5 +3 +033093 + + +7February2023 + + +22June2023 + + + +Published by the American Physical Society +2023 +authors + +Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. + + +

Laser cooling of single atoms in optical tweezers is a prerequisite for neutral atom quantum computing and simulation. Resolved sideband cooling comprises a well-established method for efficient motional ground-state preparation, but typically requires careful cancellation of light shifts in so-called magic traps. Here, we study a novel laser cooling scheme which overcomes such constraints, and applies when the ground state of a narrow cooling transition is trapped stronger than the excited state. We demonstrate our scheme, which exploits sequential addressing of red sideband transitions via frequency chirping of the cooling light, at the example of Sr88 atoms and report ground-state populations compatible with recent experiments in magic tweezers. The scheme also induces light-assisted collisions, which are key to the assembly of large atom arrays. Our work enriches the toolbox for tweezer-based quantum technology, also enabling applications for tweezer-trapped molecules and ions that are incompatible with resolved sideband cooling conditions.

 + + + + +Bundesministerium für Bildung und Forschung +10.13039/501100002347 + + + + + + + +Carl-Zeiss-Stiftung +10.13039/100007569 + + + + + + + +Vector Stiftung +10.13039/501100013912 + + + + + + + +Office of Naval Research +10.13039/100000006 + + +N00014-20-1-2513 + + + + + + + + + +INTRODUCTION +

Quantum control of individual atoms trapped in optical tweezers has seen a very rapid development in the last years [1], which has opened routes for applications such as quantum computing and simulation [2–4], precision metrology [5,6], or ultracold chemistry [7,8]. The quest for high-fidelity operations on the internal states of the atoms, for example, logic gate operations or optical clock interrogation, also requires cooling of the external motion, ideally down to the quantum mechanical ground state of the tweezer trap [9,10]. Efficient ground-state preparation is also key for assembling Hubbard-type lattice models atom by atom with optical tweezers [11,12], or for realizing ultrafast quantum gate protocols via resonant Förster interactions [13].

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Large motional ground-state occupation is typically achieved using well-established sideband-resolved cooling protocols [9,10], which more recently also became available for alkaline-earth(-like) atom arrays exploiting their narrow intercombination transitions [14–16]. Sideband laser cooling, however, poses tight constraints on the trapping condition, as it requires careful cancellation of differential light shifts for the internal atomic states involved in the cooling cycle, a situation referred to as magic trapping. Consequently, many of the applications mentioned above, including optical tweezer-based atomic clocks [5,6] or novel concepts for qubit implementations in gate-based quantum processors [17,18], are incompatible with such constraints. Novel laser cooling mechanisms that work at more general conditions thus not only expand the toolbox for neutral atom quantum technology, but may also find applications in controlling optically trapped molecules or even ions [19–21].

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In this paper, we demonstrate a method for motional ground-state cooling at the example of single trapped Sr88 atoms which is applicable in the generic situation of sizable differential light shifts between the relevant atomic states. The strategy relies on a frequency chirp of the cooling light to quench the population of initially occupied motional states towards the trap ground state. A detailed theoretical proposal and analysis thereof has been reported recently in Ref. [22]. Optimizing the cooling parameters, we measure ground-state populations exceeding 80% along the radial (more strongly confined) tweezer axis. At the same time, the cooling protocol efficiently removes pairs of atoms from the trap via light-assisted collisions resulting in a final 50% filling probability with exactly one atom, prerequisite for the systematic assembly of large atom arrays [23,24].

+ + +GROUND-STATE COOLING SCHEME FOR STATE-DEPENDENT TWEEZERS +

We consider an atom with two internal electronic states |g〉 and |e〉 trapped in the Gaussian-shaped potential formed by an optical tweezer [Fig. 1(a)]. Sufficiently close to the trap bottom, the atom is harmonically confined with state-dependent oscillator frequencies ωg and ωe, respectively. Standard sideband cooling (Raman- or single-photon scheme) requires equal AC polarizabilities αe and αg for the electronic levels, yielding identical ladders of motional states independent of the atom's internal state. Provided a narrow laser transition between |g〉 and |e〉 with linewidth γ≪ωg (festina lente regime, see Refs. [25,26] for its importance in all-optical cooling schemes to Bose-Einstein condensation), one can then cool the atomic motion in the trap by setting the laser to a fixed frequency resonant with the first red sideband, i.e., detuned by −ωg from the free-space transition frequency. Magic trapping(αe=αg) ensures, that the condition for addressing the red sideband is independent of the harmonic oscillator level. + +10.1103/PhysRevResearch.5.033093.f1 +1 + +

Chirp-cooling in state-dependent optical tweezers. (a) Radial tweezer potential for the |g〉=|1S0〉 and |e〉=|3P1〉 states of the cooling transition. The state-dependent trap depth enables transfer of excited motional states into the trap ground-state via strong red sideband transitions, reducing νg by Δν>1 vibrational quanta (arrows denote excitation and subsequent spontaneous decay). The potential curves are offset vertically so that Udiff indicates the (positive) light shift of the electronic transition between the motional ground states from the free-space resonance. (b) Numerical simulations of the cooling dynamics, demonstrating ground-state transfer when the laser detuning δ (measured from the trap shifted resonance at Udiff) is ramped from a large red detuning towards δ≈−ωg. The thickness of the yellow areas is proportional to the population in |νg〉. Arrows indicate population transfer when the resonance condition for the sideband transitions drawn in (a) is matched. (c) Experimental realization: Single Sr88 atoms are trapped in an optical tweezer (green), using a high-NA objective. The polarization of the tweezer light is indicated by the arrow in the NA cone. Chirp-cooling is realized with three pairs of counter-propagating, circularly polarized 689nm MOT beams (cooling). Additional laser beams (probe and shelving) are used for sideband thermometry. (d) Ratio of the AC polarizability (and hence of the trap depths) between |1S0〉 and the light-shifted substates |e0〉 and |e±〉 of |3P1〉 as a function of trapping wavelength. The dotted line indicates the wavelength used in the experiment.

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Such persistent cooling conditions are no longer given when the trapping potential is state-dependent (αe≠αg). Only very recently, it has been demonstrated that efficient cooling into the trap ground state can still be realized with a fixed laser frequency for αg/αe<1, i.e., when the excited state |e〉 is trapped stronger than the ground state |g〉 [27,28]. Here, we investigate the opposite case, αg/αe>1, for which cooling with a fixed frequency cannot work. This can be seen in Fig. 1(a), which depicts the situation we study in our experiment. Specifically, we employ the narrow (γ=2π×7.4kHz) 1S0 to 3P1 intercombination transition at a laser wavelength of about 689nm for in-trap cooling. It is convenient to define the laser detuning δ with respect to the trap-shifted resonance condition Udiff/ℏ [see Fig. 1(a)] for driving the electronic transition with the atom in the motional ground state, i.e., |g,νg=0〉 to |e,νe=0〉. Here, νg(νe) is the harmonic oscillator quantum number of the ground (excited) state vibrational ladder. In our scenario where ωe<ωg, the resonance condition to drive the red sideband δ≈(νg−1)ωe−νgωg now depends on νg. An attempt to cool the lowest-lying vibrational excitations requires to set δ close to the red sideband condition for νg=1, i.e., δ≈−ωg. Such a laser frequency, however, causes heating of higher-lying vibrational states. Consequently, cooling with a fixed frequency would not succeed. This problem can be resolved by using a time-dependent chirp of δ which compensates for the νg-dependence of the condition to address red sidebands one after the other. Such a frequency chirp then dissipates motional quanta without concurrent heating, since the protocol assures that higher vibrational states are first transferred to lower energy, before states closer to the trap bottom are addressed (see also Ref. [22] for a recent proposal). Note that a fixed laser frequency provides an effective repulsive energy cap in the trap, which was exploited in Ref. [14] to prevent atom loss during imaging and which was interpreted by a classical Sisyphus effect.

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Before we turn to the experimental results, we briefly analyze the chirp-cooling approach numerically. To this end, we compute the quantum dynamics of a harmonically confined and laser-coupled (Rabi frequency Ω) two-level atom in 1D including state-dependent trapping. The time evolution is obtained by integrating the Liouville-von Neumann equation for the density matrix with a finite basis set of oscillator levels accounting for decay of the excited state (decay rate γ) via the Lindblad operators (see Appendix E). Results for typical experimental parameters are shown in Fig. 1(b) for a linear (10 ms long) ramp of δ from δi/ωg=−3.7 to δf/ωg=−1 and for the atom initially prepared in |νg=4〉. The data reveal that chirping allows for efficient transfer into the motional ground state. For the chosen parameters [(ωg,ωe,Ω)=2π(150,110,20)kHz], the final ground-state population is >94%. Cooling occurs due to resonant addressing of various red sidebands during the chirp, coupling states of different motional quantum numbers νg>νe [arrows in Figs. 1(a) and 1(b)]. Note that in contrast to magic trapping conditions, where couplings between different oscillator levels with Δν=|νg−νe|>1 are strongly suppressed for tight confinement by the Lamb-Dicke effect, here, they play a vital role in the cooling dynamics due to direct wave-function overlaps 〈νg|νe〉 between levels of equal parity [29]. The simulations also reveal optimal ground-state population when δf≈−ωg, i.e., when the laser frequency chirp ends near the resonance condition for driving the first red sideband from |νg=1〉. Note that the simulation parameters for Fig. 1(b) are chosen in a way that the underlying mechanism of the cooling scheme is well visible and are not optimized for highest ground-state transfer.

+ + +COOLING AND THERMOMETRY +

Our experiments start with loading a single optical tweezer with wavelength λ=539.91nm and a waist of 564(5)nm from a Sr88 magneto-optical trap (MOT) operated on the 1S0 to 3P1 intercombination line [Fig. 1(c)] (see Appendix A) . After loading, the tweezer is typically occupied by more than one atom in the electronic ground state 1S0 (trap depth ≈0.5mK). At the trapping wavelength, the tweezer potential for 1S0 is deeper than for the excited 3P1 state of the cooling transition, i.e., realizing the situation αg/αe>1 discussed above. More precisely, we perform all experiments at nominally zero magnetic field and with a linearly polarized tweezer. In that case, the three magnetic substates of the 3P1 level (mJ=0,±1) are shifted by the AC-Stark interaction with the trap light. We label the AC-Stark-shifted eigenstates |e0〉, |e−〉, and |e+〉. The latter two are energetically degenerate (see Ref. [14] and Appendix C). The wavelength dependence of the ratio αg/αe for all three levels is shown in Fig. 1(d). This allows us to perform experiments using two different transitions with vastly different values αg/αe (∼1.13 for |e0〉 and ∼1.90 for |e±〉).

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We start investigating the cooling dynamics on the transition to |e±〉, which exhibits the stronger differential light shift, measured to be Udiff|e±〉/ℏ=2π×5.50(5)MHz for the tweezer's optical power of about 1.70(2)mW set throughout this work. The cooling protocol starts by switching on the 689nm MOT beams with an initial detuning δi=−2π×4.1MHz. We estimate the Rabi frequency to about 2π×50kHz. The beams are kept on for 100ms, during which the laser frequency is ramped linearly to a variable final detuning δf.

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The temperature after the ramp is measured via the release-and-recapture technique [30]. Briefly, the trap is turned off diabatically for a variable release time tr before it is suddenly switched on again. We then image the atoms on the 1S0−1P1 transition at 461nm by collecting fluorescence photons on a sCMOS camera. From the photon signal on the camera, we deduce the survival probability of a single atom in the trap (see Appendix B and below for more details). Exemplary data of such measurements are shown in Fig. 2(a) for different values of δf and compared to data taken without the cooling protocol (circles). A slower decay of the measured survival probability with tr is indicative for lower temperature, as hotter atoms escape faster from the trap volume. The results reveal a reduction of temperature with decreasing |δf|, i.e., when the frequency is chirped closer to the light-shifted resonance at Udiff|e±〉/ℏ. + +10.1103/PhysRevResearch.5.033093.f2 +2 + +

Thermometry after chirp-cooling via release-and-recapture. (a) Atom-survival probability as a function of the release time tr is shown for three different values of δf as indicated (triangles, squares, diamonds) after cooling on the transition to |e±〉, and compared to a measurement with no cooling applied (circles). Solid lines are fits of a classical particle trajectory simulation to the data to extract the temperature. The blue-dashed line shows the prediction of a quantum mechanical simulation for an atom in the motional ground state (see Appendix D). (b) Temperature Tcl extracted from classical trajectory simulations of data as shown in (a) as a function of the detuning δf at the end of the frequency chirp. Triangles (squares) are results for cooling on the transition to |e±〉 (|e0〉). The circle denotes the temperature without cooling applied. The vertical line indicates the resonance condition (δf=−ωg) for the lowest red sideband |g,νg=1〉→|e,νe=0〉. At this point, the minimal temperature is achieved. The dashed line depicts the temperature-equivalent Tgs of the radial zero-point motion energy in the trap. Error bars in all figures show one standard deviation and are mostly smaller than the data points.

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To extract a classical temperature Tcl at the end of the chirp from datasets as shown in Fig. 2(a), we fit classical Monte-Carlo simulations of the release-and-recapture sequence to the data, assuming a thermal energy distribution with mean energy E¯=kBTcl in each spatial direction (solid lines) [30]. Results of this analysis are plotted in Fig. 2(b) as a function of δf (triangles). The data reveal a minimum temperature of 4.17(6)µK in the vicinity of δf≈−ωg. Throughout this work ωg=2π×126(5)kHz, as measured via sideband spectroscopy (see below). Compared to the data taken without cooling, we achieve a reduction in temperature by about one order of magnitude. Chirping further down in |δf| again leads to heating, as the cooling light approaches the resonance condition for driving the carrier transition from the ground state (δ=0).

+

The minimal measured temperature is found close to the temperature-equivalent of the zero-point motion energy of the radial tweezer ground state Tgs=ℏωg/2kB=3.04µK (dashed line), indicating sizeable radial ground-state population. Since the classical analysis does not account for the zero-point motion in the trap, we also analyze the experimental data with the lowest-measured temperature quantum mechanically. As the release-and-recapture method is only weakly dependent on the axial tweezer direction, it is sufficient to model the recapture probability along the radial direction. To this end, we time-evolve the wave functions of the first few states of a 2D harmonic oscillator numerically. After time tr of free expansion, the Gaussian tweezer potential is added to deduce the probability of recapture (see Appendix D). The result of this analysis for the 2D ground state [dashed line in Fig. 2(a)] is already close to the lowest-temperature experimental data (δf=−ωg). Next, the analysis is extended to a thermal distribution of the first few 2D harmonic oscillator states. A fit of this model to the data for δf=−ωg yields 82(3)% ground-state population along one radial direction. To demonstrate the robustness of the chirp-cooling, we finally repeat the measurements on the transition to |e0〉, for which the differential light shift Udiff|e0〉/ℏ=2π×1.30(2)MHz is much weaker. We obtain very similar results [squares in Fig. 2(b)] and attribute the slightly higher minimal temperature to the higher sensitivity required for tuning δf.

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While the above results already provide evidence that the chirp-cooling method yields a large ground-state population, we complement the thermometry by resolved sideband spectroscopy on the 1S0−3P1 transition to |e0〉 along the radial direction of tweezer confinement. For spectroscopy, we apply a short (75µs) probe pulse with the laser frequency set to the vicinity of the |e0〉 resonance and the propagation direction perpendicular to the tweezer axis. Subsequently, the population transferred to |e0〉 is rapidly shelved into the metastable states 3P0 and 3P2 [Fig. 3(a)]. The sequence is repeated three times before imaging on the 1S0 - 1P1 transition to increase the signal. Results are shown in Fig. 3(b) close to the lowest temperature achieved when cooling on the transition to |e±〉 (diamonds). Compared to the case of less-deep cooling (circles), we observe strong sideband asymmetry, a hallmark for large ground-state population. Shift and broadening of the line with increasing temperature is due to the differential AC-Stark shift on the probe transition. Note that without cooling, the sidebands are completely unresolved. Such effects are absent in more conventional sideband thermometry, for which narrow optical lines with vanishing differential AC-Stark shifts are used. + +10.1103/PhysRevResearch.5.033093.f3 +3 + +

(a) Scheme for sideband thermometry via shelving into metastable states. The atom in the ground state 1S0 is excited to the 3P1 (|e0〉) state by a short probe laser pulse. The population transferred to 3P1 is hidden from the imaging cycle (1S0−1P1) by optically pumping into the metastable 3P0 and 3P2 states via 3S1. (b) Measured loss fraction from 1S0 due to shelving as a function of probe detuning δp after cooling on the transition to |e±〉 to δf=−1.2ωg (diamonds) and δf=−5.5ωg (circles). Solid lines are numerical simulations fitted to the data to extract the ground-state fraction (see text).

+ +

+

Extracting the ground-state population from our data thus requires fitting with a full numerical simulation of the spectroscopy sequence. To this end, we compute the dynamics of the trapped two-level atom density matrix with an initial thermal trap population as above. The population in |e0〉 is extracted at the end of the probe pulse and the loss fraction is determined assuming a fitted success probability ps for shelving. Trap frequencies ωg,e and the temperature of the initial state are also fit parameters (see Appendix E). Our lowest temperature data (diamonds) is found to be compatible with a ground-state fraction in the range of 73% to 97%, which is in agreement with the results extracted from release-and-recapture. This ground-state fraction is also comparable with sideband cooling in magic-wavelength tweezers reported for strontium [14,15].

+

Finally, we note that our data analysis neglects possible shifts and broadening of the carrier signal due to axial temperature orthogonal to the probe beam direction. Those can be present when probing at nonzero AC-Stark shift due to the dependence of the radial carrier transition frequency on the axial motional state, and allow us to infer also an upper limit estimate for the axial temperature. Indeed, since the observed linewidth is well compatible with our 1D analysis, we conclude that the axial temperature cannot be significantly higher than the measured radial temperature. This provides evidence that cooling acts simultaneously in axial trap direction, revealing additional information that is not accessible from release-and-recapture.

+ + +PARITY PROJECTION DURING COOLING +

Next, we study the dynamics of the atom number population in the tweezer during cooling. Most importantly, we find that chirp-cooling to low temperatures in the trap also causes light-induced losses which reliably remove pairs of atoms from the trap [31,32]. This can be readily seen from histograms of the detected photon count before and after cooling [Fig. 4(a)]. Without cooling, we observe a multipeak structure in the histogram, where the individual peaks are associated with one, two and more than two atoms in the trap. After cooling, a clean binary distribution with a one and a zero atom peak is observed. We find an approximately equal number of photon counts in the two peaks, i.e., about 50% filling with exactly one atom. In Fig. 4(b), we show how the probabilities for finding one (circles) and more than one (triangles) atom(s) in the trap evolve with the final detuning δf of the cooling ramp. Pairs of atoms are continuously lost with decreasing temperature and essentially vanish for |δf|≲2ωg. Indeed, this is expected as the rate for light-assisted collisions, which arise from coupling to a weakly bound molecular state below the 1S0−3P1 asymptote [14,33], strongly depends on the wavefunction overlap of the initial pair of atoms. Moreover, along with the decreasing multiatom signal, we find an increase in the single-atom probability, providing evidence that initial trap loading with an odd atom number ≥3 results in a single atom after cooling. Thus the chirp-cooling directly delivers parity projection, an ideal starting point for a deterministic assembly of large atom arrays [23,24]. + +10.1103/PhysRevResearch.5.033093.f4 +4 + +

Light-induced losses and parity projection during cooling. (a) Histograms of photon counts before (yellow) and after (blue) cooling on |e±〉 to δf=−1.2ωg. The vertical dashed (dotted) line indicates the lower threshold set for identifying N=1 (N≥2) atom(s). Dash-dotted lines are Gaussian fits to the data to guide the eye. The inset shows the fluorescence of a single atom imaged onto at 3x3 pixel array of the sCMOS camera. (b) Probability to detect N=1 (circles) and N≥2 (triangles) atom(s) in the tweezer after chirp-cooling to a final detuning δf. The dashed line indicates unity filling with 50% probability. Note the change in scaling of the axis of abscissas for δf/ωg>−2 (shaded area).

+ +

+ + +COOLING IN AN ARRAY OF TWEEZERS +

Finally, we demonstrate the possibility to apply our chirp-cooling scheme to an array of multiple tweezers. To this end, we generate a one-dimensional line of ten equally spaced (≈10µm) traps [Fig. 5(a)], using an acousto-optical deflector in our optical tweezer path (see Appendix A). The traps are generated by applying ten RF tones to the modulator. We equalize the trap depths to a level of ≈2% via tweezer-resolved measurements of light shifts and correction on the individual RF amplitudes. The procedure for tweezer loading and cooling is equivalent to the single-tweezer case. + +10.1103/PhysRevResearch.5.033093.f5 +5 + +

Chirp-cooling in a one-dimensional tweezer array. (a) Averaged fluorescence image of a line of ten tweezers (10 µm spacing) after cooling chirp (including parity projection). [(b)–(d)] Release-and-recapture thermometry data (cf. Fig. 2) for the three tweezers indicated with boxes after a cooling chirp to δf=−1.2ωg on the transition to |e±〉. Solid lines are fits of a classical particle trajectory simulation to extract the temperature Tcl. The blue dash-dotted line shows the prediction of a quantum mechanical simulation for an atom in the motional ground state. The red dashed line (gray dotted line) shows the classical fit to the single-tweezer data of Fig. 2 for the same cooling parameters (with no cooling applied). (e) Temperatures Tcl as a function of the tweezer index shown in (a). Again, the red dashed line depicts the single-tweezer result after cooling with the same cooling parameters, and the gray dotted line is the temperature measured in a single tweezer without cooling. The blue dash dotted line depicts the temperature-equivalent Tgs of the radial zero-point motion energy in the trap. Error bars show one standard deviation and are mostly smaller than the data points.

+ +

+

Exemplary release-and-recapture thermometry data for the two outermost and a central tweezer are shown in Figs. 5(b)–5(d) when chirp-cooling on the transition to |e±〉 with δf=−1.2ωg, i.e., close to the minimally achieved temperature obtained in a single tweezer (cf. Fig. 2). As before, the classical temperature Tcl is extracted from Monte-Carlo trajectory simulations (solid green lines), which for all ten tweezers fall almost on top of the simulation results of the single-tweezer data (red-dashed line), reported above for the same parameters. The extracted values for Tcl, shown in Fig. 5(e), lie between Tcl=3.8(2)µK and Tcl=5.7(2)µK, close to the single-tweezer result [Tcl=4.8(5)µK] reported above for the same cooling parameters. The gray-dotted line again indicates the temperature without cooling. The small variations in Tcl across the array are attributed to residual differences in the trap depths. Indeed, the absolute value of δf required for optimal cooling depends approximately linearly on the trap depth, and in our case needs to be controlled on the percent level (ℏωg/Udiff≈2%).

+ + +CONCLUSION AND OUTLOOK +

In conclusion, we have demonstrated a novel, broadly applicable ground-state cooling method for trapped atoms in optical tweezer arrays, which heavily releases constraints on magic trapping conditions for future experiments. This opens new routes for tweezer-based quantum technologies, requiring trapping wavelengths that have been so far incompatible with efficient in-trap cooling, specifically in view of the rapid developments with alkaline-earth(like) atoms [5,34], i.e., strontium [14,15] or ytterbium [16]. For example, the tweezer wavelength selected in this work provides magic trapping of the two clock states 3P0 and 3P2, which enables new concepts for qubit encoding in a neutral atom quantum computer [18]. The cooling scheme also allows operating at wavelengths that offer additional magic trapping for Rydberg states, which mitigates decoherence and in-fidelity of two-body gates [17], and is also applicable to optical lattice systems [22]. When simultaneously applied during imaging on the narrow intercombination transition, the reported technique may also allow for reaching higher scattering rates, particularly for atom detection in the so far unexplored case of a more strongly trapped ground state [28]. Finally, we expect that applying individually controlled cooling beams along radial and axial direction in a pulsed sequence should allow for reaching the full three-dimensional trap ground state, and leave a detailed investigation of the cooling dynamics along the weakly trapped tweezer axis for future work. In that context, it is also interesting to apply optimal control strategies on the frequency-chirped laser pulses to increase the cooling efficiency.

+ + + + +ACKNOWLEDGMENTS +

We thank Johannes Zeiher, Jacob Covey, and the QRydDemo team for fruitful discussions. We acknowledge funding from the Federal Ministry of Education and Research (BMBF) under the grants CiRQus and QRydDemo, the Carl Zeiss Foundation via IQST, and the Vector Foundation. MSS acknowledges support by ONR Grant No. N00014-20-1-2513. This research was supported in part through the use of University of Delaware HPC Caviness and DARWIN computing systems.

 + +TWEEZER LOADING +

Tweezer loading starts with the preparation of a six-beam blue magneto-optical trap (MOT) of Sr88 atoms in a glass cell (Japan Cell), operated on the 1S0−1P1 (λ=460.9nm) transition. The MOT is loaded from an atomic beam source (AOSense, Inc.) comprising an oven, a Zeeman slower, and a 2D-MOT for transverse cooling. After 30 ms of MOT loading at a detuning of −2π×46 MHz and with a saturation parameter of s=0.12, we decrease the detuning and saturation parameter to −2π×21 MHz and s=0.01 within 10 ms to reduce temperature. The atoms are kept in this second stage of the blue MOT for another 50 ms, before they are loaded into a narrow-line red MOT on the 1S0−3P1 (λ=689.5nm) intercombination transition. To this end, the magnetic field gradient is ramped from ≈50 G/cm to ≈2 G/cm. During the first part of the red MOT, the cooling laser is broadened to a frequency comb with a 5 MHz width and a regular 30 kHz spacing by periodically modulating the RF-frequency applied to an acousto-optical modulator (AOM) to increase the capture volume. The comb is subsequently ramped down to a single frequency with a final detuning of −2π×150 kHz while simultaneously reducing the laser intensity from s≈3700 to ≈45. After another 10 ms hold time, the MOT contains several 104 atoms at an equilibrium temperature of ≈1.4µK. This atom number is found to be well-suited for loading a single tweezer and also multitweezer arrays, so that each site is filled with one or more atoms while also avoiding too many atoms in a single trap. The latter may cause nonpairwise losses, ultimately leading to less than 50% filling after parity projection [28].

+

For generating tweezers, we employ a frequency-doubled fiber laser system providing 10 W output power at 540 nm (TOPTICA Photonics). We send the trapping light through a 2D acousto-optical deflector (AA Opto-Electronic DTSXY-400) before focusing into the MOT region with a high-NA (0.5) microscope objective (Mitutoyo G Plan Apo 50X) to a waist of 564(5)nm. This allows us to extend the studies of our cooling scheme also to multitweezer arrays (see Fig. 5). To achieve homogeneous loading over an extended array, we increase the MOT volume by a two-step ramp of the laser detuning from −2π×150 kHz to −2π×280 kHz within 25 ms and further to −2π×560 kHz within 15 ms [14]. During the second part of this detuning ramp, the tweezer intensity is ramped up to 0.85 mW for a single tweezer. Subsequently, the MOT beams and the magnetic quadrupole field are turned off, and the tweezer intensity is further ramped to its final value of 1.7 mW, at which we perform our chirp-cooling experiments. Note that we illuminate the atoms with two repumping beams resonant with the 3P0−3S1 (679 nm) and 3P2−3S1 (707 nm) transitions during the entire MOT and tweezer loading procedure.

 + +ATOM IMAGING +

For single-atom detection, we induce fluorescence on the S10−P11 transition using a separate imaging beam, pulsed on for 75 ms with s≈1.2×10−3. Prior to this, the tweezer light intensity is increased to 3.4 mW. Fluorescence photons are collected via the same objective used to focus the optical tweezers and are imaged onto a 3×3 pixel area [compare inset of Fig. 4(a)] of a sCMOS camera (Teledyne KINETIX). The atoms are kept cold during imaging by using the red MOT beams with a fixed detuning of +2π×1.2 MHz from the 1S0−3P1 free-space resonance, corresponding to a detuning from the trap-shifted transition to |e0〉 of about −2π×1.4 MHz.

+

The images are classified into three categories depending on the number Nph of detected photons: No atom (Nph≤20), single atom (20<Nph≤65), multiple atoms (Nph>65). We quantify the accuracy of this classification in a similar way as reported in Ref. [14]. Taking consecutive images of the same single atom, we find a probability of p=92.2(2)% to detect the atom in the second image conditioned on its detection in the first one, similar to the results reported in Refs. [14,15]. This atom loss during imaging also dictates the imaging fidelity. More specifically, realizations where an atom is lost from the trap result in a reduced number of detected photons, which may be below the lower threshold Nth set for identifying one atom. An upper bound for the probability of such false negative detection events may be estimated via 1−exp(ln(p)Nth/N¯), where N¯ is the mean number of detected photons for one atom [14]. For the imaging parameters used throughout this work, we find a false negative rate of about 3.6(1) %. The false positive rate for identifying a single atom (0.00(2) %) is negligible.

+

To extract the survival probability in Fig. 2(a) and the loss fraction in Fig. 3(b), a value ζc is assigned to each of the imaging classification outcomes: ζc=0 (no atom), ζc=1 (single atom), and ζc=2 for multiple atoms, which is then averaged for each data point. To improve the signal-to-noise ratio of the sideband spectroscopy data in Fig. 3(c), we take two consecutive images. The first one is taken after the shelving spectroscopy pulse. Before taking the second image, we return the shelved population from the long-lived 3P0 and 3P2 states back into the imaging cycle by applying the 679-nm and 707-nm repumping lasers. We then postselect on realizations where an atom is detected in the second image.

+

Finally, we note that the accuracy to distinguish between single and multiple atoms in the trap is much lower than between zero and one atom, since the corresponding signals in the histograms [Fig. 4(a)] strongly overlap. The characterization threshold for multiple atoms is chosen high enough that it does not affect the low-temperature measurements with a clean bimodal distribution, i.e., after successful parity projection.

 + +DIFFERENTIAL AC-STARK SHIFTS +

In Fig. 1(d), we plot the ratio of the AC polarizabilities (and hence the trap depths) between the 1S0 electronic ground state and the trap-shifted sublevels |e0〉 and |e±〉 of the 3P1 excited state. For our case of nominally zero magnetic field and a linearly polarized tweezer, the AC-Stark shift for 3P1 can be readily expressed in terms of scalar (αs) and tensor (αt) polarizabilities (the contribution of the vector polarizability vanishes in the absence of magnetic field and for linear polarization [14,35]). The polarizability of the 1S0 ground state has only a scalar contribution.

+

More specifically, we consider the time-independent AC-Stark interaction Hamiltonian with an optical field E⃗(t)=E⃗+e−iωt+E⃗−e+iωt, where E⃗+=E0ε⃗, ε⃗ being the polarization vector, and E⃗− the complex conjugate of E⃗+. Omitting the vector term, the Hamiltonian reads + +H=−αsE02−3αtJ(2J−1){E⃗+·J⃗,E⃗−·J⃗}2−J(J+1)E023.Here, J⃗=(Jx,Jy,Jz) denotes the total angular momentum operator and J the associated angular momentum quantum number. In the absence of an external magnetic field, it is convenient to define the quantization axis along the tweezer polarization, which we set (without loss of generality) along the x direction. Accordingly, we label the bare Zeeman substates of the 3P1 level as |mJ=0,±1〉, where mJ is the magnetic quantum number associated with the projection of the total angular momentum along the tweezer polarization. The AC-Stark Hamiltonian then reduces to + +Hlin=−E02αs+αt3Jx2−J(J+1)2(2J−1).Since the Hamiltonian above is diagonal in the |mJ=0,±1〉 basis defined along the x direction, the total polarizability reads [14] + +α=αs+αt3mJ2−J(J+1)J(2J−1).For J=1 (3P1) one finds α=αs−2αt for |e0〉=|mJ=0〉 and α=αs+αt for |e±〉=|mJ=±1〉. Wavelength-dependent values for αs and αt are obtained from numerical calculations as follows.

+

We evaluated the dynamic polarizabilities by solving the inhomogeneous equation in valence space [36] using the Dalgarno-Lewis [37] approach. This approach allows to account for both discrete states and the continuum. We find intermediate-state wave functions δψ± from an inhomogeneous equation, + +|δψ±〉=1Heff−E0±ω∑k|Ψk〉〈Ψk|D|Ψ0〉=1Heff−E0±ωD|Ψ0〉,where D is the z-component of the effective electric dipole operator D, Ψ0 is the wave function, and E0 is the energy of the state of interest, either 1S0 or 3P1 in the present work.

+

The wave functions are computed using the relativistic high-precision hybrid method that combines configuration interaction and coupled-cluster approaches (CI+all order) [38,39]. In this method, the energies and wave functions are determined from the time-independent multiparticle Schrödinger equation + +Heff(Ek)Ψk=EkΨk,where the effective Hamiltonian Heff includes contributions of the core states constructed using the coupled-cluster method.

+

The polarizability is given by + +αv(ω)=〈Ψ0|D0|δψ+〉+〈Ψ0|D0|δψ−〉,where v indicates that this method gives the valence contribution to the polarizability. The small core polarizability contribution is computed in the random-phase approximation. One of the challenges of the accurate polarizability computation for the 3P1 state in the region below 600 nm is strong sensitivity to the accuracy of the energy levels. To resolve this problem, we developed a code to automatically replace the theoretical energy values for low-lying dominant contributions by exact experimental values as well as use improved recommended values of the reduced matrix element where available. The substitution is done for all data points using the sum-over-states formula + +αv(ω)=2∑kEk−E0|〈Ψ0|D0|Ψk〉|2Ek−E02−ω2,improving the polarizability accuracy. The uncertainties are estimated for all polarizability values.

 + +CLASSICAL AND QUANTUM MECHANICAL RELEASE-AND-RECAPTURE ANALYSIS +

In this section, we provide details on the classical and quantum mechanical analysis of the release-and-recapture data shown in Fig. 2. Our classical analysis follows the procedure described in Ref. [30]. Specifically, we draw a Monte Carlo sample of spatial and velocity vectors from a thermal distribution of point particles in a 3D harmonic trap using our experimental parameters. For a given classical temperature Tcl, the three spatial coordinates are Gauss-distributed with a standard deviation of σi=kBTcl/mωi22, where m is the mass of the Sr88 atom, and ωi the oscillator trap frequency in direction i=(x,y,z). The Gaussian velocity distribution in each direction has a standard deviation of σv=kBTcl/m. A simulated release-and-recapture trace is obtained by propagating this ensemble in free-space for a variable time tr. The survival probability is then computed by evaluating the fraction of the ensemble that is trapped after instantaneous switch-on of the Gaussian tweezer potential. A particle is considered to be trapped, when its kinetic energy is smaller than the local (absolute) potential energy after the free propagation. Such simulated traces are then fit to the data via a chi-squared analysis with Tcl as fit parameter.

+

This analysis does not capture the zero-point motion energy of the trapped atom, and the fitted classical temperature Tcl overestimates the true quantum mechanical temperature. Effects of quantized motion in the trap are taken into account by our quantum mechanical analysis of the lowest energy data in Fig. 2(a). To this end, it is sufficient to consider only the radial dynamics, since the release-and-recapture technique is only weakly dependent on the longitudinal motion. First, we compute the free expansion of initial 2D harmonic oscillator wave functions with occupation numbers (νx,νy), where x and y denote the cartesian coordinates of the radial tweezer direction. After a variable time tr, the Gaussian tweezer potential is turned on instantaneously and the wave function is evolved inside the potential for another 100 µs. The fraction of the wave function that has remained inside the potential then yields the survival probability p(νx,νy). Assuming a thermal population of the 2D harmonic oscillator levels, the survival probability pqm for an ensemble at a quantum mechanical temperature Tqm is finally obtained by weighting p(νx,νy) with the corresponding Boltzmann factor, + +pqm=1Zp(νx,νy)e−ℏω(x,y)(νx+νy+1)/(kBTqm).Here, ω(x,y) denotes the radial trap frequency and Z the partition function of the 2D harmonic oscillator. We fit the simulated pqm to the data with Tqm as fit parameter. Finally, the fitted value for Tqm yields the ground-state population along one radial direction stated in the main paper.

 + +NUMERICAL MODELING OF CHIRP-COOLING AND SIDEBAND SPECTROSCOPY DATA +

In this section, we provide details of our method for describing the chirp-cooling and for modeling the sideband spectroscopy data. Our method is similar to the approach taken in Refs. [22,29]. To reduce the computational costs of our simulations, we neglect coupling between different spatial directions and restrict ourselves to one radial direction of the three-dimensional trap. Under this approximation, the system can be described as a driven two-level atom in a one-dimensional harmonic trap. The electronic ground state |g〉 is trapped with the frequency ωg and the excited state |e〉 with ωe. We use the levels of the harmonic oscillator with the frequency ωg as a basis for the motional state of the atom. For our simulations, we take the 30 lowest oscillator levels into account. Using the creation and annihilation operators a† and a that act on these levels, the Hamiltonian of the system reads [29,40] + +H=ℏωga†a+12+Hatom(t)+Hint.

+

To derive the electronic Hamiltonian Hatom(t) of the laser-driven atom, we change into the rotating frame of the laser. Using the rotating wave approximation, we obtain + +Hatom(t)=ℏΩ2eiη(a†+a)|e〉〈g|+ℏΩ*2e−iη(a†+a)|g〉〈e|−ℏδ(t)|e〉〈e|,with the time-dependent detuning δ(t), Rabi frequency Ω, Lamb-Dicke parameter η=kx0=2πλℏ2mω2, wavelength λ=689nm, and m being the mass of the Sr88 atom.

+

The interaction Hint between the electronic and motional states of the atom emerges from the trap frequency being state-dependent and is given by + +Hint=ℏ(ωe2−ωg2)4ωg(a+a†)2|e〉〈e|+U|e〉〈e|,where U is chosen such that the transition from |g〉 to |e〉 is driven resonantly for δ(t)=0 if the atom is in its motional ground state.

+

To incorporate the decay of the excited state with rate γ=2π×7.4kHz, we describe the system with a Lindblad master equation, ρ̇=−i(Heffρ−ρHeff)+Lρ. The decay enters the non-Hermitian Hamiltonian Heff=H−iℏγ2|e〉〈e| and the term + +Lρ=ℏγ2∫0πe−iη(a†+a)cosθ|g〉〈e|ρ|e〉〈g|×eiη(a†+a)cosθsinθdθ.This term accounts for the population returning into the ground state and for the recoil of the emitted photon projected onto the direction of the trap.

+

To illustrate the chirp-cooling approach [Fig. 1(b)], we apply a Rabi frequency of Ω=2π×20kHz and a time-dependent detuning δ(t) that is ramped linearly from −3.7ωg to −1ωg within 10ms. The trap frequencies are set to ωg=2π×150kHz and ωe=2π×110kHz. To demonstrate the cooling mechanism, we use the motionally excited state |g,4〉 as the initial state. The resulting Lindblad master equation is solved using QuTiP [41].

+

For modeling the sideband spectroscopy data, we extend our model by introducing an effective dark state. The sideband spectroscopy is a two-step process. First, we apply a probe pulse with Rabi frequency Ω and detuning δ for 75µs. Second, we transfer the resulting population in |e〉 to the dark state with a success probability ps. The probe pulse is again simulated using QuTiP, using a thermal density matrix with temperature T as initial state. In the experiment, the shelving signal is increased by repeating the spectroscopy sequence three times before imaging. To account for this in our simulation, we also compute the pulse sequence three times, and take the populations of oscillator states after each pulse (with the atom measured in |g〉) as the initial condition for the next pulse. In doing so, we include the influence of the probe pulse on the population of motional states, which is experimentally relevant since the lifetime of |e〉 (1/γ=21.5µs) is comparable to the probe pulse length. Repeating this procedure for various values of δ yields the simulated sideband spectra. Finally, we fit the simulation results to the experimentally measured data with free parameters T, ωg, and ps. The Rabi frequency Ω=2π×44.9(5)kHz is measured independently (see Fig. 6 and text below), and the excited state trap frequency ωe is computed from the calculated polarizability ratio of the transition to |e0〉. The fit yields ωg=2π×126(5)kHz, i.e., the trap frequency stated in the main paper and ps=0.33. The range of ground-state fraction reported in the main text reflects the set of simulated sideband spectra when varying T that are compatible with the data within the experimental error bars. + +10.1103/PhysRevResearch.5.033093.f6 +6 + +

Measurement of the probe Rabi frequency Ω used for sideband spectroscopy. Measured loss fraction from 1S0 due to shelving as a function of probe time for different detunings from the carrier transition, i.e., νe=νg, to |e0〉. Solid lines show numerical simulations fit to the data, which yield Ω=2π×44.9(5)kHz.

+ +

+

Finally, we discuss briefly the independent measurement of Ω, for which we apply our sideband spectroscopy sequence as before but now vary the length of the probe pulses. Measured shelving data as a function of probe time exhibits damped Rabi oscillations (see Fig. 6). To extract Ω, we fit the data with the same numerical simulations as described above, but now vary the time of the probe pulses for different values of δ.

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PRXPRXHAEPhysical Review XPhys. Rev. X2160-3308American Physical Society10.1103/PhysRevX.4.021018RESEARCH ARTICLEScond-mattCondensed Matter PhysicsmesoscopicsMesoscopicssuperconductivitySuperconductivityNon-Abelian Majorana Doublets in Time-Reversal-Invariant Topological SuperconductorsNON-ABELIAN MAJORANA DOUBLETS IN TIME- …XIONG-JUN LIU, CHRIS L. M. WONG, AND K. T. LAWLiuXiong-Jun1,2,*WongChris L. M.1LawK. T.1,†Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, ChinaInstitute for Advanced Study, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

phyliuxiongjun@gmail.com

phlaw@ust.hk

29April20141April20144202101819June201316September2013Published by the American Physical Society2014authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

The study of non-Abelian Majorana zero modes advances our understanding of the fundamental physics in quantum matter and pushes the potential applications of such exotic states to topological quantum computation. It has been shown that in two-dimensional (2D) and 1D chiral superconductors, the isolated Majorana fermions obey non-Abelian statistics. However, Majorana modes in a Z2 time-reversal-invariant (TRI) topological superconductor come in pairs due to Kramers’s theorem. Therefore, braiding operations in TRI superconductors always exchange two pairs of Majoranas. In this work, we show interestingly that, due to the protection of time-reversal symmetry, non-Abelian statistics can be obtained in 1D TRI topological superconductors and may have advantages in applications to topological quantum computation. Furthermore, we unveil an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the superconducting state. This effect provides direct measurements of the topological qubit states in such 1D TRI superconductors.

Majorana fermions, whose existence in solid-state materials appears to have left discernible footprints in recent experiments, are strange particles in a number of ways. First, they are their own antiparticles. Second, when in isolation from each other, they obey a different interparticle rule, called “non-Abelian statistics,” from the usual fermions. For example, swapping two isolated Majorana fermions in space transforms one quantum state of the solid system that hosts them to another, while the system is rather robust against quantum noise. This second property actually lies at the heart of “topological quantum computing.”

A question of fundamental interest can be asked: What interparticle rule (statistics) do pairs of bound Majorana fermions obey? This fundamental question is motivated by what has been learned about topological superconductors—a new type of quantum matter. Topological superconductors with time-reversal symmetry are predicted to be able to host pairs of bound Majorana fermions, but the statistics of such bound pairs is far from clear. In fact, since two Majorana fermions can form a usual Dirac fermion, for a long time it was thought that such bound Majorana pairs should obey the usual statistics of normal fermions. In this theoretical paper, we show, for the first time, that a new type of non-Abelian statistics for Majorana-fermion pairs emerges in systems that satisfy certain symmetry conditions.

The time-reversal-invariant topological superconductors can be realized with heterostructure devices formed by quantum nanowires and conventional (s-wave) or unconventional (p-wave, d-wave, etc.) superconductors. In the topological regime, two bound Majorana fermions are predicted to exist at each end of the quantum nanowire. Because of the protection of time-reversal symmetry, we show that swapping two Majorana pairs localized in the nanowire ends can always be reduced to two separate two-Majorana interchanges that are related by time-reversal symmetry. This result shows that such Majorana pairs actually obey a new type of non-Abelian statistics that is, at the same time, “protected” by time-reversal symmetry.

INTRODUCTION

The search for exotic non-Abelian quasiparticles has been a focus of both theoretical and experimental studies in condensed matter physics, driven by both the exploration of the fundamental physics and the promising applications of such modes to a building block for a fault-tolerant topological quantum computer [1–7]. Following this pursuit, the topological superconductors have been brought to the forefront, for they host exotic zero-energy states known as Majorana fermions [8–16]. For a two-dimensional (2D) chiral p+ip-pairing state, which breaks time-reversal symmetry, one Majorana mode exists in each vortex core [2], and for the 1D p-wave case, such a state is located at each end of the system [4]. Because of the particle-hole symmetry, Majorana fermions in a topological superconductor are self-Hermitian modes that are identical to their own antiparticles. A complex fermion, whose quantum states span the physical space in the condensed-matter system, is formed by two Majoranas that can be located far away from each other. This property allows us to encode quantum information in the nonlocal fermionic states, which are topologically stable against local perturbations. The existence of 2n Majorana zero modes leads to 2n-1-fold ground-state degeneracy, and braiding two of such isolated modes in 2D or 1D superconductors transforms one state into another, which defines the non-Abelian statistics [3,15]. Remarkably, Majorana end states have been suggestively observed through tunneling measurements [17–19] in 1D effective p-wave superconductors obtained using heterostructures formed by semiconductor nanowire and s-wave superconductor [20–22].

Recently, a new class of topological superconductors with time-reversal symmetry, referred to as a DIII-symmetry-class superconductor and classified by the Z2 topological invariant [23–27], has attracted rapidly growing efforts [24–32]. Differently from chiral superconductors, in DIII-class superconductors, the zero modes come in pairs due to Kramers’s theorem. Many interesting proposals have been studied to realize Z2 time-reversal-invariant (TRI) Majorana quantum wires using the proximity effects of d-wave, p-wave, s±-wave, or conventional s-wave superconductors. It was shown that at each end of such a quantum wire are localized two Majorana fermions that form a Kramers doublet and are protected by time-reversal symmetry [30–38].

With the practicability in realization, a fundamental question is, can the DIII-class topological superconductor be applied to topological quantum computation? The puzzle arises from the fact that braiding the end states in a DIII-class 1D superconductor always exchanges Majorana Kramers pairs rather than isolated Majorana modes. While braiding two pairs of Majoranas in chiral topological superconductors yields Abelian operations, in this work, we show interestingly that braiding Majorana end states in DIII-class topological superconductors is non-Abelian due to the protection of time-reversal symmetry. We further unveil an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the superconducting state, which provides direct measurements of all topological qubit states in the DIII-class 1D superconductors.

The article is organized as follows. In Sec. II, we briefly introduce how to engineer the DIII-class 1D topological superconductor by inducing p-wave superconductivity in a conducting wire in proximity to a noncentrosymmetric superconductor. Then, in Sec. III, we turn to a detailed study of the non-Abelian Majorana doublets in the DIII-class 1D topological superconductor. Section IV is devoted to investigating the Josephson effect, which shows an interesting strategy to read out topological qubit states in TRI superconductors. Finally, the conclusions are given in Sec. V.

TOPOLOGICAL SUPERCONDUCTOR OF DIII CLASS BY PROXIMITY EFFECT

Several interesting proposals have been considered to realize DIII-class 1D topological superconductors, including to use the proximity effects of d-wave, p-wave, s±-wave, and conventional s-wave superconductors [30–33]. Here, we briefly introduce how to engineer such a Z2 topological superconductor by depositing a conducting quantum wire on a noncentrosymmetric superconductor thin film that can induce s-and p-wave pairings in the wire by the proximity effect [39,40], as illustrated in Fig. 1. The total Hamiltonian of the heterostructure system reads H=HSC+Hwire+Ht, where HSC, Hwire, and Ht represent the Hamiltonians for the substrate superconductor, the conducting wire, and the tunneling at the interface, respectively. Because of the lack of inversion symmetry, a noncentrosymmetric superconductor has both s-wave and p-wave pairings [39]. For convenience, we denote the pairings in the substrate superconductor as Δs(0) and Δp(0), respectively. The Bogoliubov de Gennes Hamiltonian for the 2D noncentrosymmetric superconductor is given by HSC=∑kx,ky[ε0(kx,ky)τz+αR(0)sinkyσx-αR(0)sinkxσyτz+Δp(0)sinkyσzτx-Δp(0)sinkxτy-Δs(0)σyτy],where ε0(kx,ky)=-2t(0)(coskx+cosky)-μ(0) is the normal dispersion relation, with t(0) the hopping coefficient in the superconductor; σj and τj (j=x, y, z) are the Pauli matrices acting on the spin and Nambu spaces, respectively; αR(0) is the spin-orbit-coupling coefficient; and μ(0) is the chemical potential. The pairing order parameters can be reorganized by Δ^=(Δs(0)+d·σ)(iσy), with the d vectors defined as d=Δp(0)(-sinky,sinkx,0).

110.1103/PhysRevX.4.021018.f1

The DIII-class 1D topological superconductor realized by depositing a conducting nanowire (NW) on the top of a noncentrosymmetric superconductor (SC). Both s- and p-wave pairings can be induced in the conducting wire by tunneling couplings through the interface between the NW and the substrate superconductor.

A single-channel 1D conducting quantum wire, being put along the x axis, can be described by the following Hamiltonian: Hwire=∑kx(-2twcoskx-μw)τz,with tw the hopping coefficient and μw the chemical potential in the wire. It is noteworthy that the intrinsic spin-orbit interaction is not needed to reach the TRI topological superconducting phase, while the proximity effect can induce an effective spin-orbit interaction in the nanowire. Now, we give the tunneling Hamiltonian Ht for the interface. For simplicity, we consider that at the interface, the coupling between the substrate superconductor and the nanowire is uniform, and thus the momentum kx is still a good quantum number. Then, the tunneling Hamiltonian can be written down as Ht=-t⊥∑kx,σcσ†(kx)diy0,σ(kx)+H.c.,where σ=↑, ↓ are the spin indices, t⊥ denotes the tunneling coefficient between the nanowire and substrate superconductor, and cσ†, cσ and dσ†, dσ are the creation and annihilation operators of electrons for the quantum nanowire and the superconductor, respectively. The site number iy0 characterizes where the heterostructure is located on the y axis in the noncentrosymmetric superconductor.

The induced superconductivity in the wire can be obtained by integrating out the degree of freedom of the superconductor substrate. We perform the integration in two steps. First, for the uniform noncentrosymmetric superconductor, we can determine its Green’s function Gs(kx,iy0) with momentum kx and at the site iy0 below the nanowire by the standard recursive method [41]. Then, the coupling of the nanowire to the superconductor can be reduced to the coupling to the site iy0 below the wire and described by the Green’s function Gs(kx,iy0). Integrating out the degree of freedom of the sites in the superconductor right below the nanowire yields a self-energy for the Green’s function of the nanowire, which gives rise to the proximity effect. The effective Green’s function of the nanowire takes the form Gwire(iω,kx)=1iω-εw(kx)τz-Σ(iω),where εw=-2twcoskx-μw and the self-energy reads Σ(iω)=t⊥2Gs(kx,iy0). Finally, the spectral function is determined by A(ω,kx)=-12πIm{Tr[τzGwire(ω+i0+,kx)]},with Im taking the imaginary part, Tr denoting the trace over the spin and Nambu spaces, and 0+ a positive infinitesimal. The spectral function determines the bulk band structure, which is numerically shown in Fig. 2 with different chemical potentials of the nanowire. In particular, from the numerical results, we find that the nanowire is in the topologically nontrivial regime when |μw|<2|tw| and |Δp(0)|>|Δs(0)|, which leads to the induced pairings in the wire |Δp|>|Δs|, while it is in the trivial regime when |Δs(0)|>|Δp(0)| or |μw|>2|tw| (i.e., the chemical potential is tuned out of the band of the wire). When tuning the chemical potential down to the band bottom, the bulk gap in the nanowire is reduced and closes right at the bottom, implying the critical value of the chemical potential μwc=-2tw (similar results can be obtained around 2tw, the top of the band) [Figs. 2(a)–2(c)]. In the topological regime at each end of the nanowire are localized two Majorana zero modes γj and γ˜j (j=L, R) that form a Kramers’s doublet, with their wave functions shown in Fig. 3. Further lowering the chemical potential reopens the bulk gap, and the system is driven into a trivial phase [Fig. 2(d)].

210.1103/PhysRevX.4.021018.f2

The logarithmic plot of the spectral function for the nanowire or noncentrosymmetric superconductor heterostructure. The dotted yellow curves show the bulk band structure of the nanowire system. The solid red areas (in the upper and lower positions of each panel) represent the bulk states of the substrate superconductor. (a) Topological regime with the chemical potential μw set as -2tw+5|Δp(0)| in the nanowire. In this regime, at each end of the wire are localized two Majorana zero modes (Fig. 3). (b) Topological regime with a reduced bulk gap by tuning μw=-2tw+|Δp(0)| close to the band bottom. (c) Critical point μwc=-2tw for the topological phase transition with the bulk gap closed. (d) Trivial phase regime for the nanowire with μ=-2tw-|Δp(0)|. Other parameters are taken that t⊥=0.5tw=0.5t(0), |Δs(0)|=0.5|Δp(0)|, and αR(0)=|Δp(0)|.

310.1103/PhysRevX.4.021018.f3

(a) Two Majorana bound modes exist at each end of the nanowire in the topological regime with μ=-2tw+|Δp(0)|, as considered in Fig. 2(b). (b),(c) The wave functions of two Majorana modes γL(R) and γ˜L(R) at the same end have exactly the same spatial profile, with ξ the coherence length in the nanowire.

It is interesting that the phase diagram in the nanowire does not depend on parameter details of the couplings between the nanowire and the substrate superconductor, and for |Δp(0)|>|Δs(0)|, the topological regime in the nanowire can be obtained in a large parameter range that -2tw<μw<2tw. This result enables a feasible way to engineer the DIII-class topological states in the experiment by tuning μw to be below or above the band bottom of the nanowire.

We note that the time-reversal symmetry is essential for the existence of the Majorana doublets in the topological phase. If time-reversal symmetry is broken, e.g., by introducing a Zeeman term Mzσz, the two Majorana modes at the same end will couple to each other and open a gap. On the other hand, while we consider here the DIII-class 1D topological superconductor through the proximity effect of noncentrosymmetric superconductors, the non-Abelian statistics predicted in this work are fundamental physics that are proposal independent and can also be studied with all other realistic setups for the 1D TRI topological superconductor, as proposed in recent works [30–38]. In particular, the recent proposal based on the proximity effect of s-wave conventional superconductors may be feasible for the experimental studies [33].

NON-ABELIAN STATISTICS

In this section, we show in detail that Majorana Kramers’s doublets obey non-Abelian statistics due to the protection of time-reversal symmetry. In the previous section, we have demonstrated that for the topological phase, at each end of the Z2 Majorana quantum wire are localized two Majorana modes γj and γ˜j (j=1, 2), transformed by a time-reversal operator that T-1γjT=γ˜j and T-1γ˜jT=-γj [24]. To prove the non-Abelian statistics, we first show a new result that in a DIII-class topological superconductor at zero temperature, the fermion parity is conserved for each time-reversed sector of the system. With this result, we then get that braiding Majorana doublets can generically reduce to two independent processes of exchanging, respectively, two pairs of Majoranas belonging to two different time-reversed sectors, which leads to the symmetry-protected non-Abelian statistics.

Fermi-parity conservation

Fermion parity measures the even and odd numbers of the fermions in a quantum system. Note that the non-Abelian statistics are properties of the ground-state subspace of a topological superconductor. We only need to consider the superconductor at zero temperature, in which case no thermally excited quasiparticles exist and the superconductor is characterized as a condensate. The fermion number of a ground state can only vary by pairs due to the presence of a pairing gap, which leads to the fermion-parity conservation for the condensate. For the DIII-class 1D topological superconductor, we prove here a central result that by grouping all the fermionic modes into two sectors that are time-reversed partners of each other, the fermion parity is conserved for each sector of the condensate, not only for the entire system. It is trivial to know that this result is true if the DIII-class topological superconductor is composed of two decoupled copies (e.g., corresponding to the spin-up and spin-down, respectively) of 1D chiral p-wave superconductors. For the generic case, the proof is equivalent to showing that in a TRI Majorana quantum wire, which has four degenerate ground states |n1n˜1⟩ (n1, n˜1=0, 1), the four topological qubit states are decoupled from each other with the presence of finite TRI perturbations. (The change in the fermion parity for each sector of the condensate necessitates the transition between |01˜⟩ and |10˜⟩ or between |00˜⟩ and |11˜⟩.) The coupling Hamiltonian, assumed to depend on a manipulatable parameter λ, should take the generic TRI form V(λ)=iE1(λ)(γ1γ2-γ˜1γ˜2)+iE2(λ)(γ1γ˜2-γ2γ˜1), which splits the two even-parity eigenstates |00˜⟩ and |11˜⟩ by an energy E(λ)=2E12+E22. Since |10˜⟩ and |01˜⟩ form a Kramers’s doublet at an arbitrary λ value, the transition between them is forbidden by time-reversal symmetry. Then, the fermion-parity conservation requires that the following adiabatic condition be satisfied in the manipulation: |⟨11˜|λ˙∂λ|00˜⟩|≪2E(λ), where λ˙=∂λ/∂t. This criterion is followed by R˜≡12E(λ)|∂λ∂t∂θ∂λ|≪1,θ=tan-1E1E2.We show below that the above condition is generically satisfied under realistic conditions.

According to the the previous section, the proximity effect induces p-wave and s-wave superconducting pairings in the nanowire. The effective tight-binding Hamiltonian of the DIII-class Majorana nanowire in the generic case can be written as Hwireeff=∑⟨i,j⟩,σtijciσ†cjσ+∑⟨i,j⟩(tijSOci↑†cj↓+H.c.)+∑⟨i,j⟩(Δijpci↑cj↑+Δijp*ci↓cj↓+H.c.)+∑j(Δscj↑cj↓+H.c.)-μ∑j,σnjσ,where the hopping coefficients and the chemical potential are generically renormalized by the proximity effect, with the spin-conserved and the spin-orbit-coupled hopping terms satisfying tij=tji=t and tijSO=-tjiSO=tSO. For the case with uniform pairing orders, the parameters Δs and Δp can be taken as real. On the other hand, for the present 1D system, one can verify that the phases in the (spin-orbit) hopping coefficients can always be absorbed into electron operators. Therefore, below, we consider that all the parameters in Hwireeff are real numbers. Then, in terms of the electron operators, the Majorana bound modes take the following general forms: γ1=∑j[u↑(1)(xj)c↑(xj)+u↓(1)(xj)c↓(xj)+u↑(1)(xj)c↑†(xj)+u↓(1)(xj)c↓†(xj)],γ2=i∑j[u↑(2)(xj)c↑(xj)+u↓(2)(xj)c↓(xj)-u↑(2)(xj)c↑†(xj)-u↓(2)(xj)c↓†(xj)],and γ˜j=TγjT-1. The coupling energies between the Majorana modes at the left (γ1, γ˜1) and right (γ2, γ˜2) ends are calculated by E1=i⟨γ1|Hwireeff|γ2⟩=-i⟨γ˜1|Hwireeff|γ˜2⟩ and E2=i⟨γ1|Hwireeff|γ˜2⟩=-i⟨γ˜1|Hwireeff|γ2⟩.

It can be found that the coefficients E1,2 are proportional to the overlapping integrals of the left- and right-end Majorana wave functions, which decay exponentially with the distance d between the Majorana modes. Since γj and γ˜j are connected by a T transformation, their wave functions have exactly the same spatial profile, which leads to the same exponential form of the coefficients E1,2(λ)=α1,2(λ)e-d/ξ, with ξ the coherence length in the nanowire. The prefactors αj(λ) depend on the local couplings, i.e., the hopping terms and pairings in Hwireeff, between electrons belonging to the same (for j=1) or different (for j=2) sectors of the time-reversal partners. For the realistic conditions, we consider that the chemical potential in the nanowire is far below the half-filling condition and thus the Fermi momentum satisfying kFa≪1, and the coherence length (in the order of 1.0  μm) is typically much larger than the lattice constant ξ≫a (a∼0.5  nm). Under these conditions, we can verify that to E1,2, the contributions of the spin-orbit-coupling and p-wave-pairing terms in Hwireeff vanish, and we find (details can be found in the Appendix) E1≃t∑⟨i,j⟩σuσ(1)(xi)uσ(2)(xj),E2≃Δs∑⟨i,j⟩σuσ(1)(xi)uσ(2)(xj).Therefore, while the magnitudes of E1,2 can vary with d and the bulk gap, their ratio E1/E2 is nearly a constant, and we always have ∂λθ≈0, which validates the adiabatic condition. The above results are consistent with the fact that when Δs=0, the original Hamiltonian (7) can be block diagonalized, and then E2≡0. The adiabatic condition is clearly confirmed with the numerical results in Fig. 4. The fermion-parity conservation for each sector of the condensate shows that an isolated DIII-class 1D Majorana wire should stay in one of the four fermion-parity eigenstates germinated by nonlocal complex fermion operators fj and f˜j, given that time-reversal symmetry is not broken. In particular, one can always prepare a nanowire initially in the ground state |00˜⟩ or |11˜⟩ by controlling the initial couplings E1,2(λ) and then manipulate the states adiabatically.

410.1103/PhysRevX.4.021018.f4

Adiabatic condition and fermion-parity conservation for each sector of the superconductor. (a),(b) The couplings E1,2 between Majorana modes are manipulated (a) by tuning the chemical that changes the bulk gap (λ=Ebulk) in the nanowire and (b) by varying the length (λ=d) of a trivial (gray) region that separates the two pairs of Majoranas. (c),(d) The energy splitting E between |00˜⟩ and |11˜⟩ (red curves) and the ratio R˜ (blue curves), (c) as functions of Ebulk and (d) versus the trivial region distance d. The parameters in the nanowire are taken such that the proximity-induced p-wave pairing Δp=1.0  meV, the s-wave pairing Δs=0.5  meV, and the spin-orbit-coupling energy ESO=0.1  meV. In the numerical simulation, we assume that the coupling energy E is tuned from 0 to 1.0 meV in the time 1.0  μs. We also numerically confirm the adiabatic condition R˜≪1 with other different parameter regimes.

It is noteworthy that for a realistic system at finite temperature, the quasiparticle poisoning may exist, which can change fermion parity and lead to the decoherence of Majorana qubit states. At low temperature, the dominant effect in the quasiparticle poisoning comes from the single-electron tunneling between the nanowire and the substrate superconductor [42]. The decoherence time in the chiral Majorana nanowires ranges from 10 ns to 0.1 ms, depending on the parameter details [42]. We note that in realizing chiral topological superconductors, an external magnetic field is required to drive the system into the topological phase [20–22], and such a field suppresses the proximity-induced order parameter in the nanowires. Moreover, the topological gap in the nanowire depends on the ratio of the spin-orbit-coupling strength over the Zeeman energy and is further reduced by the magnetic field. As a result, in the experiments on chiral Majorana nanowires, the topological gap is typically much smaller than the s-wave-pairing gap in the proximate superconductor. For the DIII-class nanowires, without suppression of the external magnetic field, the proximity-induced gap in the similar parameter regime is expected to be larger compared with that in chiral nanowires, which suggests a longer decoherence time in the DIII-class Majorana nanowires [30–33].

To ensure that the decoherence effect induced by quasiparticle poisoning does not lead to serious problems, one requires that the adiabatic manipulation time for Majorana modes should be much less than the decoherence time. For the DIII-class Majorana nanowires, the adiabatic time depends on the two characteristic time scales. One is determined by the bulk gap τ1ad=h/Eg, and another τ2ad corresponds to the fermion-parity conservation for each time-reversal sector. The time scale τ1ad with an induced superconducting gap 0.1 meV in the DIII-class wire is about 0.1 ns, which is much less than the decoherence time. Furthermore, if using the parameter regime in Fig. 4, one can estimate that τ2ad<1.0  ns. On the other hand, for the proposals considered in Refs. [30,31,33], the effective Hamiltonian has no s-wave-pairing order, and the time scale τ2ad then renders the magnitude of τ1ad. These estimates imply that the adiabatic manipulation of Majorana modes may be reached in DIII-class 1D topological superconductors.

On the other hand, when a weak time-reversal-breaking term is present, e.g., in the presence of Zeeman couplings induced by a stray field, the decoherence effect may also result due to the couplings between qubit states with the same total fermion parity. Note that Majorana doublets in a DIII-class Majorana wire are of Ising type in the spin degree of freedom; thus, the time-reversal-breaking couplings can be induced by a stray field only along specific directions, depending on the concrete setup used in the experimental realization [35,37]. For the semiconductor nanowires with large Landé factors, e.g., the InAs wire that has a Landé factor g≈20 [22], one can verify that the resulting decoherence time is over 1.6 ns with a fluctuating stray field of strength 0.01 T. This result implies that the decoherence effect is negligible when the field strength is several times less than 0.01 T. In general, using semiconductor nanowires with small g factors should be preferred in realizing the DIII-class topological superconductors, in which case the braiding operations can be stable against relatively stronger stray fields.

Braiding statistics

Note that braiding Majorana end modes is not well defined for a single 1D nanowire and, as first recognized by Alicea et al., the minimum setup for braiding requires a trijunction, e.g., a T junction composed of two nanowire segments [15]. The braiding can be performed by transporting the Majorana zero modes following the steps as illustrated in Figs. 5(a)–5(d).

510.1103/PhysRevX.4.021018.f5

(a)–(d) Braiding Majorana end modes through gating a T junction following the study by Alicea et al. [15]. The dark (light) gray area of the nanowires depicts the topological (trivial) region, which can be controlled by tuning the chemical potential in the nanowire. The arrows depict the direction that the Majorana fermions are transported to in the braiding process.

The fermion-parity conservation for each sector shown in the above subsection implies that the exchange of Majorana end modes in a DIII-class topological superconductor generically reduces to two independent processes of braiding Majoranas of two different sectors, respectively. The reason is because, first of all, braiding adiabatically the Majorana pairs, e.g., γ1, γ˜1 and γ2, γ˜2 in Fig. 5, does not affect the bulk states, which are gapped. Furthermore, assuming that other Majorana modes are located far away from γ1,2 and γ˜1,2, the braiding evolves only the Majoranas that are exchanged. Finally, due to the fermion-parity conservation for the condensate, in the braiding, the qubit modes corresponding to f1 and f˜1 are decoupled and their evolution can be derived independently. By a detailed derivative, we show that after braiding, the topological qubit states evolve according to (see the Supplemental Material [43]) |11˜⟩final=|11˜⟩initial,|10˜⟩final=i|10˜⟩initial,|01˜⟩final=-i|01˜⟩initial,|00˜⟩final=|00˜⟩initial.We therefore obtain the braiding matrix by U12(T,T˜)=exp[(π/4)γ1γ2]exp[(π/4)γ˜1γ˜2], which is time-reversal invariant. This braiding operator can also be obtained in a different generic way. Because of the time-reversal symmetry, we have U12(T,T˜)|10˜⟩=eiϕ|10˜⟩ and U12(T,T˜)|01˜⟩=e-iϕ|01˜⟩, while |11˜⟩ and |00˜⟩ are unchanged. For the present DIII-class superconductor, one can show that U124=1, which is followed by ϕ=0, π/2, or π. On the other hand, in the special situation that the DIII-class topological superconductor is composed of two decoupled copies of 1D chiral p-wave superconductors, it can be derived straightforwardly that ϕ=π/2 [43]. Furthermore, the Hamiltonian of a generic DIII-class Majorana wire can be deformed continuously from the special one of two decoupled copies, while the phase factor ϕ only takes discrete values and cannot vary continuously. Therefore, the result ϕ=π/2 is valid for the generic case, completing the proof. Note that the oppositely handed braiding process of U12 reads U12†(T,T˜)=exp[-(π/4)γ1γ2)exp(-(π/4)γ˜1γ˜2], which describes a process in which one first transports γ2 and γ˜2 to the end of the vertical wire, then transports the two modes γ1 and γ˜1 to the right-hand end, and finally, transports the two modes γ2 and γ˜2 to the left-hand end of the horizontal wire. The braiding matrix U12 reflects that the two Majorana pairs γ1, γ˜1 and γ2, γ˜2 are braided independently, which leads to the symmetry-protected non-Abelian statistics as presented below.

We consider two DIII-class wire segments with eight Majorana modes γ1,…,4 and γ˜1,…,4 [Fig. 6(a)] that define four complex fermion modes by f1=12(γ1+iγ2), f2=12(γ3+iγ4), and f˜1,2=T-1f1,2T. The Hilbert space of the four complex fermions is spanned by 16 qubit states |n1n˜1⟩L|n2n˜2⟩R (n1,2, n˜1,2=0, 1), where L (R) represents the left (right) nanowire segment. If the initial state of the system is |00˜⟩L|00˜⟩R, for instance, by braiding the two pairs of Majoranas γ2, γ˜2 and γ3, γ˜3, we get straightforwardly U23(T,T˜)|00˜⟩L|00˜⟩R=12(|00˜⟩L|00˜⟩R+|11˜⟩L|11˜⟩R+i|10˜⟩L|10˜⟩R-i|01˜⟩L|01˜⟩R).Here, the operator U23(T,T˜)=exp(π4γ2γ3)exp(π4γ˜2γ˜3) takes the same form as U12, while γ2, γ˜2 and γ3, γ˜3 are separated by a trivial region. Actually, the braiding operator is independent of the existence of γ1,4 and γ˜1,4. We can therefore fuse them by connecting the left-hand end of the L wire and the right-hand end of the R wire, and then U23(T,T˜) is simply equivalent to exchanging two Majorana pairs for a single wire segment [15]. It is interesting that the above state is generically a four-particle-entangled state, which shows the natural advantage in generating a multiparticle-entangled state using DIII-class topological superconductors. Furthermore, a full braiding, i.e., braiding twice γ2, γ˜2 and γ3, γ˜3, yields the final state |11˜⟩L|11˜⟩R, which distinguishes from the initial state in that each copy of the p-wave superconductor changes fermion parity. After braiding four times the two pairs of Majoranas, the ground state returns to the original state. On the other hand, it is also straightforward to verify that U12U23≠U23U12, implying the noncommutability of the braiding processes. These results demonstrate the non-Abelian statistics obeyed by Majorana doublets.

610.1103/PhysRevX.4.021018.f6

Symmetry-protected non-Abelian statistics in a DIII-class 1D topological superconductor. (a) Majorana end modes γ2, γ˜2 and γ3, γ˜3 are braided through similar processes to those shown in Fig. 5. (b) Braiding Majorana modes in a DIII-class superconductor are equivalent to two independent processes of exchanging γ2, γ3 and γ˜2, γ˜3, respectively. Majorana modes of one time-reversed sector do not experience the branch cuts of Majoranas belonging to another sector. In the depicted process, γ2 (γ˜2) crosses only the branch cut of γ3 (γ˜3) and therefore acquires a minus sign after braiding. (c) In contrast, if braiding two Majorana pairs in a chiral superconductor, for the depicted process, γ2 (γ2′) crosses the branch cuts of both γ3 and γ3′, and then no sign change occurs for the Majorana operators after braiding [3]. Therefore, by braiding twice two Majorana pairs the system always returns to the original state.

From the above discussion, we find that in the braiding, the Majorana modes γj are unaffected by their time-reversal partners γ˜j, which is an essential difference from the situation in exchanging two pairs of Majoranas in a chiral superconductor and makes the braiding operator in the TRI topological superconductor nontrivial. This property can be pictorialized by assigning branch cuts for the Majorana modes braided through the junction [15], as illustrated in Figs. 6(b) and 6(c). Majorana modes of one time-reversed sector do not experience the branch cuts of Majoranas belonging to another sector. When exchanging Majorana modes γ2, γ3 and γ˜2, γ˜3 in the DIII-class superconductor, γ2 (γ˜2) crosses only the branch cut of γ3 (γ˜3) and therefore acquires a minus sign after braiding. In contrast, if braiding two Majorana pairs in a chiral superconductor, for the process in Fig. 6(c), γ2 (γ2′) crosses the branch cuts of both γ3 and γ3′, and then no sign change occurs for the Majorana operators after braiding [3]. Therefore, a full braiding of two Majorana pairs always returns to the original state.

It is worthwhile to note that realizing a DIII-class superconductor applies no external magnetic field, which might be helpful to construct a realistic Majorana network to implement braiding operations. In comparison, for the chiral topological superconductor observed in a spin-orbit-coupled semiconductor nanowire using the s-wave superconducting proximity effect [20–22], the external magnetic field should be applied perpendicular to the spin-quantization axis by spin-orbit interaction, driving optimally the nanowire into the topological phase [20,22,44]. It is shown that for a network formed by multiple nanowire segments, such an optimal condition cannot be reached for all segments without inducing detrimental orbital effects, which creates further experimental challenges in braiding Majoranas [44]. It is clear that such an intrinsic difficulty is absent in the present DIII-class TRI topological superconductor, and one might have more flexibility in constructing 2D and even 3D Majorana networks for topological quantum computation.

JOSEPHSON EFFECT IN DIII-CLASS TOPOLOGICAL SUPERCONDUCTOR

It is important to study how to detect the topological qubit states in a DIII-class Majorana quantum wire. The ground states of a single DIII-class Majorana quantum wire include two even- (|00˜⟩ and |11˜⟩) and two odd- (|01˜⟩ and |10˜⟩) parity eigenstates. In a chiral topological superconductor, the ground states of the same fermion parity are not distinguishable. On other hand, in the generic case, the two different time-reversal sectors do not correspond to different measurable good quantum numbers. Therefore, the two qubit states with the same total fermion parity, e.g., |01˜⟩ and |10˜⟩, cannot be distinguished via direct quantum-number measurements. However, according to the fermion-parity conservation shown in Sec. III A for each time-reversed sector of the condensate, in a 1D TRI topological superconductor, the two even- or odd-parity states are decoupled due to time-reversal symmetry, implying that such two states should be distinguishable. We show in this section that all four topological qubit states can be measured by the Josephson effect in DIII-class topological superconductors.

We consider a Josephson junction illustrated in Fig. 7(a) formed by DIII-class superconductor. As derived in the Appendix, the effective coupling Hamiltonian of the Josephson junction is given by Heff(ϕ)=iΓ0cosϕ2(γLγR-γ˜Lγ˜R)+iΓ1sinϕ(γLγ˜L-γRγ˜R),where ϕ is the phase difference across the junction, and L (R) represents the left-hand (right-hand) lead of the junction. The Γ0 term in Heff represents the first-order direct coupling between Majorana fermions at different junction leads. It can be seen that the direct coupling term is of 4π periodicity, which can be understood in the following way. When the phase difference across the junction advances 2π, the Cooper-pair wave function changes 2π across the junction, while for single-electron operators, the phase varies only π. This property implies that the periodic coupling coefficients also change the π phase and thus reverse sign, leading to the 4π periodicity of the direct coupling term. The Γ1 term results from the second-order perturbation of the tunneling process, and this term vanishes if the s-wave pairing Δs=0. The reason is because the couplings such as iγjγ˜j (j=L, R) break time-reversal symmetry, while the direct coupling between γj and γ˜j does not experience the phase difference across the junction and should preserve time-reversal symmetry. Actually, a uniform pairing phase in one end of the junction can be removed by a constant gauge transformation. Therefore, the coupling between Majorana fermions at the same end can only be induced by electron tunneling and the minimum requirement is to consider the second-order tunneling process. In the second-order perturbation, γL and γ˜L (γR and γ˜R) couple to the electron modes cR and c˜R in the right-hand end (cL and c˜L in the left-hand end), respectively. When a nonzero s-wave pairing is present in the nanowires, the electrons cL(R) and c˜L(R) form a Cooper pair and condense. This process leads to the effective coupling between Majorana zero modes localized at the same end, with the coupling strength proportional to the s-wave order parameter. Finally, note that the system restores time-reversal symmetry at ϕ=mπ, which explains why the Γ1 term is proportional to sinϕ and has 2π periodicity. All these properties have been confirmed with numerical results.

710.1103/PhysRevX.4.021018.f7

Josephson measurement of the topological qubit states in a DIII-class 1D topological superconductor. (a) The sketch of a Josephson junction with phase difference ϕ. (b) The single-particle Andreev bound-state spectra versus the phase difference ϕ. (c) The energy spectra of the four qubit states |n1n˜1⟩ (n1, n˜1=0, 1) according to the results in (b). (d) The Josephson currents (in units of 2eΔp/ℏ) for different topological qubit states. Parameters used in the numerical calculation are taken such that Δp=1.0  meV, Δs=0.25  meV, ESO=0.1  meV, and the width of the junction d=0.5ξ, and in the middle trivial (gray) region of the junction, the chemical potential is set to be at the band bottom.

Redefining the Majorana bases by γ1′=γL+γ˜R and γ2′=γR+γ˜L, we recast the above Hamiltonian into Heff=i(Γ0cosϕ/2+Γ1sinϕ)γ1′γ2′-i(Γ0cosϕ/2-Γ1sinϕ)γ˜1′γ˜2′. The Andreev bound-state spectra are obtained straight forwardly by Enf′,n˜f′(ϕ)=(Γ0cosϕ2+Γ1sinϕ)(2nf′-1)+(Γ0cosϕ2-Γ1sinϕ)(2nf˜′-1),which is shown numerically in Figs. 7(b) and 7(c). Here, nf′,f˜′ are complex fermion-number operators for f′ and f˜′ modes, respectively. The Josephson currents are obtained by the slope of the Andreev bound-state spectra. In particular, we have that the Josephson currents Jϕeven=±eℏΓ0sinϕ2 for the even-parity states |00˜⟩ and |11˜⟩ and Jϕodd=±eℏΓ1cosϕ for the odd-parity states |01˜⟩ and |10˜⟩, respectively [Fig. 7(d)].

It is remarkable that the currents for odd-parity states are of 2π periodicity, half of those for even-parity states [Fig. 7(d)]. This result reflects that Jϕeven is contributed from the direct Majorana coupling induced by first-order single-electron tunneling [4], while Jϕodd is a consequence of the second-order tunneling process, which corresponds to the Cooper-pair tunneling. This nontrivial property is essentially different from the the Josephson physics with multiple Majorana end modes studied by Sticlet et al. in the BDI-class Majorana chains [45], where a multicopy version of the fractional Josephson effect with 4π periodicity is investigated. The reason is because in a BDI-class topological superconductor, the time-reversal-symmetry operator T2=1 and the different copies of the superconductor are not related by time-reversal symmetry (nor by any other symmetry), while in the DIII-class topological superconductor, the two copies are related by T symmetry. The present result is also consistent with the fact that the time-reversal symmetry is restored with |01˜⟩ and |10˜⟩, forming a Kramers’s doublet at ϕ=mπ, which necessitates the 2π periodicity in their spectra. Furthermore, the two qubit states with the same total parity (e.g., |00˜⟩ and 11˜⟩) are distinguished by the direction of the currents. The qualitative difference in the Josephson currents implies that the four topological qubit states can be measured in the experiment.

CONCLUSIONS

In summary, we have shown that Majorana doublets obtained in the DIII-class 1D topological superconductors obey non-Abelian statistics, due to the protection of time-reversal symmetry. The key results are that the fermion parity is conserved for each copy of the Z2 TRI topological superconductor, and the exchange of Majorana end modes can generically reduce to two independent processes of braiding Majoranas of two different copies, respectively. These results lead to the symmetry-protected non-Abelian statistics for the Majorana doublets, and the braiding statistics are protected by time-reversal symmetry. Furthermore, we unveiled an intriguing phenomenon in the Josephson effect, that the periodicity of Josephson currents depends on the fermion parity of the 1D TRI topological superconductors. We found that this effect can provide direct measurements of the topological qubit states in the DIII-class Majorana quantum wires. Our results will motivate further studies in both theory and experiments on the braiding statistics and nontrivial Josephson effects in the wide classes of symmetry-protected topological superconductors.

ACKNOWLEDGMENTS

We appreciate the very helpful discussions with P. A. Lee, J. Alicea, L. Fu, Z.-X. Liu, A. Potter, Z.-C. Gu, M. Cheng, C. Wang, and X. G. Wen. The authors thank HKRGC for support through DAG12SC01, Grants No. 602813, No. 605512, and No. HKUST3/CRF/13G.

APPENDIXFermi-parity conservation

We consider a single Majorana quantum wire, which hosts four Majorana end modes denoted by γ1,2 and γ˜1,2 and is transformed via T-1γjT=γ˜j and T-1γ˜jT=-γj. With the four Majorana states, we can define two nonlocal complex fermions by f1=12(γ1+iγ2) and f˜1=12(γ˜1-iγ˜2), which germinate four topological qubit states |n1n˜1⟩ with n1, n˜1=0, 1. Note that at zero temperature, the superconductor stays in the ground-state manifold. The proof of fermion-parity conservation for each sector of the condensate is equivalent to showing that the four topological qubit states |n1n˜1⟩ are generically decoupled from each other in the presence of TRI perturbations.

Note that the coupling between the Majorana modes localized at the same end of the nanowire γj and γ˜j breaks time-reversal symmetry. The coupling Hamiltonian in terms of Majorana end modes should take the following generic TRI form: V(λ)=iE1(λ)(γ1γ2-γ˜1γ˜2)+iE2(λ)(γ1γ˜2-γ2γ˜1),where we assume that the couplings coefficients E1,2(λ) depend on an experimentally manipulatable parameter λ (e.g., the bulk gap in the nanowire or the distance between the Majorana modes). The above Hamiltonian can be rewritten in the block-diagonal form with new Majorana bases such that V(λ)=iE(λ)[γ(1)γ(2)-γ˜(1)γ˜(2)],where γ(1)=γ1, γ˜(1)=γ˜1, γ(2)=sinθγ2+cosθγ˜2, γ˜(2)=sinθγ˜2-cosθγ2, and E=E12+E22. The mixing angle θ is defined via tanθ=E1/E2. The complex fermions f(1) and f˜(1) in the eigenbasis are then defined by f(1)=12[γ(1)+iγ(2)],f˜(1)=12[γ˜(1)-iγ˜(2)].It is easy to know that the even-parity eigenstates |00˜⟩ and |11˜⟩ germinated by f(1) and f˜(1) acquire an energy splitting 2E(λ), while the odd-parity states |01˜⟩ and |10˜⟩ are still degenerate due to time-reversal symmetry. To prove the fermion-parity conservation for each sector, we need to confirm that all four topological qubit states |n1n˜1⟩ can evolve adiabatically when the coupling Hamiltonian V(λ) varies with the parameter λ. Since |10˜⟩ and |01˜⟩ form a Kramers’s doublet, the transition between them is forbidden by the time-reversal symmetry. Therefore, we only need to consider the adiabatic condition for the two even-parity states. The fermion-parity conservation for each sector is guaranteed when the following adiabatic condition is satisfied in the manipulation: |⟨11˜|∂λ∂t∂∂λ|00˜⟩|≪2|E(λ)|.It should be noted that the adiabatic condition needs to be justified only in the presence of finite couplings. When E(λ)→0, the couplings between Majorana end modes vanish and then all the topological qubit states are automatically decoupled from each other. One can verify that ∂f(1)∂λ=i2∂θ∂λ(cosθγ2-sinθγ˜2)=12∂θ∂λ(f˜(1)†-f˜(1)),∂f˜(1)∂λ=-i2∂θ∂λ(cosθγ˜2+sinθγ2)=-12∂θ∂λ(f(1)†-f(1)).With some calculation, one can show that in the above formulas, the derivatives of the bases γj, γ˜j with respect to λ will not contribute to the left-hand side of Eq. (A4) and are therefore neglected. The condition (A4) then reads |∂λ∂t∂θ∂λ|≪4|E(λ)|.We show below that the above condition is generically satisfied in the realistic materials.

With the proximity-induced p-wave and s-wave superconducting pairings, the effective tight-binding Hamiltonian in the nanowire can be generically written as Hwireeff=∑⟨i,j⟩,σtijciσ†cjσ+∑⟨i,j⟩(tijSOci↑†cj↓+H.c.)+∑⟨i,j⟩(Δijpci↑cj↑+Δijp*ci↓cj↓+H.c.)+∑j(Δscj↑cj↓+H.c.)-μ∑j,σnjσ+∑j,σVjdisnjσ,where the hopping coefficients and the chemical potential are generically renormalized by the proximity effect. Without loss of generality, in the above Hamiltonian, we have taken into account the spin-orbit interaction described by the tijSO term and the random on-site disorder potential Vjdis with ⟨Vjdis⟩=0. For the case with uniform pairing orders, the parameters Δs and Δp can be taken as real. On the other hand, for the present 1D system, one can verify that the phases in the (spin-orbit) hopping coefficients can always be absorbed into electron operators. Therefore, in the following study, we consider that all the parameters in Hwireeff are real numbers.

In the topological regime, at each end of the wire, we obtain two Majorana zero modes that are transformed to each other by a time-reversal operator. In terms of the electron operators, these bound modes take the form γ1=∑j[u↑(1)(xj)c↑(xj)+u↓(1)(xj)c↓(xj)+u↑(1)*(xj)c↑†(xj)+u↓(1)*(xj)c↓†(xj)],γ˜1=∑j[u↑(1)*(xj)c↓(xj)-u↓(1)*(xj)c↑(xj)+u↑(1)(xj)c↓†(xj)-u↓(1)(xj)c↑†(xj)],γ2=i∑j[u↑(2)(xj)c↑(xj)+u↓(2)(xj)c↓(xj)-u↑(2)*(xj)c↑†(xj)-u↓(2)*(xj)c↓†(xj)],γ˜2=i∑j[u↑(2)*(xj)c↓(xj)-u↓(2)*(xj)c↑(xj)-u↑(2)(xj)c↓†(xj)+u↓(2)(xj)c↑†(xj)].Note that the coefficients in Hwireeff are real, and we have that u↑,↓(1,2)=u↑,↓(1,2)*. The coupling energies between the Majorana modes at the left (γ1, γ˜1) and right (γ2, γ˜2) ends are calculated by E1=i⟨γ1|Hwireeff|γ2⟩=-i⟨γ˜1|Hwireeff|γ˜2⟩ and E2=i⟨γ1|Hwireeff|γ˜2⟩=-i⟨γ˜1|Hwireeff|γ2⟩. Using the relations cj↑≃u↑(1)(xj)γ1-u↓(1)(xj)γ˜1-iu↑(2)(xj)γ2+iu↓(2)(xj)γ˜2,cj↓≃u↓(1)(xj)γ1+u↑(1)(xj)γ˜1-iu↓(2)(xj)γ2-iu↑(2)(xj)γ˜2,we obtain that E1=∑⟨i,j⟩σtijuσ(1)(xi)uσ(2)(xj)+∑⟨i,j⟩σΔijpuσ(1)(xi)uσ(2)(xj)+∑⟨i,j⟩tijSO[u↑(1)(xi)u↓(2)(xj)+u↓(1)(xj)u↑(2)(xi)]+∑jΔs[u↑(1)(xj)u↓(2)(xj)-u↓(1)(xj)u↑(2)(xj)]+∑j,σVjdisuσ(1)(xj)uσ(2)(xj),E2=∑⟨i,j⟩tij[u↑(1)(xi)u↓(2)(xj)-u↓(1)(xi)u↑(2)(xj)]+∑⟨i,j⟩Δijp[u↑(1)(xi)u↓(2)(xj)-u↓(1)(xi)u↑(2)(xj)]+∑⟨i,j⟩tijSO[u↑(1)(xi)u↑(2)(xj)-u↓(1)(xj)u↓(2)(xi)]+∑jΔs[u↑(1)(xj)u↑(2)(xj)+u↓(1)(xj)u↓(2)(xj)]+∑j,σVjdis[u↑(1)(xj)u↓(2)(xj)-u↓(1)(xj)u↑(2)(xj)].Note that tijSO=-tjiSO due to time-reversal symmetry, and for a uniform nanowire, we have that ∑⟨i,j⟩u↑(1)(xi)u↓(2)(xj)=∑⟨i,j⟩u↑(2)(xi)u↓(1)(xj) and ∑ju↑(1)(xj)u↓(2)(xj)=∑ju↑(2)(xj)u↓(1)(xj). With these properties, we find that in E1, the terms corresponding to tijSO and Δs vanish, while in E2, the terms for tij, Δp, and Vjdis vanish. We then have E1=∑⟨i,j⟩σtijuσ(1)(xi)uσ(2)(xj)+∑⟨i,j⟩σΔijpuσ(1)(xi)uσ(2)(xj)+∑j,σVjdisuσ(1)(xj)uσ(2)(xj),E2=∑⟨i,j⟩σtijSOuσ(1)(xi)uσ(2)(xj)+∑j,σΔsuσ(1)(xj)uσ(2)(xj).

The wave functions of Majorana bound modes decay exponentially as a function of the distance from the end of the nanowire, multiplying by an oscillatory function with the oscillating period equal to the Fermi wavelength in the nanowire. This property implies that uσ(1)∝sin(kFx)e-x/ξ and uσ(2)∝sine-(L-x)/ξ, where ξ is the effective coherence length of the wire. In the realistic material, we consider that the chemical potential in the nanowire is far below the half-filling condition, and thus kFa≪1. In this way, we have uσ(1)(xj)uσ(2)(xj)≈uσ(1)(xj)uσ(2)(xj±1)e∓a/ξ. Furthermore, the coherence length (in the order of 1.0  μm) is typically much larger than the lattice constant ξ≫a (a∼0.5  nm), and we can further approximate that uσ(1)(xj)uσ(2)(xj)≈uσ(1)(xj)uσ(2)(xj±1). Bearing this result in mind, we get E1=∑⟨i,j⟩σtijuσ(1)(xi)uσ(2)(xj)+∑⟨i,j⟩σΔijpuσ(1)(xi)uσ(2)(xj)+∑j,σVjdisuσ(1)(xj)uσ(2)(xj),E2=∑⟨i,j⟩σtijSOuσ(1)(xi)uσ(2)(xj)+∑⟨i,j⟩σΔsuσ(1)(xi)uσ(2)(xj).The spin-orbit hopping coefficient tijSO=-tjiSO and the p-wave pairing Δijp=-Δjip are staggered parameters. In the limit that kFa≪1 and ξ≫a, the summation for such two terms in E1 and E2 also turns out to be 0. On the other hand, the spin-conserved hopping is a constant, and we denote tij=tji=t. Finally, if the the random potential Vjdis with ⟨Vjdis⟩=0 is distributed homogeneously in the nanowire, we expect that the last term in E1 gives V0∑j,σuσ(1)(xj)uσ(2)(xj), with the constant factor V0 depending on the specific disorder profile and much less than the amplitude of the disorder potential. The couplings E1,2 become E1≃(t+V0)∑⟨i,j⟩σuσ(1)(xi)uσ(2)(xj),E2≃Δs∑⟨i,j⟩σuσ(1)(xi)uσ(2)(xj).From the above result, we find that E2/E1≈Δs/(t+V0), which is consistent with the fact that when Δs=0, the original Hamiltonian (A8) can be block diagonalized, and then E2≡0. These results show that in the realistic nanowire materials, while the magnitudes of E1,2(λ) depend on λ, which determines the overlapping between the wave functions of Majorana bound modes at the left and right ends, their ratio is nearly a constant. Therefore, we always have ∂λθ≈0,which validates the adiabatic condition. The results in Eqs. (A21) and (A22) can be interpreted by an intuitive physical picture. Being proportional to the overlapping between the wave functions of Majorana bound modes at different ends, the coupling coefficients E1,2 are exponential decaying functions of the nanowire length. Since the Majorana modes γj and γ˜j are connected by a T transformation, their wave functions have exactly the same spatial profile, which leads to the same exponential form for the coefficients E1,2(λ)=α1,2(λ)e-d/ξ, with d the distance between the left and right Majorana end modes. The prefactors αj(λ) depend on the local couplings, i.e., the hopping coefficients and pairings between electrons belonging to the same (for j=1) or different (for j=2) sectors of the time-reversal partners. For the case with constant and homogeneous local couplings, we have that their ratio α1/α2 is proportional to the ratio of couplings between electrons of the same and different sectors and is nearly a constant, justifying the adiabatic condition. The above derivative is clearly confirmed with numerical results in the realistic systems with the presence of random on-site disorder scattering, as shown in Fig. 8.

810.1103/PhysRevX.4.021018.f8

Adiabatic condition and fermion-parity conservation for each sector of time-reversal partners in the presence of disorder scattering. The energy splitting E between |00˜⟩ and |11˜⟩ (red curves) and the ratio R˜ (blue curves) versus (a) the bulk gap that varies by tuning the chemical potential and (b) the distance between the Majorana end modes. In the numerical simulation, the random on-site disorder potential is considered, with the potential amplitude Vdis∼1.0  meV. Other parameters in the nanowire are taken such that Δp=1.0  meV, Δs=0.5  meV, and ESO=0.1   meV. The coupling energy E is tuned from 0 to 1.0 meV within the time 1.0  μs.

It is worthwhile to note that for a fixed parameter λ, the physics of the fermion-parity conservation for each sector can be easily understood in another way. For a DIII-class topological superconductor, the helical p-wave pairings occur between two electrons belonging to the same sector of the time-reversal partners. While the change by 1 in the fermion number of each sector conserves the total fermion parity of the system, it changes fermion parity for each sector and thus breaks a p-wave Cooper pair in each sector. This process costs finite energy and is completely suppressed by the p-wave-pairing gap at zero temperature, if Δp dominates over Δs and the time-reversal symmetry is not broken. The previous study in this section further proves this conservation law when the couplings between Majorana end modes are allowed and adjusted adiabatically. To simplify the notations in the further discussion, we relabel the block-diagonal Majorana modes γj(1), γ˜j(1) as γj, γ˜j. Accordingly, the diagonal complex fermion modes are redefined as fj, f˜j.

Josephson effect in the DIII-class 1D topological superconductor

Now, we study how to measure the topological qubit states with the Josephson effect. It has been predicted that in the chiral 1D topological superconductor, the Josephson current has 4π periodicity [4], and the topological qubit states for a single wire |0⟩ and |1⟩ can be read out from the direction of Josephson currents in the junction [8,15]. In this section, we predict a novel phenomenon in the Josephson effect of the DIII-class 1D topological superconductor, which provides a feasible scheme to read out the topological qubit states in a TRI Majorana quantum wire.

Effective coupling Hamiltonian

We consider a Josephson junction formed by two Majorana nanowire ends with a phase difference ϕ=ϕR-ϕL, as illustrated in Fig. 9(a), and derive the effective coupling Hamiltonian for the Majorana zero modes localized at the left (L) and right (R) ends. The electron tunneling process in the junction is described by HT=ϒcL,N†cR,1+ϒc˜L,N†c˜R,1+H.c.,where cL,N, c˜L,N and cR,1, c˜R,1 represent the electron operators for the Nth site at the left end and the first site at the right end of the junction, respectively, and ϒ is the tunneling coefficient across the junction. The Majorana end modes can be generically expanded in terms of electron operators γL=∑j(uL,jcL,j+uL,j*cL,j†),γ˜L=∑j(u˜L,jc˜L,j+u˜L,j*c˜L,j†),γR=∑j(uR,jcR,j+uR,j*cR,j†),γ˜R=∑j(u˜R,jc˜R,j+u˜R,j*c˜R,j†),where uL(R),j=u˜L(R),j* if ϕL(R)=0. Note that cj and c˜j represent electron operators of a general time-reversal pair at the jth site, not necessarily corresponding to spin-up and spin-down, since the spin is not a good quantum number when spin-orbit coupling and s-wave order are present. From the above formulas, we can solve the electron operators in terms of Majorana and nonzero-energy Bogoliubov quasiparticle operators. Reexpressing the Bogoliubov quasiparticles in terms of electron operators, we can interpret cL,N, c˜L,N and cR,1, c˜R,1 by cL,N=uL,N*γL-∑j=1NaL,jcL,j-∑j=1NbL,j*cL,j†,c˜L,N=u˜L,N*γL-∑j=1Na˜L,jc˜L,j-∑j=1Nb˜L,j*c˜L,j†,cR,1=uR,1*γR-∑j=1NaR,jcR,j-∑j=1NbR,j*cR,j†,c˜R,1=u˜R,1*γR-∑j=1Na˜R,jc˜R,j-∑j=1Nb˜R,j*c˜R,j†,with a constant normalization factor neglected. Here, aL(R),j, a˜L(R),j and bL(R),j, b˜L(R),j are expansion coefficients, originated from the quasiparticle operators other than the corresponding Majorana mode. Substituting these results into the tunneling Hamiltonian HT yields that HT=ϒ(uL,NγL-∑j=1NaL,j*cL,j†-∑j=1NbL,jcL,j)(uR,1*γR-∑j=1NaR,jcR,j-∑j=1NbR,j*cR,j†)+ϒ(u˜L,Nγ˜L-∑j=1Na˜L,j*c˜L,j†-∑j=1Nb˜L,jc˜L,j)(u˜R,1*γ˜R-∑j=1Na˜R,jc˜R,j-∑j=1Nb˜R,j*c˜R,j†)+H.c.≈H(0)+H(1),where H(0)=ϒuL,NuR,1*γLγR+ϒu˜L,Nu˜R,1*γ˜Lγ˜R+H.c.,H(1)=-ϒuL,NγL(∑j=1NaR,jcR,j+∑j=1NbR,j*cR,j†)-ϒuR,1γR(∑j=1NaL,jcL,j+∑j=1NbL,j*cL,j†)-ϒu˜L,Nγ˜L(∑j=1Na˜R,jc˜R,j+∑j=1Nb˜R,j*c˜R,j†)-ϒu˜R,1γ˜R(∑j=1Na˜L,jc˜L,j+∑j=1Nb˜L,j*c˜L,j†)+H.c.In the second equation of the formula (A31), we have neglected the higher-order irrelevant terms. The term H(0) represents the direct coupling between Majorana modes at different junction ends, which gives the first term of the effective Hamiltonian Heff in the main text. This result can be seen by noticing that uL,N=i|uL,N|eiϕL/2,uR,1=|uR,1|eiϕR/2,u˜L,N=-i|u˜L,N|eiϕL/2,u˜R,1=|u˜R,1|eiϕR/2,with which we can recast H(0) into H(0)=iΓ0cosϕ2(γLγR-γ˜Lγ˜R),Γ0=2ϒ|uL,NuR,1|.

910.1103/PhysRevX.4.021018.f9

Josephson effect in a DIII-class 1D topological superconductor with the inclusion of random disorder scattering. (a) The sketch of a Josephson junction with phase difference ϕ. (b) The single-particle Andreev bound-state spectra versus the phase difference ϕ. (c) The energy spectra of the four qubit states |n1n˜1⟩ (n1, n˜1=0, 1) according to the results in (b). (d) The Josephson currents (in units of 2eΔp/ℏ) for different topological qubit states. In the numerical simulation, the amplitude of the random on-site disorder potential is set as Vdis∼1.0  meV. Other parameters are taken such that Δp=1.0  meV, Δs=0.25  meV, ESO=0.1  meV, and the width of the junction d=0.75ξ, and in the middle trivial (gray) region of the junction, the chemical potential is set to be at the band bottom.

On the other hand, for H(1), we shall calculate up to the second-order perturbation, which is responsible for the second term of Heff in the main text. From H(1), we know that Majorana modes at one end (e.g., the left end) also couple to the electron modes at another end (the right end). In the second-order perturbation, γL and γ˜L (γR and γ˜R) couple to cR and c˜R (cL and c˜L), respectively. When a nonzero s-wave pairing is present in the quantum wires, the electrons cL(R) and c˜L(R) form a Cooper pair and condense. This process leads to an effective coupling between Majorana zero modes localized at the same end. Therefore, up to the second-order perturbation in the tunneling process, we obtain that Heff(1)=12ϒ2uL,Nu˜L,NγLγ˜L[∑j=1NaR,ja˜R,j∫dτ⟨TτcR,j(τ)c˜R,j(0)⟩+∑j=1NbR,j*b˜R,j*∫dτ⟨TτcR,j†(τ)c˜R,j†(0)⟩]+12ϒ2uR,1u˜R,1γRγ˜R[∑j=1NaL,ja˜L,j∫dτ⟨TτcL,j(τ)c˜L,j(0)⟩+∑j=1NbL,j*b˜L,j*∫dτ⟨cL,j†(τ)c˜L,j†(0)⟩]+H.c.Here, ∫dτ⟨Tτ⋯⟩ represents a time-ordered integral. Assuming that the superconducting pairings are uniform in the Majorana nanowires, we obtain from the above formula that Heff(1)=12ϒ2uL,Nu˜L,NγLγ˜L[∑j=1NaR,ja˜R,j∑kΔs,R*ER2(Δs,R;Δp,R;k)-∑j=1NbR,j*b˜R,j*∑kΔs,RER2(Δs,R;Δp,R;k)]+12ϒ2uR,1u˜R,1γRγ˜R[∑j=1NaL,ja˜L,j∑kΔs,L*EL2(Δs,L;Δp,L;k)-∑j=1NbL,j*b˜L,j*∑kΔs,LEL2(Δs,L;Δp,L;k)]+H.c.=iϒL(ϕ)γLγ˜L+iϒ˜R(ϕ)γRγ˜R,with EL(R)(Δs,R;Δp,R;k) the bulk excitation spectra in the left (for L) and right (for R) wires of the junction, respectively. The coupling coefficients read ϒL(R)(ϕ)=-i12ϒ2uL(R),N/1u˜L(R),N/1[∑j=1NaR(L),ja˜R(L),j∑kΔs,R(L)*ER(L)2(Δs,R(L);Δp,R(L);k)-∑j=1NbR(L),j*b˜R(L),j*∑kΔs,R(L)ER(L)2(Δs,R(L);Δp,R(L);k)]-c.c.With the relations obtained in Eqs. (A27)–(A30), we have that aR(L),ja˜R(L),j=|aR(L),ja˜R(L),j| and bR(L),jb˜R(L),j=|bR(L),jb˜R(L),j|ei2ϕR(L). Together with the results in Eq. (A32), we can simplify ϒL(R)(ϕ) to be ϒL(ϕ)=-iΓ1eiϕ-c.c.=Γ1sinϕ,ϒR(ϕ)=-iΓ1e-iϕ-c.c.=-Γ1sinϕ,and the effective coupling Hamiltonian for Majorana fermions at the same end takes the following form: Heff(1)=iΓ1sinϕ(γLγ˜L-γRγ˜R).The coupling constant Γ1 is calculated by Γ1=ϒ2|uL,Nu˜L,N|[∑j=1N|aR,ja˜R,j|∑k|Δs,R|ER2(Δs,R;Δp,R;k)-∑j=1N|bR,j*b˜R,j*|∑k|Δs,R|ER2(Δs,R;Δp,R;k)].We have assumed the uniformity of the parameters in the left and right wires of the junction such that |Δs,L|=|Δs,R| and |Δp,L|=|Δp,R|, and therefore |uL,Nu˜L,N|=|uR,1u˜R,1| and EL(Δs,L;Δp,L;k)=ER(Δs,R;Δp,R;k). We note that this condition is typically satisfied in the realistic systems. It is clear that the Γ1 term vanishes when the s-wave pairing Δs,L(R) is absent in the wires.

To this end, we combine H(0) and Heff(1) to finally reach the effective Hamiltonian for a Josephson junction formed by DIII-class topological superconductors that Heff(ϕ)=iΓ0cosϕ2(γLγR-γ˜Lγ˜R)+iΓ1sinϕ(γLγ˜L-γRγ˜R).Note that if treating ϕ as a fixed parameter, the Γ1 term in the above formula breaks time-reversal symmetry. This property reflects that the leading-order contribution to the coupling between Majoranas at the same end (iγjγ˜j) should come from the second-order perturbation in the tunneling process. Actually, the direct coupling between γj and γ˜j does not experience the phase difference across the junction and should preserve time-reversal symmetry. The reason is because a uniform pairing phase in one end of the junction can be removed by a constant gauge transformation. Therefore, the coupling between Majorana fermions at the same end can only be induced by electron tunneling across the junction, and the minimum requirement is to consider the second-order tunneling process. Furthermore, the system restores time-reversal symmetry at ϕ=mπ, which explains why the Γ1 term is proportional to sinϕ, and has 2π periodicity.

Josephson current

The Hamiltonian (A39) can be block diagonalized by a constant transformation in the Majorana bases that γ1′=γL+γ˜R, γ2′=γR+γ˜L, and γ˜1,2′=Tγ1,2′T-1, which sends Heff to be Heff=i(Γ0cosϕ/2+Γ1sinϕ)γ1′γ2′-i(Γ0cosϕ/2-Γ1sinϕ)γ˜1′γ˜2′. The Andreev bound-state spectra are obtained straightforwardly by Enf′,n˜f′(ϕ)=(Γ0cosϕ/2+Γ1sinϕ)(2nf′-1)+(Γ0cosϕ/2-Γ1sinϕ)(2nf˜′-1), which are doubly degenerate at ϕ=mπ, reflecting the time-reversal symmetry at these points. Here, nf′,f˜′ are complex fermion-number operators for the f′ and f˜′ modes, respectively. The Josephson current then reads Jϕ=(eΓ02ℏsinϕ2+eΓ12ℏcosϕ)(2nf′-1)+(eΓ02ℏsinϕ2-eΓ12ℏcosϕ)(2nf˜′-1).

From Eq. (A40), we find that for the even-parity states (|00˜⟩ and |11˜⟩), the Josephson currents Jϕeven=±(e/ℏ)Γ0sinϕ2, which are of 4π periodicity, while for the odd-parity states (|01˜⟩ and |10˜⟩), Jϕodd=±(e/ℏ)Γ1cosϕ exhibit 2π periodicity. The difference in the periodicity reflects different mechanisms for Jϕeven, odd. The currents Jϕeven are contributed from the Γ0 term in the effective coupling Hamiltonian, which is due to the direct coupling between Majorana modes at different ends of the junction. Therefore, the currents Jϕeven are a consequence of the single-electron tunneling process and have 4π periodicity. On the other hand, as contributed from the Γ1 term, the Josephson currents Jϕodd result from the second-order tunneling process, which corresponds to the Cooper-pair tunneling, therefore being of 2π periodicity. Furthermore, the currents Jϕodd are nonzero even for ϕ=0, which reflects the fact that the odd-parity states violate the time-reversal symmetry that even Heff preserves at ϕ=mπ.

The 4π periodicity of the Josephson currents for even-parity states can also be understood in the following way. When the phase difference across the junction advances 2π, the Cooper-pair wave function changes 2π across the junction, while for single-electron operators, the phase varies only π. Therefore, the coupling coefficients also change π phase and thus reverse sign, leading to the 4π periodicity of the direct coupling term. The generality of this argument implies that the 4π periodicity of Jϕeven is stable against the disorder scattering without breaking time-reversal symmetry. On the other hand, for odd-parity states, the twofold degeneracy at ϕ=mπ is protected by time-reversal symmetry, which shows that the qualitative properties of the Josephson currents Jϕodd are also stable against the TRI disorder scattering. The numerical results are shown in Fig. 9.

With the above results, we can have different strategies in the experiment to distinguish Jϕeven and Jϕodd. For instance, one can measure the periodicity of the Josephson currents or measure the currents at ϕ=π/2, where Jϕeven=±(e/2ℏ)Γ0 and Jϕodd=0. Furthermore, the two qubit states of the same total parity are distinguished by current directions. The qualitative difference in the Josephson measurements provides direct detection of the four topological qubit states.

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diff --git a/tests/data/aps/PhysRevX.4.021018_expected.yml b/tests/data/aps/PhysRevX.4.021018_expected.yml new file mode 100644 index 0000000..694a3d4 --- /dev/null +++ b/tests/data/aps/PhysRevX.4.021018_expected.yml @@ -0,0 +1,1322 @@ +abstract: The study of non-Abelian Majorana zero modes advances our understanding of + the fundamental physics in quantum matter and pushes the potential applications + of such exotic states to topological quantum computation. It has been shown that + in two-dimensional (2D) and 1D chiral superconductors, the isolated Majorana fermions + obey non-Abelian statistics. However, Majorana modes in a Z2 + time-reversal-invariant (TRI) topological superconductor come in pairs due to + Kramers’s theorem. Therefore, braiding operations in TRI superconductors always + exchange two pairs of Majoranas. In this work, we show interestingly that, due + to the protection of time-reversal symmetry, non-Abelian statistics can be obtained + in 1D TRI topological superconductors and may have advantages in applications + to topological quantum computation. Furthermore, we unveil an intriguing phenomenon + in the Josephson effect, that the periodicity of Josephson currents depends on + the fermion parity of the superconducting state. This effect provides direct measurements + of the topological qubit states in such 1D TRI superconductors. +artid: 021018 +publication_date: 2014-04-01 +year: 2014 +title: Non-Abelian Majorana Doublets in Time-Reversal-Invariant Topological Superconductors +journal_issue: "2" +journal_volume: "4" +publisher: American Physical Society +journal_title: Physical Review X +dois: +- doi: 10.1103/PhysRevX.4.021018 + material: publication +authors: +- raw_affiliations: + - source: American Physical Society + value: Department of Physics, Hong Kong University of Science and Technology, + Clear Water Bay, Hong Kong, China + - source: American Physical Society + value: Institute for Advanced Study, Hong Kong University of Science and + Technology, Clear Water Bay, Hong Kong, China + full_name: Liu, Xiong-Jun +- raw_affiliations: + - source: American Physical Society + value: Department of Physics, Hong Kong University of Science and Technology, + Clear Water Bay, Hong Kong, China + full_name: Wong, Chris L.M. +- raw_affiliations: + - source: American Physical Society + value: Department of Physics, Hong Kong University of Science and Technology, + Clear Water Bay, Hong Kong, China + full_name: Law, K.T. +copyright_holder: authors +copyright_statement: Published by the American Physical Society +copyright_year: 2014 +number_of_pages: 16 +document_type: article +material: publication +license_url: http://creativecommons.org/licenses/by/3.0/ +license_statement: Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +article_type: research-article +is_conference_paper: false +documents: +- key: PhysRevX.4.021018.xml + url: http://example.org/PhysRevX.4.021018.xml + source: American Physical Society + fulltext: true + hidden: true +references: +- reference: + publication_info: + journal_volume: B360 + page_start: '362' + year: 1991 + artid: '362' + journal_title: Nucl. Phys. + authors: + - inspire_role: author + full_name: Moore, G. + - inspire_role: author + full_name: Read, N. + title: + title: Nonabelions in the Fractional Quantum Hall Effect + label: '1' + dois: + - 10.1016/0550-3213(91)90407-O + raw_refs: + - source: American Physical Society + value: 1G. Moore and N. + Read, Nonabelions in the Fractional + Quantum Hall Effect, Nucl. Phys. 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B + authors: + - inspire_role: author + full_name: Liu, X.-J. + - inspire_role: author + full_name: Lobos, A.M. + title: + title: Manipulating Majorana Fermions in Quantum Nanowires with Broken Inversion + Symmetry + label: '44' + dois: + - 10.1103/PhysRevB.87.060504 + raw_refs: + - source: American Physical Society + value: 44X.-J. Liu and A. M. + Lobos, Manipulating Majorana Fermions + in Quantum Nanowires with Broken Inversion Symmetry, Phys. + Rev. B 87, 060504(R) (2013).PRBMDO1098-012110.1103/PhysRevB.87.060504 + schema: JATS +- reference: + publication_info: + journal_volume: '87' + year: 2013 + artid: '104509' + journal_title: Phys. Rev. B + authors: + - inspire_role: author + full_name: Sticlet, D. + - inspire_role: author + full_name: Bena, C. + - inspire_role: author + full_name: Simon, P. + title: + title: Josephson Effect in Superconducting Wires Supporting Multiple Majorana + Edge States + label: '45' + dois: + - 10.1103/PhysRevB.87.104509 + raw_refs: + - source: American Physical Society + value: 45D. Sticlet, C. + Bena, and P. Simon, + Josephson Effect in Superconducting Wires Supporting Multiple + Majorana Edge States, Phys. Rev. B 87, + 104509 (2013).PRBMDO1098-012110.1103/PhysRevB.87.104509 + schema: JATS diff --git a/tests/data/aps/PhysRevX.7.021021.xml b/tests/data/aps/PhysRevX.7.021021.xml new file mode 100644 index 0000000..3057f89 --- /dev/null +++ b/tests/data/aps/PhysRevX.7.021021.xml @@ -0,0 +1,562 @@ + + +
+ + + PRX + PRXHAE + + Physical Review X + Phys. Rev. X + + 2160-3308 + + American Physical Society + + + + 10.1103/PhysRevX.7.021021 + + + RESEARCH ARTICLES + + + + quantum + Quantum Physics + + + quantum-info + Quantum Information + + + + + R2 + CODE PROPERTIES FROM HOLOGRAPHIC GEOMETRIES + FERNANDO PASTAWSKI AND JOHN PRESKILL + + + + + Pastawski + Fernando + + + + Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany + + + + + Preskill + John + + + + Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA + + + 15 + May + 2017 + + + 1 + April + 2017 + + 7 + 2 + 021021 + + + 5 + January + 2017 + + + + Published by the American Physical Society + 2017 + authors + + Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. + + + +

Almheiri, Dong, and Harlow [J. High Energy Phys.04 (2015) 163.JHEPFG1029-847910.1007/JHEP04(2015)163] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius.

+ + +

A deep link might exist between two seemingly disparate but far-reaching ideas in physics: quantum error correction and the holographic principle. Quantum error correction concerns using redundant encoding to protect quantum information from damage, and it has been studied extensively because of its relevance to reliable operation of noisy quantum computers. The holographic principle, meanwhile, states that all information about a volume of space can be encoded on the surface area of that volume, much as a hologram encodes a 3D image on a 2D surface. Recent research suggests that how our physical space is structured may also correspond to redundantly represented information. We further explore this connection between redundant information and geometry.

+

Specifically, we look at connections between quantum error-correcting codes and the “holographic correspondence,” which asserts that a suitably chosen quantum theory, without gravity, can be precisely equivalent to a theory of quantum gravity in a negatively curved spacetime. We analyze the properties of holographic quantum codes, quantum error-correcting codes that capture the essential features of the holographic correspondence. These codes provide an information-theoretic interpretation for physical notions such as points in space, black holes, and spacetime curvature.

+

Our work provides a new paradigm for designing quantum error-correction schemes and secret sharing codes, from which we expect many new constructions, and it also clarifies the information-theoretic foundations of the holographic correspondence.

+ + + + + National Science Foundation + http://dx.doi.org/10.13039/100000001 + NSF + http://sws.geonames.org/6252001/ + NSF + + NSF PHY-1125915NSF PHY-1125915 + + + + Gordon and Betty Moore Foundation + http://dx.doi.org/10.13039/100000936 + Gordon E. and Betty I. Moore Foundation + http://sws.geonames.org/6252001/ + + + + + Simons Foundation + http://dx.doi.org/10.13039/100000893 + http://sws.geonames.org/6252001/ + + + + + H2020 European Research Council + http://dx.doi.org/10.13039/100010663 + H2020 Excellent Science - European Research Council + European Research Council + ERC + http://sws.geonames.org/2802361/ + + + + + + + + + + + + INTRODUCTION +

Quantum error correction and the holographic principle are two of the most far-reaching ideas in contemporary physics. Quantum error correction provides a basis for believing that scalable quantum computers can be built and operated in the foreseeable future. The AdS/CFT holographic correspondence is currently our best tool for understanding nonperturbative quantum gravity. In a remarkable paper [1], Almheiri, Dong, and Harlow suggested that these two deep ideas are closely related.

+

The AdS/CFT correspondence is an exact duality between two quantum theories—quantum gravity in (D+1)-dimensional anti–de Sitter space and a conformally invariant quantum field theory (without gravity) defined on its D-dimensional boundary. The observables of the two theories are related by a complex dictionary, which maps local operators supported deep inside the bulk spacetime to highly nonlocal operators acting on the boundary CFT. Almheiri et al. proposed interpreting this dictionary as the encoding map of a quantum error-correcting code, where the code subspace is the low-energy sector of the CFT. Bulk local operators are regarded as “logical” operators that map the code subspace HC to itself, and they are well protected against erasure of portions of the boundary. The holographic dictionary is an encoding map that embeds the logical system inside the physical Hilbert space H of the CFT. This proposal provides a rich and enticing new perspective on the relationship between the emergent bulk geometry and the entanglement structure of the CFT.

+

To model holography faithfully, the quantum error-correcting code must have special properties that invite a geometrical interpretation. Code constructions that realize the ideas in Ref. [1], based on tensor networks that cover the associated bulk geometry, were constructed in Ref. [2] and extended in Ref. [3]. Importantly, it was shown [3] that codes can have holographic properties even when the underlying bulk geometry does not have negative curvature; this insight may broaden our perspective on how AdS space is special.

+

Our goal in this paper is to develop these ideas further. Our motivation is twofold. On one hand, holographic codes have opened a new avenue in quantum coding theory, and it is worthwhile to explore more deeply how geometric insights can provide new methods for deriving code properties. On the other hand, holographic codes provide a useful tool for sharpening the connections between holographic duality and quantum information theory. Specifically, as emphasized in Ref. [1], holographic codes are best described and analyzed using the language of operator algebra quantum error correction [4–7]. This powerful framework deserves to be better known, and much of this paper is devoted to amplifying and applying it.

+

We view our work here as a step along the road toward answering a fundamental question about quantum gravity and holography: What is the bulk? The AdS/CFT correspondence has bestowed many blessings but should be regarded as a crutch that must eventually be discarded to clear the way for future progress in quantum cosmology. We think that strengthening the ties between geometric and algebraic properties will be empowering and that operator algebra quantum error correction can help us reach this goal. As examples, we provide an algebraic characterization of a point in the bulk spacetime and discuss criteria for local correctability of the boundary theory. We also elaborate on the notion of uberholography, in which bulk physics can be reconstructed on a boundary subsystem with fractal geometry.

+ + + Outline +

In Sec. II, we review the formalism of operator algebra quantum error correction (OAQEC). We explain how the notion of code distance can be applied to a subalgebra of a quantum code’s logical algebra, and we introduce the complementary notion of the price of a logical subalgebra, the size of the minimal subsystem of the physical Hilbert space which supports the logical subalgebra. We also derive some inequalities relating distance and price, and note that distance and price are equal for the logical subalgebra supported on a bulk point. In Sec. III, we review the connection between holography and quantum error correction, emphasizing the role of OAQEC in the analysis of holographic codes. We formulate the entanglement wedge hypothesis, a geometric criterion that determines whether a bulk logical subalgebra can be reconstructed on a specified boundary region, and work out some of its implications. We also discuss properties of punctures in the bulk geometry, which provide a crude description of black holes inside the bulk.

+

In Sec. IV, we explain the idea of uberholography and compute the universal fractal dimension, which determines how price and distance scale with system size for a holographic code defined on a hyperbolic disk. In Sec. V, we investigate the conditions for local correctability in a holographic code, where by “local” we mean that the erasure of a small connected boundary region R can be corrected by a recovery map that acts only in a slightly larger region containing R. We explain that holographic codes are locally correctable when the bulk geometry is negatively curved asymptotically but not for asymptotic flat or positive curvature. We interpret this property as a signal of nonlocal physics on the boundary in the flat and positively curved cases, and we also relate properties of black holes to features of holographic codes with positive curvature. In Sec. VI, we use geometrical and entropic arguments to prove a strong quantum Singleton bound for holographic codes, which constrains the price and distance of a logical subalgebra. Section VII contains some concluding comments.

+ + + + + OPERATOR ALGEBRA QUANTUM ERROR CORRECTION +

In this section, we briefly review the principles of OAQEC [4–7], providing a foundation for the discussion of holographic codes. We explain how the notion of code distance can be generalized to the OAQEC setting. We also introduce a related but complementary notion, the price of a code and of a logical operator algebra. In a holographic context, the distance of a bulk logical algebra characterizes how well the bulk degrees of freedom are protected against erasure of portions of the boundary, while its price characterizes the minimal boundary region on which the bulk degrees of freedom can be reconstructed.

+ + + von Neumann algebras +

Since we formulate quantum error correction in an operator algebra framework, we begin by reviewing the structure of finite-dimensional von Neumann algebras. For a finite-dimensional complex Hilbert space H, a von Neumann algebra on H is a complex vector space of linear operators acting on H, which is closed under multiplication and Hermitian adjoint. Any such algebra A can be characterized in the following way. The Hilbert space H contains a subspace with a direct sum decomposition, such that each summand is a product of two tensor factors: H⊇⨁αHα⊗Hα¯,where Hα has dimension dα and Hα¯ has dimension dα¯. The von Neumann algebra A can be expressed as A=⨁αMα⊗Iα¯,where Mα denotes the algebra of dα×dα matrices and Iα¯ denotes the dα¯×dα¯ identity matrix. The commutant A′ of A contains all operators on H, which commute with all operators in A, and can be expressed as A′=⨁αIα⊗Mα¯.The center Z(A) of A, which is also the center of its commutant, contains all elements of the form ⨁αmαIα⊗Iα¯;note that the center is Abelian.

+

A nontrivial von Neumann algebra (with more than one summand) describes a quantum system with superselection sectors. We may regard α as a label that specifies a sector with a specified value of a locally conserved charge. By focusing on A, we focus our attention on operators that preserve α. To interpret the decomposition, Eq. (1), we imagine a system shared by two parties, Alice and Bob, where in each α sector, the parties have equal and opposite charges. The algebras A and A′ capture the charge-preserving operations that can be applied by Alice and Bob, respectively. Equivalently, we may say that a nontrivial von Neumann algebra describes a system that encodes both classical and quantum information, where operators in the center Z(A)=Z(A′) act only on the classical data (the label α), while Mα acts on the quantum data in the sector labeled by α.

+

In OAQEC, we consider HC to be a code subspace of a larger physical Hilbert space; hence, A and A′ are algebras of logical operators that preserve the code subspace. In the case where there is a single summand and Mα¯ is one dimensional, A is the complete algebra of logical operators. This is the standard setting of quantum error-correcting codes. If there is a single summand and Mα¯ is nontrivial, then A is the algebra of “bare” logical operators in a subsystem code. In this setting, the code subspace HC=Hα⊗Hα¯has a decomposition into a protected tensor factor Hα and a “gauge” factor Hα¯, and A acts only on the protected system.

+

The more general setting, with a nontrivial sum over α, arises naturally in the context of holographic duality, where the code subspace corresponds to the low-energy sector of a conformal field theory whose gravitational dual is a bulk system with emergent gauge symmetry. The Abelian center Z(A) of A can, for example, encode classical data of the bulk geometry (see Ref. [8] for a recent tensor network interpretation). An important example of such a classical variable contained in A is the area operator (see Ref. [9]) that arises in the Ryu-Takayanagi formula relating boundary entropy to bulk geometry.

+

Another reason the OAQEC formalism is convenient in discussions of holography is that we can formulate the notion of complementary recovery [10] using this language. If the physical (boundary) Hilbert space has a decomposition as a product RRc of two subsystems, we may ask whether a subalgebra A acting on the code space can be “reconstructed” as an algebra of physical operators with support on R, in which case we may say that erasure of Rc can be corrected for the algebra A. We say that the code exhibits complementary recovery if the logical subalgebra A can be reconstructed on R and its commutant A′ can be reconstructed on Rc. Equivalently, complementary recovery means that erasure of the physical subsystem Rc is correctable with respect to A and erasure of the complementary physical subsystem R is correctable with respect to A′.

+ + + + Correctability +

Quantum error correction is a way of protecting properly encoded quantum states from the potentially damaging effects of noise with suitable properties. The noise can be described by a completely positive trace-preserving map (CPTP map), also called a quantum channel. A channel is a linear map that takes density operators to density operators; saying that the channel is “completely” positive means that the positivity of the density operator is preserved even when the channel acts on a system that is entangled with other systems.

+

A channel N has an operator sum representation (also called a Kraus representation) of the form N(ρ)=∑aNaρNa†,where the condition ∑aNa†Na=Iensures that tr[N(ρ)]=tr[ρ]. The operators {Na} appearing in the Kraus representation are called Kraus operators. If there is only one Kraus operator in the sum, then the map is unitary, taking pure states to pure states. If there are two or more linearly independent Kraus operators, the map N describes a decoherence process, in which pure states can evolve to mixed states.

+

Equation (6) is the Schrödinger picture description of the channel, in which N maps states to states. Since we are particularly interested in whether operators (rather than states) are well protected against noise, we find it more convenient to consider the Heisenberg picture description in which states are fixed and operators evolve. In this picture, the noise acts on the operator X according to N†(X)=∑aNa†XNa.We say that N† is the dual map of N, also called the adjoint map of N. The condition (7) ensures that N† maps the identity operator to itself.

+

We consider a quantum system with Hilbert space H and a noise channel N acting on the system. Quantum error correction is a process that reverses the effect of N. This error-correction process is itself a channel, called the recovery channel and denoted R. Unless N is unitary, error correction is not possible for arbitrary states of the system. Instead, we consider a subspace HC of H, which is called a quantum error-correcting code (QECC), and we settle for a recovery channel that corrects N acting on states of HC. We say that R corrects N on code subspace HC if, for any density operator ρ supported on HC, (R∘N)(ρ)=ρ,and we say that the noise channel N is correctable on HC if there exists a recovery operator R that corrects N.

+

In the Heisenberg picture language, we may consider an algebra of logical operators that act on the code space. We denote the set of linear operators mapping H to H by L(H). If P denotes the orthogonal projector from H to HC, then an operator X∈L(H) is logical if [X,P]=0; hence, X maps HC=PH to itself: XHC=XPH=PXH⊆HC.It is clear from this definition that if X is logical, so is its Hermitian adjoint X† (because P=P†); furthermore, a linear combination of logical operators is logical and so is a product of logical operators. Hence, the set of all logical operators forms an algebra, which we call the complete logical operator of the code. The theory of operator algebra quantum error correction addresses whether a subalgebra of this complete logical algebra can be protected against noise.

+

Sometimes, we are only interested in how a logical operator X acts on the code space, so we consider the corresponding operator PXP, which has support only on HC. Operators of this type are also closed under multiplication because, if X and Y both commute with P, then PXP·PYP=P(XY)P.In other words, if A is an algebra of logical operators in L(H), then PAP is an algebra of logical operators in L(HC). Two logical operators X and X˜ might differ as elements of L(H) yet act on the code space in the same way because PXP=PX˜P. In that case, we say that X and X˜ are logically equivalent, denoted X∼PX˜.

+

Now we can formulate the notion of error correction in the Heisenberg picture.

(correctability). The noise channel N is correctable on the code space HC=PH with respect to the operator X∈L(H) if and only if there exists a recovery channel R such that P(R∘N)†(X)P=PXP.

+

This means that the operator X and the recovered operator (R∘N)†(X) act on the code space in the same way, though they may act differently on state vectors outside the code space. Because this condition is linear in X, the operators with respect to which N is correctable form a linear space.

+

In an important series of works [4,11,12], culminating in Refs. [5,6], the necessary and sufficient conditions for correctability of a logical algebra were derived.

(criterion for correctability). Given code subspace HC=PH and logical subalgebra A, the noise channel N with Kraus operators {Na} is correctable with respect to A if and only if [PNa†NbP,X]=0for all X∈A and each pair of Kraus operators Na, Nb.

+

If A is the code’s complete logical algebra, Eq. (13) becomes PNa†NbP=cabP,which is the well-known Knill-Laflamme error-correction condition [13]. More generally, Eq. (13) says that PNa†NbP lies in the commutant A′ of A. Invoking the general structure of von Neumann algebras reviewed in Sec. II A, we see from Eq. (3) that PNa†NbP is supported on the second factor of each summand. In effect, Eq. (13) means that the Knill-Laflamme condition is satisfied in each superselection sector of the logical algebra.

+ + + + Erasure and reconstruction +

A noise channel of particular interest is the erasure channel. To define the erasure channel, we consider a decomposition of the Hilbert space H as a product of two tensor factors, H=HR⊗HRc;we will sometimes express this decomposition more succinctly as RRc. Anticipating the geometrical interpretation of holographic codes, we call R a region and say that Rc is its complementary region. We say that R is erased when the quantum information in R is lost while the information in Rc is retained. A noise channel describing this process is ΔR(ρ)=σR⊗trR(ρ),which is called the erasure map on R, or the depolarizing map on R; it throws away the state of R and replaces it by the fixed state σR.

+

As for any noise channel, we say that the erasure channel N=ΔR is correctable with respect to the operator X if there is a recovery operator R satisfying Eq. (12). As a convenient shorthand, we say that the subsystem R is correctable if erasure of R is correctable:

(correctable subsystem). Given a code subspace HC=PH and a logical subalgebra A, a subsystem R of H is correctable with respect to A if erasure of R is a correctable map with respect to A.

+

Whether R is correctable does not depend on how we choose the state σR in Eq. (16); if R recovers from ΔR, then R∘ΔR recovers from ΔR′, where ΔR′(ρ)=σR′⊗trR(ρ).

+

For the special case of erasure, the criterion for correctability in Theorem 1 simplifies. We may choose σR∝IR, in which case the Kraus operators realizing ΔR may be chosen to be (up to normalization) the complete set of Pauli operators supported on R, which constitute a complete basis for operators acting on R. More generally, we may realize ΔR by taking the Kraus operators in Eq. (6) as a Haar average over the unitary operators supported on R. We conclude as follows:

(criterion for correctability of a subsystem). Given code subspace HC=PH and logical subalgebra A, subsystem R of H is correctable with respect to A if and only if [PYP,X]=0for all X∈A and every operator Y supported on R.

+

Thus, erasure of R is correctable with respect to a logical algebra A if and only if PYP lies in the commutant A′ of A for any operator Y supported on R. Because X is logical ([X,P]=0), this criterion can also be written as P[Y,X]P=0; that is, the commutator [Y,X] maps the code space to its orthogonal complement. If A is the code’s complete logical algebra, the criterion for correctability of erasure becomes PYP=cP,the Knill-Laflamme criterion for erasure correction [13].

+

If erasure of R is correctable with respect to logical operator X, then it is possible to find an operator X˜ that is logically equivalent to X (PXP=PX˜P) such that X˜ is supported on the complementary subsystem Rc. Borrowing the language of the AdS/CFT correspondence, we may say that X can be “reconstructed” on Rc. In the quantum information literature, one says that X can be “cleaned” on R, meaning that there is an equivalent logical operator that acts trivially on the correctable set.

+

To see why this reconstruction is possible, we may consider the dual ΔR† of the erasure map ΔR, which satisfies tr(XΔR(ρ))=tr(ΔR†(X)ρ).By the definition of correctability, Eq. (12), if R is correctable with respect to X, then there is a recovery map RR which corrects erasure, such that P(RR∘ΔR)†(X)P=P(ΔR†∘RR†)(X)P=PXP.Furthermore, the dual map ΔR† takes any (not necessarily logical) operator Y to an operator that acts trivially on R: ΔR†(Y)=IR⊗Y˜Rc.We see that (ΔR†∘RR†)(X) is logically equivalent to X and supported on Rc; that is, it is a reconstruction of X on the complement of the erased subsystem.

+

To understand Eq. (21), we argue as follows. Consider a unitary map supported on R, under which ρ↦ρ′=(UR⊗IRc)ρ(UR†⊗IRc).Hence, ρRc=ρRc′, and therefore, ΔR(ρ)=ΔR(ρ′), from which we infer that tr(ΔR†(Y)ρ)=tr(ΔR†(Y)ρ′)=tr((UR†⊗IRc)ΔR†(Y)(UR⊗IRc)ρ).If it holds for any state ρ, Eq. (23) implies ΔR†(Y)=(UR†⊗IRc)ΔR†(Y)(UR⊗IRc)for any unitary UR. Equation (21) then follows. Thus, we have shown the following:

(reconstruction). Given code subspace HC=PH and logical subalgebra A, if subsystem R of H is correctable with respect to A, then A can be reconstructed on the complementary subsystem Rc. In other words, for each logical operator in A, there is a logically equivalent operator supported on Rc.

+ + + + Distance and price +

In the standard theory of quantum error correction, we consider the physical Hilbert space H to have a natural decomposition as a tensor product of small subsystems, for example, a decomposition into n qubits (two-level systems); n is the length of the code. This decomposition is “natural” in the sense of being motivated by the underlying physics—e.g., each qubit might be carried by a separate particle, where the particles interact pairwise. Typically, we suppose that the code subspace HC also has a decomposition into “logical” qubits, in other words, that the dimension of the code space is 2k, where k is a positive integer. We may define the distance d of the code as the size of the smallest set R of physical qubits for which erasure of R is not correctable. Equivalently, d is the size of the smallest region that supports observables capable of distinguishing among distinct logical states. We use the notation [[n,k,d]] for a code with n physical qubits, k logical qubits, and distance d. For a given n, it is desirable for k and d to be as large as possible, but there is a trade-off: Larger k means smaller d and vice versa. This standard theory can be generalized in some obvious ways; for example, the dimension of the code subspace might not be a power of 2, or the physical Hilbert space might be decomposed into higher-dimensional subsystems rather than qubits.

+

The distance d loosely characterizes the error-correcting power of the code. But if some encoded degrees of freedom are better protected than others, then a more refined characterization can be useful since the distance captures only the worst case. In holographic codes, in particular, bulk degrees of freedom far from the boundary are better protected than bulk degrees of freedom near the boundary. To describe the performance of a holographic code more completely, we may assign a distance value to each of the code’s logical subalgebras.

+

As in the standard theory, we assume the physical Hilbert space is uniformly factorizable, H=H0⊗n, where H0 is finite dimensional. In applications to quantum field theory, then, H is the Hilbert space of a suitably regulated theory; for example, if the theory is defined on a spatial lattice, a subsystem with Hilbert space H0 resides at each lattice site. Guided by this picture, we refer to the elementary subsystem as a “site.” By a “region” R we mean a subset of the n sites, and the number of sites it contains is called the size of R, denoted |R|. We may now define the distance of a logical algebra A.

(distance). Given code subspace HC=PH and logical subalgebra A, the distance d(A) is the size of the smallest region R that is not correctable with respect to A.

+

If A is the code’s complete logical algebra, then d(A) coincides with the standard definition of distance for a subspace code. In the case of a subsystem code, if A is the algebra of “bare” logical operators that act nontrivially on the protected subsystem and trivially on the gauge subsystem, d(A) is the size of the smallest region that supports a nontrivial “dressed” logical operator, one that acts nontrivially on the protected subsystem and might act on the gauge subsystem as well. In that case, d(A) coincides with the standard definition of distance for a subsystem code. More generally, we might want to consider multiple ways of decomposing HC into a protected subsystem and its complement, and our definition assigns a distance to each of these protected subsystems.

+

For a given code HC and logical algebra A, we may also consider the smallest possible region R such that all operators in A are supported on R. We call the size of this region the price of the algebra.

(price). Given code subspace HC=PH and logical subalgebra A, the price p(A) is the size of the smallest region R such that, for every operator X∈A, there is a logically equivalent operator X˜ that is supported on R.

+

As already noted, if region R is correctable with respect to operator X, then an operator logically equivalent to X can be reconstructed on the complementary region Rc. In this sense, the notions of distance and price are dual to one another. The relation between distance and price can be formulated more precisely with some simple lemmas.

(complementarity). Given code subspace HC=PH and logical subalgebra A, where H contains n sites, the distance and price of A obey p(A)+d(A)≤n+1.

Consider a region R that is correctable with respect to A and also unextendable, meaning R has the property that adding any additional site makes it noncorrectable. Then, there are noncorrectable sets with |R|+1 sites, and therefore d(A)≤|R|+1. Furthermore, since R is correctable, all operators in A can be reconstructed on its complement Rc; hence, p(A)≤|Rc|=n-|R|. Adding these two inequalities yields Eq. (25). □

+

We may anticipate that if a region R supports a nontrivial logical algebra, then erasing R inflicts an irreversible logical error. This intuition is correct if the algebra is non-Abelian. Let us say that a logical subalgebra is non-Abelian if it contains two logical operators X and Y such that PXP and PYP are noncommuting. Then, we have the following:

(no free lunch). Given code subspace HC=PH and non-Abelian logical subalgebra A, the distance and price of A obey d(A)≤p(A).

Consider two logical operators X and Y in A (both commuting with P), such that PXP and PYP are noncommuting. By the definition of p(A), there is a region R with |R|=p(A) such that an operator Y˜ logically equivalent to Y is supported on R; hence, 0≠[PYP,PXP]=[PY˜P,PXP]=[PY˜P,X].This result means that region R does not satisfy the criterion for correctability in Lemma 1 and therefore is not correctable with respect to A. By the definition of distance, d(A)≤|R|=p(A), and Eq. (26) follows. □

+

If A is Abelian, then Eq. (26) need not apply. Consider, for example, the three-qubit quantum repetition code, spanned by the states |000⟩ and |111⟩, and the logical algebra generated by Z¯=|000⟩⟨000|-|111⟩⟨111|.This algebra has price p=1 because the operator Z⊗I⊗I, supported only on the first qubit, is logically equivalent to Z¯. On the other hand, the distance is d=3; because the logical algebra can be supported on any one of the three physical qubits, it is protected against the erasure of any two qubits. Note that p+d=4, saturating Eq. (25).

+

For a traditional subspace code, we may define the price of the code as the price of its complete logical algebra, just as we define the code’s distance to be the distance of its complete logical algebra. The price and distance of a code are constrained by an inequality, which can be derived from the subadditivity of von Neumann entropy. This constraint on price is a corollary to the following theorem.

(constraint on correctable regions). Consider a code subspace HC=PH, where H contains n sites, and let k=logdimHC/logdimH0. Suppose that R1 and R2 are two disjoint correctable regions. Then, n-k≥|R1|+|R2|.

Let A denote the code block H0⊗n, let T denote a reference system, and let |Φ⟩ denote a state of AT in which T is maximally entangled with the code space. The criterion for correctability says that if R is a correctable region, then for any operator Y supported on R, PYP=cP; therefore, if Y is supported on R and X is supported on T, ⟨Φ|Y⊗X|Φ⟩=⟨Φ|PYP⊗X|Φ⟩=c⟨Φ|P⊗X|Φ⟩=c⟨Φ|I⊗X|Φ⟩=⟨Φ|Y⊗I|Φ⟩⟨Φ|I⊗X|Φ⟩.Because ⟨Φ|Y⊗X|Φ⟩ factorizes for any Y supported on R and X supported on T, we conclude that the marginal density operator of RT factorizes, ρRT=ρR⊗ρT,if R is correctable.

To proceed, we use properties of the entropy S(ρ)=-tr(ρlogρ),where, for convenience, we define entropy using logarithms with base dimH0. Because R1 and R2 are both correctable, R1T and R2T are product states; therefore, S(R1T)=S(R1)+S(T),S(R2T)=S(R2)+S(T).Denoting by Rc the region of the code block complementary to R1R2, and noting that the overall state of R1R2RcT is pure, we have S(R1Rc)=S(R2T)=S(R2)+S(T),S(R2Rc)=S(R1T)=S(R1)+S(T);adding these equations yields S(T)=12(S(R1Rc)+S(R2Rc)-S(R1)-S(R2))=S(Rc)-12(I(R1;Rc)+I(R2;Rc)).Since the mutual information I(R;Rc) is non-negative (subadditivity of entropy), S(T)=k, and S(Rc)≤|Rc|=n-|R1|-|R2|, we obtain Eq. (29). □

(strong quantum Singleton bound). Consider a code subspace HC=PH, where H contains n sites, and where k=logdimHc/logdimH0. Then, the distance d and price p of the code obey p-k≥d-1.

In Eq. (29), choose R1 to be the complement of the smallest region that supports the logical algebra of the code (hence, |R1|=n-p), and choose R2 to be any set of d-1 qubits not contained in R1. Then, Eq. (38) follows. □

(quantum Singleton bound). Consider a code subspace HC=PH, where H contains n sites, and where k=logdimHc/logdimH0. Then, n-k≥2(d-1),where d is the code distance.

Combine Corollary 1 and Lemma 3. □

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Because of its resemblance to the Singleton bound n-k≥d-1,satisfied by classical [n,k,d] codes, Eq. (39) is called the quantum Singleton bound. We therefore call Eq. (38) the strong quantum Singleton bound. This bound is saturated, for example, by the [[7,1,3]] Steane code. In that case, the logical Pauli operators X¯ and Z¯ can both be supported on a set of three qubits; therefore, the price is p=3, and the bound becomes 1=k≤p-d+1=3-3+1=1.

+

This strong quantum Singleton bound constrains the distance and price of a traditional subspace code, and it is natural to wonder what we can say about similar constraints on the distance and price of a logical subalgebra. In Sec. VI, we will see that for holographic codes, using more sophisticated entropic arguments, we can derive an operator algebra version of the strong quantum Singleton bound.

+ + + + + HOLOGRAPHY AND QUANTUM ERROR CORRECTION +

The AdS/CFT correspondence [14] is a remarkable proposed equivalence between two theories—quantum gravity in the bulk of a (D+1)-dimensional asymptotically anti–de Sitter spacetime, and conformally invariant quantum field theory (CFT), without gravity, residing on the D-dimensional boundary of the spacetime. A very complex dictionary relates operators acting in the bulk theory to the corresponding operators in the boundary theory. This dictionary is only partially understood, but it is known that local operators acting deep inside the bulk correspond to highly nonlocal operators acting on the boundary. Much evidence indicates that geometrical properties of the bulk theory are intimately related to the structure of quantum entanglement in the boundary theory [15,16]. Further elucidation of this relationship should help to clarify how spacetime geometry can arise as an emergent property of a nongravitational theory.

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A puzzling feature of the correspondence is that a single bulk operator can be faithfully represented by a boundary operator in multiple ways. In a very insightful paper, Almheiri, Dong, and Harlow [1] suggested interpreting this ambiguity using the language of quantum error correction. According to their proposal, the low-energy sector of the boundary CFT can be viewed as a code subspace of the CFT Hilbert space, corresponding to weakly perturbed AdS geometry in the bulk, and the local operators acting on the bulk can be regarded as the logical operators acting on this code subspace. Local operators in the bulk can be reconstructed on the boundary in multiple ways, reflecting the property of quantum error-correcting codes that operators acting differently on the physical Hilbert space H may be logically equivalent when acting on the code subspace HC. High-energy states of the CFT, which are outside the code space, correspond to large black holes in the bulk.

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In Ref. [2], holographic codes were constructed, which capture the features envisioned in Ref. [1]. Such codes provide a highly idealized lattice regularization of the AdS/CFT correspondence, with bulk and boundary lattice sites. The code subspace, or bulk Hilbert space, is (disregarding some caveats expressed below) a tensor product of finite-dimensional Hilbert spaces, one associated with each bulk site, and likewise, the boundary Hilbert space is a tensor product of finite-dimensional Hilbert spaces, one associated with each boundary site. The code defines an embedding of the bulk Hilbert space inside the boundary Hilbert space. The embedding map can be realized by a tensor network construction based on a uniform tiling of the negatively curved bulk geometry. This tensor network provides an explicit holographic dictionary, in particular, mapping each (logical) bulk local operator (with support on a single bulk site) to a corresponding physical nonlocal operator on the boundary (acting on many boundary sites).

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From the perspective of quantum coding theory, holographic codes are a family of quantum codes in which logical degrees of freedom have a pleasing geometrical interpretation, and as emphasized in Ref. [3], this connection between coding and geometry can be extended beyond anti–de Sitter space. From the perspective of the AdS/CFT correspondence, holographic codes strengthen our intuition regarding how quantum error correction relates to emergent geometry. Both perspectives provide ample motivation for further developing these ideas.

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The precise sense in which the low-energy sector of a CFT realizes a quantum code remains rather murky. But loosely speaking, the logical operators are CFT operators that map low-energy states to other low-energy states. Operators that are logically equivalent act on the low-energy states in the same way, but they act differently on the high-energy states that are outside the code space. The algebra of logical operators needs to be truncated because acting on a state with a product of too many logical operators may raise the energy too high, and thus, the resulting state leaves the code space.

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From the bulk point of view, there is a logical algebra Ax associated with each bulk site x, and formally, the complete logical algebra of the code is, in a first approximation, A=⊗xAx,where the tensor product is over all bulk sites. However, implicitly, the number of bulk local operators needs to be small enough so that the backreaction on the geometry can be safely neglected. If so many bulk operators are applied that the bulk geometry is significantly perturbed, then the code space will need to be enlarged, as explained Sec. III B. A further complication is that gauge symmetry in the bulk may prevent the bulk algebra from factorizing as in Eq. (42).

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The holographic dictionary determines how the logical operator subalgebra supported on a region in the bulk (a set of logical bulk sites) can be mapped to an operator algebra supported on a corresponding region on the boundary (a set of physical boundary sites). The geometrical interpretation of this relation between the bulk and boundary operator algebras will be elaborated in the following subsections.

+ + + Entanglement wedge reconstruction +

For holographic codes, whether a specified subsystem of the physical Hilbert space H is correctable with respect to a particular logical subalgebra can be formulated as a question about the bulk geometry. This connection between correctability and geometry is encapsulated by the entanglement wedge hypothesis [17–20], which holds in AdS/CFT [21,22]. This hypothesis specifies the largest bulk region whose logical subalgebra can be represented on a given boundary region.

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The entanglement wedge hypothesis can be formulated for dynamical spacetimes, but for our purposes, it will suffice to consider a special case. We consider a smooth Riemannian manifold B, which may be regarded as a spacelike slice through a static bulk spacetime. Somewhat more generally, we may imagine that B is a slice through a Lorentzian manifold, which is invariant under time reversal about B. Any B can be locally extended to such a Lorentzian manifold, which solves the Einstein field equation without matter sources. To formulate the entanglement wedge hypothesis for this case, we need the concept of a minimal bulk surface embedded in B. We denote the boundary of B by ∂B and consider a boundary region R⊆∂B.

(Minimal surface). Given a Riemannian manifold B with boundary ∂B, the minimal surface χR associated with a boundary region R⊆∂B is the minimum area co-dimension-one surface in B which separates R from its boundary complement Rc (see Fig. 1 for some examples).

+ + 1 + 10.1103/PhysRevX.7.021021.f1 + + +

Geometric notions of minimal surface and entanglement wedge. In each diagram, we highlight a boundary region R with a crayon stroke; the corresponding minimal surface χR is indicated, and the entanglement wedge E[R] is shaded in green. On the left, B is a hyperboloid whose boundary ∂B has two connected components, where R is one of those components (the one on the right). The minimal surface cuts the hyperboloid at its waist, and the entanglement wedge is everything to the right of χR. In the central diagram, B is the interior of a Euclidean ellipse; the boundary region R=R1⊔R2 has two connected components, and χR also has two connected components. As shown, the connected components of χR need not be homologous to R1 and R2, allowing E[R1⊔R2] to be significantly larger than E[R1]⊔E[R2]. On the right, B is the Poincaré disc, portraying an infinite hyperbolic geometry. The minimal surface is a geodesic in the bulk with end points on ∂B.

+ + + +

For the most part, we assume that the minimal surface χR is unique and geometrically well defined. Some choices of geometry B and boundary region R admit more than one minimal surface, but one can usually slightly alter the choice of R in order to make the minimal surface χR unique. Note that, according to Definition 5, R and Rc have the same minimal surface.

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Now, we can define the entanglement wedge.

(entanglement wedge). Given a boundary region R⊆∂B, the entanglement wedge of R is a bulk region E[R]⊆B, whose boundary is ∂E[R]≔χR∪R, where χR is the minimal surface for R (see Fig. 1 for some examples).

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Note that under the uniqueness assumption for the minimal surface χR, the entanglement wedge E(R) of a boundary region R and the entanglement wedge E(Rc) of its boundary complement Rc cover the full bulk manifold B, and they intersect exclusively at the minimal surface.

(geometric complementarity). Given a region R⊆∂B and its boundary complement Rc, we have that χR=χRc=E[R]∩E[Rc] and E[R]∪E[Rc]=B.

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As we will see, this geometric statement, which holds for a generic manifold B and boundary region R, leads to very strong code-theoretic guarantees under the entanglement wedge hypothesis.

+

For a holographic code, the entanglement wedge hypothesis states a sufficient condition for a boundary region to be correctable with respect to the logical subalgebra supported at a site in the bulk. Because of Lemma 2, this condition also informs us that the logical subalgebra can be reconstructed on the complementary boundary region. Evoking the continuum limit of the regulated bulk theory, we will sometimes refer to a bulk site as a point in the bulk, though it will be implicit that associated logical subalgebra is finite dimensional and slightly smeared in space.

(entanglement wedge hypothesis). If the bulk point x is contained in the entanglement wedge E[R] of boundary region R, then the complementary boundary region Rc is correctable with respect to the logical bulk subalgebra Ax. Thus, for each operator in Ax, there is a logically equivalent operator supported on R.

+

This connection between holographic duality and operator algebra quantum error correction has many implications worth exploring.

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For a holographic code corresponding to a regulated boundary theory, there are a finite number of boundary sites, each describing a finite-dimensional subsystem. Thus, we can speak of the length n of the code, meaning the number of boundary sites, as well as the distance d and price p of the code (or of any logical subalgebra), which also take integer values. It is convenient, though, to imagine taking a formal continuum limit of the boundary theory in which the total boundary volume stays fixed as n→∞, while maintaining a uniform number of boundary sites per unit boundary volume as determined by the bulk induced metric. Without intending to place restrictions on the dimension of B, from now on, we use the term area when speaking about the size of a boundary region, and we save the term volume for describing the size of a bulk region. In the continuum limit, we may still mention n, d, and p, but now taking real values; n becomes the total area of the boundary, while d(A) is the area of the smallest boundary region that is not correctable with respect to logical subalgebra A, and p(A) is the area of the smallest boundary region that supports A. For now, to ensure that backreaction on the bulk geometry is negligible, we suppose that the bulk algebra A has support on a constant number of points. In the formal continuum limit, then, the logical algebra has negligible dimension, in effect defining a k≈0 code if the size of the logical system is expressed in geometrical units.

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The entanglement wedge hypothesis has notable consequences for the logical subalgebra Ax supported at a bulk point x. Consider the distance of Ax. For any boundary region R with boundary complement Rc, if x∈E[Rc], then R is correctable with respect to Ax. On the other hand, if x∈E[R], then Ax can be reconstructed on R; arguing as in the proof of Lemma 4, R cannot be correctable with respect to Ax if R supports Ax and Ax is non-Abelian. By geometric complementarity, either x∈E[Rc] or x∈E[R]; we conclude that R is correctable with respect to Ax if and only if x∈E[Rc]. By the definition of distance, then, d(Ax)=minR⊆∂B:x∈E(Rc)|R|,assuming Ax is non-Abelian.

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Now consider the price of Ax. According to the entanglement wedge hypothesis, Ax can be reconstructed on R if x∈E[R]. On the other hand, if x∈E[R], then x∈E[Rc] by geometric complementarity, and therefore, Ax can be reconstructed on Rc. Because operators supported on R commute with operators supported on Rc, it is not possible for Ax to be reconstructed on both R and Rc if Ax is non-Abelian. We conclude that Ax can be reconstructed on R if and only if x∈E[R]. By the definition of price, then, p(Ax)=minR⊆∂B:x∈E(R)|R|,assuming Ax is non-Abelian.

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Geometric complementarity says that x∈E[R] if and only if x∈E[Rc]. Therefore, by comparing Eqs. (43) and (44), we see that the expressions for the distance and the price are identical. Thus, we have shown the following:

(price equals distance for a point). For a holographic code, let Ax be the non-Abelian logical algebra associated with a bulk point x. Then, p(Ax)=d(Ax).

+

Thus, in a holographic code, the bound p(Ax)≥d(Ax) in Lemma 4 is saturated by the logical subalgebra of a point. It is intriguing that a geometrical point admits this simple algebraic characterization, suggesting how geometrical properties might be ascribed to logical subalgebras in a broader setting.

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We can extend this reasoning to a bulk region X that contains a finite number of bulk points, continuing to assume that the number of operator insertions is sufficiently small that backreaction on the bulk geometry can be neglected and that the logical subalgebra factorizes as in Eq. (42). In that case, a boundary region R is correctable with respect to the algebra AX if it is correctable with respect to Ax for each bulk point x in X. Therefore, d(AX)=minx∈Xd(Ax).Equation (44) can also be extended to a bulk region X: p(AX)≔minR⊆∂B:X⊆E[R]|R|,assuming that each nontrivial operator in AX fails to commute with some other operator in AX. If all points of X are contained in E[R], it follows that p(AX)≤|R|.

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We emphasize again that these properties apply not only to AdS bulk geometry but also to other quantum code constructions satisfying geometric complementarity and the entanglement wedge hypothesis. Such codes were constructed in Ref. [2] for tensor networks associated with tilings of bulk geometries having nonpositive curvature. These results were extended to arbitrary graph connectivity in Ref. [3], where a discrete generalization of the entanglement wedge hypothesis was found to be valid in the limit of large bond dimension. Taking a suitable limit, these codes can be viewed as regularized approximations to underlying smooth geometries.

+ + + + Punctures in the bulk +

In quantum gravity, there is an upper limit on the dimension of the Hilbert space that can be encoded in a physical region known as the Bousso bound [23]; the log of this maximal dimension is proportional to the surface area of the region. When one attempts to surpass this limit, a black hole forms, with entropy proportional to the area of its event horizon.

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This feature of bulk quantum gravity can be captured by holographic codes, rather crudely, if we allow punctures in the bulk. A subsystem of the code space HC resides along the edge of each such puncture, and the holographic tensor network provides an isometric embedding of this logical subsystem in the physical Hilbert space H that resides on the exterior boundary of the bulk geometry. This picture is crude because, in an actual gravitational theory, a black hole in the bulk would carry mass and modify the bulk curvature outside the black hole. For our purposes, this impact on the curvature associated with a puncture will not be particularly relevant, and we will, for the most part, ignore it here.

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In the continuum limit, we associate the holographic code with a Riemannian bulk manifold B as in Sec. III A, but now the boundary ∂B is the union of two components: the exterior (physical) boundary, denoted Φ, and the interior (logical) boundary, denoted Λ. The logical boundary is the union of the boundaries of all punctures. For the physical region R⊆Φ, we now need to distinguish between its boundary complement ∂B\R and its physical complement Φ\R. The entanglement wedge hypothesis continues to apply, where now it is understood that the minimal surface χR separates R from its boundary complement. In AdS/CFT, this is called the homology constraint, meaning that R⊔χR is the boundary of a bulk region.

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As we take the continuum limit n→∞ with the bulk geometry fixed, we assume as before that the density of sites per unit area on ∂B is uniform, with the same density on both the physical boundary and the logical boundary. If we assume as in Sec. III A that the bulk logical algebra outside of the punctures is supported on a bounded number of bulk points, then the size k of the logical system is determined by the area of the logical boundary: k/n=|Λ|/|Φ|.

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A holographic code without punctures obeys the celebrated Ryu-Takayanagi formula [15], which asserts that for a boundary region R, the entanglement entropy of R with its physical complement Rc≔Φ\R is the area |χR| of the minimal surface separating R from Rc, with area measured in the same units used to define |Φ| and |Λ|. To extend this formula to a manifold B with punctures, we imagine introducing a reference system T that is maximally entangled with the logical system Λ so that the joint state of T⊔Φ is pure. For a physical boundary region R⊆Φ, the area |χR| of the minimal surface separating R from its boundary complement Λ⊔(Φ\R) is the entanglement entropy of R with T⊔(Φ\R) in this pure state.

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One way to visualize the purifying reference system T is inspired by the thermofield double construction used in AdS/CFT [24]. Given a manifold B with physical boundary Φ and logical boundary Λ, we introduce a second copy B˜ of the manifold, with physical boundary Φ˜ and logical boundary Λ˜ (see Fig. 2). Then, we join Λ and Λ˜, obtaining manifold BB˜, whose physical boundary ΦΦ˜ has two connected components. This construction describes two holographic codes, whose logical systems are maximally entangled; the second copy of the code provides the reference system purifying the first copy. The combined manifold BB˜ has no punctures, and we can apply the original formulation of the Ryu-Takayanagi formula to BB˜. For a boundary region R⊆Φ, the minimal surface χR separating R from Φ˜⊔(Φ\R) never reaches into B˜, and therefore, it coincides with the minimal surface separating R from Λ⊔(Φ\R). [We note that if the logical boundary Λ represents the event horizon of a static (2+1)-dimensional black hole, then, because a geodesic outside the black hole is the spatial trajectory of a light ray, χR either fully contains Λ or avoids it entirely. For a nonstatic geometry, it is possible for χR to include only part of Λ.]

+ + 2 + 10.1103/PhysRevX.7.021021.f2 + + +

The left diagram illustrates the thermofield double construction, in which a bulk manifold B with logical boundary Λ is extended to two copies of B with their logical boundaries identified. This doubled manifold BB˜ describes two holographic codes whose logical systems are maximally entangled. The other two diagrams illustrate that, for a boundary region R contained in the physical boundary Φ of B, the corresponding minimal surface lies in B.

+ + + +

The Ryu-Takayanagi formula relating entanglement entropy S(R) to |χR| is actually the leading term in a systematic expansion [25], in which the next correction arises from entanglement among bulk degrees of freedom, specifically the bulk entanglement of E(R) with its bulk complement. In fact, the division of S(R) into the geometrical contribution |χR| and the bulk entanglement contribution is not a renormalization group invariant; the geometrical contribution dominates in the extreme low-energy limit of the boundary theory, and the bulk entropy becomes more important as we probe the boundary theory at shorter and shorter distances. We implicitly work in the low-energy limit in which geometrical entanglement is dominant. Even in this framework, it is possible to include bulk entanglement in our discussion, in accordance with the so-called ER=EPR principle [26], which identifies entanglement with wormhole connectedness. For example, we can consider two punctures of equal size in the bulk and identify their logical boundaries Λ1 and Λ2, so the two logical systems are maximally entangled. If R is sufficiently large, the minimal surface χR in the bulk might pass in between the two punctures and include (say) the logical boundary Λ1 in order to satisfy the homology constraint. Then, the entanglement entropy of R includes a contribution |Λ1| due to inclusion of the puncture in its entanglement wedge E[R]. In this way, the geometrical entanglement of R can capture the bulk entanglement shared by the punctures.

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Up until now, we have implicitly assumed that the holographic code provides an isometric embedding of the logical system Λ into the physical system Φ. This requirement places restrictions on the geometry of the puncture(s). The Ryu-Takayanagi formula provides one possible way to understand the restriction. Suppose that the reference system T is maximally entangled with the logical boundary Λ so that the state of T⊔Φ is pure, and the entanglement entropy of Φ is |Λ|. This must agree with the geometrical entropy given by |χΦ|=|χΛ|, the area of the minimal surface separating Φ and Λ. Holographic tensor network constructions suggest a stronger constraint, χΦ=χΛ=Λ,since in that case, the tensor network provides an explicit isometric map from Λ into Φ. This consistency condition is illustrated in Fig. 3. The constraint equation (49) ensures that the geometrical entanglement entropy S(Λ) is compatible with the Bekenstein-Hawking entropy |Λ| of a black hole with event horizon at Λ, as we should expect when the microstates of the black hole are maximally entangled with a reference system. If several such black holes approach one another, they must coalesce into a larger black hole in order to enforce Eq. (49). See Fig. 3.

+ + 3 + 10.1103/PhysRevX.7.021021.f3 + + +

Necessary condition χΛ=Λ for the interior boundary of a Riemannian manifold B to be identified as a logical system. In both diagrams, the physical Hilbert space H resides on the exterior boundary Φ of B, and Λ is the boundary of the punctures in the bulk, which are shaded in black. The green region is the entanglement wedge E[Φ], bounded by Φ and the minimal surface χΦ=χΛ separating Φ from Λ; the gray region is B\E[Φ]. For purposes of illustration, we assume the bulk metric is Euclidean. On the left, we have χΛ=Λ, and the interpretation of Λ as a logical system is consistent. On the right, we have χΛ≠Λ, and two possible reasons for this are illustrated. First, a connected component of Λ may fail to be convex. Second, the union Λ1 of several connected components of Λ may be encapsulated by a surface Λ˜1 with smaller area than Λ1, in which case the logical system resides on Λ˜1 rather than Λ1. The emergence of this new logical system is reminiscent of the merging of small black holes to form a larger black hole.

+ + + +

For a holographic code with punctures, we may consider the logical subalgebra associated with a bulk region X⊆B, where now X may include pieces of Λ. Because we are considering the low-energy regime in which geometric entanglement dominates bulk entanglement and backreaction on the geometry due to bulk fields is negligible, we ignore the contribution of bulk points to the logical subalgebra, just as in Eq. (48). Therefore, if X does intersect with Λ, then the effective size of the logical system is given by kX=|Λ∩X|,the area of the portion of Λ contained in X. (In the language of holographic tensor network codes [2], we are assuming that most bulk tensors carry no logical indices, so nearly all of the bulk logical indices contained in the bulk region X are located on the logical boundary.)

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For bulk region X, we consider a reference system T that is maximally entangled with the logical subsystem residing in X⊆B. Then, when we say that AX can be reconstructed on boundary region R⊆Φ, we mean that R contains a subsystem that is maximally entangled with T. If we apply the entanglement wedge hypothesis to a manifold B with a logical boundary, we can use the same reasoning as in Sec. III A to obtain a geometrical expression for the price of the logical algebra: p(AX)≔minR⊆Φ:X⊆E[R]|R|.On the other hand, a boundary region R⊆Φ will be correctable with respect to AX if AX is supported on the physical complement Φ\R of R, and the expression for distance becomes d(AX)≔minR⊆Φ:X⊆E[Φ\R]|R|.It is important to notice that the minimization in d(AX) is over X not contained in the entanglement wedge of the physical complement, rather than the boundary complement, of R. In particular, when X is a single point {x} in the bulk, the expressions for price and distance are not identical because the entanglement wedges E[R] and E[Φ\R] are not complementary regions of the bulk. Therefore, Lemma 5 does not apply to the case of a bulk manifold B with logical boundaries.

+

The geometrical interpretations for price and distance of AX allow us to prove a version of the strong quantum Singleton bound that applies to subalgebras with nonvanishing kX in the continuum limit. This will be explained in Sec. VI.

+ + + + + NEGATIVE CURVATURE AND UBERHOLOGRAPHY +

Next, we discuss a general property of holographic codes defined on bulk manifolds with asymptotically uniform negative curvature, which we call uberholography. The essence of uberholography is that both the distance and price of a logical subalgebra scale sublinearly with the length n of the holographic code. In the formal continuum limit n→∞, the logical subalgebra can be supported on a fractal subset of the boundary, with fractal dimension strictly less than the dimension of the boundary. This fractal dimension is a universal feature of the code, in the sense that it does not depend on which logical subalgebra we consider. Uberholography is intriguing, as it suggests that (D+1)-dimensional bulk geometry can emerge, not just from an underlying D-dimensional system, but also from a system of even lower dimension.

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Though uberholography applies more generally, to be concrete, we consider the bulk to have a two-dimensional hyperbolic geometry with radius of curvature L. Now the boundary is one dimensional, and the minimal “surface” χR associated with connected boundary region R is really a bulk geodesic, whose “area” is actually the geodesic’s length. For our purpose, we need to know only one feature of the bulk geometry: For an interval R on the boundary with length |R|, the length of the bulk geodesic χR separating R from its boundary complement is |χR|=2Llog(|R|/a).Here, a is a short-distance cutoff, which we may think of as a lattice spacing for the boundary theory, so |R|/a is the number of boundary sites contained in R. Applying the Ryu-Takayanagi formula, we conclude that the entanglement entropy S(R) scales logarithmically with the size of R, which is the expected result for the vacuum state of a CFT in one spatial dimension.

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For some bulk region X, we compute the distance of the logical subalgebra AX associated with X. This distance d(AX) is the size of the smallest boundary region R which is not correctable with respect to X. Pick a point x in X, and choose a connected boundary region R such that E[R] contains x, but just barely—if we choose a slightly smaller connected boundary region R′⊂R, then E[R′] will not contain x. Since x∈E(R), we know that R is not correctable with respect to Ax, and therefore d(AX)≤|R|. We could get a tighter upper bound on d(AX) if we find a smaller boundary region R′⊂R whose entanglement wedge still contains x. There may be no such connected boundary region, but can we find a disconnected R′⊂R such that x∈E[R′]?

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Let us try punching a hole in R. In other words, we divide R into three consecutive disjoint intervals R1HR2, where |R1|=|R2|=(r2)|R|,|H|=(1-r)|R|,0<r<1,and then remove the middle (hole) interval H, leaving the disconnected region R′=R1R2=R\H. There are two possible ways to choose bulk geodesics that separate R′ from its complement (illustrated in Fig. 4), either χR1⊔χR2 or χR⊔χH; the minimal surface χR′ is the smaller of these two. Thus, if |χR1|+|χR2|>|χR|+|χH|,we get E[R′]=E[R]\E[H];removing H from R has the effect of removing E[H] from E[R]. Therefore, E[R′] still contains x, and hence d(AX)≤|R′|.

+ + 4 + 10.1103/PhysRevX.7.021021.f4 + + +

Two possible geometries for the entanglement wedge E[R′] of a boundary region R′=R1⊔R2 with two connected components separated by the interval H. In the left diagram, the minimal surface is χR′=χR1⊔χR2, and the entanglement wedge is E[R′]=E[R1]⊔E[R2]. In the right diagram, the minimal surface is χR′=χR⊔χH, where R=R1HR2, and the entanglement wedge is E[R′]=E[R]\E[H].

+ + + +

If we choose H as large as possible, while respecting Eq. (55), then Eq. (53) implies |R1|·|R2|=|R|·|H|⇒r2/4=(1-r),which is satisfied by r/2=2-1.Each connected component of R′ is smaller than R by this factor.

+

Now we repeat this construction recursively. In each round of the procedure, we start with a disconnected region R˜ such that E[R˜] contains x, where R˜ is the union of many connected components of equal size. Then, we punch a hole out of each connected component to obtain a new region R˜′ such that E[R˜′] still contains x. Punching the holes increases the number of connected components by a factor of 2 and reduces the size of each component by a factor of r/2.

+

The procedure halts when the connected components are reduced in size to the lattice spacing a, which occurs after m rounds, where a=(r/2)m|R|.The remaining region Rmin has 2m components, each containing one lattice site. so that d(AX)≤|Rmin|/a=2m=(|R|/a)α,where α=log2log(2/r)=1log2(2+1)≈0.786.The initial interval R is surely no larger than the whole boundary, so the distance is bounded above by nα for any logical subalgebra, where d and n are expressed as a number of boundary sites (rather than length along the boundary).

+

We can also consider codes with punctures in the bulk. To be specific, suppose B is a hyperbolic disk of proper radius rout, with a single puncture at the center of radius rin. The code length n is proportional to the circumference of the outer boundary, and the size k of the logical system is proportional to the circumference of the inner boundary. Because the circumference of a circle with radius r is 2πLer/L, the rate of the code is k/n=e(rin-rout)/L.We may choose an interval R on the boundary, such that χR is tangent to the inner boundary at a single point. The length of this geodesic is essentially twice the difference between the inner and outer boundaries, so Eq. (53) implies rout-rin=Llog(|R|/a).Using the recursive construction to repeatedly carve holes out of R, we obtain the bound Eq. (60) on the code distance, which becomes d≤(|R|/a)α=(e(rout-rin)/L)α=(n/k)α(with the code distance expressed as a number of boundary sites). This scaling of the code distance, with α≈0.786, compares favorably with the bound [27] on local commuting projector codes defined on a two-dimensional Euclidean lattice, for which α=1/2.

+

The scaling p(AX)∼nα applies to price as well as distance. Once we have found a sufficiently large boundary region R such that E[R] contains the bulk region X, we can proceed to hollow out R recursively until we reach the much smaller region Rmin such that |Rmin|/a=(|R|/a)α, where E[Rmin] still contains X, and hence AX is supported on Rmin. The resulting region Rmin, with fractal dimension α, has a geometry reminiscent of the Cantor set, as illustrated in Fig. 5.

+ + 5 + 10.1103/PhysRevX.7.021021.f5 + + +

This figure illustrates uberholography for the case of a two-dimensional hyperbolic bulk geometry. The inner logical boundary is contained inside the entanglement wedge, shaded in blue, of a boundary region R. By repeatedly punching holes of decreasing size out of this boundary region, we obtain a much smaller region Rmin whose entanglement wedge still contains the logical boundary. Thus, the logical algebra is supported on a fractal boundary set, whose geometry is reminiscent of the Cantor set.

+ + + +

It is interesting to compare this universal exponent α for planar uberholography with the scaling laws for distance and price realized by other code families, such as holographic tensor network codes and concatenated quantum codes. In fact, it can be quite challenging to obtain tight lower bounds on distance and price for tensor network code constructions; see the Appendix for further discussion.

+ + + + QUANTUM MARKOV CONDITION AND LOCAL CORRECTABILITY +

For a holographic code, consider (as in Sec. IV) a connected region R=R1HR2, which is the disjoint union of three adjoining intervals. Imagine that the middle interval H is erased. If H is correctable, there is a recovery map R that corrects this erasure error. However, now we ask whether a stronger condition is satisfied: Is it possible to choose a recovery map taking R′=R1R2 to R so that RR′→R′H “fills in” the erased hole H? If the erasure of H can be corrected by a map that acts only on a somewhat larger region containing H (larger by a constant factor independent of system size), then we say that erasure is locally correctable.

+

The quantum Markov condition provides a criterion for local correctability [28]. We say that the state ρABC of three disjoint regions A, B, C obeys the quantum Markov condition (also called quantum conditional independence) if 0=I(A;C|B)=S(AB)+S(BC)-S(ABC)-S(B),which is equivalent to saying that the strong subadditivity inequality is saturated (satisfied as an equality). If the Markov condition is satisfied, then ρABC can be reconstructed from the marginal state ρAB using a map RB→BC, which maps B→BC: RB→BC:  ρAB↦ρABC,known as the Petz recovery map [29]. See Ref. [30] for a construction of a map that is robust to condition (65) holding only approximately. Likewise, in view of the symmetry of the condition under the interchange of A and C, ρABC can be reconstructed from ρBC by a map from B to AB.

+

In fact, Eq. (65) implies that B has a decomposition as a direct sum of tensor products of Hilbert spaces, HB=⨁jHBj=⨁jHBjL⊗HBjR,and that the state of ABC has the block-diagonal form ρABC=⨁jpjρABjL⊗ρBjRC.Evidently, we can recover ρABC from ρAB by replacing each ρBjR by ρBjRC, without touching the system A.

+

To apply the Markov condition to our holographic setting, consider a holographic code with no punctures, where the state of the physical boundary is pure. We choose A, B, C to be three disjoint regions whose union is the complete boundary, namely, A=Rc,B=R′,C=H,where Rc denotes the boundary region complementary to R. Because the state of the complete boundary is pure, S(ABC)=0 and S(H)=S(Hc); therefore, the condition Eq. (65) becomes S(AB)+S(BC)=S(B)⇔S(H)+S(R)=S(R′).When this condition is satisfied, R′ can be divided into two subsystems, where one purifies the state of H and the other purifies the state of Rc. To correct the erasure of H, we need only restore the entanglement between R′ and H, and for this purpose, there is no need to venture outside R.

+ + + Hyperbolic bulk +

Using the Ryu-Takayanagi formula, this statement Eq. (70) about entropy would follow from a statement about minimal surfaces: χR′=χH⊔χR,which is the same as the condition (discussed in Sec. IV) for the entanglement wedge E]R′] to be E[R]\E[H]. For the case in which the bulk is a hyperbolic disk, the calculation in Sec. IV shows that erasure of H can be corrected by a recovery map that acts on region R containing H, where |R|/|H|=(1-r)-1=3+22≈5.828. Thus, the erasure error is locally correctable. This local correctability is a general feature of holographic codes with asymptotically uniform negative bulk curvature.

+

We may also consider the case of a manifold with punctures, where the logical boundary Λ is maximally entangled with a reference system. In that case, the entropy of the physical boundary matches S(Λ), and the Markov condition is satisfied provided that |χR′|+|χΛ|=|χHc|+|χR|,which holds if χR′=χH⊔χR,χHc=χH⊔χΛ.As for the case without punctures, Eq. (73) will be satisfied if H is a sufficiently small interval on the physical boundary of the hyperbolic disk and R is an interval containing H, where |R| is larger than |H| by a constant factor. The interpretation is the same as before; Eq. (73) implies that R′ contains a subsystem that purifies H, so there is no need to reach outside of R to recover from the erasure of H.

+

It is also notable that if the Markov condition Eq. (65) is approximately satisfied, then a local recovery map can be constructed which approximately corrects the erasure of H. This is important because in realistic AdS/CFT, the Markov condition is not exactly satisfied because of the small corrections to the Ryu-Takayanagi formula, which we have neglected. Local correctability in the approximate setting has been discussed recently in Refs. [28,31,32].

+

While holographic codes based on tensor networks can successfully reproduce the Ryu-Takayanagi relation satisfied by the von Neumann entanglement entropy of the boundary theory [2,3], they do not correctly capture the properties of Rényi entropies [33–35] and therefore do not provide a fully satisfactory description of conformal field theories with dual geometries. It is fortunate that the Markov condition Eq. (65), and its approximate version [36], is stated in terms of von Neumann entanglement entropies. We therefore expect that holographic tensor network codes can provide a reasonable picture of local correctability in realistic holography.

+ + + + Flat bulk +

The criterion for local correctability is satisfied by generic negatively curved bulk geometries but not by bulk geometries that are flat or positively curved. Consider, for example, a Euclidean two-dimensional disk with unit radius. For an interval R on the boundary that subtends angle θ, the geodesic χR is a chord of the boundary circle with length |χR|=2sin(θ/2). Suppose we erase a hole H that subtends angle 2δ and correct the erasure by acting in a larger region R that contains H. If R=R1HR2 subtends angle 2ϕ, where |R1|=|R2|, the Markov condition can be satisfied only if |χR1|+|χR2|=4sin((ϕ-δ)/2)≥|χR|+|χH|=2sin(ϕ)+2sin(δ).If δ and ϕ are small, this condition becomes, to leading order in small quantities, δ≤ϕ3/16.Thus, when |H| is small, |R|∼|H|1/3 is far larger; the erasure is not locally correctable. The same will be true, even more so, for a positively curved bulk geometry.

+

The failure of local correctability for holographic codes associated with flat and positively curved bulk manifolds suggests that, in these cases, the physics of the boundary system is highly nonlocal; in particular, the boundary state is not likely to be the ground state of a local Hamiltonian. This conclusion is reinforced by the observation that, according to the Ryu-Takayanagi formula, the entanglement entropy of a small connected region on the boundary of a flat ball scales linearly with the boundary volume of the region; this strong violation of the entanglement area law would not be expected in the ground state if the Hamiltonian is local.

+

That flat bulk geometry implies nonlocal boundary physics also teaches us a valuable lesson about AdS/CFT. A holographic tensor network provides not just an isometric map from the logical boundary Λ to the physical boundary Φ of the manifold B, but also a map from Λ to the boundary ∂X of a bulk region X that contains Λ. If the bulk geometry of X is flat or nearly flat, the entanglement structure of holographic codes indicates that the system supported on ∂X should exhibit flagrant violations of bulk locality. This picture suggests that black holes in the bulk that are small compared to the AdS curvature scale ought to have highly nonlocal dynamics, as seems necessary for these small black holes to be fast scramblers of quantum information [37,38].

+ + + + Positively curved bulk +

This nonlocality of boundary physics is even more pronounced for holographic codes defined on positively curved manifolds. Consider the extreme case of a two-dimensional hemisphere B, with the boundary ∂B at its equator. Geodesics on the sphere are great circles, lying in a plane that passes through the sphere’s center. For any boundary region R with |R|≠|∂B|/2, there is a unique minimal surface χR, which lies in ∂B; for |R|<|∂B|/2, we have χR=R, while for |R|>|∂B|/2, we have χR=∂B\R. Invoking the Ryu-Takayanagi formula, we see that as R increases in size, the entropy S(R) rises linearly as |R| until R occupies half of the boundary; it then decreases linearly thereafter. This behavior is the same as for a Haar-random pure state [39].

+

For |R|<|∂B|/2, the entanglement wedge E[R] contains only R, while for |R|>|∂B|/2, we have R⊔χR=∂B, and the entanglement wedge E[R] is all of the hemisphere B. Accordingly, for any region X in the bulk with associated bulk logical algebra AX, the price p(AX) and distance d(AX) are both given by |∂B|/2. This also mimics the behavior of a Haar-random pure state.

+

If we regulate bulk and boundary by introducing a lattice spacing a, then the number n of boundary sites in the code block is n=|∂B|/a=2πr/a,where r is the radius of the sphere. It is noteworthy that the area of the hemisphere, expressed in lattice units, is |B|/a2=2πr2/a2=n2/2π,which is quadratic in n. Since the hemisphere is the surface of smallest area whose boundary reproduces the entanglement structure of a Haar-random state, it is tempting to interpret the area n2/2π as a measure of the circuit complexity of preparing this state, in accordance with the complexity-equals-action conjecture [40]. Indeed, a random geometrically local circuit in the bulk containing O(n2) gates can closely approximate a unitary 2-design, which prepares a state with the desired properties [41].

+

As noted in Sec. III B, we can crudely model a black hole in the bulk by punching a hole in the bulk, with the microstates of the black hole residing on the logical boundary Λ of the puncture. If we introduce a reference system that is maximally entangled with Λ, then the black hole microstates are maximally mixed, and the entanglement entropy S(Λ)=|Λ| counts these microstates, in agreement with the black hole’s Bekenstein-Hawking entropy. Alternatively, we might wish to describe a black hole in a typical pure state, rather than a highly mixed state. Our observations about the properties of a holographic code defined on a hemisphere suggest how this can be done. Instead of entangling its boundary with a reference system, we fill the puncture with a hemispherical cap. The tensor network filling this cap realizes the minimal geometrically local procedure for preparing the black hole’s highly scrambled pure state.

+ + + + + HOLOGRAPHIC STRONG QUANTUM SINGLETON BOUND +

In Sec. II D, we discussed the strong quantum Singleton bound, Corollary 1, which relates p, d, and k for a code subspace, and we left open whether this bound can be extended to more general logical operator algebras. Here, we will see that, for holographic codes, such an extension is possible.

+

We consider the case of a holographic code with punctures in the bulk; hence, there is a physical boundary Φ and a logical boundary Λ as discussed in Sec. III B. In the formal continuum limit, the code parameters p, d, and k are measured in units of area, and the contribution to the area from O(1) boundary sites can be neglected; hence, Eq. (38) becomes k≤p-d.We would like to show that this constraint applies to logical subalgebras of a holographic code.

+

We consider a region X in the bulk, and its associated logical subalgebra AX. The region X may contain a portion of the logical boundary Λ, as well as some additional isolated points in the bulk. We denote the intersection X∩Λ of X with the logical boundary by ΛX; if ΛX is nonempty, then the bulk points are a negligible portion of the subalgebra AX, whose size is therefore kX=|ΛX|.

+

In what follows, for the sake of clarity, we denote the minimal surface associated with boundary region R by χ(R), in place of the subscript notation χR used earlier. We also use the notation Rc for the physical complement Φ\R, and p, d, k as a shorthand for p(AX), d(AX), kX.

+

Let Rp be a region of the physical boundary Φ such that |Rp|=p(AX) and E[Rp] contains X; this means that the associated minimal surface χ(Rp) must contain ΛX. Let Rd⊆Rp be a subset of Rp such that, in the regulated theory with a nonzero lattice spacing, Rd contains one less boundary site than the distance of AX; therefore, Rd is surely correctable with respect to AX, and in the continuum limit (where a single site has negligible size), |Rd|=d(AX). Because Rd is correctable, the entanglement wedge of its physical complement Rdc contains X, which means that χ(Rdc) contains ΛX.

+

We may consider gradually “growing” a boundary region from Rd to Rp, obtaining an inequality by observing that the corresponding minimal surface cannot grow faster than the boundary surface itself: |Rp|-|Rd|≥∫RdRpdRd|χ(R)|dR=|χ(Rp)|-|χ(Rd)|,|Rdc|-|Rpc|≥∫RpcRdcdRd|χ(R)|dR=|χ(Rdc)|-|χ(Rpc)|.Together with p-d=|Rp|-|Rd|=|Rdc|-|Rpc|, this implies p-d≥|χ(Rp)|-|χ(Rd)|+|χ(Rdc)|-|χ(Rpc)|2.

+

Now recall that χ(Rp) contains ΛX. Hence, the rest of the minimal surface χ(Rp), excluding ΛX, is χ(Rp∪ΛX), or in other words, χ(Rp)=χ(Rp∪ΛX)∪ΛX⇒|χ(Rp)|=|χ(Rp∪ΛX)|+|ΛX|.Likewise, χ(Rdc) contains ΛX, which implies |χ(Rdc)|=|χ(Rdc∪ΛX)|+|ΛX|.Plugging this into Eq. (80) yields p-d-|ΛX|=p-d-k≥12(|χ(Rp∪ΛX)|+|χ(Rdc∪ΛX)|-|χ(Rd)|-|χ(Rpc)|).

+

Now we can use the property that two complementary boundary regions share the same minimal bulk surface (where by the “complement” we mean the boundary complement rather than the physical complement; that is, we are simultaneously taking the complement with respect to the logical and physical boundaries). Let us denote by ΛXc the complement of ΛX with respect to the logical boundary so that Λ=ΛXΛXc. Then, χ(Rdc∪ΛX)=χ(Rd∪ΛXc),χ(Rpc)=χ(Rp∪ΛXΛXc),and hence, p-d-k≥|χ(Rp∪ΛX)|/2+|χ(Rd∪ΛXc)|/2-|χ(Rd)|/2-|χ(Rp∪ΛXΛXc)|/2.Using the Ryu-Takayanagi relation between entropy and area, and identifying AB=Rp∪ΛX,BC=Rd∪ΛXc,B=Rd,ABC=Rp∪ΛXΛXc,the right-hand side of Eq. (86) is proportional to S(AB)+S(BC)-S(B)-S(ABC),which is non-negative by strong subadditivity of entropy. This completes the holographic proof of the strong quantum Singleton bound.

(holographic strong quantum Singleton bound). Consider a holographic code with logical boundary Λ, and a logical subalgebra AX associated with bulk region X, where kX=|X∩Λ|. Then, the price and distance of AX obey kX≤p(AX)-d(AX).

+

It is intriguing that we used strong subadditivity of entropy in this holographic proof, which applies to logical subalgebras, while the proof of Corollary 1, which applies to the price and distance of a traditional code subspace, used only subadditivity. We have not found a proof of the strong quantum Singleton bound that applies to logical subalgebras and that does not use holographic reasoning; it is an open question whether Eq. (88) holds beyond the setting of holographic codes.

+ + + + DISCUSSION AND OUTLOOK +

Our studies of holographic codes have only scratched the surface of this subject. More in-depth studies are needed, including searches, guided by geometrical intuition, for codes with improved parameters and investigations of the efficiency of decoding.

+

Regarding the implications of holographic codes for quantum gravity, we have uncovered several hints that may help steer future research. We have seen that positive curvature of the bulk manifold can improve properties such as the code distance but at a cost—increasing distance is accompanied by enhanced nonlocality of the boundary system. The observation that the logical algebra of a bulk point has price equal to distance is a step toward characterizing bulk geometry using algebraic ideas, and we anticipate further advances in that direction. Uberholography, in bulk spacetimes with asymptotically negative curvature, illustrates how notions from quantum coding can elucidate the emergence of bulk geometry beyond the appearance of just one extra spatial dimension.

+

Following Refs. [1,10], we have discussed boundary reconstruction of bulk physics using the formalism of OAQEC [5,6], which captures salient features of holography. To make firmer contact with realistic AdS/CFT, this discussion should be extended to the setting of approximate OAQEC [42]. First steps in this direction have already been taken in Ref. [31], a study of approximate erasure correction in the (1+1)-dimensional Ising CFT at very low temperatures, and in Ref. [32], an investigation of approximate local correctability in a MERA network, which has polynomially decaying correlations on its boundary.

+

We are encouraged by recent progress connecting quantum error correction and quantum gravity, but much remains unclear. Most obviously, our discussion of the entanglement wedge and bulk reconstruction applies only to static spacetimes or very special spatial slices through dynamical spacetimes. Applying the principles of quantum coding to more general dynamical spacetimes is an important goal, which poses serious unresolved challenges.

+ + + + + ACKNOWLEDGMENTS +

F. P. would like to thank Nicolas Delfosse, Henrik Wilming, and Jens Eisert for helpful discussions and comments. F. P. gratefully acknowledges funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center, with support from the Gordon and Betty Moore Foundation, as well as the Simons Foundation through the It from Qubit program and the FUB through the ERC project (TAQ). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.

+ + + + + COMPARISON OF UBERHOLOGRAPHY DIMENSIONAL EXPONENT WITH EXPLICIT CODES +

In Sec. IV, we computed the universal fractal dimension α≈0.786 for a holographic code defined on the Poincaré disk, assuming that geometric complementarity and the entanglement wedge hypothesis are precisely satisfied. We may also define a fractal dimension for other code families, such as concatenated quantum codes or holographic tensor network codes. Choosing a particular logical algebra A (for example, the algebra of a logical qubit at the center of the bulk), let αp≔limℓ→∞logpℓlognℓ,αd≔limℓ→∞logdℓlognℓ,where pℓ and dℓ are the price and distance of the logical algebra after ℓ levels of concatenation or equivalent iteration. (If A is the algebra of local operators at the center of the bulk, then we may think of ℓ as the radial distance from the center of the bulk to its boundary.) The “no free lunch” lemma ensures that αp≥αd.

+

By a concatenated code, we mean a recursive hierarchy of codes within codes; these can be constructed in many ways. In the simplest case, we consider an [[n,1,d]] code C1, with just one logical qubit, which has an encoder isometrically mapping one qubit to a block of n physical qubits. The code C2, which has k=1 and length n2=n2, is obtained by applying this encoder to each of the n physical qubits in C1; likewise, the code Cℓ, with length nℓ, is obtained by applying the encoder to each of the nℓ-1 qubits in the code Cℓ-1. The corresponding tensor network, with one logical qubit at its center, is a branching tree extending radially outward, in which each branch has n descendants.

+

Suppose that n is odd and the code C1 has the largest possible distance d=(n+1)/2. The complementarity bound Eq. (25) then implies that the price is p=d; the full logical algebra can be supported on d of the n qubits, and all nontrivial logical Pauli operators have weight d. Therefore, all nontrivial logical operators of Cℓ can be supported on dℓ qubits, and all have weight dℓ. We conclude that αp=αd=logdlogn.As n increases, αp and αd approach 1 from below. Although the tensor network can be embedded in a plane, it does not approximate the geometry of the Poincaré disk, and its price and distance obey a different scaling law than we found in Sec. IV.

+

A more complicated recursive encoding scheme, based on an [[n,k,d]] code C1 with k>1, is depicted in Fig. 6. In this case, the C1 encoder maps k qubits to a block of n qubits. To build the code Cℓ, we first assemble k copies of the code Cℓ-1, and then apply the k→n encoder for C1 altogether nℓ-1 times, where each encoder acts on k qubits drawn from the k distinct copies. Each time we add another layer to the code, the number of encoded qubits increases by a factor of k, and the number of physical qubits increases by a factor of n; therefore, nℓ=nℓ,kℓ=kℓ.However, in this case, the price and distance of Cℓ are not so easy to calculate, though we can derive some simple bounds. When we add an additional layer to the code, each nontrivial logical operator of Cℓ-1 maps to a logical operator of Cℓ whose weight is at least d times larger; furthermore, if all logical operators of Cℓ-1 can be supported on w physical qubits, then at most pw qubits are needed to support all logical operators of Cℓ. We therefore have dℓ≤dℓ≤pℓ≤pℓ.However, to make a more precise statement about the price and distance of Cℓ, we need more information about the structure of C1.

+ + 6 + 10.1103/PhysRevX.7.021021.f6 + + +

A recursive coding network to which Eqs. (A3) and (A4) apply, with logical qubits at the top and physical qubits at the bottom. To derive that the distance is bounded below by dℓ and the price is bounded above by pℓ, it suffices to observe that there is a unique “causal” path connecting each physical qubit to each logical qubit. Here, a [[4,2,2]] code (with the encoder drawn as a parallelogram) is concatenated to obtain a [[16,4,4]] code, and a causal path connecting a physical qubit to a logical qubit is highlighted.

+ + + +

We may also consider the price and distance of holographic tensor network codes, which capture some of the features of full-blown AdS/CFT duality [2]. For example, we can tile the Poincaré disk with pentagons and associate a six-index “perfect tensor” with each pentagon, where each pentagon carries a single logical qubit. For this pentagon code, where A is the logical algebra of the central pentagon, we find that the price pℓ(A) and distance dℓ(A) are badly mismatched (where ℓ denotes the graph distance from the central pentagon to the physical boundary of the tensor network). In fact, the distance dℓ(A)=4 is a constant independent of ℓ, as explained in Sec. 5.6 of Ref. [2]; in other words, there are logical operators of weight 4 that act nontrivially on the central qubit. In contrast, we expect the price to scale as pℓ(A)=nℓαp5 with 0.786<αp5<1 (i.e., with an exponent larger than the value attained in idealized holography). This upper bound may be derived using a discrete version of the hole-punching approach, thereby explicitly constructing a subset of the physical boundary qubits for which the greedy algorithm of Ref. [2] reaches the central tensor. It is more difficult to obtain lower bounds on pℓ, as these cannot be witnessed by examples.

+

A nontrivial scaling exponent for the distance dℓ(A) can be obtained if we thin out the logical qubits, replacing pentagons in the bulk by hexagons that carry no logical qubit index. (In such codes, the central qubit is well protected against erasure of a randomly chosen nonzero fraction of all the physical boundary qubits, as shown in Ref. [2].) Codes with relatively sparse bulk logical qubits are better suited than the pentagon code for illustrating the ideas we have explored in this paper, where we have focused on the regime in which geometric entanglement dominates bulk entanglement. We may anticipate that holographic tensor network code families that mimic the geometry of the Poincaré disk will have a price-scaling exponent αp that approximates α≈0.786 from above and a distance-scaling exponent αd that approximates α from below. We have confirmed this expectation by studying some examples, though we have no rigorous general argument.

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diff --git a/tests/data/aps/PhysRevX.7.021021_expected.yml b/tests/data/aps/PhysRevX.7.021021_expected.yml new file mode 100644 index 0000000..b812d61 --- /dev/null +++ b/tests/data/aps/PhysRevX.7.021021_expected.yml @@ -0,0 +1,1208 @@ +abstract: Almheiri, Dong, and Harlow [J. High Energy Phys.04 (2015) 163.] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius. +artid: "021021" +publication_date: 2017-04-01 +year: 2017 +title: R2 +journal_issue: "2" +journal_volume: "7" +publisher: American Physical Society +journal_title: Physical Review X +dois: +- doi: 10.1103/PhysRevX.7.021021 + material: publication +authors: +- full_name: Pastawski, Fernando + raw_affiliations: + - source: American Physical Society + value: Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany +- full_name: Preskill, John + raw_affiliations: + - source: American Physical Society + value: Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA +copyright_holder: authors +copyright_statement: Published by the American Physical Society +copyright_year: 2017 +number_of_pages: 20 +document_type: article +material: publication +license_url: https://creativecommons.org/licenses/by/4.0/ +license_statement: Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +article_type: research-article +is_conference_paper: false +documents: +- key: PhysRevX.7.021021.xml + url: http://example.org/PhysRevX.7.021021.xml + source: American Physical Society + fulltext: true + hidden: true +references: +- reference: + publication_info: + journal_volume: '2015' + page_start: '163' + journal_issue: '04' + artid: '163' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Almheiri, A. + - inspire_role: author + full_name: Dong, X. + - inspire_role: author + full_name: Harlow, D. + title: + title: Bulk Locality and Quantum Error Correction in AdS/CFT + label: '1' + dois: + - 10.1007/JHEP04(2015)163 + raw_refs: + - source: American Physical Society + value: "\n \n 1A. Almheiri, X. Dong,\ + \ and D. Harlow, Bulk\ + \ Locality and Quantum Error Correction in AdS/CFT, J.\ + \ High Energy Phys.04 (2015) 163.JHEPFG1029-847910.1007/JHEP04(2015)163\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '2015' + page_start: '149' + journal_issue: '06' + artid: '149' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Pastawski, F. + - inspire_role: author + full_name: Yoshida, B. + - inspire_role: author + full_name: Harlow, D. + - inspire_role: author + full_name: Preskill, J. + title: + title: 'Holographic Quantum Error-Correcting Codes: Toy Models for the Bulk/Boundary + Correspondence' + label: '2' + dois: + - 10.1007/JHEP06(2015)149 + raw_refs: + - source: American Physical Society + value: "\n \n 2F. Pastawski, B. Yoshida,\ + \ D. Harlow, and J. Preskill,\ + \ Holographic Quantum Error-Correcting Codes: Toy Models for\ + \ the Bulk/Boundary Correspondence, J. High Energy Phys.06\ + \ (2015) 149.JHEPFG1029-847910.1007/JHEP06(2015)149\n \ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '2016' + page_start: 009 + journal_issue: '11' + artid: 009 + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Hayden, P. + - inspire_role: author + full_name: Nezami, S. + - inspire_role: author + full_name: Qi, X.-L. + - inspire_role: author + full_name: Thomas, N. + - inspire_role: author + full_name: Walter, M. + - inspire_role: author + full_name: Yang, Z. + title: + title: Holographic Duality from Random Tensor Networks + label: '3' + dois: + - 10.1007/JHEP11(2016)009 + raw_refs: + - source: American Physical Society + value: "\n \n 3P. Hayden, S. Nezami,\ + \ X.-L. Qi, N. Thomas,\ + \ M. Walter, and Z. Yang,\ + \ Holographic Duality from Random Tensor Networks,\ + \ J. High Energy Phys.11 (2016)\ + \ 009.JHEPFG1029-847910.1007/JHEP11(2016)009\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '94' + year: 2005 + artid: '180501' + journal_title: Phys. Rev. Lett. + authors: + - inspire_role: author + full_name: Kribs, D.W. + - inspire_role: author + full_name: Laflamme, R. + - inspire_role: author + full_name: Poulin, D. + title: + title: Unified and Generalized Approach to Quantum Error Correction + label: '4' + dois: + - 10.1103/PhysRevLett.94.180501 + raw_refs: + - source: American Physical Society + value: "\n \n 4D. W. Kribs, R. Laflamme,\ + \ and D. Poulin, Unified\ + \ and Generalized Approach to Quantum Error Correction, Phys.\ + \ Rev. Lett.94, 180501 (2005).PRLTAO0031-900710.1103/PhysRevLett.94.180501\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '76' + year: 2007 + artid: '042303' + journal_title: Phys. Rev. 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Lett. + authors: + - inspire_role: author + full_name: Bény, C. + - inspire_role: author + full_name: Kempf, A. + - inspire_role: author + full_name: Kribs, D.W. + title: + title: Generalization of Quantum Error Correction via the Heisenberg Picture + label: '6' + dois: + - 10.1103/PhysRevLett.98.100502 + raw_refs: + - source: American Physical Society + value: "\n \n 6C. Bény, A. Kempf, and\ + \ D. W. Kribs, Generalization\ + \ of Quantum Error Correction via the Heisenberg Picture, Phys.\ + \ Rev. Lett.98, 100502 (2007).PRLTAO0031-900710.1103/PhysRevLett.98.100502\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '75' + year: 2007 + artid: '064304' + journal_title: Phys. Rev. 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High Energy Phys. + authors: + - inspire_role: author + full_name: Almheiri, A. + - inspire_role: author + full_name: Dong, X. + - inspire_role: author + full_name: Swingle, B. + title: + title: Linearity of Holographic Entanglement Entropy + label: '9' + dois: + - 10.1007/JHEP02(2017)074 + raw_refs: + - source: American Physical Society + value: "\n \n 9A. Almheiri, X. Dong,\ + \ and B. Swingle, Linearity\ + \ of Holographic Entanglement Entropy, J. High Energy\ + \ Phys.02 (2017) 074.JHEPFG1029-847910.1007/JHEP02(2017)074\n\ + \ " + schema: JATS +- reference: + authors: + - inspire_role: author + full_name: Harlow, D. + title: + title: The Ryu-Takayanagi Formula from Quantum Error Correction + label: '10' + arxiv_eprint: '1607.03901' + raw_refs: + - source: American Physical Society + value: "\n \n 10D. 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Phys.346, 397\ + \ (2016).CMPHAY0010-361610.1007/s00220-016-2706-8\n\ + \ " + schema: JATS +- reference: + authors: + - inspire_role: author + full_name: Bény, C. + title: + title: Conditions for the Approximate Correction of Algebras + label: '42' + arxiv_eprint: '0907.4207' + raw_refs: + - source: American Physical Society + value: "\n \n 42C. Bény, Conditions\ + \ for the Approximate Correction of Algebras, arXiv:0907.4207.\n " + schema: JATS diff --git a/tests/data/aps/PhysRevX.7.021022.xml b/tests/data/aps/PhysRevX.7.021022.xml new file mode 100644 index 0000000..382592c --- /dev/null +++ b/tests/data/aps/PhysRevX.7.021022.xml @@ -0,0 +1,562 @@ + + +
+ + + PRX + PRXHAE + + Physical Review X + Phys. Rev. X + + 2160-3308 + + American Physical Society + + + + 10.1103/PhysRevX.7.021022 + + + RESEARCH ARTICLES + + + + quantum + Quantum Physics + + + quantum-info + Quantum Information + + + + + Code Properties from Holographic Geometries + CODE PROPERTIES FROM HOLOGRAPHIC GEOMETRIES + FERNANDO PASTAWSKI AND JOHN PRESKILL + + + + + Pastawski + Fernando + + + + Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany + + + + + Preskill + John + + + + Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA + + + 15 + May + 2017 + + + 1 + April + 2017 + + 7 + 2 + 021022 + + + 5 + January + 2017 + + + + Published by the American Physical Society + 2017 + authors + + Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. + + + +

Almheiri, Dong, and Harlow [J. High Energy Phys.04 (2015) 163.JHEPFG1029-847910.1007/JHEP04(2015)163] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius.

+ + +

A deep link might exist between two seemingly disparate but far-reaching ideas in physics: quantum error correction and the holographic principle. Quantum error correction concerns using redundant encoding to protect quantum information from damage, and it has been studied extensively because of its relevance to reliable operation of noisy quantum computers. The holographic principle, meanwhile, states that all information about a volume of space can be encoded on the surface area of that volume, much as a hologram encodes a 3D image on a 2D surface. Recent research suggests that how our physical space is structured may also correspond to redundantly represented information. We further explore this connection between redundant information and geometry.

+

Specifically, we look at connections between quantum error-correcting codes and the “holographic correspondence,” which asserts that a suitably chosen quantum theory, without gravity, can be precisely equivalent to a theory of quantum gravity in a negatively curved spacetime. We analyze the properties of holographic quantum codes, quantum error-correcting codes that capture the essential features of the holographic correspondence. These codes provide an information-theoretic interpretation for physical notions such as points in space, black holes, and spacetime curvature.

+

Our work provides a new paradigm for designing quantum error-correction schemes and secret sharing codes, from which we expect many new constructions, and it also clarifies the information-theoretic foundations of the holographic correspondence.

+ + + + + National Science Foundation + http://dx.doi.org/10.13039/100000001 + NSF + http://sws.geonames.org/6252001/ + NSF + + NSF PHY-1125915NSF PHY-1125915 + + + + Gordon and Betty Moore Foundation + http://dx.doi.org/10.13039/100000936 + Gordon E. and Betty I. Moore Foundation + http://sws.geonames.org/6252001/ + + + + + Simons Foundation + http://dx.doi.org/10.13039/100000893 + http://sws.geonames.org/6252001/ + + + + + H2020 European Research Council + http://dx.doi.org/10.13039/100010663 + H2020 Excellent Science - European Research Council + European Research Council + ERC + http://sws.geonames.org/2802361/ + + + + + + + + + + + + INTRODUCTION +

Quantum error correction and the holographic principle are two of the most far-reaching ideas in contemporary physics. Quantum error correction provides a basis for believing that scalable quantum computers can be built and operated in the foreseeable future. The AdS/CFT holographic correspondence is currently our best tool for understanding nonperturbative quantum gravity. In a remarkable paper [1], Almheiri, Dong, and Harlow suggested that these two deep ideas are closely related.

+

The AdS/CFT correspondence is an exact duality between two quantum theories—quantum gravity in (D+1)-dimensional anti–de Sitter space and a conformally invariant quantum field theory (without gravity) defined on its D-dimensional boundary. The observables of the two theories are related by a complex dictionary, which maps local operators supported deep inside the bulk spacetime to highly nonlocal operators acting on the boundary CFT. Almheiri et al. proposed interpreting this dictionary as the encoding map of a quantum error-correcting code, where the code subspace is the low-energy sector of the CFT. Bulk local operators are regarded as “logical” operators that map the code subspace HC to itself, and they are well protected against erasure of portions of the boundary. The holographic dictionary is an encoding map that embeds the logical system inside the physical Hilbert space H of the CFT. This proposal provides a rich and enticing new perspective on the relationship between the emergent bulk geometry and the entanglement structure of the CFT.

+

To model holography faithfully, the quantum error-correcting code must have special properties that invite a geometrical interpretation. Code constructions that realize the ideas in Ref. [1], based on tensor networks that cover the associated bulk geometry, were constructed in Ref. [2] and extended in Ref. [3]. Importantly, it was shown [3] that codes can have holographic properties even when the underlying bulk geometry does not have negative curvature; this insight may broaden our perspective on how AdS space is special.

+

Our goal in this paper is to develop these ideas further. Our motivation is twofold. On one hand, holographic codes have opened a new avenue in quantum coding theory, and it is worthwhile to explore more deeply how geometric insights can provide new methods for deriving code properties. On the other hand, holographic codes provide a useful tool for sharpening the connections between holographic duality and quantum information theory. Specifically, as emphasized in Ref. [1], holographic codes are best described and analyzed using the language of operator algebra quantum error correction [4–7]. This powerful framework deserves to be better known, and much of this paper is devoted to amplifying and applying it.

+

We view our work here as a step along the road toward answering a fundamental question about quantum gravity and holography: What is the bulk? The AdS/CFT correspondence has bestowed many blessings but should be regarded as a crutch that must eventually be discarded to clear the way for future progress in quantum cosmology. We think that strengthening the ties between geometric and algebraic properties will be empowering and that operator algebra quantum error correction can help us reach this goal. As examples, we provide an algebraic characterization of a point in the bulk spacetime and discuss criteria for local correctability of the boundary theory. We also elaborate on the notion of uberholography, in which bulk physics can be reconstructed on a boundary subsystem with fractal geometry.

+ + + Outline +

In Sec. II, we review the formalism of operator algebra quantum error correction (OAQEC). We explain how the notion of code distance can be applied to a subalgebra of a quantum code’s logical algebra, and we introduce the complementary notion of the price of a logical subalgebra, the size of the minimal subsystem of the physical Hilbert space which supports the logical subalgebra. We also derive some inequalities relating distance and price, and note that distance and price are equal for the logical subalgebra supported on a bulk point. In Sec. III, we review the connection between holography and quantum error correction, emphasizing the role of OAQEC in the analysis of holographic codes. We formulate the entanglement wedge hypothesis, a geometric criterion that determines whether a bulk logical subalgebra can be reconstructed on a specified boundary region, and work out some of its implications. We also discuss properties of punctures in the bulk geometry, which provide a crude description of black holes inside the bulk.

+

In Sec. IV, we explain the idea of uberholography and compute the universal fractal dimension, which determines how price and distance scale with system size for a holographic code defined on a hyperbolic disk. In Sec. V, we investigate the conditions for local correctability in a holographic code, where by “local” we mean that the erasure of a small connected boundary region R can be corrected by a recovery map that acts only in a slightly larger region containing R. We explain that holographic codes are locally correctable when the bulk geometry is negatively curved asymptotically but not for asymptotic flat or positive curvature. We interpret this property as a signal of nonlocal physics on the boundary in the flat and positively curved cases, and we also relate properties of black holes to features of holographic codes with positive curvature. In Sec. VI, we use geometrical and entropic arguments to prove a strong quantum Singleton bound for holographic codes, which constrains the price and distance of a logical subalgebra. Section VII contains some concluding comments.

+ + + + + OPERATOR ALGEBRA QUANTUM ERROR CORRECTION +

In this section, we briefly review the principles of OAQEC [4–7], providing a foundation for the discussion of holographic codes. We explain how the notion of code distance can be generalized to the OAQEC setting. We also introduce a related but complementary notion, the price of a code and of a logical operator algebra. In a holographic context, the distance of a bulk logical algebra characterizes how well the bulk degrees of freedom are protected against erasure of portions of the boundary, while its price characterizes the minimal boundary region on which the bulk degrees of freedom can be reconstructed.

+ + + von Neumann algebras +

Since we formulate quantum error correction in an operator algebra framework, we begin by reviewing the structure of finite-dimensional von Neumann algebras. For a finite-dimensional complex Hilbert space H, a von Neumann algebra on H is a complex vector space of linear operators acting on H, which is closed under multiplication and Hermitian adjoint. Any such algebra A can be characterized in the following way. The Hilbert space H contains a subspace with a direct sum decomposition, such that each summand is a product of two tensor factors: H⊇⨁αHα⊗Hα¯,where Hα has dimension dα and Hα¯ has dimension dα¯. The von Neumann algebra A can be expressed as A=⨁αMα⊗Iα¯,where Mα denotes the algebra of dα×dα matrices and Iα¯ denotes the dα¯×dα¯ identity matrix. The commutant A′ of A contains all operators on H, which commute with all operators in A, and can be expressed as A′=⨁αIα⊗Mα¯.The center Z(A) of A, which is also the center of its commutant, contains all elements of the form ⨁αmαIα⊗Iα¯;note that the center is Abelian.

+

A nontrivial von Neumann algebra (with more than one summand) describes a quantum system with superselection sectors. We may regard α as a label that specifies a sector with a specified value of a locally conserved charge. By focusing on A, we focus our attention on operators that preserve α. To interpret the decomposition, Eq. (1), we imagine a system shared by two parties, Alice and Bob, where in each α sector, the parties have equal and opposite charges. The algebras A and A′ capture the charge-preserving operations that can be applied by Alice and Bob, respectively. Equivalently, we may say that a nontrivial von Neumann algebra describes a system that encodes both classical and quantum information, where operators in the center Z(A)=Z(A′) act only on the classical data (the label α), while Mα acts on the quantum data in the sector labeled by α.

+

In OAQEC, we consider HC to be a code subspace of a larger physical Hilbert space; hence, A and A′ are algebras of logical operators that preserve the code subspace. In the case where there is a single summand and Mα¯ is one dimensional, A is the complete algebra of logical operators. This is the standard setting of quantum error-correcting codes. If there is a single summand and Mα¯ is nontrivial, then A is the algebra of “bare” logical operators in a subsystem code. In this setting, the code subspace HC=Hα⊗Hα¯has a decomposition into a protected tensor factor Hα and a “gauge” factor Hα¯, and A acts only on the protected system.

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The more general setting, with a nontrivial sum over α, arises naturally in the context of holographic duality, where the code subspace corresponds to the low-energy sector of a conformal field theory whose gravitational dual is a bulk system with emergent gauge symmetry. The Abelian center Z(A) of A can, for example, encode classical data of the bulk geometry (see Ref. [8] for a recent tensor network interpretation). An important example of such a classical variable contained in A is the area operator (see Ref. [9]) that arises in the Ryu-Takayanagi formula relating boundary entropy to bulk geometry.

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Another reason the OAQEC formalism is convenient in discussions of holography is that we can formulate the notion of complementary recovery [10] using this language. If the physical (boundary) Hilbert space has a decomposition as a product RRc of two subsystems, we may ask whether a subalgebra A acting on the code space can be “reconstructed” as an algebra of physical operators with support on R, in which case we may say that erasure of Rc can be corrected for the algebra A. We say that the code exhibits complementary recovery if the logical subalgebra A can be reconstructed on R and its commutant A′ can be reconstructed on Rc. Equivalently, complementary recovery means that erasure of the physical subsystem Rc is correctable with respect to A and erasure of the complementary physical subsystem R is correctable with respect to A′.

+ + + + Correctability +

Quantum error correction is a way of protecting properly encoded quantum states from the potentially damaging effects of noise with suitable properties. The noise can be described by a completely positive trace-preserving map (CPTP map), also called a quantum channel. A channel is a linear map that takes density operators to density operators; saying that the channel is “completely” positive means that the positivity of the density operator is preserved even when the channel acts on a system that is entangled with other systems.

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A channel N has an operator sum representation (also called a Kraus representation) of the form N(ρ)=∑aNaρNa†,where the condition ∑aNa†Na=Iensures that tr[N(ρ)]=tr[ρ]. The operators {Na} appearing in the Kraus representation are called Kraus operators. If there is only one Kraus operator in the sum, then the map is unitary, taking pure states to pure states. If there are two or more linearly independent Kraus operators, the map N describes a decoherence process, in which pure states can evolve to mixed states.

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Equation (6) is the Schrödinger picture description of the channel, in which N maps states to states. Since we are particularly interested in whether operators (rather than states) are well protected against noise, we find it more convenient to consider the Heisenberg picture description in which states are fixed and operators evolve. In this picture, the noise acts on the operator X according to N†(X)=∑aNa†XNa.We say that N† is the dual map of N, also called the adjoint map of N. The condition (7) ensures that N† maps the identity operator to itself.

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We consider a quantum system with Hilbert space H and a noise channel N acting on the system. Quantum error correction is a process that reverses the effect of N. This error-correction process is itself a channel, called the recovery channel and denoted R. Unless N is unitary, error correction is not possible for arbitrary states of the system. Instead, we consider a subspace HC of H, which is called a quantum error-correcting code (QECC), and we settle for a recovery channel that corrects N acting on states of HC. We say that R corrects N on code subspace HC if, for any density operator ρ supported on HC, (R∘N)(ρ)=ρ,and we say that the noise channel N is correctable on HC if there exists a recovery operator R that corrects N.

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In the Heisenberg picture language, we may consider an algebra of logical operators that act on the code space. We denote the set of linear operators mapping H to H by L(H). If P denotes the orthogonal projector from H to HC, then an operator X∈L(H) is logical if [X,P]=0; hence, X maps HC=PH to itself: XHC=XPH=PXH⊆HC.It is clear from this definition that if X is logical, so is its Hermitian adjoint X† (because P=P†); furthermore, a linear combination of logical operators is logical and so is a product of logical operators. Hence, the set of all logical operators forms an algebra, which we call the complete logical operator of the code. The theory of operator algebra quantum error correction addresses whether a subalgebra of this complete logical algebra can be protected against noise.

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Sometimes, we are only interested in how a logical operator X acts on the code space, so we consider the corresponding operator PXP, which has support only on HC. Operators of this type are also closed under multiplication because, if X and Y both commute with P, then PXP·PYP=P(XY)P.In other words, if A is an algebra of logical operators in L(H), then PAP is an algebra of logical operators in L(HC). Two logical operators X and X˜ might differ as elements of L(H) yet act on the code space in the same way because PXP=PX˜P. In that case, we say that X and X˜ are logically equivalent, denoted X∼PX˜.

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Now we can formulate the notion of error correction in the Heisenberg picture.

(correctability). The noise channel N is correctable on the code space HC=PH with respect to the operator X∈L(H) if and only if there exists a recovery channel R such that P(R∘N)†(X)P=PXP.

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This means that the operator X and the recovered operator (R∘N)†(X) act on the code space in the same way, though they may act differently on state vectors outside the code space. Because this condition is linear in X, the operators with respect to which N is correctable form a linear space.

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In an important series of works [4,11,12], culminating in Refs. [5,6], the necessary and sufficient conditions for correctability of a logical algebra were derived.

(criterion for correctability). Given code subspace HC=PH and logical subalgebra A, the noise channel N with Kraus operators {Na} is correctable with respect to A if and only if [PNa†NbP,X]=0for all X∈A and each pair of Kraus operators Na, Nb.

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If A is the code’s complete logical algebra, Eq. (13) becomes PNa†NbP=cabP,which is the well-known Knill-Laflamme error-correction condition [13]. More generally, Eq. (13) says that PNa†NbP lies in the commutant A′ of A. Invoking the general structure of von Neumann algebras reviewed in Sec. II A, we see from Eq. (3) that PNa†NbP is supported on the second factor of each summand. In effect, Eq. (13) means that the Knill-Laflamme condition is satisfied in each superselection sector of the logical algebra.

+ + + + Erasure and reconstruction +

A noise channel of particular interest is the erasure channel. To define the erasure channel, we consider a decomposition of the Hilbert space H as a product of two tensor factors, H=HR⊗HRc;we will sometimes express this decomposition more succinctly as RRc. Anticipating the geometrical interpretation of holographic codes, we call R a region and say that Rc is its complementary region. We say that R is erased when the quantum information in R is lost while the information in Rc is retained. A noise channel describing this process is ΔR(ρ)=σR⊗trR(ρ),which is called the erasure map on R, or the depolarizing map on R; it throws away the state of R and replaces it by the fixed state σR.

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As for any noise channel, we say that the erasure channel N=ΔR is correctable with respect to the operator X if there is a recovery operator R satisfying Eq. (12). As a convenient shorthand, we say that the subsystem R is correctable if erasure of R is correctable:

(correctable subsystem). Given a code subspace HC=PH and a logical subalgebra A, a subsystem R of H is correctable with respect to A if erasure of R is a correctable map with respect to A.

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Whether R is correctable does not depend on how we choose the state σR in Eq. (16); if R recovers from ΔR, then R∘ΔR recovers from ΔR′, where ΔR′(ρ)=σR′⊗trR(ρ).

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For the special case of erasure, the criterion for correctability in Theorem 1 simplifies. We may choose σR∝IR, in which case the Kraus operators realizing ΔR may be chosen to be (up to normalization) the complete set of Pauli operators supported on R, which constitute a complete basis for operators acting on R. More generally, we may realize ΔR by taking the Kraus operators in Eq. (6) as a Haar average over the unitary operators supported on R. We conclude as follows:

(criterion for correctability of a subsystem). Given code subspace HC=PH and logical subalgebra A, subsystem R of H is correctable with respect to A if and only if [PYP,X]=0for all X∈A and every operator Y supported on R.

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Thus, erasure of R is correctable with respect to a logical algebra A if and only if PYP lies in the commutant A′ of A for any operator Y supported on R. Because X is logical ([X,P]=0), this criterion can also be written as P[Y,X]P=0; that is, the commutator [Y,X] maps the code space to its orthogonal complement. If A is the code’s complete logical algebra, the criterion for correctability of erasure becomes PYP=cP,the Knill-Laflamme criterion for erasure correction [13].

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If erasure of R is correctable with respect to logical operator X, then it is possible to find an operator X˜ that is logically equivalent to X (PXP=PX˜P) such that X˜ is supported on the complementary subsystem Rc. Borrowing the language of the AdS/CFT correspondence, we may say that X can be “reconstructed” on Rc. In the quantum information literature, one says that X can be “cleaned” on R, meaning that there is an equivalent logical operator that acts trivially on the correctable set.

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To see why this reconstruction is possible, we may consider the dual ΔR† of the erasure map ΔR, which satisfies tr(XΔR(ρ))=tr(ΔR†(X)ρ).By the definition of correctability, Eq. (12), if R is correctable with respect to X, then there is a recovery map RR which corrects erasure, such that P(RR∘ΔR)†(X)P=P(ΔR†∘RR†)(X)P=PXP.Furthermore, the dual map ΔR† takes any (not necessarily logical) operator Y to an operator that acts trivially on R: ΔR†(Y)=IR⊗Y˜Rc.We see that (ΔR†∘RR†)(X) is logically equivalent to X and supported on Rc; that is, it is a reconstruction of X on the complement of the erased subsystem.

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To understand Eq. (21), we argue as follows. Consider a unitary map supported on R, under which ρ↦ρ′=(UR⊗IRc)ρ(UR†⊗IRc).Hence, ρRc=ρRc′, and therefore, ΔR(ρ)=ΔR(ρ′), from which we infer that tr(ΔR†(Y)ρ)=tr(ΔR†(Y)ρ′)=tr((UR†⊗IRc)ΔR†(Y)(UR⊗IRc)ρ).If it holds for any state ρ, Eq. (23) implies ΔR†(Y)=(UR†⊗IRc)ΔR†(Y)(UR⊗IRc)for any unitary UR. Equation (21) then follows. Thus, we have shown the following:

(reconstruction). Given code subspace HC=PH and logical subalgebra A, if subsystem R of H is correctable with respect to A, then A can be reconstructed on the complementary subsystem Rc. In other words, for each logical operator in A, there is a logically equivalent operator supported on Rc.

+ + + + Distance and price +

In the standard theory of quantum error correction, we consider the physical Hilbert space H to have a natural decomposition as a tensor product of small subsystems, for example, a decomposition into n qubits (two-level systems); n is the length of the code. This decomposition is “natural” in the sense of being motivated by the underlying physics—e.g., each qubit might be carried by a separate particle, where the particles interact pairwise. Typically, we suppose that the code subspace HC also has a decomposition into “logical” qubits, in other words, that the dimension of the code space is 2k, where k is a positive integer. We may define the distance d of the code as the size of the smallest set R of physical qubits for which erasure of R is not correctable. Equivalently, d is the size of the smallest region that supports observables capable of distinguishing among distinct logical states. We use the notation [[n,k,d]] for a code with n physical qubits, k logical qubits, and distance d. For a given n, it is desirable for k and d to be as large as possible, but there is a trade-off: Larger k means smaller d and vice versa. This standard theory can be generalized in some obvious ways; for example, the dimension of the code subspace might not be a power of 2, or the physical Hilbert space might be decomposed into higher-dimensional subsystems rather than qubits.

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The distance d loosely characterizes the error-correcting power of the code. But if some encoded degrees of freedom are better protected than others, then a more refined characterization can be useful since the distance captures only the worst case. In holographic codes, in particular, bulk degrees of freedom far from the boundary are better protected than bulk degrees of freedom near the boundary. To describe the performance of a holographic code more completely, we may assign a distance value to each of the code’s logical subalgebras.

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As in the standard theory, we assume the physical Hilbert space is uniformly factorizable, H=H0⊗n, where H0 is finite dimensional. In applications to quantum field theory, then, H is the Hilbert space of a suitably regulated theory; for example, if the theory is defined on a spatial lattice, a subsystem with Hilbert space H0 resides at each lattice site. Guided by this picture, we refer to the elementary subsystem as a “site.” By a “region” R we mean a subset of the n sites, and the number of sites it contains is called the size of R, denoted |R|. We may now define the distance of a logical algebra A.

(distance). Given code subspace HC=PH and logical subalgebra A, the distance d(A) is the size of the smallest region R that is not correctable with respect to A.

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If A is the code’s complete logical algebra, then d(A) coincides with the standard definition of distance for a subspace code. In the case of a subsystem code, if A is the algebra of “bare” logical operators that act nontrivially on the protected subsystem and trivially on the gauge subsystem, d(A) is the size of the smallest region that supports a nontrivial “dressed” logical operator, one that acts nontrivially on the protected subsystem and might act on the gauge subsystem as well. In that case, d(A) coincides with the standard definition of distance for a subsystem code. More generally, we might want to consider multiple ways of decomposing HC into a protected subsystem and its complement, and our definition assigns a distance to each of these protected subsystems.

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For a given code HC and logical algebra A, we may also consider the smallest possible region R such that all operators in A are supported on R. We call the size of this region the price of the algebra.

(price). Given code subspace HC=PH and logical subalgebra A, the price p(A) is the size of the smallest region R such that, for every operator X∈A, there is a logically equivalent operator X˜ that is supported on R.

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As already noted, if region R is correctable with respect to operator X, then an operator logically equivalent to X can be reconstructed on the complementary region Rc. In this sense, the notions of distance and price are dual to one another. The relation between distance and price can be formulated more precisely with some simple lemmas.

(complementarity). Given code subspace HC=PH and logical subalgebra A, where H contains n sites, the distance and price of A obey p(A)+d(A)≤n+1.

Consider a region R that is correctable with respect to A and also unextendable, meaning R has the property that adding any additional site makes it noncorrectable. Then, there are noncorrectable sets with |R|+1 sites, and therefore d(A)≤|R|+1. Furthermore, since R is correctable, all operators in A can be reconstructed on its complement Rc; hence, p(A)≤|Rc|=n-|R|. Adding these two inequalities yields Eq. (25). □

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We may anticipate that if a region R supports a nontrivial logical algebra, then erasing R inflicts an irreversible logical error. This intuition is correct if the algebra is non-Abelian. Let us say that a logical subalgebra is non-Abelian if it contains two logical operators X and Y such that PXP and PYP are noncommuting. Then, we have the following:

(no free lunch). Given code subspace HC=PH and non-Abelian logical subalgebra A, the distance and price of A obey d(A)≤p(A).

Consider two logical operators X and Y in A (both commuting with P), such that PXP and PYP are noncommuting. By the definition of p(A), there is a region R with |R|=p(A) such that an operator Y˜ logically equivalent to Y is supported on R; hence, 0≠[PYP,PXP]=[PY˜P,PXP]=[PY˜P,X].This result means that region R does not satisfy the criterion for correctability in Lemma 1 and therefore is not correctable with respect to A. By the definition of distance, d(A)≤|R|=p(A), and Eq. (26) follows. □

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If A is Abelian, then Eq. (26) need not apply. Consider, for example, the three-qubit quantum repetition code, spanned by the states |000⟩ and |111⟩, and the logical algebra generated by Z¯=|000⟩⟨000|-|111⟩⟨111|.This algebra has price p=1 because the operator Z⊗I⊗I, supported only on the first qubit, is logically equivalent to Z¯. On the other hand, the distance is d=3; because the logical algebra can be supported on any one of the three physical qubits, it is protected against the erasure of any two qubits. Note that p+d=4, saturating Eq. (25).

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For a traditional subspace code, we may define the price of the code as the price of its complete logical algebra, just as we define the code’s distance to be the distance of its complete logical algebra. The price and distance of a code are constrained by an inequality, which can be derived from the subadditivity of von Neumann entropy. This constraint on price is a corollary to the following theorem.

(constraint on correctable regions). Consider a code subspace HC=PH, where H contains n sites, and let k=logdimHC/logdimH0. Suppose that R1 and R2 are two disjoint correctable regions. Then, n-k≥|R1|+|R2|.

Let A denote the code block H0⊗n, let T denote a reference system, and let |Φ⟩ denote a state of AT in which T is maximally entangled with the code space. The criterion for correctability says that if R is a correctable region, then for any operator Y supported on R, PYP=cP; therefore, if Y is supported on R and X is supported on T, ⟨Φ|Y⊗X|Φ⟩=⟨Φ|PYP⊗X|Φ⟩=c⟨Φ|P⊗X|Φ⟩=c⟨Φ|I⊗X|Φ⟩=⟨Φ|Y⊗I|Φ⟩⟨Φ|I⊗X|Φ⟩.Because ⟨Φ|Y⊗X|Φ⟩ factorizes for any Y supported on R and X supported on T, we conclude that the marginal density operator of RT factorizes, ρRT=ρR⊗ρT,if R is correctable.

To proceed, we use properties of the entropy S(ρ)=-tr(ρlogρ),where, for convenience, we define entropy using logarithms with base dimH0. Because R1 and R2 are both correctable, R1T and R2T are product states; therefore, S(R1T)=S(R1)+S(T),S(R2T)=S(R2)+S(T).Denoting by Rc the region of the code block complementary to R1R2, and noting that the overall state of R1R2RcT is pure, we have S(R1Rc)=S(R2T)=S(R2)+S(T),S(R2Rc)=S(R1T)=S(R1)+S(T);adding these equations yields S(T)=12(S(R1Rc)+S(R2Rc)-S(R1)-S(R2))=S(Rc)-12(I(R1;Rc)+I(R2;Rc)).Since the mutual information I(R;Rc) is non-negative (subadditivity of entropy), S(T)=k, and S(Rc)≤|Rc|=n-|R1|-|R2|, we obtain Eq. (29). □

(strong quantum Singleton bound). Consider a code subspace HC=PH, where H contains n sites, and where k=logdimHc/logdimH0. Then, the distance d and price p of the code obey p-k≥d-1.

In Eq. (29), choose R1 to be the complement of the smallest region that supports the logical algebra of the code (hence, |R1|=n-p), and choose R2 to be any set of d-1 qubits not contained in R1. Then, Eq. (38) follows. □

(quantum Singleton bound). Consider a code subspace HC=PH, where H contains n sites, and where k=logdimHc/logdimH0. Then, n-k≥2(d-1),where d is the code distance.

Combine Corollary 1 and Lemma 3. □

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Because of its resemblance to the Singleton bound n-k≥d-1,satisfied by classical [n,k,d] codes, Eq. (39) is called the quantum Singleton bound. We therefore call Eq. (38) the strong quantum Singleton bound. This bound is saturated, for example, by the [[7,1,3]] Steane code. In that case, the logical Pauli operators X¯ and Z¯ can both be supported on a set of three qubits; therefore, the price is p=3, and the bound becomes 1=k≤p-d+1=3-3+1=1.

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This strong quantum Singleton bound constrains the distance and price of a traditional subspace code, and it is natural to wonder what we can say about similar constraints on the distance and price of a logical subalgebra. In Sec. VI, we will see that for holographic codes, using more sophisticated entropic arguments, we can derive an operator algebra version of the strong quantum Singleton bound.

+ + + + + HOLOGRAPHY AND QUANTUM ERROR CORRECTION +

The AdS/CFT correspondence [14] is a remarkable proposed equivalence between two theories—quantum gravity in the bulk of a (D+1)-dimensional asymptotically anti–de Sitter spacetime, and conformally invariant quantum field theory (CFT), without gravity, residing on the D-dimensional boundary of the spacetime. A very complex dictionary relates operators acting in the bulk theory to the corresponding operators in the boundary theory. This dictionary is only partially understood, but it is known that local operators acting deep inside the bulk correspond to highly nonlocal operators acting on the boundary. Much evidence indicates that geometrical properties of the bulk theory are intimately related to the structure of quantum entanglement in the boundary theory [15,16]. Further elucidation of this relationship should help to clarify how spacetime geometry can arise as an emergent property of a nongravitational theory.

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A puzzling feature of the correspondence is that a single bulk operator can be faithfully represented by a boundary operator in multiple ways. In a very insightful paper, Almheiri, Dong, and Harlow [1] suggested interpreting this ambiguity using the language of quantum error correction. According to their proposal, the low-energy sector of the boundary CFT can be viewed as a code subspace of the CFT Hilbert space, corresponding to weakly perturbed AdS geometry in the bulk, and the local operators acting on the bulk can be regarded as the logical operators acting on this code subspace. Local operators in the bulk can be reconstructed on the boundary in multiple ways, reflecting the property of quantum error-correcting codes that operators acting differently on the physical Hilbert space H may be logically equivalent when acting on the code subspace HC. High-energy states of the CFT, which are outside the code space, correspond to large black holes in the bulk.

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In Ref. [2], holographic codes were constructed, which capture the features envisioned in Ref. [1]. Such codes provide a highly idealized lattice regularization of the AdS/CFT correspondence, with bulk and boundary lattice sites. The code subspace, or bulk Hilbert space, is (disregarding some caveats expressed below) a tensor product of finite-dimensional Hilbert spaces, one associated with each bulk site, and likewise, the boundary Hilbert space is a tensor product of finite-dimensional Hilbert spaces, one associated with each boundary site. The code defines an embedding of the bulk Hilbert space inside the boundary Hilbert space. The embedding map can be realized by a tensor network construction based on a uniform tiling of the negatively curved bulk geometry. This tensor network provides an explicit holographic dictionary, in particular, mapping each (logical) bulk local operator (with support on a single bulk site) to a corresponding physical nonlocal operator on the boundary (acting on many boundary sites).

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From the perspective of quantum coding theory, holographic codes are a family of quantum codes in which logical degrees of freedom have a pleasing geometrical interpretation, and as emphasized in Ref. [3], this connection between coding and geometry can be extended beyond anti–de Sitter space. From the perspective of the AdS/CFT correspondence, holographic codes strengthen our intuition regarding how quantum error correction relates to emergent geometry. Both perspectives provide ample motivation for further developing these ideas.

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The precise sense in which the low-energy sector of a CFT realizes a quantum code remains rather murky. But loosely speaking, the logical operators are CFT operators that map low-energy states to other low-energy states. Operators that are logically equivalent act on the low-energy states in the same way, but they act differently on the high-energy states that are outside the code space. The algebra of logical operators needs to be truncated because acting on a state with a product of too many logical operators may raise the energy too high, and thus, the resulting state leaves the code space.

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From the bulk point of view, there is a logical algebra Ax associated with each bulk site x, and formally, the complete logical algebra of the code is, in a first approximation, A=⊗xAx,where the tensor product is over all bulk sites. However, implicitly, the number of bulk local operators needs to be small enough so that the backreaction on the geometry can be safely neglected. If so many bulk operators are applied that the bulk geometry is significantly perturbed, then the code space will need to be enlarged, as explained Sec. III B. A further complication is that gauge symmetry in the bulk may prevent the bulk algebra from factorizing as in Eq. (42).

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The holographic dictionary determines how the logical operator subalgebra supported on a region in the bulk (a set of logical bulk sites) can be mapped to an operator algebra supported on a corresponding region on the boundary (a set of physical boundary sites). The geometrical interpretation of this relation between the bulk and boundary operator algebras will be elaborated in the following subsections.

+ + + Entanglement wedge reconstruction +

For holographic codes, whether a specified subsystem of the physical Hilbert space H is correctable with respect to a particular logical subalgebra can be formulated as a question about the bulk geometry. This connection between correctability and geometry is encapsulated by the entanglement wedge hypothesis [17–20], which holds in AdS/CFT [21,22]. This hypothesis specifies the largest bulk region whose logical subalgebra can be represented on a given boundary region.

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The entanglement wedge hypothesis can be formulated for dynamical spacetimes, but for our purposes, it will suffice to consider a special case. We consider a smooth Riemannian manifold B, which may be regarded as a spacelike slice through a static bulk spacetime. Somewhat more generally, we may imagine that B is a slice through a Lorentzian manifold, which is invariant under time reversal about B. Any B can be locally extended to such a Lorentzian manifold, which solves the Einstein field equation without matter sources. To formulate the entanglement wedge hypothesis for this case, we need the concept of a minimal bulk surface embedded in B. We denote the boundary of B by ∂B and consider a boundary region R⊆∂B.

(Minimal surface). Given a Riemannian manifold B with boundary ∂B, the minimal surface χR associated with a boundary region R⊆∂B is the minimum area co-dimension-one surface in B which separates R from its boundary complement Rc (see Fig. 1 for some examples).

+ + 1 + 10.1103/PhysRevX.7.021022.f1 + + +

Geometric notions of minimal surface and entanglement wedge. In each diagram, we highlight a boundary region R with a crayon stroke; the corresponding minimal surface χR is indicated, and the entanglement wedge E[R] is shaded in green. On the left, B is a hyperboloid whose boundary ∂B has two connected components, where R is one of those components (the one on the right). The minimal surface cuts the hyperboloid at its waist, and the entanglement wedge is everything to the right of χR. In the central diagram, B is the interior of a Euclidean ellipse; the boundary region R=R1⊔R2 has two connected components, and χR also has two connected components. As shown, the connected components of χR need not be homologous to R1 and R2, allowing E[R1⊔R2] to be significantly larger than E[R1]⊔E[R2]. On the right, B is the Poincaré disc, portraying an infinite hyperbolic geometry. The minimal surface is a geodesic in the bulk with end points on ∂B.

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For the most part, we assume that the minimal surface χR is unique and geometrically well defined. Some choices of geometry B and boundary region R admit more than one minimal surface, but one can usually slightly alter the choice of R in order to make the minimal surface χR unique. Note that, according to Definition 5, R and Rc have the same minimal surface.

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Now, we can define the entanglement wedge.

(entanglement wedge). Given a boundary region R⊆∂B, the entanglement wedge of R is a bulk region E[R]⊆B, whose boundary is ∂E[R]≔χR∪R, where χR is the minimal surface for R (see Fig. 1 for some examples).

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Note that under the uniqueness assumption for the minimal surface χR, the entanglement wedge E(R) of a boundary region R and the entanglement wedge E(Rc) of its boundary complement Rc cover the full bulk manifold B, and they intersect exclusively at the minimal surface.

(geometric complementarity). Given a region R⊆∂B and its boundary complement Rc, we have that χR=χRc=E[R]∩E[Rc] and E[R]∪E[Rc]=B.

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As we will see, this geometric statement, which holds for a generic manifold B and boundary region R, leads to very strong code-theoretic guarantees under the entanglement wedge hypothesis.

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For a holographic code, the entanglement wedge hypothesis states a sufficient condition for a boundary region to be correctable with respect to the logical subalgebra supported at a site in the bulk. Because of Lemma 2, this condition also informs us that the logical subalgebra can be reconstructed on the complementary boundary region. Evoking the continuum limit of the regulated bulk theory, we will sometimes refer to a bulk site as a point in the bulk, though it will be implicit that associated logical subalgebra is finite dimensional and slightly smeared in space.

(entanglement wedge hypothesis). If the bulk point x is contained in the entanglement wedge E[R] of boundary region R, then the complementary boundary region Rc is correctable with respect to the logical bulk subalgebra Ax. Thus, for each operator in Ax, there is a logically equivalent operator supported on R.

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This connection between holographic duality and operator algebra quantum error correction has many implications worth exploring.

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For a holographic code corresponding to a regulated boundary theory, there are a finite number of boundary sites, each describing a finite-dimensional subsystem. Thus, we can speak of the length n of the code, meaning the number of boundary sites, as well as the distance d and price p of the code (or of any logical subalgebra), which also take integer values. It is convenient, though, to imagine taking a formal continuum limit of the boundary theory in which the total boundary volume stays fixed as n→∞, while maintaining a uniform number of boundary sites per unit boundary volume as determined by the bulk induced metric. Without intending to place restrictions on the dimension of B, from now on, we use the term area when speaking about the size of a boundary region, and we save the term volume for describing the size of a bulk region. In the continuum limit, we may still mention n, d, and p, but now taking real values; n becomes the total area of the boundary, while d(A) is the area of the smallest boundary region that is not correctable with respect to logical subalgebra A, and p(A) is the area of the smallest boundary region that supports A. For now, to ensure that backreaction on the bulk geometry is negligible, we suppose that the bulk algebra A has support on a constant number of points. In the formal continuum limit, then, the logical algebra has negligible dimension, in effect defining a k≈0 code if the size of the logical system is expressed in geometrical units.

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The entanglement wedge hypothesis has notable consequences for the logical subalgebra Ax supported at a bulk point x. Consider the distance of Ax. For any boundary region R with boundary complement Rc, if x∈E[Rc], then R is correctable with respect to Ax. On the other hand, if x∈E[R], then Ax can be reconstructed on R; arguing as in the proof of Lemma 4, R cannot be correctable with respect to Ax if R supports Ax and Ax is non-Abelian. By geometric complementarity, either x∈E[Rc] or x∈E[R]; we conclude that R is correctable with respect to Ax if and only if x∈E[Rc]. By the definition of distance, then, d(Ax)=minR⊆∂B:x∈E(Rc)|R|,assuming Ax is non-Abelian.

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Now consider the price of Ax. According to the entanglement wedge hypothesis, Ax can be reconstructed on R if x∈E[R]. On the other hand, if x∈E[R], then x∈E[Rc] by geometric complementarity, and therefore, Ax can be reconstructed on Rc. Because operators supported on R commute with operators supported on Rc, it is not possible for Ax to be reconstructed on both R and Rc if Ax is non-Abelian. We conclude that Ax can be reconstructed on R if and only if x∈E[R]. By the definition of price, then, p(Ax)=minR⊆∂B:x∈E(R)|R|,assuming Ax is non-Abelian.

+

Geometric complementarity says that x∈E[R] if and only if x∈E[Rc]. Therefore, by comparing Eqs. (43) and (44), we see that the expressions for the distance and the price are identical. Thus, we have shown the following:

(price equals distance for a point). For a holographic code, let Ax be the non-Abelian logical algebra associated with a bulk point x. Then, p(Ax)=d(Ax).

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Thus, in a holographic code, the bound p(Ax)≥d(Ax) in Lemma 4 is saturated by the logical subalgebra of a point. It is intriguing that a geometrical point admits this simple algebraic characterization, suggesting how geometrical properties might be ascribed to logical subalgebras in a broader setting.

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We can extend this reasoning to a bulk region X that contains a finite number of bulk points, continuing to assume that the number of operator insertions is sufficiently small that backreaction on the bulk geometry can be neglected and that the logical subalgebra factorizes as in Eq. (42). In that case, a boundary region R is correctable with respect to the algebra AX if it is correctable with respect to Ax for each bulk point x in X. Therefore, d(AX)=minx∈Xd(Ax).Equation (44) can also be extended to a bulk region X: p(AX)≔minR⊆∂B:X⊆E[R]|R|,assuming that each nontrivial operator in AX fails to commute with some other operator in AX. If all points of X are contained in E[R], it follows that p(AX)≤|R|.

+

We emphasize again that these properties apply not only to AdS bulk geometry but also to other quantum code constructions satisfying geometric complementarity and the entanglement wedge hypothesis. Such codes were constructed in Ref. [2] for tensor networks associated with tilings of bulk geometries having nonpositive curvature. These results were extended to arbitrary graph connectivity in Ref. [3], where a discrete generalization of the entanglement wedge hypothesis was found to be valid in the limit of large bond dimension. Taking a suitable limit, these codes can be viewed as regularized approximations to underlying smooth geometries.

+ + + + Punctures in the bulk +

In quantum gravity, there is an upper limit on the dimension of the Hilbert space that can be encoded in a physical region known as the Bousso bound [23]; the log of this maximal dimension is proportional to the surface area of the region. When one attempts to surpass this limit, a black hole forms, with entropy proportional to the area of its event horizon.

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This feature of bulk quantum gravity can be captured by holographic codes, rather crudely, if we allow punctures in the bulk. A subsystem of the code space HC resides along the edge of each such puncture, and the holographic tensor network provides an isometric embedding of this logical subsystem in the physical Hilbert space H that resides on the exterior boundary of the bulk geometry. This picture is crude because, in an actual gravitational theory, a black hole in the bulk would carry mass and modify the bulk curvature outside the black hole. For our purposes, this impact on the curvature associated with a puncture will not be particularly relevant, and we will, for the most part, ignore it here.

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In the continuum limit, we associate the holographic code with a Riemannian bulk manifold B as in Sec. III A, but now the boundary ∂B is the union of two components: the exterior (physical) boundary, denoted Φ, and the interior (logical) boundary, denoted Λ. The logical boundary is the union of the boundaries of all punctures. For the physical region R⊆Φ, we now need to distinguish between its boundary complement ∂B\R and its physical complement Φ\R. The entanglement wedge hypothesis continues to apply, where now it is understood that the minimal surface χR separates R from its boundary complement. In AdS/CFT, this is called the homology constraint, meaning that R⊔χR is the boundary of a bulk region.

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As we take the continuum limit n→∞ with the bulk geometry fixed, we assume as before that the density of sites per unit area on ∂B is uniform, with the same density on both the physical boundary and the logical boundary. If we assume as in Sec. III A that the bulk logical algebra outside of the punctures is supported on a bounded number of bulk points, then the size k of the logical system is determined by the area of the logical boundary: k/n=|Λ|/|Φ|.

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A holographic code without punctures obeys the celebrated Ryu-Takayanagi formula [15], which asserts that for a boundary region R, the entanglement entropy of R with its physical complement Rc≔Φ\R is the area |χR| of the minimal surface separating R from Rc, with area measured in the same units used to define |Φ| and |Λ|. To extend this formula to a manifold B with punctures, we imagine introducing a reference system T that is maximally entangled with the logical system Λ so that the joint state of T⊔Φ is pure. For a physical boundary region R⊆Φ, the area |χR| of the minimal surface separating R from its boundary complement Λ⊔(Φ\R) is the entanglement entropy of R with T⊔(Φ\R) in this pure state.

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One way to visualize the purifying reference system T is inspired by the thermofield double construction used in AdS/CFT [24]. Given a manifold B with physical boundary Φ and logical boundary Λ, we introduce a second copy B˜ of the manifold, with physical boundary Φ˜ and logical boundary Λ˜ (see Fig. 2). Then, we join Λ and Λ˜, obtaining manifold BB˜, whose physical boundary ΦΦ˜ has two connected components. This construction describes two holographic codes, whose logical systems are maximally entangled; the second copy of the code provides the reference system purifying the first copy. The combined manifold BB˜ has no punctures, and we can apply the original formulation of the Ryu-Takayanagi formula to BB˜. For a boundary region R⊆Φ, the minimal surface χR separating R from Φ˜⊔(Φ\R) never reaches into B˜, and therefore, it coincides with the minimal surface separating R from Λ⊔(Φ\R). [We note that if the logical boundary Λ represents the event horizon of a static (2+1)-dimensional black hole, then, because a geodesic outside the black hole is the spatial trajectory of a light ray, χR either fully contains Λ or avoids it entirely. For a nonstatic geometry, it is possible for χR to include only part of Λ.]

+ + 2 + 10.1103/PhysRevX.7.021022.f2 + + +

The left diagram illustrates the thermofield double construction, in which a bulk manifold B with logical boundary Λ is extended to two copies of B with their logical boundaries identified. This doubled manifold BB˜ describes two holographic codes whose logical systems are maximally entangled. The other two diagrams illustrate that, for a boundary region R contained in the physical boundary Φ of B, the corresponding minimal surface lies in B.

+ + + +

The Ryu-Takayanagi formula relating entanglement entropy S(R) to |χR| is actually the leading term in a systematic expansion [25], in which the next correction arises from entanglement among bulk degrees of freedom, specifically the bulk entanglement of E(R) with its bulk complement. In fact, the division of S(R) into the geometrical contribution |χR| and the bulk entanglement contribution is not a renormalization group invariant; the geometrical contribution dominates in the extreme low-energy limit of the boundary theory, and the bulk entropy becomes more important as we probe the boundary theory at shorter and shorter distances. We implicitly work in the low-energy limit in which geometrical entanglement is dominant. Even in this framework, it is possible to include bulk entanglement in our discussion, in accordance with the so-called ER=EPR principle [26], which identifies entanglement with wormhole connectedness. For example, we can consider two punctures of equal size in the bulk and identify their logical boundaries Λ1 and Λ2, so the two logical systems are maximally entangled. If R is sufficiently large, the minimal surface χR in the bulk might pass in between the two punctures and include (say) the logical boundary Λ1 in order to satisfy the homology constraint. Then, the entanglement entropy of R includes a contribution |Λ1| due to inclusion of the puncture in its entanglement wedge E[R]. In this way, the geometrical entanglement of R can capture the bulk entanglement shared by the punctures.

+

Up until now, we have implicitly assumed that the holographic code provides an isometric embedding of the logical system Λ into the physical system Φ. This requirement places restrictions on the geometry of the puncture(s). The Ryu-Takayanagi formula provides one possible way to understand the restriction. Suppose that the reference system T is maximally entangled with the logical boundary Λ so that the state of T⊔Φ is pure, and the entanglement entropy of Φ is |Λ|. This must agree with the geometrical entropy given by |χΦ|=|χΛ|, the area of the minimal surface separating Φ and Λ. Holographic tensor network constructions suggest a stronger constraint, χΦ=χΛ=Λ,since in that case, the tensor network provides an explicit isometric map from Λ into Φ. This consistency condition is illustrated in Fig. 3. The constraint equation (49) ensures that the geometrical entanglement entropy S(Λ) is compatible with the Bekenstein-Hawking entropy |Λ| of a black hole with event horizon at Λ, as we should expect when the microstates of the black hole are maximally entangled with a reference system. If several such black holes approach one another, they must coalesce into a larger black hole in order to enforce Eq. (49). See Fig. 3.

+ + 3 + 10.1103/PhysRevX.7.021022.f3 + + +

Necessary condition χΛ=Λ for the interior boundary of a Riemannian manifold B to be identified as a logical system. In both diagrams, the physical Hilbert space H resides on the exterior boundary Φ of B, and Λ is the boundary of the punctures in the bulk, which are shaded in black. The green region is the entanglement wedge E[Φ], bounded by Φ and the minimal surface χΦ=χΛ separating Φ from Λ; the gray region is B\E[Φ]. For purposes of illustration, we assume the bulk metric is Euclidean. On the left, we have χΛ=Λ, and the interpretation of Λ as a logical system is consistent. On the right, we have χΛ≠Λ, and two possible reasons for this are illustrated. First, a connected component of Λ may fail to be convex. Second, the union Λ1 of several connected components of Λ may be encapsulated by a surface Λ˜1 with smaller area than Λ1, in which case the logical system resides on Λ˜1 rather than Λ1. The emergence of this new logical system is reminiscent of the merging of small black holes to form a larger black hole.

+ + + +

For a holographic code with punctures, we may consider the logical subalgebra associated with a bulk region X⊆B, where now X may include pieces of Λ. Because we are considering the low-energy regime in which geometric entanglement dominates bulk entanglement and backreaction on the geometry due to bulk fields is negligible, we ignore the contribution of bulk points to the logical subalgebra, just as in Eq. (48). Therefore, if X does intersect with Λ, then the effective size of the logical system is given by kX=|Λ∩X|,the area of the portion of Λ contained in X. (In the language of holographic tensor network codes [2], we are assuming that most bulk tensors carry no logical indices, so nearly all of the bulk logical indices contained in the bulk region X are located on the logical boundary.)

+

For bulk region X, we consider a reference system T that is maximally entangled with the logical subsystem residing in X⊆B. Then, when we say that AX can be reconstructed on boundary region R⊆Φ, we mean that R contains a subsystem that is maximally entangled with T. If we apply the entanglement wedge hypothesis to a manifold B with a logical boundary, we can use the same reasoning as in Sec. III A to obtain a geometrical expression for the price of the logical algebra: p(AX)≔minR⊆Φ:X⊆E[R]|R|.On the other hand, a boundary region R⊆Φ will be correctable with respect to AX if AX is supported on the physical complement Φ\R of R, and the expression for distance becomes d(AX)≔minR⊆Φ:X⊆E[Φ\R]|R|.It is important to notice that the minimization in d(AX) is over X not contained in the entanglement wedge of the physical complement, rather than the boundary complement, of R. In particular, when X is a single point {x} in the bulk, the expressions for price and distance are not identical because the entanglement wedges E[R] and E[Φ\R] are not complementary regions of the bulk. Therefore, Lemma 5 does not apply to the case of a bulk manifold B with logical boundaries.

+

The geometrical interpretations for price and distance of AX allow us to prove a version of the strong quantum Singleton bound that applies to subalgebras with nonvanishing kX in the continuum limit. This will be explained in Sec. VI.

+ + + + + NEGATIVE CURVATURE AND UBERHOLOGRAPHY +

Next, we discuss a general property of holographic codes defined on bulk manifolds with asymptotically uniform negative curvature, which we call uberholography. The essence of uberholography is that both the distance and price of a logical subalgebra scale sublinearly with the length n of the holographic code. In the formal continuum limit n→∞, the logical subalgebra can be supported on a fractal subset of the boundary, with fractal dimension strictly less than the dimension of the boundary. This fractal dimension is a universal feature of the code, in the sense that it does not depend on which logical subalgebra we consider. Uberholography is intriguing, as it suggests that (D+1)-dimensional bulk geometry can emerge, not just from an underlying D-dimensional system, but also from a system of even lower dimension.

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Though uberholography applies more generally, to be concrete, we consider the bulk to have a two-dimensional hyperbolic geometry with radius of curvature L. Now the boundary is one dimensional, and the minimal “surface” χR associated with connected boundary region R is really a bulk geodesic, whose “area” is actually the geodesic’s length. For our purpose, we need to know only one feature of the bulk geometry: For an interval R on the boundary with length |R|, the length of the bulk geodesic χR separating R from its boundary complement is |χR|=2Llog(|R|/a).Here, a is a short-distance cutoff, which we may think of as a lattice spacing for the boundary theory, so |R|/a is the number of boundary sites contained in R. Applying the Ryu-Takayanagi formula, we conclude that the entanglement entropy S(R) scales logarithmically with the size of R, which is the expected result for the vacuum state of a CFT in one spatial dimension.

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For some bulk region X, we compute the distance of the logical subalgebra AX associated with X. This distance d(AX) is the size of the smallest boundary region R which is not correctable with respect to X. Pick a point x in X, and choose a connected boundary region R such that E[R] contains x, but just barely—if we choose a slightly smaller connected boundary region R′⊂R, then E[R′] will not contain x. Since x∈E(R), we know that R is not correctable with respect to Ax, and therefore d(AX)≤|R|. We could get a tighter upper bound on d(AX) if we find a smaller boundary region R′⊂R whose entanglement wedge still contains x. There may be no such connected boundary region, but can we find a disconnected R′⊂R such that x∈E[R′]?

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Let us try punching a hole in R. In other words, we divide R into three consecutive disjoint intervals R1HR2, where |R1|=|R2|=(r2)|R|,|H|=(1-r)|R|,0<r<1,and then remove the middle (hole) interval H, leaving the disconnected region R′=R1R2=R\H. There are two possible ways to choose bulk geodesics that separate R′ from its complement (illustrated in Fig. 4), either χR1⊔χR2 or χR⊔χH; the minimal surface χR′ is the smaller of these two. Thus, if |χR1|+|χR2|>|χR|+|χH|,we get E[R′]=E[R]\E[H];removing H from R has the effect of removing E[H] from E[R]. Therefore, E[R′] still contains x, and hence d(AX)≤|R′|.

+ + 4 + 10.1103/PhysRevX.7.021022.f4 + + +

Two possible geometries for the entanglement wedge E[R′] of a boundary region R′=R1⊔R2 with two connected components separated by the interval H. In the left diagram, the minimal surface is χR′=χR1⊔χR2, and the entanglement wedge is E[R′]=E[R1]⊔E[R2]. In the right diagram, the minimal surface is χR′=χR⊔χH, where R=R1HR2, and the entanglement wedge is E[R′]=E[R]\E[H].

+ + + +

If we choose H as large as possible, while respecting Eq. (55), then Eq. (53) implies |R1|·|R2|=|R|·|H|⇒r2/4=(1-r),which is satisfied by r/2=2-1.Each connected component of R′ is smaller than R by this factor.

+

Now we repeat this construction recursively. In each round of the procedure, we start with a disconnected region R˜ such that E[R˜] contains x, where R˜ is the union of many connected components of equal size. Then, we punch a hole out of each connected component to obtain a new region R˜′ such that E[R˜′] still contains x. Punching the holes increases the number of connected components by a factor of 2 and reduces the size of each component by a factor of r/2.

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The procedure halts when the connected components are reduced in size to the lattice spacing a, which occurs after m rounds, where a=(r/2)m|R|.The remaining region Rmin has 2m components, each containing one lattice site. so that d(AX)≤|Rmin|/a=2m=(|R|/a)α,where α=log2log(2/r)=1log2(2+1)≈0.786.The initial interval R is surely no larger than the whole boundary, so the distance is bounded above by nα for any logical subalgebra, where d and n are expressed as a number of boundary sites (rather than length along the boundary).

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We can also consider codes with punctures in the bulk. To be specific, suppose B is a hyperbolic disk of proper radius rout, with a single puncture at the center of radius rin. The code length n is proportional to the circumference of the outer boundary, and the size k of the logical system is proportional to the circumference of the inner boundary. Because the circumference of a circle with radius r is 2πLer/L, the rate of the code is k/n=e(rin-rout)/L.We may choose an interval R on the boundary, such that χR is tangent to the inner boundary at a single point. The length of this geodesic is essentially twice the difference between the inner and outer boundaries, so Eq. (53) implies rout-rin=Llog(|R|/a).Using the recursive construction to repeatedly carve holes out of R, we obtain the bound Eq. (60) on the code distance, which becomes d≤(|R|/a)α=(e(rout-rin)/L)α=(n/k)α(with the code distance expressed as a number of boundary sites). This scaling of the code distance, with α≈0.786, compares favorably with the bound [27] on local commuting projector codes defined on a two-dimensional Euclidean lattice, for which α=1/2.

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The scaling p(AX)∼nα applies to price as well as distance. Once we have found a sufficiently large boundary region R such that E[R] contains the bulk region X, we can proceed to hollow out R recursively until we reach the much smaller region Rmin such that |Rmin|/a=(|R|/a)α, where E[Rmin] still contains X, and hence AX is supported on Rmin. The resulting region Rmin, with fractal dimension α, has a geometry reminiscent of the Cantor set, as illustrated in Fig. 5.

+ + 5 + 10.1103/PhysRevX.7.021022.f5 + + +

This figure illustrates uberholography for the case of a two-dimensional hyperbolic bulk geometry. The inner logical boundary is contained inside the entanglement wedge, shaded in blue, of a boundary region R. By repeatedly punching holes of decreasing size out of this boundary region, we obtain a much smaller region Rmin whose entanglement wedge still contains the logical boundary. Thus, the logical algebra is supported on a fractal boundary set, whose geometry is reminiscent of the Cantor set.

+ + + +

It is interesting to compare this universal exponent α for planar uberholography with the scaling laws for distance and price realized by other code families, such as holographic tensor network codes and concatenated quantum codes. In fact, it can be quite challenging to obtain tight lower bounds on distance and price for tensor network code constructions; see the Appendix for further discussion.

+ + + + QUANTUM MARKOV CONDITION AND LOCAL CORRECTABILITY +

For a holographic code, consider (as in Sec. IV) a connected region R=R1HR2, which is the disjoint union of three adjoining intervals. Imagine that the middle interval H is erased. If H is correctable, there is a recovery map R that corrects this erasure error. However, now we ask whether a stronger condition is satisfied: Is it possible to choose a recovery map taking R′=R1R2 to R so that RR′→R′H “fills in” the erased hole H? If the erasure of H can be corrected by a map that acts only on a somewhat larger region containing H (larger by a constant factor independent of system size), then we say that erasure is locally correctable.

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The quantum Markov condition provides a criterion for local correctability [28]. We say that the state ρABC of three disjoint regions A, B, C obeys the quantum Markov condition (also called quantum conditional independence) if 0=I(A;C|B)=S(AB)+S(BC)-S(ABC)-S(B),which is equivalent to saying that the strong subadditivity inequality is saturated (satisfied as an equality). If the Markov condition is satisfied, then ρABC can be reconstructed from the marginal state ρAB using a map RB→BC, which maps B→BC: RB→BC:  ρAB↦ρABC,known as the Petz recovery map [29]. See Ref. [30] for a construction of a map that is robust to condition (65) holding only approximately. Likewise, in view of the symmetry of the condition under the interchange of A and C, ρABC can be reconstructed from ρBC by a map from B to AB.

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In fact, Eq. (65) implies that B has a decomposition as a direct sum of tensor products of Hilbert spaces, HB=⨁jHBj=⨁jHBjL⊗HBjR,and that the state of ABC has the block-diagonal form ρABC=⨁jpjρABjL⊗ρBjRC.Evidently, we can recover ρABC from ρAB by replacing each ρBjR by ρBjRC, without touching the system A.

+

To apply the Markov condition to our holographic setting, consider a holographic code with no punctures, where the state of the physical boundary is pure. We choose A, B, C to be three disjoint regions whose union is the complete boundary, namely, A=Rc,B=R′,C=H,where Rc denotes the boundary region complementary to R. Because the state of the complete boundary is pure, S(ABC)=0 and S(H)=S(Hc); therefore, the condition Eq. (65) becomes S(AB)+S(BC)=S(B)⇔S(H)+S(R)=S(R′).When this condition is satisfied, R′ can be divided into two subsystems, where one purifies the state of H and the other purifies the state of Rc. To correct the erasure of H, we need only restore the entanglement between R′ and H, and for this purpose, there is no need to venture outside R.

+ + + Hyperbolic bulk +

Using the Ryu-Takayanagi formula, this statement Eq. (70) about entropy would follow from a statement about minimal surfaces: χR′=χH⊔χR,which is the same as the condition (discussed in Sec. IV) for the entanglement wedge E]R′] to be E[R]\E[H]. For the case in which the bulk is a hyperbolic disk, the calculation in Sec. IV shows that erasure of H can be corrected by a recovery map that acts on region R containing H, where |R|/|H|=(1-r)-1=3+22≈5.828. Thus, the erasure error is locally correctable. This local correctability is a general feature of holographic codes with asymptotically uniform negative bulk curvature.

+

We may also consider the case of a manifold with punctures, where the logical boundary Λ is maximally entangled with a reference system. In that case, the entropy of the physical boundary matches S(Λ), and the Markov condition is satisfied provided that |χR′|+|χΛ|=|χHc|+|χR|,which holds if χR′=χH⊔χR,χHc=χH⊔χΛ.As for the case without punctures, Eq. (73) will be satisfied if H is a sufficiently small interval on the physical boundary of the hyperbolic disk and R is an interval containing H, where |R| is larger than |H| by a constant factor. The interpretation is the same as before; Eq. (73) implies that R′ contains a subsystem that purifies H, so there is no need to reach outside of R to recover from the erasure of H.

+

It is also notable that if the Markov condition Eq. (65) is approximately satisfied, then a local recovery map can be constructed which approximately corrects the erasure of H. This is important because in realistic AdS/CFT, the Markov condition is not exactly satisfied because of the small corrections to the Ryu-Takayanagi formula, which we have neglected. Local correctability in the approximate setting has been discussed recently in Refs. [28,31,32].

+

While holographic codes based on tensor networks can successfully reproduce the Ryu-Takayanagi relation satisfied by the von Neumann entanglement entropy of the boundary theory [2,3], they do not correctly capture the properties of Rényi entropies [33–35] and therefore do not provide a fully satisfactory description of conformal field theories with dual geometries. It is fortunate that the Markov condition Eq. (65), and its approximate version [36], is stated in terms of von Neumann entanglement entropies. We therefore expect that holographic tensor network codes can provide a reasonable picture of local correctability in realistic holography.

+ + + + Flat bulk +

The criterion for local correctability is satisfied by generic negatively curved bulk geometries but not by bulk geometries that are flat or positively curved. Consider, for example, a Euclidean two-dimensional disk with unit radius. For an interval R on the boundary that subtends angle θ, the geodesic χR is a chord of the boundary circle with length |χR|=2sin(θ/2). Suppose we erase a hole H that subtends angle 2δ and correct the erasure by acting in a larger region R that contains H. If R=R1HR2 subtends angle 2ϕ, where |R1|=|R2|, the Markov condition can be satisfied only if |χR1|+|χR2|=4sin((ϕ-δ)/2)≥|χR|+|χH|=2sin(ϕ)+2sin(δ).If δ and ϕ are small, this condition becomes, to leading order in small quantities, δ≤ϕ3/16.Thus, when |H| is small, |R|∼|H|1/3 is far larger; the erasure is not locally correctable. The same will be true, even more so, for a positively curved bulk geometry.

+

The failure of local correctability for holographic codes associated with flat and positively curved bulk manifolds suggests that, in these cases, the physics of the boundary system is highly nonlocal; in particular, the boundary state is not likely to be the ground state of a local Hamiltonian. This conclusion is reinforced by the observation that, according to the Ryu-Takayanagi formula, the entanglement entropy of a small connected region on the boundary of a flat ball scales linearly with the boundary volume of the region; this strong violation of the entanglement area law would not be expected in the ground state if the Hamiltonian is local.

+

That flat bulk geometry implies nonlocal boundary physics also teaches us a valuable lesson about AdS/CFT. A holographic tensor network provides not just an isometric map from the logical boundary Λ to the physical boundary Φ of the manifold B, but also a map from Λ to the boundary ∂X of a bulk region X that contains Λ. If the bulk geometry of X is flat or nearly flat, the entanglement structure of holographic codes indicates that the system supported on ∂X should exhibit flagrant violations of bulk locality. This picture suggests that black holes in the bulk that are small compared to the AdS curvature scale ought to have highly nonlocal dynamics, as seems necessary for these small black holes to be fast scramblers of quantum information [37,38].

+ + + + Positively curved bulk +

This nonlocality of boundary physics is even more pronounced for holographic codes defined on positively curved manifolds. Consider the extreme case of a two-dimensional hemisphere B, with the boundary ∂B at its equator. Geodesics on the sphere are great circles, lying in a plane that passes through the sphere’s center. For any boundary region R with |R|≠|∂B|/2, there is a unique minimal surface χR, which lies in ∂B; for |R|<|∂B|/2, we have χR=R, while for |R|>|∂B|/2, we have χR=∂B\R. Invoking the Ryu-Takayanagi formula, we see that as R increases in size, the entropy S(R) rises linearly as |R| until R occupies half of the boundary; it then decreases linearly thereafter. This behavior is the same as for a Haar-random pure state [39].

+

For |R|<|∂B|/2, the entanglement wedge E[R] contains only R, while for |R|>|∂B|/2, we have R⊔χR=∂B, and the entanglement wedge E[R] is all of the hemisphere B. Accordingly, for any region X in the bulk with associated bulk logical algebra AX, the price p(AX) and distance d(AX) are both given by |∂B|/2. This also mimics the behavior of a Haar-random pure state.

+

If we regulate bulk and boundary by introducing a lattice spacing a, then the number n of boundary sites in the code block is n=|∂B|/a=2πr/a,where r is the radius of the sphere. It is noteworthy that the area of the hemisphere, expressed in lattice units, is |B|/a2=2πr2/a2=n2/2π,which is quadratic in n. Since the hemisphere is the surface of smallest area whose boundary reproduces the entanglement structure of a Haar-random state, it is tempting to interpret the area n2/2π as a measure of the circuit complexity of preparing this state, in accordance with the complexity-equals-action conjecture [40]. Indeed, a random geometrically local circuit in the bulk containing O(n2) gates can closely approximate a unitary 2-design, which prepares a state with the desired properties [41].

+

As noted in Sec. III B, we can crudely model a black hole in the bulk by punching a hole in the bulk, with the microstates of the black hole residing on the logical boundary Λ of the puncture. If we introduce a reference system that is maximally entangled with Λ, then the black hole microstates are maximally mixed, and the entanglement entropy S(Λ)=|Λ| counts these microstates, in agreement with the black hole’s Bekenstein-Hawking entropy. Alternatively, we might wish to describe a black hole in a typical pure state, rather than a highly mixed state. Our observations about the properties of a holographic code defined on a hemisphere suggest how this can be done. Instead of entangling its boundary with a reference system, we fill the puncture with a hemispherical cap. The tensor network filling this cap realizes the minimal geometrically local procedure for preparing the black hole’s highly scrambled pure state.

+ + + + + HOLOGRAPHIC STRONG QUANTUM SINGLETON BOUND +

In Sec. II D, we discussed the strong quantum Singleton bound, Corollary 1, which relates p, d, and k for a code subspace, and we left open whether this bound can be extended to more general logical operator algebras. Here, we will see that, for holographic codes, such an extension is possible.

+

We consider the case of a holographic code with punctures in the bulk; hence, there is a physical boundary Φ and a logical boundary Λ as discussed in Sec. III B. In the formal continuum limit, the code parameters p, d, and k are measured in units of area, and the contribution to the area from O(1) boundary sites can be neglected; hence, Eq. (38) becomes k≤p-d.We would like to show that this constraint applies to logical subalgebras of a holographic code.

+

We consider a region X in the bulk, and its associated logical subalgebra AX. The region X may contain a portion of the logical boundary Λ, as well as some additional isolated points in the bulk. We denote the intersection X∩Λ of X with the logical boundary by ΛX; if ΛX is nonempty, then the bulk points are a negligible portion of the subalgebra AX, whose size is therefore kX=|ΛX|.

+

In what follows, for the sake of clarity, we denote the minimal surface associated with boundary region R by χ(R), in place of the subscript notation χR used earlier. We also use the notation Rc for the physical complement Φ\R, and p, d, k as a shorthand for p(AX), d(AX), kX.

+

Let Rp be a region of the physical boundary Φ such that |Rp|=p(AX) and E[Rp] contains X; this means that the associated minimal surface χ(Rp) must contain ΛX. Let Rd⊆Rp be a subset of Rp such that, in the regulated theory with a nonzero lattice spacing, Rd contains one less boundary site than the distance of AX; therefore, Rd is surely correctable with respect to AX, and in the continuum limit (where a single site has negligible size), |Rd|=d(AX). Because Rd is correctable, the entanglement wedge of its physical complement Rdc contains X, which means that χ(Rdc) contains ΛX.

+

We may consider gradually “growing” a boundary region from Rd to Rp, obtaining an inequality by observing that the corresponding minimal surface cannot grow faster than the boundary surface itself: |Rp|-|Rd|≥∫RdRpdRd|χ(R)|dR=|χ(Rp)|-|χ(Rd)|,|Rdc|-|Rpc|≥∫RpcRdcdRd|χ(R)|dR=|χ(Rdc)|-|χ(Rpc)|.Together with p-d=|Rp|-|Rd|=|Rdc|-|Rpc|, this implies p-d≥|χ(Rp)|-|χ(Rd)|+|χ(Rdc)|-|χ(Rpc)|2.

+

Now recall that χ(Rp) contains ΛX. Hence, the rest of the minimal surface χ(Rp), excluding ΛX, is χ(Rp∪ΛX), or in other words, χ(Rp)=χ(Rp∪ΛX)∪ΛX⇒|χ(Rp)|=|χ(Rp∪ΛX)|+|ΛX|.Likewise, χ(Rdc) contains ΛX, which implies |χ(Rdc)|=|χ(Rdc∪ΛX)|+|ΛX|.Plugging this into Eq. (80) yields p-d-|ΛX|=p-d-k≥12(|χ(Rp∪ΛX)|+|χ(Rdc∪ΛX)|-|χ(Rd)|-|χ(Rpc)|).

+

Now we can use the property that two complementary boundary regions share the same minimal bulk surface (where by the “complement” we mean the boundary complement rather than the physical complement; that is, we are simultaneously taking the complement with respect to the logical and physical boundaries). Let us denote by ΛXc the complement of ΛX with respect to the logical boundary so that Λ=ΛXΛXc. Then, χ(Rdc∪ΛX)=χ(Rd∪ΛXc),χ(Rpc)=χ(Rp∪ΛXΛXc),and hence, p-d-k≥|χ(Rp∪ΛX)|/2+|χ(Rd∪ΛXc)|/2-|χ(Rd)|/2-|χ(Rp∪ΛXΛXc)|/2.Using the Ryu-Takayanagi relation between entropy and area, and identifying AB=Rp∪ΛX,BC=Rd∪ΛXc,B=Rd,ABC=Rp∪ΛXΛXc,the right-hand side of Eq. (86) is proportional to S(AB)+S(BC)-S(B)-S(ABC),which is non-negative by strong subadditivity of entropy. This completes the holographic proof of the strong quantum Singleton bound.

(holographic strong quantum Singleton bound). Consider a holographic code with logical boundary Λ, and a logical subalgebra AX associated with bulk region X, where kX=|X∩Λ|. Then, the price and distance of AX obey kX≤p(AX)-d(AX).

+

It is intriguing that we used strong subadditivity of entropy in this holographic proof, which applies to logical subalgebras, while the proof of Corollary 1, which applies to the price and distance of a traditional code subspace, used only subadditivity. We have not found a proof of the strong quantum Singleton bound that applies to logical subalgebras and that does not use holographic reasoning; it is an open question whether Eq. (88) holds beyond the setting of holographic codes.

+ + + + DISCUSSION AND OUTLOOK +

Our studies of holographic codes have only scratched the surface of this subject. More in-depth studies are needed, including searches, guided by geometrical intuition, for codes with improved parameters and investigations of the efficiency of decoding.

+

Regarding the implications of holographic codes for quantum gravity, we have uncovered several hints that may help steer future research. We have seen that positive curvature of the bulk manifold can improve properties such as the code distance but at a cost—increasing distance is accompanied by enhanced nonlocality of the boundary system. The observation that the logical algebra of a bulk point has price equal to distance is a step toward characterizing bulk geometry using algebraic ideas, and we anticipate further advances in that direction. Uberholography, in bulk spacetimes with asymptotically negative curvature, illustrates how notions from quantum coding can elucidate the emergence of bulk geometry beyond the appearance of just one extra spatial dimension.

+

Following Refs. [1,10], we have discussed boundary reconstruction of bulk physics using the formalism of OAQEC [5,6], which captures salient features of holography. To make firmer contact with realistic AdS/CFT, this discussion should be extended to the setting of approximate OAQEC [42]. First steps in this direction have already been taken in Ref. [31], a study of approximate erasure correction in the (1+1)-dimensional Ising CFT at very low temperatures, and in Ref. [32], an investigation of approximate local correctability in a MERA network, which has polynomially decaying correlations on its boundary.

+

We are encouraged by recent progress connecting quantum error correction and quantum gravity, but much remains unclear. Most obviously, our discussion of the entanglement wedge and bulk reconstruction applies only to static spacetimes or very special spatial slices through dynamical spacetimes. Applying the principles of quantum coding to more general dynamical spacetimes is an important goal, which poses serious unresolved challenges.

+ + + + + ACKNOWLEDGMENTS +

F. P. would like to thank Nicolas Delfosse, Henrik Wilming, and Jens Eisert for helpful discussions and comments. F. P. gratefully acknowledges funding provided by the Institute for Quantum Information and Matter, a NSF Physics Frontiers Center, with support from the Gordon and Betty Moore Foundation, as well as the Simons Foundation through the It from Qubit program and the FUB through the ERC project (TAQ). This research was supported in part by the National Science Foundation under Grant No. NSF PHY-1125915.

+ + + + + COMPARISON OF UBERHOLOGRAPHY DIMENSIONAL EXPONENT WITH EXPLICIT CODES +

In Sec. IV, we computed the universal fractal dimension α≈0.786 for a holographic code defined on the Poincaré disk, assuming that geometric complementarity and the entanglement wedge hypothesis are precisely satisfied. We may also define a fractal dimension for other code families, such as concatenated quantum codes or holographic tensor network codes. Choosing a particular logical algebra A (for example, the algebra of a logical qubit at the center of the bulk), let αp≔limℓ→∞logpℓlognℓ,αd≔limℓ→∞logdℓlognℓ,where pℓ and dℓ are the price and distance of the logical algebra after ℓ levels of concatenation or equivalent iteration. (If A is the algebra of local operators at the center of the bulk, then we may think of ℓ as the radial distance from the center of the bulk to its boundary.) The “no free lunch” lemma ensures that αp≥αd.

+

By a concatenated code, we mean a recursive hierarchy of codes within codes; these can be constructed in many ways. In the simplest case, we consider an [[n,1,d]] code C1, with just one logical qubit, which has an encoder isometrically mapping one qubit to a block of n physical qubits. The code C2, which has k=1 and length n2=n2, is obtained by applying this encoder to each of the n physical qubits in C1; likewise, the code Cℓ, with length nℓ, is obtained by applying the encoder to each of the nℓ-1 qubits in the code Cℓ-1. The corresponding tensor network, with one logical qubit at its center, is a branching tree extending radially outward, in which each branch has n descendants.

+

Suppose that n is odd and the code C1 has the largest possible distance d=(n+1)/2. The complementarity bound Eq. (25) then implies that the price is p=d; the full logical algebra can be supported on d of the n qubits, and all nontrivial logical Pauli operators have weight d. Therefore, all nontrivial logical operators of Cℓ can be supported on dℓ qubits, and all have weight dℓ. We conclude that αp=αd=logdlogn.As n increases, αp and αd approach 1 from below. Although the tensor network can be embedded in a plane, it does not approximate the geometry of the Poincaré disk, and its price and distance obey a different scaling law than we found in Sec. IV.

+

A more complicated recursive encoding scheme, based on an [[n,k,d]] code C1 with k>1, is depicted in Fig. 6. In this case, the C1 encoder maps k qubits to a block of n qubits. To build the code Cℓ, we first assemble k copies of the code Cℓ-1, and then apply the k→n encoder for C1 altogether nℓ-1 times, where each encoder acts on k qubits drawn from the k distinct copies. Each time we add another layer to the code, the number of encoded qubits increases by a factor of k, and the number of physical qubits increases by a factor of n; therefore, nℓ=nℓ,kℓ=kℓ.However, in this case, the price and distance of Cℓ are not so easy to calculate, though we can derive some simple bounds. When we add an additional layer to the code, each nontrivial logical operator of Cℓ-1 maps to a logical operator of Cℓ whose weight is at least d times larger; furthermore, if all logical operators of Cℓ-1 can be supported on w physical qubits, then at most pw qubits are needed to support all logical operators of Cℓ. We therefore have dℓ≤dℓ≤pℓ≤pℓ.However, to make a more precise statement about the price and distance of Cℓ, we need more information about the structure of C1.

+ + 6 + 10.1103/PhysRevX.7.021022.f6 + + +

A recursive coding network to which Eqs. (A3) and (A4) apply, with logical qubits at the top and physical qubits at the bottom. To derive that the distance is bounded below by dℓ and the price is bounded above by pℓ, it suffices to observe that there is a unique “causal” path connecting each physical qubit to each logical qubit. Here, a [[4,2,2]] code (with the encoder drawn as a parallelogram) is concatenated to obtain a [[16,4,4]] code, and a causal path connecting a physical qubit to a logical qubit is highlighted.

+ + + +

We may also consider the price and distance of holographic tensor network codes, which capture some of the features of full-blown AdS/CFT duality [2]. For example, we can tile the Poincaré disk with pentagons and associate a six-index “perfect tensor” with each pentagon, where each pentagon carries a single logical qubit. For this pentagon code, where A is the logical algebra of the central pentagon, we find that the price pℓ(A) and distance dℓ(A) are badly mismatched (where ℓ denotes the graph distance from the central pentagon to the physical boundary of the tensor network). In fact, the distance dℓ(A)=4 is a constant independent of ℓ, as explained in Sec. 5.6 of Ref. [2]; in other words, there are logical operators of weight 4 that act nontrivially on the central qubit. In contrast, we expect the price to scale as pℓ(A)=nℓαp5 with 0.786<αp5<1 (i.e., with an exponent larger than the value attained in idealized holography). This upper bound may be derived using a discrete version of the hole-punching approach, thereby explicitly constructing a subset of the physical boundary qubits for which the greedy algorithm of Ref. [2] reaches the central tensor. It is more difficult to obtain lower bounds on pℓ, as these cannot be witnessed by examples.

+

A nontrivial scaling exponent for the distance dℓ(A) can be obtained if we thin out the logical qubits, replacing pentagons in the bulk by hexagons that carry no logical qubit index. (In such codes, the central qubit is well protected against erasure of a randomly chosen nonzero fraction of all the physical boundary qubits, as shown in Ref. [2].) Codes with relatively sparse bulk logical qubits are better suited than the pentagon code for illustrating the ideas we have explored in this paper, where we have focused on the regime in which geometric entanglement dominates bulk entanglement. We may anticipate that holographic tensor network code families that mimic the geometry of the Poincaré disk will have a price-scaling exponent αp that approximates α≈0.786 from above and a distance-scaling exponent αd that approximates α from below. We have confirmed this expectation by studying some examples, though we have no rigorous general argument.

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diff --git a/tests/data/aps/PhysRevX.7.021022_expected.yml b/tests/data/aps/PhysRevX.7.021022_expected.yml new file mode 100644 index 0000000..cdb5bf1 --- /dev/null +++ b/tests/data/aps/PhysRevX.7.021022_expected.yml @@ -0,0 +1,1208 @@ +abstract: Almheiri, Dong, and Harlow [J. High Energy Phys.04 (2015) 163.] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius. +artid: "021022" +publication_date: 2017-04-01 +year: 2017 +title: Code Properties from Holographic Geometries +journal_issue: "2" +journal_volume: "7" +publisher: American Physical Society +journal_title: Physical Review X +dois: +- doi: 10.1103/PhysRevX.7.021022 + material: publication +authors: +- full_name: Pastawski, Fernando + raw_affiliations: + - source: American Physical Society + value: Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany +- full_name: Preskill, John + raw_affiliations: + - source: American Physical Society + value: Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA +copyright_holder: authors +copyright_statement: Published by the American Physical Society +copyright_year: 2017 +number_of_pages: 20 +document_type: article +material: publication +license_url: https://creativecommons.org/licenses/by/4.0/ +license_statement: Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. +article_type: research-article +is_conference_paper: false +documents: +- key: PhysRevX.7.021022.xml + url: http://example.org/PhysRevX.7.021022.xml + source: American Physical Society + fulltext: true + hidden: true +references: +- reference: + publication_info: + journal_volume: '2015' + page_start: '163' + journal_issue: '04' + artid: '163' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Almheiri, A. + - inspire_role: author + full_name: Dong, X. + - inspire_role: author + full_name: Harlow, D. + title: + title: Bulk Locality and Quantum Error Correction in AdS/CFT + label: '1' + dois: + - 10.1007/JHEP04(2015)163 + raw_refs: + - source: American Physical Society + value: "\n \n 1A. Almheiri, X. Dong,\ + \ and D. Harlow, Bulk\ + \ Locality and Quantum Error Correction in AdS/CFT, J.\ + \ High Energy Phys.04 (2015) 163.JHEPFG1029-847910.1007/JHEP04(2015)163\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '2015' + page_start: '149' + journal_issue: '06' + artid: '149' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Pastawski, F. + - inspire_role: author + full_name: Yoshida, B. + - inspire_role: author + full_name: Harlow, D. + - inspire_role: author + full_name: Preskill, J. + title: + title: 'Holographic Quantum Error-Correcting Codes: Toy Models for the Bulk/Boundary + Correspondence' + label: '2' + dois: + - 10.1007/JHEP06(2015)149 + raw_refs: + - source: American Physical Society + value: "\n \n 2F. Pastawski, B. Yoshida,\ + \ D. Harlow, and J. Preskill,\ + \ Holographic Quantum Error-Correcting Codes: Toy Models for\ + \ the Bulk/Boundary Correspondence, J. 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Fawzi,\ + \ and R. Renner, Universal\ + \ Recovery Map for Approximate Markov Chains, Proc.\ + \ R. Soc. Lond A472, 623\ + \ (2016).1364-502110.1098/rspa.2015.0623\n " + schema: JATS +- reference: + publication_info: + journal_volume: '2007' + page_start: '120' + journal_issue: 09 + artid: '120' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Hayden, P. + - inspire_role: author + full_name: Preskill, J. + title: + title: 'Black Holes as Mirrors: Quantum Information in Random Subsystems' + label: '37' + dois: + - 10.1088/1126-6708/2007/09/120 + raw_refs: + - source: American Physical Society + value: "\n \n 37P. Hayden and J. Preskill,\ + \ Black Holes as Mirrors: Quantum Information in Random Subsystems,\ + \ J. High Energy Phys.09 (2007)\ + \ 120.JHEPFG1029-847910.1088/1126-6708/2007/09/120\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '2008' + page_start: '065' + journal_issue: '10' + artid: '065' + journal_title: J. High Energy Phys. + authors: + - inspire_role: author + full_name: Sekino, Y. + - inspire_role: author + full_name: Susskind, L. + title: + title: Fast Scramblers + label: '38' + dois: + - 10.1088/1126-6708/2008/10/065 + raw_refs: + - source: American Physical Society + value: "\n \n 38Y. Sekino and L. Susskind,\ + \ Fast Scramblers, J. High Energy Phys.10\ + \ (2008) 065.JHEPFG1029-847910.1088/1126-6708/2008/10/065\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '71' + page_start: '1291' + year: 1993 + artid: '1291' + journal_title: Phys. Rev. Lett. + authors: + - inspire_role: author + full_name: Page, D.N. + title: + title: Average Entropy of a Subsystem + label: '39' + dois: + - 10.1103/PhysRevLett.71.1291 + raw_refs: + - source: American Physical Society + value: "\n \n 39D. N. Page, Average\ + \ Entropy of a Subsystem, Phys. Rev. Lett.71,\ + \ 1291 (1993).PRLTAO0031-900710.1103/PhysRevLett.71.1291\n \ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '116' + year: 2016 + artid: '191301' + journal_title: Phys. Rev. Lett. + authors: + - inspire_role: author + full_name: Brown, A.R. + - inspire_role: author + full_name: Roberts, D.A. + - inspire_role: author + full_name: Susskind, L. + - inspire_role: author + full_name: Swingle, B. + - inspire_role: author + full_name: Zhao, Y. + title: + title: Holographic Complexity Equals Bulk Action? + label: '40' + dois: + - 10.1103/PhysRevLett.116.191301 + raw_refs: + - source: American Physical Society + value: "\n \n 40A. R. Brown, D. A. Roberts,\ + \ L. Susskind, B. Swingle,\ + \ and Y. Zhao, Holographic\ + \ Complexity Equals Bulk Action?, Phys. Rev. Lett.116,\ + \ 191301 (2016).PRLTAO0031-900710.1103/PhysRevLett.116.191301\n\ + \ " + schema: JATS +- reference: + publication_info: + journal_volume: '346' + page_start: '397' + year: 2016 + artid: '397' + journal_title: Commun. Math. Phys. + authors: + - inspire_role: author + full_name: Brandão, F.G.S.L. + - inspire_role: author + full_name: Harrow, A.W. + - inspire_role: author + full_name: Horodecki, M. + title: + title: Local Random Quantum Circuits are Approximate Polynomial-Designs + label: '41' + dois: + - 10.1007/s00220-016-2706-8 + raw_refs: + - source: American Physical Society + value: "\n \n 41F. G. S. L. Brandão, A. W.\ + \ Harrow, and M. Horodecki,\ + \ Local Random Quantum Circuits are Approximate Polynomial-Designs,\ + \ Commun. Math. Phys.346, 397\ + \ (2016).CMPHAY0010-361610.1007/s00220-016-2706-8\n\ + \ " + schema: JATS +- reference: + authors: + - inspire_role: author + full_name: Bény, C. + title: + title: Conditions for the Approximate Correction of Algebras + label: '42' + arxiv_eprint: '0907.4207' + raw_refs: + - source: American Physical Society + value: "\n \n 42C. Bény, Conditions\ + \ for the Approximate Correction of Algebras, arXiv:0907.4207.\n " + schema: JATS diff --git a/tests/data/aps/Physics.15.168.xml b/tests/data/aps/Physics.15.168.xml new file mode 100644 index 0000000..85bcf86 --- /dev/null +++ b/tests/data/aps/Physics.15.168.xml @@ -0,0 +1,141 @@ + + +
+ + + PHYSICS + PHYSGM + + Physics + Physics + Physics + + 1943-2879 + + American Physical Society + + + + 10.1103/Physics.15.168 + + + VIEWPOINTS + + + + quantum-info + Quantum Information + + + + + quantum + Quantum Physics + + + + + Chirping toward a Quantum RAM + + + + + Pla + Jarryd + + + + +

Jarryd Pla is a quantum engineer at the University of New South Wales, Sydney. He works on problems related to quantum information processing and more broadly to quantum technologies. Pla was instrumental in demonstrating the first quantum bits made from the electron and nucleus of a single impurity atom inside a silicon chip. His current research interests span spin-based quantum computation, superconducting quantum circuits, and hybrid quantum technologies. He is focused on developing new quantum technologies to aid the scaling of quantum computers and to advance capabilities in spectroscopy and sensing.

+ + + 1 + + + School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia + + + 07 + November + 2022 + + 15 + + 168 + + © 2022 American Physical Society + 2022 + American Physical Society + + + +

A new quantum random-access memory device reads and writes information using a chirped electromagnetic pulse and a superconducting resonator, making it significantly more hardware-efficient than previous devices.

+ + + + + + + + + 10.1103/Physics.15.168.f1 + 1 + + +

Researchers have developed a RAM device from a superconducting circuit resonator and a silicon chip embedded with bismuth atoms. Chirped microwave pulses transfer quantum information back and forth between the resonator and the bismuth atoms, where the information is stored in the atoms’ spin states.

+ + + APS/Carin Cain + +

Random-access memory (or RAM) is an integral part of a computer, acting as a short-term memory bank from which information can be quickly recalled. Applications on your phone or computer use RAM so that you can switch between tasks in the blink of an eye. Researchers working on building future quantum computers hope that such systems might one day operate with analogous quantum RAM elements, which they envision could speed up the execution of a quantum algorithm [1, 2] or increase the density of information storable in a quantum processor. Now James O’Sullivan of the London Centre for Nanotechnology and colleagues have taken an important step toward making quantum RAM a reality, demonstrating a hardware-efficient approach that uses chirped microwave pulses to store and retrieve quantum information in atomic spins [3].

+

Just like quantum computers, experimental demonstrations of quantum memory devices are in their early days. One leading chip-based platform for quantum computation uses circuits made from superconducting metals. In this system, the central processing is done with superconducting qubits, which send and receive information via microwave photons. At present, however, there exists no quantum memory device that can reliably store these photons for long times. Luckily, scientists have a few ideas.

+

One of those ideas is to use the spins of impurity atoms embedded in the superconducting circuit’s chip. Spin is one of the fundamental quantum properties of an atom. It acts like an internal compass needle, aligning with or against an applied magnetic field. These two alignments are analogous to the 0 and 1 of a classical bit and can be used to store quantum information [4, 5]. If the chip contains many impurity atoms, the atoms’ spins can act as a “multimode” memory device—one that can simultaneously store the information contained in numerous photons.

+

For atomic spins, the information-storage times can be orders of magnitude longer than those of superconducting qubits. Researchers have shown, for example, that bismuth atoms placed inside silicon chips can store quantum information for times longer than a second [6]. One might ask: why not use spin qubits in place of superconducting qubits? Indeed, there are research groups working on atom-based quantum computers, but the control and measurement of atomic spins presents its own unique challenges. A hybrid approach is to use superconducting qubits for processing and atomic spins for storage, but here the challenge has been how to transfer information between the two systems using microwave photons. While researchers have already demonstrated the absorption and retrieval of information from microwave photons by an atomic spin ensemble, those demonstrations required the use of strong magnetic-field gradients or specialized superconducting circuitry, both of which add complexity to the quantum memory hardware [7, 8].

+

O’Sullivan and his colleagues offer an elegant solution to microwave photon information storage and retrieval that uses a hardware-efficient approach. The team’s device consists of a superconducting circuit resonator that sits on a silicon chip embedded with bismuth atoms (Fig. 1). The team sent into the resonator weak microwave excitations containing approximately 1000 photons, which were absorbed by the spins of the bismuth atoms. They then hit the resonator with electromagnetic microwave pulses that had a frequency that ramped up over time, an effect known as chirping. Because of that, the quantum information contained in the photons imprinted on the spins with a unique “phase” identifier, which captured the relative pointing positions of neighboring spins. The team then retrieved this information, transferring photons back to the superconducting circuit, by hitting the spin ensemble with an identical pulse, which they found reversed this imprinted phase.

+

O’Sullivan and colleagues show that their memory device is able to simultaneously store multiple pieces of photonic information in the form of four weak microwave pulses. Importantly, they also demonstrate that the information can be read back in any order, making their device a true RAM.

+

In this first demonstration, the team reports a 3% efficiency, indicating that most of the information is lost by the memory. Thus, their device is still some way from the faithful storage and retrieval required for a future quantum computer. However, an analysis of the potential sources of this low efficiency indicates that it does not come from the transfer process but instead arises from potentially resolvable limitations of the device. The team thinks that by increasing the number of spins they could significantly improve the device’s efficiency.

+

As well as storing information, quantum RAM elements could help in increasing the density of qubits in a quantum processor. In September, IBM introduced Project Goldeneye, a large dilution refrigerator [9]. This ultracold behemoth has a volume larger than that of three home refrigerators and will host IBM’s next-generation superconducting quantum computer. Current superconducting quantum computers have a qubit density of less than 100 per square millimeter—classical computer chips contain 100 million transistors per square millimeter—so it is understandable why IBM needs such a large refrigerator. O’Sullivan and colleagues’ spin-based quantum memory device could, in principle, store multiple qubit states in the space currently occupied by just one, which might one day help alleviate this size problem.

+ + + + + + 1V. Giovannetti, “Quantum random access memory,” Phys. Rev. Lett. 100, 160501 (2008)10.1103/PhysRevLett.100.160501. + + + + 2J. Biamonte, “Quantum machine learning,” Nature 549, 195 (2017)10.1038/nature23474. + + + + 3J. O’Sullivan, “Random-access quantum memory using chirped pulse phase encoding,” Phys. Rev. X 12, 04101410.1103/PhysRevX.12.041014 (2022). + + + + 4Daniel Loss and D. P. DiVincenzo, “Quantum computation with quantum dots,” Phys. Rev. A 57, 120 (1998)10.1103/PhysRevA.57.120. + + + + 5B. E. Kane, “A silicon-based nuclear spin quantum computer,” Nature 393, 133 (1998)10.1038/30156. + + + + 6G. Wolfowicz, “Atomic clock transitions in silicon-based spin qubits,” Nat. Nanotechnol. 8, 561 (2013)10.1038/nnano.2013.117. + + + + 7H. Wu, “Storage of multiple coherent microwave excitations in an electron spin ensemble,” Phys. Rev. Lett. 105, 140503 (2010)10.1103/PhysRevLett.105.140503. + + + + 8C. Grezes, “Multimode storage and retrieval of microwave fields in a spin ensemble,” Phys. Rev. X 4, 021049 (2014)10.1103/PhysRevX.4.021049. + + + + 9P. Gumann and J. Chow, “IBM scientists cool down the world’s largest quantum-ready cryogenic concept system,” IBM blog, 8 Sept. 2022. + + + +
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The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized." +imprints: '2018' diff --git a/tests/data/crossref/9781316535783.011.json b/tests/data/crossref/9781316535783.011.json new file mode 100644 index 0000000..1e13663 --- /dev/null +++ b/tests/data/crossref/9781316535783.011.json @@ -0,0 +1,127 @@ +{ + "status":"ok", + "message-version":"1.0.0", + "message":{ + "DOI":"10.1017/9781316535783.011", + "subtitle":[ + + ], + "issued":{ + "date-parts":[ + [] + ] + }, + "publisher-location":"Cambridge", + "short-title":[ + + ], + "prefix":"10.1017", + "relation":{ + + }, + "ISBN":[ + "9781316535783" + ], + "author":[ + { + "affiliation":[ + + ], + "given":"Chris", + "family":"Smeenk" + } + ], + "reference-count":0, + "member":"56", + "source":"Crossref", + "score":1.0, + "deposited":{ + "timestamp":1523597812000, + "date-time":"2018-04-13T05:36:52Z", + "date-parts":[ + [ + 2018, + 4, + 13 + ] + ] + }, + "editor":[ + { + "affiliation":[ + + ], + "given":"Khalil", + "family":"Chamcham" + }, + { + "affiliation":[ + + ], + "given":"Joseph", + "family":"Silk" + }, + { + "affiliation":[ + + ], + "given":"John D.", + "family":"Barrow" + }, + { + "affiliation":[ + + ], + "given":"Simon", + "family":"Saunders" + } + ], + "indexed":{ + "timestamp":1524067539055, + "date-time":"2018-04-18T16:05:39Z", + "date-parts":[ + [ + 2018, + 4, + 18 + ] + ] + }, + "type":"book-chapter", + "URL":"http://dx.doi.org/10.1017/9781316535783.011", + "is-referenced-by-count":0, + "references-count":0, + "original-title":[ + + ], + "short-container-title":[ + + ], + "publisher":"Cambridge University Press", + "content-domain":{ + "domain":[ + + ], + "crossmark-restriction":"False" + }, + "created":{ + "timestamp":1492423296000, + "date-time":"2017-04-17T10:01:36Z", + "date-parts":[ + [ + 2017, + 4, + 17 + ] + ] + }, + "title":[ + "Testing Inflation" + ], + "container-title":[ + "The Philosophy Of Cosmology" + ], + "page":"206-227" + }, + "message-type":"work" +} \ No newline at end of file diff --git a/tests/data/crossref/9781316535783.011_expected.yml b/tests/data/crossref/9781316535783.011_expected.yml new file mode 100644 index 0000000..7826b8c --- /dev/null +++ b/tests/data/crossref/9781316535783.011_expected.yml @@ -0,0 +1,14 @@ +--- +dois: +- doi: 10.1017/9781316535783.011 + material: publication +title: Testing Inflation +authors: +- full_name: Smeenk, Chris +page_start: '206' +page_end: '227' +year: +material: publication +parent_isbn: '9781316535783' +document_type: book chapter +imprints: diff --git a/tests/data/crossref/PhysRevB.33.3547.2.json b/tests/data/crossref/PhysRevB.33.3547.2.json new file mode 100644 index 0000000..0ebca9b --- /dev/null +++ b/tests/data/crossref/PhysRevB.33.3547.2.json @@ -0,0 +1,159 @@ +{ + "status":"ok", + "message-version":"1.0.0", + "message":{ + "DOI":"10.1103/physrevb.33.3547.2", + "subtitle":[ + + ], + "issued":{ + "date-parts":[ + [ + 1986, + 3, + 1 + ] + ] + }, + "short-title":[ + + ], + "prefix":"10.1103", + "relation":{ + + }, + "subject":[ + "Condensed Matter Physics" + ], + "author":[ + { + "affiliation":[ + + ], + "given":"Marcos H.", + "family":"Degani" + }, + { + "affiliation":[ + + ], + "given":"Oscar", + "family":"Hipólito" + } + ], + "reference-count":0, + "ISSN":[ + "0163-1829" + ], + "member":"16", + "source":"Crossref", + "score":1.0, + "deposited":{ + "timestamp":1491471917000, + "date-time":"2017-04-06T09:45:17Z", + "date-parts":[ + [ + 2017, + 4, + 6 + ] + ] + }, + "indexed":{ + "timestamp":1508802763449, + "date-time":"2017-10-23T23:52:43Z", + "date-parts":[ + [ + 2017, + 10, + 23 + ] + ] + }, + "type":"journal-article", + "published-online":{ + "date-parts":[ + [ + 1986, + 3, + 1 + ] + ] + }, + "URL":"http://dx.doi.org/10.1103/physrevb.33.3547.2", + "is-referenced-by-count":1, + "volume":"33", + "issn-type":[ + { + "type":"print", + "value":"0163-1829" + } + ], + "link":[ + { + "URL":"http://link.aps.org/article/10.1103/PhysRevB.33.3547.2", + "intended-application":"syndication", + "content-version":"vor", + "content-type":"unspecified" + }, + { + "URL":"http://harvest.aps.org/v2/journals/articles/10.1103/PhysRevB.33.3547.2/fulltext", + "intended-application":"similarity-checking", + "content-version":"vor", + "content-type":"unspecified" + } + ], + "references-count":0, + "original-title":[ + + ], + "short-container-title":[ + "Phys. 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Oceanogr. + authors: + - inspire_role: author + full_name: Amon + dois: + - 10.4319/lo.1996.41.1.0041 + title: + title: Bacterial utilization of different size classes of dissolved organic + matter +- reference: + publication_info: + journal_volume: '255' + page_start: '1561' + year: 1992 + journal_title: Science + authors: + - inspire_role: author + full_name: Benner + dois: + - 10.1126/science.255.5051.1561 + title: + title: Bulk chemical characteristics of dissolved organic matter in the ocean +- reference: + publication_info: + journal_volume: '5' + page_start: '477' + year: 1983 + journal_title: J. Plankton Res. + authors: + - inspire_role: author + full_name: Chrost + dois: + - 10.1093/plankt/5.4.477 + title: + title: 'Organic carbon release by phytoplankton: its composition and utilization by bacterioplankton' +- reference: + publication_info: + journal_volume: '41' + page_start: '698' + year: 1996 + journal_title: Limnol. Oceanogr. + authors: + - inspire_role: author + full_name: Graneli + dois: + - 10.4319/lo.1996.41.4.0698 + title: + title: Photo-oxidative production of dissolved inorganic carbon in lakes of different humic content +- reference: + publication_info: + journal_volume: '8' + page_start: '229' + year: 1962 + journal_title: Can. J. Microbiol. + authors: + - inspire_role: author + full_name: Guillard + dois: + - 10.1139/m62-029 + title: + title: Studies of marine planktonic diatoms I. Cyclotella nana Hustedt and Detonula confervacea (Cleve) Gran +- reference: + publication_info: + journal_volume: '341' + page_start: '637' + year: 1989 + journal_title: Nature + authors: + - inspire_role: author + full_name: Kieber + dois: + - 10.1038/341637a0 + title: + title: 'Photochemical source of biological substrates in sea water: implications for carbon cycling' +- reference: + publication_info: + journal_volume: '40' + page_start: '195' + year: 1995 + journal_title: Limnol. Oceanogr. + authors: + - inspire_role: author + full_name: Lindell + dois: + - 10.4319/lo.1995.40.1.0195 + title: + title: Enhanced bacterial growth in response to photochemical transformation of dissolved organic matter +- reference: + publication_info: + journal_volume: '382' + page_start: '445' + year: 1996 + journal_title: Nature + authors: + - inspire_role: author + full_name: Lovley + dois: + - 10.1038/382445a0 + title: + title: Humic substances as electron acceptors for microbial respiration +- reference: + publication_info: + journal_volume: '22' + page_start: '417' + year: 1995 + journal_title: Geophys. Res. Lett. + authors: + - inspire_role: author + full_name: Miller + dois: + - 10.1029/94GL03344 + title: + title: 'Photochemical production of dissolved inorganic carbon from terrestrial organic matter: significance to the oceanic organic carbon cycle' +- reference: + publication_info: + page_start: '31' + year: 1970 + authors: + - inspire_role: author + full_name: Menzel + title: + title: Distribution and cycling of organic matter in the oceans +- reference: + publication_info: + journal_volume: '353' + page_start: '60' + year: 1991 + journal_title: Nature + authors: + - inspire_role: author + full_name: Mopper + dois: + - 10.1038/353060a0 + title: + title: Photochemical degradation of dissolved organic carbon and its impact on the oceanic carbon cycle +- reference: + publication_info: + journal_volume: '40' + page_start: '1381' + year: 1995 + journal_title: Limnol. Oceanogr. + authors: + - inspire_role: author + full_name: Morris + dois: + - 10.4319/lo.1995.40.8.1381 + title: + title: The attenuation of solar UV radiation in lakes and the role of dissolved organic carbon +- raw_refs: + - source: Crossref + value: Murakami, Y., 1999. Effects of a bacterial siderophore on microalgal proliferation. MSc thesis, School of Biosphere Science, Hiroshima University. + schema: text +- reference: + publication_info: + journal_volume: '53' + page_start: '435' + year: 1997 + journal_title: J. Oceanogr. + authors: + - inspire_role: author + full_name: Naganuma + title: + title: Abundance, production and viability of bacterioplankton in the Seto Inland Sea, Japan +- reference: + publication_info: + journal_volume: '135' + page_start: '309' + year: 1996 + journal_title: Mar. Ecol. Prog. Ser. + authors: + - inspire_role: author + full_name: Naganuma + dois: + - 10.3354/meps135309 + title: + title: Photodegradation or photoalteration? 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A theoretical explanation is that the inversion transformation, (Σ:t→−1/t, x→x/t), leaves the Euler equations of fluid dynamics, with standard polytropic exponent, invariant. This implies that the kinematical invariance group of the Euler equations is larger than the Galilei group. In this paper we determine, in a systematic manner, the maximal invariance group G of general fluid dynamics and show that it is a semi-direct product G=SL(2, R)⋌G, where the SL(2, R) group contains the time-translations, dilations, and the inversion Σ, and G is the static (nine-parameter) Galilei group. A subtle aspect of the inclusion of viscosity fields is discussed and it is shown that the Navier–Stokes assumption of constant viscosity breaks the SL(2, R) group to a two-parameter group of time translations and dilations in a tensorial way. The 12-parameter group G is also known to be the maximal invariance group of the free Schrödinger equation. It originates in the free Hamilton–Jacobi equation which is central to both fluid dynamics and the Schrödinger equation.ReferencesREFERENCES1Supernova hydrodynamics up close: Science and technology review, January/February, 2000;, http://www.llnl.gov/str.2IHachisuAstrophys. J.3681991L273HSakagamiKNishiharaPhys. Fluids B2199027154LO'C DruryJ.TMendonçaPhys. Plasmas7200051485SWeinbergGravitation and Cosmology1972WileyNew York6E.C.GSudarshanNMukundaClassical Dynamics: A Modern Perspective1974WileyNew York7aMHassaineP.AHorvathyAnn. Phys.2822000218bMHassaineP.AHorvathyPhys. Lett. A2792001215cA.MGrundlandLLalagueCan. J. Phys.721994362dA.MGrundlandLLalagueCan. J. Phys.731995463eR. Jackiw, physics/0010042.8aUNiedererHelv. Phys. Acta451972802bC.RHagenPhys. Rev. D51972377cRJackiwPhys. Today251972239L.DLandauE.MLifshitzFluid Mechanics1959PergamonElmsford10aAClebschJ. Reine Angew. Math.5618591bHLambHydrodynamics1942Cambridge Univ. PressCambridgecfor related work see, S. Deser, R. Jackiw, and, A. P. Polychronakos, physics/0006056;dR. Jackiw, V. P. Nair, and, So-Young, Pi, hep-th/0004084. diff --git a/tests/data/elsevier/aphy.2001.6176_expected.yml b/tests/data/elsevier/aphy.2001.6176_expected.yml new file mode 100644 index 0000000..cd553d2 --- /dev/null +++ b/tests/data/elsevier/aphy.2001.6176_expected.yml @@ -0,0 +1,329 @@ +abstract: It has recently been found that supernova explosions can be simulated in the laboratory by implosions induced in a plasma by intense lasers. A theoretical explanation is that the inversion transformation, (Σ:t→−1/t, x→x/t), leaves the Euler equations of fluid dynamics, with standard polytropic exponent, invariant. This implies that the kinematical invariance group of the Euler equations is larger than the Galilei group. In this paper we determine, in a systematic manner, the maximal invariance group G of general fluid dynamics and show that it is a semi-direct product G=SL(2, R)⋌G, where the SL(2, R) group contains the time-translations, dilations, and the inversion Σ, and G is the static (nine-parameter) Galilei group. A subtle aspect of the inclusion of viscosity fields is discussed and it is shown that the Navier–Stokes assumption of constant viscosity breaks the SL(2, R) group to a two-parameter group of time translations and dilations in a tensorial way. The 12-parameter group G is also known to be the maximal invariance group of the free Schrödinger equation. It originates in the free Hamilton–Jacobi equation which is central to both fluid dynamics and the Schrödinger equation. +copyright_holder: Academic Press +copyright_statement: Copyright © 2001 Academic Press All rights reserved. +copyright_year: 2001 +document_type: article +license_url: '' +license_statement: '' +keywords: [] +article_type: full-length article +journal_title: Annals of Physics +material: publication +publisher: Elsevier Inc. +year: 2001 +authors: +- full_name: O'Raifeartaigh, L. + raw_affiliations: + - value: School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 + Burlington Road, Dublin, 4, Ireland + source: Elsevier Inc. +- full_name: Sreedhar, V.V + raw_affiliations: + - value: School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 + Burlington Road, Dublin, 4, Ireland + source: Elsevier Inc. +artid: '96176' +title: The Maximal Kinematical Invariance Group of Fluid Dynamics and Explosion–Implosion Duality +dois: +- material: publication + doi: 10.1006/aphy.2001.6176 +journal_volume: '293' +journal_issue: '' +is_conference_paper: false +publication_date: '2001-11-01' +collaborations: [] +documents: +- key: aphy.2001.6176.xml + url: http://example.org/aphy.2001.6176.xml + source: Elsevier Inc. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: 'Supernova + hydrodynamics up close: Science and technology review, January/February, 2000;, + http://www.llnl.gov/str.' + source: Elsevier Inc. + reference: + urls: + - value: http://www.llnl.gov/str + label: '1' + misc: + - 'Supernova hydrodynamics up close: Science and technology review, January/February, + 2000' +- raw_refs: + - schema: Elsevier + value: IHachisu<maintitle>Astrophys. + J.</maintitle>3681991L27 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Astrophys. J. + journal_volume: '368' + year: 1991 + page_start: L27 + label: '2' + authors: + - full_name: Hachisu, I. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: HSakagamiKNishihara<maintitle>Phys. + Fluids B</maintitle>219902715 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Phys. Fluids B + journal_volume: '2' + year: 1990 + page_start: '2715' + label: '3' + authors: + - full_name: Sakagami, H. + inspire_role: author + - full_name: Nishihara, K. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: LO'C DruryJ.TMendonça<maintitle>Phys. Plasmas</maintitle>720005148 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Phys. Plasmas + journal_volume: '7' + year: 2000 + page_start: '5148' + label: '4' + authors: + - full_name: Drury, L. O'C + inspire_role: author + - full_name: Mendonça, J.T + inspire_role: author +- raw_refs: + - schema: Elsevier + value: SWeinberg<maintitle>Gravitation + and Cosmology</maintitle>1972WileyNew + York + source: Elsevier Inc. + reference: + publication_info: + parent_title: Gravitation and Cosmology + year: 1972 + label: '5' + authors: + - full_name: Weinberg, S. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'E.C.GSudarshanNMukunda<maintitle>Classical + Dynamics: A Modern Perspective</maintitle>1974WileyNew + York' + source: Elsevier Inc. + reference: + publication_info: + parent_title: 'Classical Dynamics: A Modern Perspective' + year: 1974 + label: '6' + authors: + - full_name: Sudarshan, E.C.G + inspire_role: author + - full_name: Mukunda, N. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: MHassaineP.AHorvathy<maintitle>Ann. + Phys.</maintitle>2822000218 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Ann. Phys. + journal_volume: '282' + year: 2000 + page_start: '218' + label: 7a + authors: + - full_name: Hassaine, M. + inspire_role: author + - full_name: Horvathy, P.A + inspire_role: author +- raw_refs: + - schema: Elsevier + value: MHassaineP.AHorvathy<maintitle>Phys. + Lett. A</maintitle>2792001215 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Phys. Lett. A + journal_volume: '279' + year: 2001 + page_start: '215' + label: b + authors: + - full_name: Hassaine, M. + inspire_role: author + - full_name: Horvathy, P.A + inspire_role: author +- raw_refs: + - schema: Elsevier + value: A.MGrundlandLLalague<maintitle>Can. + J. Phys.</maintitle>721994362 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Can. J. Phys. + journal_volume: '72' + year: 1994 + page_start: '362' + label: c + authors: + - full_name: Grundland, A.M + inspire_role: author + - full_name: Lalague, L. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: A.MGrundlandLLalague<maintitle>Can. + J. Phys.</maintitle>731995463 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Can. J. Phys. + journal_volume: '73' + year: 1995 + page_start: '463' + label: d + authors: + - full_name: Grundland, A.M + inspire_role: author + - full_name: Lalague, L. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: R. + Jackiw, physics/0010042. + source: Elsevier Inc. + reference: + label: e + misc: + - R. Jackiw, physics/0010042 +- raw_refs: + - schema: Elsevier + value: UNiederer<maintitle>Helv. + Phys. Acta</maintitle>451972802 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Helv. Phys. Acta + journal_volume: '45' + year: 1972 + page_start: '802' + label: 8a + authors: + - full_name: Niederer, U. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: C.RHagen<maintitle>Phys. + Rev. D</maintitle>51972377 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Phys. Rev. D + journal_volume: '5' + year: 1972 + page_start: '377' + label: b + authors: + - full_name: Hagen, C.R + inspire_role: author +- raw_refs: + - schema: Elsevier + value: RJackiw<maintitle>Phys. + Today</maintitle>25197223 + source: Elsevier Inc. + reference: + publication_info: + journal_title: Phys. Today + journal_volume: '25' + year: 1972 + page_start: '23' + label: c + authors: + - full_name: Jackiw, R. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: L.DLandauE.MLifshitz<maintitle>Fluid + Mechanics</maintitle>1959PergamonElmsford + source: Elsevier Inc. + reference: + publication_info: + parent_title: Fluid Mechanics + year: 1959 + label: '9' + authors: + - full_name: Landau, L.D + inspire_role: author + - full_name: Lifshitz, E.M + inspire_role: author +- raw_refs: + - schema: Elsevier + value: AClebsch<maintitle>J. + Reine Angew. Math.</maintitle>5618591 + source: Elsevier Inc. + reference: + publication_info: + journal_title: J. Reine Angew. Math. + journal_volume: '56' + year: 1859 + page_start: '1' + label: 10a + authors: + - full_name: Clebsch, A. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: HLamb<maintitle>Hydrodynamics</maintitle>1942Cambridge + Univ. PressCambridge + source: Elsevier Inc. + reference: + publication_info: + parent_title: Hydrodynamics + year: 1942 + label: b + authors: + - full_name: Lamb, H. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: for + related work see, S. Deser, R. Jackiw, and, A. P. Polychronakos, physics/0006056; + source: Elsevier Inc. + reference: + label: c + misc: + - for related work see, S. Deser, R. Jackiw, and, A. P. Polychronakos, physics/0006056 +- raw_refs: + - schema: Elsevier + value: R. + Jackiw, V. P. Nair, and, So-Young, Pi, hep-th/0004084. + source: Elsevier Inc. + reference: + label: d + misc: + - R. Jackiw, V. P. Nair, and, So-Young, Pi, hep-th/0004084 diff --git a/tests/data/elsevier/j.aim.2021.107831.xml b/tests/data/elsevier/j.aim.2021.107831.xml new file mode 100644 index 0000000..72e95c5 --- /dev/null +++ b/tests/data/elsevier/j.aim.2021.107831.xml @@ -0,0 +1 @@ +application/xmlLp-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequalityPhilip T. GressmanRadon-like operatorsBrascamp-Lieb inequalitiesAdvances in Mathematics 387 (2021). doi:10.1016/j.aim.2021.107831journalAdvances in Mathematics© 2021 Elsevier Inc. All rights reserved.Elsevier Inc.0001-870838727 August 20212021-08-2710.1016/j.aim.2021.107831http://dx.doi.org/10.1016/j.aim.2021.107831doi:10.1016/j.aim.2021.107831107831JournalsS300.1YAIMA107831107831S0001-8708(21)00270-X10.1016/j.aim.2021.107831Elsevier Inc.Fig. 1Block structure of ns × ns matrices M whose determinants span the space of invariant polynomials of Brascamp-Lieb data satisfying (22), (23), and (24) for sΦ = s. Here each ci1i2i3 is a scalar.Fig. 1Fig. 2Illustration of the structure of the matrix M(π1,…,πn).Fig. 2☆This work was partially supported by NSF Grant DMS-1764143.Lp-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequalityPhilip T.Gressmangressman@math.upenn.eduCommunicated by C. FeffermanAbstractThis paper considers the problem of establishing Lp-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29], and on recent results for geometric nonconcentration inequalities [11]. As an initial application of this new approach, this paper establishes sharp restricted strong type Lp-improving inequalities for certain model quadratic submanifolds in the range k<n≤2k.KeywordsRadon-like operatorsBrascamp-Lieb inequalities1Introduction1.1Background and statement of resultsLp-improving estimates for Radon-like operators have been a fundamental object of study in harmonic analysis for many decades and find applications in a number of interesting problems in PDEs and elsewhere (see, e.g., [17]). Since the late 1990s, a favored approach has been a combinatorial one, pioneered by Christ [7], who was inspired by Bourgain [5,6], Wolff [26,27], and Schlag [21], as well as others. As this approach is commonly executed, it involves the construction of a so-called “inflation map” which iterates the geometry of the operator in much the same way that a TT⁎ argument would. A key feature of the inflation map is that the dimension of its domain (usually comprised of products of fibers) and its target space must generally match and, when they do, the map must have a Jacobian determinant which is nonzero on a dense open set. The difficulty of completing a proof, once the inflation map has been obtained, boils down to a delicate understanding of how the degeneracy of the Jacobian determinant leads to certain integral inequalities.A principal limitation of this approach is that inflation maps are often difficult to construct or analyze unless the dimension and the codimension of the underlying submanifolds happen to satisfy simple numerical relationships, e.g., when one is an integer multiple of the other. For this reason, there are many gaps in the literature for Radon-like operators of intermediate dimension (being neither curves nor hypersurfaces) when the dimension and codimension are generically chosen.In this paper, we introduce a new approach to this problem which allows one to circumvent the need for an explicit inflation map. The overall philosophy of the proof is still fundamentally combinatorial and very deeply connected to earlier approaches, but incorporates recent ideas including the so-called Kakeya-Brascamp-Lieb inequality, proved by Zhang [28] and further developed by Zorin-Kranich [29], and nonconcentration inequalities [11]. The result is a significant shift in the structure of the argument which removes a number of important barriers and gives a unified framework which applies across a number of situations with wildly different inflation maps (or no known inflation map at all).Central to this approach is a new understanding of the Brascamp-Lieb constant. To define it in a form which is most suitable for the present purposes, let m, n, and k be positive integers with n>k and suppose that π1,…,πm are linear maps from Rn to Rn−k. Fix p:=nm(n−k). Let W({πj}j=1m), which will be called the Brascamp-Lieb weight associated to the maps {πj}j=1m, be defined to equal the largest nonnegative real number such that(1)W({πj}j=1m)∫Rn[∏j=1mfj(πjx)]pdx≤[∏j=1m∫Rn−kfj]p holds for all nonnegative measurable functions fj on Rn−k, j=1,…,m.At the greatest level of generality, the results of this paper are simplest to state for Radon-like operators which are defined in terms of an incidence relation Σ which is itself understood to be the zero set of a defining function ρ. More precisely, let Ω⊂Rn×Rn, and let ρ:Ω→Rn−k be a smooth function such that at every point (x,y)∈Ω such that ρ(x,y)=(ρ1(x,y),…,ρn−k(x,y))=0, the matrices(2)Dxρ:=[∂ρ1∂x1⋯∂ρ1∂xn⋮⋱⋮∂ρn−k∂x1⋯∂ρn−k∂xn] and Dyρ:=[∂ρ1∂y1⋯∂ρ1∂yn⋮⋱⋮∂ρn−k∂y1⋯∂ρn−k∂yn] (which will be called the left and right derivative matrices of ρ, respectively) both have full rank n−k. We call the set Σ:={(x,y)∈Ω|ρ(x,y)=0} the incidence relation associated to ρ and call ρ a defining function of the incidence relation Σ⊂Ω. By virtue of the Implicit Function Theorem, the setsΣx:={y∈Rn|(x,y)∈Ω and ρ(x,y)=0} andΣy:={x∈Rn|(x,y)∈Ω and ρ(x,y)=0} are embedded k-dimensional submanifolds of Rn for any values of the parameters x or y, respectively. The incidence relation Σ will be called left-algebraic of degree d when for each y such that Σy is nonempty, Σy is contained in a k-dimensional affine algebraic variety of degree at most d (where we do not distinguish between affine algebraic sets and affine algebraic varieties and do not require irreducibility). It is also important to define a canonical measure dσ on each Σx by means of the formula(3)∫Σxfdσ:=∫Σxf(y)dHk(y)det⁡(Dyρ(x,y)(Dyρ(x,y))T)1/2, where dHk is the usual k-dimensional Hausdorff measure restricted to Σx. Analogous measures on Σy may be defined as well, but will not be needed.The first main result of this paper is the following continuum version of the Kakeya-Brascamp-Lieb inequality: Theorem 1Suppose Σ is a left-algebraic incidence relation of degree d with defining function ρ. Then for any nonnegative Lebesgue integrable functionsf1,…,fmonRn,(4)∫Rn[∫Σx⋯∫Σx[W({Dxρ(x,yj)}j=1m)]1p∏j=1mfj(yj)dσ(y1)⋯dσ(ym)]pdx≤C∏j=1m(∫fj)pfor someC<∞depending only on n, m, and d, whereW({Dxρ(x,yj)}j=1m)is the constant as defined by(1)whenπj:=Dxρ(x,yj)for eachj=1,…,m.The inequality (4) is the main new tool of this paper for studying the Lp-improving properties of Radon-like operators in intermediate dimensions. When combined with recent new machinery regarding nonconcentration functionals [11], the inequality (4) can be used as a direct replacement for an inflation map construction and the associated degenerate change of variables formula. This overcomes some significant limitations of that approach in the regime of intermediate dimensions. The most general result of this paper concerning Lp-improving properties is the following: Theorem 2SupposeΣ⊂Ωis a left-algebraic incidence relation with defining function ρ. Suppose also that there existsc>0for which(5)supy1,…,ym∈F⁡[W({Dxρ(x,yj)}j=1m)]1p≥c(σ(F∩xΣ))sfor allx∈Rnand all Borel subsetsF⊂xΣ, where σ is the measure(3). Then the Radon-like transform(6)Tf(x):=∫Σxfdσsatisfies the inequality(7)(∫|TχE(x)|n(m+s)(n−k)mdx)m(n−k)n(m+s)≤C|E|mm+sfor all Borel setsE⊂Rnwith constant C which depends only onn,k,m,s,c, and the degree of Σ. Here|E|denotes the Lebesgue measure of E. We call (7) a restricted strong type (m+sm,n(m+s)(n−k)m) inequality for T following usual conventions, e.g. [1]. In Section 5.1 we give several examples of how one can verify the main hypothesis (5) in a number of important special cases. The broadest of these applications is: Theorem 3For any integersn,ksatisfyingk<n≤2k, consider the Radon-like operator acting on functions onRngiven by(8)Tf(x):=∫Rkf(x1+t1,…,xk+tk,xk+1+12∑i=1kλ1iti2,…,xn+12∑i=1kλ(n−k)iti2)dt,whereλjiis a(n−k)×kmatrix whose minors satisfy the constraintdet⁡[λ1i⋯λ1(i+n−k−1)⋮⋱⋮λ(n−k)i⋯λ(n−k)(i+n−k−1)]≠0for all i (interpreting the columns as periodic with period k to make sense of the indexi+n−k−1wheni+n−k−1>k). Then for all Borel setsE⊂Rn,(9)||TχE||L2n−kn−k(Rn)≤C|E|n2n−kfor someC<∞independent of E. A standard Knapp-type argument shows that the exponents in the conclusion (9) cannot be improved; as such, Theorem 3 can be regarded as an extension of work of by D. Oberlin [16] concerning “model surface” quadratic submanifolds. We note that it is understood through work of Ricci [19] that quadratic model surfaces exist with dimension k much less than n/2 when n is large; the restriction n≤k/2 present in Theorem 3 is not a fundamental limitation of the method; in particular, Section 5.6 illustrates how the method can be applied to a canonical non-translation-invariant quadratic Radon-like operator which integrates over submanifolds of dimension k and codimension k2.1.2Outline and notationThe remainder of this paper is organized as follows: Section 2 contains the proof of Theorem 1, which is derived from a discrete inequality of Zhang and Zorin-Kranich using a host of essentially standard limiting arguments. Section 3 proves a number of important new results about the nature of the Brascamp-Lieb constant. In the context of Theorem 2, the most important of these is Lemma 2, which establishes the comparability of the Brascamp-Lieb constant and a supremum of certain invariant polynomials. The approach is to observe a deep connection between the Brascamp-Lieb constant and the field of Geometric Invariant Theory. Lemma 3 also gives important insight into the family of these invariant polynomials, and in particular establishes that each such polynomial can be expressed as the determinant of a matrix with certain simple block structure, which is particularly useful when seeking to apply Theorem 2. Section 4 gives the proof of Theorem 2. The proof is a relatively straightforward combination of Theorem 1, Lemma 2 and Proposition 4, which is itself a generalization of a result which was central to the study of nonconcentration inequalities [11]. Section 5 provides a number of sample applications of Theorem 2 which include the moment curve case studied by Christ [7], Theorem 3, and some non-translation-invariant extensions. Finally, Section 6 is an appendix which provides some elementary quantitative versions of the Inverse and Implicit Function Theorems which are needed in the proof of Theorem 1.The remainder of this paper employs the notation ≲ as is now rather commonly done: the statement A≲B will mean that there exists a finite nonnegative constant C such that A≤CB holds uniformly over some range of parameters of A and B. When those parameters are not readily apparent, they will be explicitly identified, e.g., “Aj≲Bj uniformly for all j.” The notation A≳B is defined analogously, and A≈B will be used to indicate that both A≲B and A≳B hold simultaneously.Another important piece of space-saving notation which will be used heavily is the following: for any objects p1,…,pm, the notation {pj}j=1m will denote the m-tuple (p1,…,pm).2Continuous Kakeya-Brascamp-Lieb: proof of Theorem 1The core result of this section is the proof of Theorem 1. Our derivation is based directly on the Kakeya-Brascamp-Lieb inequality of Zorin-Kranich [29], which is a natural evolution of an earlier result of Zhang [28]. Zhang's result was itself inspired by Guth's approach to endpoint multilinear Kakeya [12], which was prompted by and built upon work of Bennett, Carbery, and Tao in the non-endpoint case [4].2.1Reduction to smooth functionsThe first step in the proof of Theorem 1 is to show that it suffices to prove (4) for nonnegative smooth functions fj of compact support. This follows by standard arguments, but as p will generally be less than one, it is reasonable to proceed carefully nevertheless. The auxiliary result needed is that for any nonnegative Lebesgue integrable function f on Rn and any δ>0, there is a pointwise nondecreasing sequence fℓ of nonnegative smooth functions of compact support such thatf(x)≤limℓ→∞⁡fℓ(x) for all x∈Rn (as opposed to merely almost everywhere) such that∫fℓ≤δ+∫f for all ℓ. To establish this auxiliary result, let η>0 be a positive real number satisfying(1+η)∫f≤δ3+∫f and let Fj:={x∈Rn|(1+η)j−1<f(x)≤(1+η)j}. By definition of these sets, one has the trivial inequalityf(x)≤∑j=−∞∞(1+η)jχFj(x) for every x∈Rn (where the sum is interpreted as an extended real number). Next, for each j∈Z, let Oj be an open set containing Fj, each chosen so that∑j(1+η)j|Oj∖Fj|≤δ3. Decompose each Oj into nonoverlapping dyadic boxes Qjk (i.e., boxes of the form [k12ℓ,(k1+1)2ℓ]×⋯×[kn2ℓ,(kn+1)2ℓ] for integers k1,…,kn and ℓ), and for each dyadic box, select a smooth nonnegative function of compact support φjk which is identically 1 on Qjk in such a way that the entire ensemble of functions satisfies∑j,k(1+η)j∫Rn∖Qjkφjk≤δ3. To bound f everywhere by the limit of an appropriate nondecreasing sequence fℓ, one may simply select some ordering of the countably many dyadic boxes Qjk and let fn be the sequence of partial sums of (1+η)jφjk. The conclusion that limℓ→∞⁡fℓ(x) is greater than f(x) at every point follows directly from the fact that φjk≥1 on Qjk and the union of the Qjk's contains Fj for each j. Similarly,limℓ→∞⁡∫fℓ=∫∑j,k(1+η)jφjk≤∑j,k(1+η)j[|Qjk|+∫Rn∖Qjkφjk]≤δ3+∑j(1+η)j|Oj|≤δ3+∑j(1+η)j[|Fj|+|Oj∖Fj|]≤2δ3+(1+η)∑j(1+η)j−1|Fj|≤2δ3+(1+η)∫f≤δ+∫f. Assuming that (4) holds for all m-tuples of smooth nonnegative functions of compact support, the passage to general integrable functions is achieved by an application of the Monotone Convergence Theorem (which applies because p>0) for the particular approximating sequences just constructed, one for each of the m functions appearing in (4), and then letting δ→0+.2.2Kakeya-Brascamp-Lieb for functions of varietiesAfter restricting attention to smooth functions of compact support, the next significant step in the proof of (4) builds on the following special case of the Kakeya-Brascamp-Lieb inequality as established by Zorin-Kranich [29], which is itself a generalization of the closely related Theorem 8.1 of Zhang [28]: TheoremTheorem 1.7 of [29]LetQbe the collection of all boxes[j1,j1+1]×⋯×[jn,jn+1]for integersj1,…,jnand suppose thatH1,…,Hmare affine algebraic varieties inRnwithdim⁡Hj=k. Then(10)∑Q∈Q(∫∏j=1m(Hj∩Q)[W({TxjHj}j=1m)]1pdHk(x1)⋯dHk(xm))p≤Cn∏j=1m(deg⁡Hj)p.Here p and W are as in(1), and for each smooth pointxjofHj,TxjHjdenotes the orthogonal projection fromRnonto the orthogonal complement of the tangent space ofHjatxj. The constantCndepends only on n.The proof of Theorem 1 proceeds by deducing some self-improvements of the above theorem which generalize it first to a discrete weighted version of Theorem 1 and then to a continuous analogue. These refinements are the contents of the upcoming Propositions 1 and 2, respectively.For convenience in the arguments that follow, let Q0 be the box [−1/2,1/2]n and let Qx=x+Q0 for all x∈Rn. The norm |⋅| on Rn will denote the ℓ∞ norm in the standard coordinate basis. Furthermore, given x∈Rn and an m-tuple {Hj}j=1m of affine algebraic varieties in Rn, define(11)ωx({Hj}j=1m):=∫∏j=1m(Hj∩Qx)[W({TxjHj}j=1m)]1pdHk(x1)⋯dHk(xm).Proposition 1IfE1,…,Emare finite sets of k-dimensional varieties inRnand ifNj:Ej→R≥0for eachj=1,…,m, then(12)∫Rn[∑H1∈E1,…,Hm∈Em(∏j=1mNj(Hj))ωx({Hj}j=1m)]pdx≤Cn∏j=1m[∑H∈EjNj(H)deg⁡H]p,where p andCnare the same as in(10).ProofThe first step of this proposition is to replace the sum over Q∈Q in (10) by an integral as in [28]. To do this, let x∈Rn be fixed and apply (10) to the shifted varieties {−x+Hj}j=1m; note that shifting does not change degree. For any Q∈Q,∫(−x+H1)∩Q⋯∫(−x+Hm)∩Q[W({Txj(−x+Hj)}j=1m)]1pdHk(xm)⋯dHk(x1)=∫H1∩(x+Q)⋯∫Hm∩(x+Q)[W({TxjHj}j=1m)]1pdHk(xm)⋯dHk(x1) by translation-invariance of Hausdorff measure. Since the sum of this quantity over Q∈Q is bounded by Cn∏j(deg⁡Hj)p for all x∈Rn, it follows that∑ℓ∈Zn[ωx+ℓ({Hj}j=1m)]p≤Cn∏j=1m(deg⁡Hj)p for all x∈Rn. Integrating x over [0,1]n gives(13)∫Rn[ωx({Hj}j=1m)]pdx≤Cn∏j=1m(deg⁡Hj)p for any m-tuple of affine varieties H1,…,Hm.The next step is to introduce the weights Nj. To that end, suppose initially that Nj is any nonnegative integer-valued function on Ej for each j=1,…,m. For any fixed δ∈(0,1) and each j=1,…,m, let H˜j be a union of varieties of the form uji+(1−δ)Hj as Hj ranges over all varieties in Ej with Nj(Hj)>0 and as i ranges over {1,…,Nj(Hj)}. Assume also that the shifts uji satisfy |uji|<δ/2 and are chosen so that no two of the varieties uji+(1−δ)Hj are equal. The key idea in the proof of this proposition is to apply (13) to the varieties H˜j. First observe that ωx({H˜j}j=1m) expands as a sum of terms of the form ωx({uji+(1−δ)Hj}j=1m), where for each j, uji+(1−δ)Hj is one of the varieties just described whose union is H˜j. Each such term ωx({uji+(1−δ)Hj}j=1m) is itself an integral over ((u1i+(1−δ)H1)∩Qx)×⋯((umi+(1−δ)Hm)∩Qx) of the corresponding weight W1/p generated by the orthogonal projections onto the orthogonal complement of the tangent spaces Txj(uji+(1−δ)Hj). Observe that (uji+(1−δ)Hj)∩Qx=uji+((1−δ)Hj)∩Qx−uji=uji+(1−δ)(Hj∩(1−δ)−1Qx−uji) and that (1−δ)−1Qx−uji⊃Q(1−δ)−1x. To see this last fact, note that(1−δ)−1x+y=(1−δ)−1(x−uji)+(1−δ)−1uji+y, and when |y|<1/2, it must follow that |(1−δ)−1uj+y|≤δ(1−δ)−1/2+1/2=(1−δ)−1/2, so that(1−δ)−1x+y=(1−δ)−1(x−uji+y˜) for some |y˜|≤1/2. These elementary observations combined with a sequence of changes of variables imply that∫∏j=1m((uji+Hj)∩Qx)[W({Txj(uji+(1−δ)Hj)}j=1m)]1pdHk(x1)⋯dHk(xm)≥∫∏j=1m(1−δ)(Hj∩Q(1−δ)−1x)[W({Txj((1−δ)Hj)}j=1m)]1pdHk(x1)⋯dHk(xm)=(1−δ)mk∫∏j=1m(Hj∩Q(1−δ)−1x)[W({TxjHj}j=1m)]1pdHk(x1)⋯dHk(xm), i.e.,ω(1−δ)−1x({Hj}j=1m)≤(1−δ)−kmωx({uji+(1−δ)Hj}j=1m). Summing over the varieties forming each H˜j gives(14)∑H1∈E1,…,Hm∈EmNj(Hj)ω(1−δ)−1x({Hj}j=1m)≤(1−δ)−kmωx({H˜j}j=1m). Since deg⁡H˜j≤∑Hj∈EjNj(Hj)deg⁡Hj, applying (13) to the varieties H˜j, invoking the inequality (14), applying a change of variables in x, and sending the spacing parameter δ→0+ gives the conclusion of this proposition when Nj is integer-valued.Because both sides of this inequality are homogeneous of degree p with respect to each Nj, multiplying each Nj by a nonzero real number preserves both sides of the inequality, meaning the inequality remains true when each Nj is a positive real multiple of a nonnegative integer-valued function. However, every nonnegative real-valued function Nj is uniformly comparable to such a function with constants which are as close as desired to 1. Therefore the proposition must be true in the general case of each Nj being an arbitrary nonnegative real-valued function.□Proposition 2For eachj=1,…,m, letUj⊂Rnbe an open set and letHjbe a mapping fromUjinto the set of k-dimensional varieties onRnof degree at mostDjsuch thatHj(y)depends smoothly on y. For any nonnegative measurable functionsfjonUj,(15)∫Rn[∫(∏j=1mfj(yj))ωx({Hj(yj)}j=1m)dy1⋯dym]pdx≤Cn∏j=1m[Dj∫fj]p,where p andωxare as above. The constantCnis the same as inProposition 1.ProofBecause Hj(yj) depends smoothly on y, ωx({Hj(yj)}j=1m is known to be a continuous function of y1,…,ym as a result work by Bennett, Bez, Cowling, and Flock [2]. For any δ>0, decompose Rn into a nonoverlapping union of boxes of side length δ. Fix arbitrary compact sets Kj⊂Uj and let Qj(δ) be a finite collection of these cubes which covers Kj. For each j, let Ej be the collection of varieties given byEj:={H|H=Hj(y) for y at the center of a cube Q′∈Qj(δ)}. For convenience, let Hj(Q′) also denote the variety Hj(y) when y is taken to be the center of Q′. Fix any nonnegative measurable functions fj on Uj and letNj(H):=∑Q′∈Qj(δ)Hj(Q′)=H∫Q′∩Kjfj. The left-hand side of (12) is exactly equal to(16)∫[∫Km⋯∫K1∏j=1mfj(yj)ωx({Hj(yj′)}j=1m)dy1⋯dym]pdx, where yj′ is the center of the cube Q′∈Qj(δ) containing yj (which is uniquely defined for a.e. yj). By Monotone Convergence and continuity of the reciprocal of the Brascamp-Lieb constant,∫[∫Km⋯∫K1∏j=1mfj(yj)ωx({Hj(yj)}j=1m)dy1⋯dym]pdx=∫limδ→0+⁡[∫Km⋯∫K1∏j=1mfj(yj)inf|zj−yj|≤δ⁡ωx({Hj(zj)}j=1m)dy1⋯dym]pdx=limδ→0+⁡∫[∫Km⋯∫K1∏j=1mfj(yj)inf|zj−yj|≤δ⁡ωx({Hj(zj)}j=1m)dy1⋯dym]pdx. For each δ>0,inf|zj−yj|≤δ⁡ωx({Hj(zj)}j=1m)≤ωx({Hj(yj′)}j=1m) because |y−y′|≤δ. But then by (12), this means that the limit of (16) as δ→0+ is dominated byCn∏j=1m[Dj∑Q′∈Qj(δ)∫Q′∩Kjfj]p=Cn∏j=1m[Dj∫Kjfj]p as desired. Because each Kj is arbitrary, a second application of Monotone Convergence establishes the proposition.□2.3Deduction of Theorem 1 from Proposition 2Proof of Theorem 1As already observed, it suffices to assume each fj is smooth and compactly supported. As the submanifolds Σy depend smoothly on y, it follows from Proposition 2 that for any δ>0,∫[∫⋯∫∏j=1mfj(yj)ωx({δ−1Σyj}j=1m)dy1⋯dym]pdx≲∏j=1m(∫fj)p for some implicit constant depending only on n and the maximum degree of any Σy. After a change of variables x↦δ−1x,∫[δ−m(n−k)∫⋯∫∏j=1mfj(yj)ωδ−1x({δ−1Σyj}j=1m)dy1⋯dym]pdx≲∏j=1m(∫fj)p uniformly for all positive δ, where the factor δ−m(n−k)p=δ−n arises as the Jacobian determinant of the change of variables.By Lemma 5 from the Appendix, it is possible to use an alternate defining function ρ˜ which exhibits better uniformity properties than ρ itself might. In particular, for the defining function ρ˜ constructed there, the matrices Dxρ˜ are exactly the orthogonal projections onto the orthogonal complement of the tangent space of Σx at x and smallness of |ρ˜(x,y)| implies proximity of x to Σy in a uniform way: |ρ˜(x,y)|≤δκn for sufficiently small δ implies that the set x+(−δ,δ)n intersects Σy in a set of k-dimensional Hausdorff measure at least comparable to δk. To proceed, one first observes that Txjδ−1Σyj is the projection from Rn onto the orthogonal complement of the tangent space at xj∈δ−1Σyj. By rescaling, the tangent plane of δ−1Σy at xj is simply a shift of the tangent plane at δxj of Σyj, so Txjδ−1Σyj=Dxjρ˜(δxj,yj). Consequently, if Qxδ denotes the set x+[−δ/2,δ/2]n, it follows thatωδ−1x({δ−1Σyj}j=1m)=∫∏j=1m((δ−1Σyj)∩Qδ−1x)[W({Txjδ−1Σyj}j=1m)]1pdHk(x1)⋯dHk(xm)≥infx1∈(δ−1Σy1)∩Qδ−1x,…xm∈(δ−1Σym)∩Qδ−1x[W({Dxjρ˜(δxj,yj)}j=1m)]1p∏j=1mHk(δ−1Σyj∩Qδ−1x)=infx1∈Σy1∩Qxδ,…xm∈Σym∩Qxδ[W({Dxjρ˜(xj,yj)}j=1m)]1pδ−km∏j=1mHk(Σyj∩Qxδ). By Lemma 5, then, it follows that for any compact subset K⊂Σ, there is some open set U containing K such that whenever δ is sufficiently small,ωδ−1x({δ−1Σyj}j=1m)≥cnminfx1∈Σy1∩Qxδ,…xm∈Σym∩Qxδ[W({Dxjρ˜(xj,yj)}j=1m)]1p∏j=1mχ|ρ˜(x,yj)|<δκn/2 provided (x,yj)∈U for all j=1,…,m.Now the coarea formula dictates that for any continuous function fj∫f(yj)χ|ρ˜(x,yj)|<δκn/2dyj=∫[−δκn/2,δκn/2]n−k∫ρ˜(x,⋅)=ufj(yj)dσu(yj)du, where dσu is a measure of continuous density with respect to k-dimensional Hausdorff measure on the level set {yj∈Rn|ρ˜(x,yj)=u}, which is a well-defined k-dimensional submanifold of Rn when u is sufficiently small. In the special case u=0, σ0 is exactly the measure dσ on Σx which was defined in (3) (assuming that ρ there is replaced by ρ˜). Since everything is continuous as a function of δ when f is assumed to be continuous with compact support, the limit as δ→0+ of the quantityδ−m(n−k)∫⋯∫∏j=1mfj(yj)infx1∈Σy1∩Qxδ,…xm∈Σym∩Qxδ[W({Dxjρ˜(xj,yj)}j=1m)]1p⋅∏j=1mχ|ρ˜(x,yj)|<δκn/2dy1⋯dym exists and equals a constant times∫Σx⋯∫Σx[W({Dxρ˜(x,yj)}j=1m)]1p∏j=1mfj(yj)dσ(y1)⋯dσ(ym). Thus∫[∫Σx⋯∫Σx[W({Dxρ˜(x,yj)}j=1m)]1p∏j=1mfj(yj)dσ(y1)⋯dσ(ym)]pdx≤limsupδ→0+∫[1δm(n−k)∫⋯∫ωδ−1x({δ−1Σyj}j=1m)∏j=1mfj(yj)dy1⋯dym]pdx≲∏j=1m(∫fj)p, which is the desired inequality (4) with ρ replaced by ρ˜.To revert from ρ˜ back to ρ, it simply remains to assume that switching the defining function in this way leaves the left-hand side of (4) unchanged. This follows from the identify[W({Mjπj}j=1m)]1p=[W({πj}j=1m)]1p∏j=1m|det⁡Mj| for Brascamp-Lieb constants, where Mj are any invertible matrices. The inequality is easily proved by replacing each fj(u) with fj(Mju) in (1). Since ρ˜ differs from any fixed defining function ρ by multiplication on the left by an invertible matrix, it follows by Lemma 5 that[W({Dxρ(x,yj)}j=1m)]1pdσ(y1)⋯dσ(yn) is unchanged when defined using ρ˜ instead of ρ itself because the extra factors of det⁡(Dxρ(Dxρ)T) arising from the Brascamp-Lieb constant are exactly cancelled by the extra factors arising from the measure dσ. This completes the proof.□3The Brascamp-Lieb constant and geometric invariant theoryThe next major task is to establish several general facts about the Brascamp-Lieb constant and its connection to Geometric Invariant Theory. These facts play a central role in understanding and verifying the main hypothesis (5) of Theorem 2. Throughout this section, for each j=1,…,m, each πj:Rn→Rnj will be an arbitrary linear map and each pj will be a real number in [0,1]. Following the usual convention, let the Brascamp-Lieb constant BL({πj,pj}j=1m) be defined to equal the smallest nonnegative real number such that(17)∫Rn∏j=1m(fj(πj(x)))pjdx≤BL({πj,pj}j=1m)∏j=1m(∫Rnjfj)pj for all nonnegative measurable functions fj∈L1(Rnj). When p1=⋯=pm=nm(n−k) and n1=⋯=nm=n−k, note that the Brascamp-Lieb constant is merely the reciprocal of the already-defined Brascamp-Lieb weight (1). This special case will of course be the most important one for the purposes of Theorem 2, but throughout most of the section the pj's will be allowed to differ.The overall goal of this section is to establish the existence of certain invariant polynomials in the entries of the πj's which give meaningful quantitative information about the Brascamp-Lieb constant. These polynomials should be thought of as generalizations of the determinant. For this description to be useful, it will be critical to show not only existence of such polynomials, but also to provide a means by which they may be explicitly constructed, so that they can be used as computational tools.3.1Brascamp-Lieb and minimum vectorsThe first major result of this section is the following lemma, which establishes an identity for the Brascamp-Lieb constant involving an infimum analogous to the one relating to minimum vectors in the sense of Kempf and Ness [13]: Lemma 1Suppose that the exponentspjand dimensionsnjsatisfy(18)∑j=1mpjnjn=1.(Note: it is well-known and can be seen from scaling that(18)is necessary for the finiteness of the Brascamp-Lieb constant.) ThenBL({πj,pj}j=1m)satisfies(19)[BL({πj,pj}j=1m)]−1=infA1∈SLn1,…,Am∈SLnm,A∈SLn∏j=1mnj−pjnj2|||AjπjA⁎|||pjnj,where|||⋅|||denotes the Hilbert-Schmidt norm computed with respect to the standard bases andSLnjis the Lie group of invertiblenj×njreal matrices with determinant 1.Before proceeding to the proof, it is worth observing that the direct link between the computation of the Brascamp-Lieb constant and Geometric Invariant Theory given by (19) provides a rather immediate interpretation of the work of Garg, Gurvits, Oliveira, and Wigderson [9]. Geometric Brascamp-Lieb data as they define it is exactly the set of data which are critical points of the functional on the right-hand side of (19) when A1,…,An,A are all identity matrices (i.e., geometric Brascamp-Lieb data correspond to minimum vectors in GIT). The functional can be shown to be convex along flows (A1,…,Am,A):=(exp⁡(tM1),…,exp⁡(tMm),exp⁡(tM)), t∈R, so critical points are automatically global minima. The iterative method in [9] to compute the Brascamp-Lieb constant approximates the argument of the infimum (argmin) of (19) when it exists by alternately computing the argmin (A1,…,Am) for fixed A in one step and the argmin A for fixed (A1,…,Am) in the subsequent step. (Also note that when the data is merely semi-stable and no global minimum exists, the algorithm instead produces a minimum vector with closed orbit contained in the original non-closed orbit.)ProofLieb [14] established that any Brascamp-Lieb inequality has an extremizing sequence of Gaussians, which implies that[BL({πj,pj}j=1m)]−1=infA1∈GLn1,…,Am∈GLnm[det⁡(∑j=1mpjπj⁎Aj⁎Ajπj)∏j=1m(det⁡Aj⁎Aj)pj]12. For any matrix A∈SLn,∑j=1mpj|||AjπjA⁎|||2=∑j=1mpjtr(Aπj⁎Aj⁎AjπjA⁎)=tr(∑j=1mpjAπj⁎Aj⁎AjπjA⁎). Both the trace and determinant of the matrix ∑j=1mpjAπj⁎Aj⁎AjπjA⁎ can be expressed in terms of its eigenvalues, all of which are nonnegative. By the inequality of arithmetic and geometric means, abbreviated as the AM-GM inequality, applied to the eigenvalues, it follows that|∑j=1mpjn|||AjπjA⁎|||2|n≥det⁡∑j=1mpjAπj⁎Aj⁎AjπjA⁎=det⁡∑j=1mpjπj⁎Aj⁎Ajπj. When the infimum of the left-hand side is taken over all A∈SLn, the inequality must be equality; to see this, fix M:=∑j=1mpjπj⁎Aj⁎Ajπj. When M is invertible, equality must hold when A:=M−1/2(det⁡M)1/(2n); if M has a kernel of dimension ℓ>0, let P be orthogonal projection onto the kernel. Equality holds in the limit t→∞ when At:=t1/ℓP+t−1/(n−ℓ)(I−P). Therefore[BL({πj,pj}j=1m)]−1=infA1∈GLn1,…,Am∈GLnm,A∈SLn[∑j=1mpj|||AjπjA⁎|||2n∏j=1m|det⁡Aj|2pjn]n2. A similar application of the AM-GM inequality also gives thatinft1>0,…,tm>0t1−2p1n1n⋯tm−2pmnmn∑j=1mpjnjntj2|||AjπjA⁎|||2nj=∏j=1m(|||AjπjA⁎|||2nj)pjnjn. To see this, the left-hand side can be seen to be greater than or equal to the right-hand side by using the version of AM-GM inequality which raises the termt1−2p1n1n⋯tm−2pmnmntj2|||AjπjA⁎|||2nj to the power pjnj/n, which is allowed precisely because (18) guarantees that the exponents sum to 1. The reverse inequality can be established by fixing tj:=(|||AjπjA⁎|||2/nj)−1/2 when all such constants are well-defined or by an appropriate limiting argument if any such tj happens to be infinite. Writing each matrix Aj as a nonzero constant times a matrix of determinant 1 then gives that[BL({πj,pj}j=1m)]−1=infA1∈SLn1,…,Am∈SLnm,A∈SLn∏j=1mnj−pjnj2|||AjπjA⁎|||pjnj. This is exactly (19).□Before continuing, it will be helpful record an important calculation relating to Lemma 1 which will be useful later. As it relates to the hypothesis (5) of Theorem 2, Lemma 1 establishes that(20)[W({πj}j=1m)]1p=infA1,…,Am∈SLn−kA∈SLn1(n−k)m(n−k)2∏j=1m|||AjπjA⁎|||n−k when each πj is an (n−k)×n matrix and 1/p=m(n−k)/n.The next step in this section is to give an abstract proof of the existence of invariant polynomials in the entries of the πj's which strongly quantify the magnitude of the Brascamp-Lieb constant. Following this, we will consider the question of how to more explicitly find these polynomials.A few minor reductions are in order. The first is that attention will be restricted to only those cases in which each pj is rational. By Theorem 1.13 of Bennett, Carbery, Christ, and Tao [3], the extreme points of the convex setP:={{pj}j=1m∈[0,1]m|BL({πj,pj}j=1m)<∞} all have rational exponents {pj}j=1m, and likewise rational exponents play a central role in Theorem 2. It may also be assumed that no pj equals zero since the inequality (17) will be trivially independent of πj for any index j such that pj=0, meaning that one can simply reduce m and consider the Brascamp-Lieb inequality for a strictly smaller number of πj's.The expression (19) has deep connections to the theory of minimum vectors in Geometric Invariant Theory. Pursuing this analogy, it is natural to make a connection between BL({πj,pj}j=1m) and polynomials invariant under the underlying group representation ρ of SLn1×⋯×SLnm×SLn defined by(21)ρ(A1,…,Am,A)({πj}j=1m):={AjπjA⁎}j=1m. Let Φ be any nonzero polynomial function of the matrices {πj}j=1m which is homogeneous of degree dj>0 in each πj and is ρ-invariant, i.e.,(22)Φ({λjπj}j=1m)=λ1d1⋯λmdmΦ({πj}j=1m) for all λ1,…,λm∈R and(23)Φ({AjπjA⁎}j=1m)=Φ({πj}j=1m) whenever det⁡A1=⋯=det⁡Am=1=det⁡A. If |||Φ||| is the maximum of |Φ| on all m-tuples {π˜j}j=1m such that |||π˜j|||≤1 for all j=1,…,m, then scaling dictates that|||Φ|||∏j=1m|||AjπjA⁎|||dj≥|Φ({AjπjA⁎}j=1m)|=|Φ({πj}j=1m)| for all inputs {πj}j=1m. If each degree dj happens to satisfy(24)p1n1d1=⋯=pmnmdm=1sΦ for some real number sΦ, then (19) implies that(25)[BL({πj,pj}j=1m)]−1≥(∏j=1mnj−pjnj2)|||Φ|||−1sΦ|Φ({πj}j=1m)|1sΦ. In the specific case relating to Theorem 2, the constraint (24) is trivially satisfied whenever d=d1=⋯=dm and (20) yields the inequality(26)[W({πj}j=1m)]1p≥(n−k)−m(n−k)2|||Φ|||−n−kd|Φ({πj}j=1m)|n−kd. The following lemma establishes that the collection of all such invariant polynomials can be used to compute the order of magnitude of the Brascamp-Lieb constant: Lemma 2Suppose that the exponents{pj}j=1m∈(0,1]mare rational and satisfy(18). Let IP be the collection of all nonzero invariant polynomials Φ satisfying(22),(23), and(24). Then(27)[BL({πj,pj}j=1m)]−1≈supΦ∈IP⁡|||Φ|||−1sΦ|Φ({πj}j=1m)|1sΦwith implicit constants that are independent of{πj}j=1m(where the supremum is understood to be zero ifIP=∅). Moreover, there exists a finite subsetIP0⊂IPsuch thatsupΦ∈IP⁡|||Φ|||−1sΦ|Φ({πj}j=1m)|1sΦ≈supΦ∈IP0⁡|||Φ|||−1sΦ|Φ({πj}j=1m)|1sΦ.ProofThe lower bound follows immediately from (25). The upper bound will be proved by contradiction. Without loss of generality, it may be assumed that data exists such that the left-hand side of (27) is strictly positive. Suppose for each positive integer N, there is some data {πjN}j=1m such that(28)[BL({πjN,pj}j=1m)]−1>NsupΦ∈IP⁡|||Φ|||−1sΦ|Φ({πjN}j=1m)|1sΦ. By homogeneity of both sides in the data {πjN}j=1m, it may be assumed that BL({πjN,pj}j=1m)=1 for each N, and by replacing each tuple {πjN}j=1m with ρ(A1N,…,AmN,AN)({πjN}j=1m) for some choice of A1N,…,AmN and AN for each N which tend to minimizers of the right-hand side of (19) as N→∞, it may further be assumed that∏j=1m|||πjN|||pjnj→∏j=1mnjpjnj2 as N→∞. Once again, noting that both sides of (28) are homogeneous in πj for each j, rescaling individual πj's as necessary allows one to assume that |||πjN|||→nj1/2 as N→∞ for each j=1,…,m. By passing to a subsequence in N, this means that πjN converges to some limiting data for each j=1,…,m. Let this limit data be denoted {πj∞}j=1m. Now for any matrices A1,…,Am,A, by Lemma 1,∏j=1m|||Ajπj∞A⁎|||pjnj=limN→∞⁡∏j=1m|||AjπjNA⁎|||pjnj≥limsupN→∞(∏j=1mnjpjnj2)[BL({πjN,pj}j=1m)]−1=∏j=1mnjpjnj2, so taking an infimum over all A1,…,Am,A gives that BL({πj∞,pj}j=1m)≤1. In fact, this inequality must be an equality, which can be seen by simply taking each Aj and A to be the identity. Now for any Φ∈IP,1=[BL({πjN,pj}j=1m)]−1≥N|||Φ|||−1sΦ|Φ({πjN}j=1m)|1sΦ, which means that Φ({πjN}j=1m)→0 as N→∞. By continuity of each Φ, it follows that(29)[BL({πj∞,pj}j=1m)]−1=1 and supΦ∈IP⁡|||Φ|||−1sΦ|Φ({πj∞}j=1m)|1sΦ=0.Since each exponent pj is rational and nonzero, it must be possible to find positive integers q1,…,qm and q such that pjnj=qj/q for each j. Now suppose thatΠ({xi1,yi1}i=1q1,…,{xim,yim}i=1qm) is any real-valued map which is linear in each xij∈Rnj and each yij∈Rn for i=1,…,qj and j=1,…,m. The group SLn1×⋯×SLnm×SLn acts on the vector space V of all such Π by definingρ(A1,…,Am,A)Π({xi1,yi1}i=1q1,…,{xim,yim}i=1qm):=Π({A1⁎xi1,A⁎yi1}i=1q1,…,{Am⁎xim,A⁎yim}i=1qm). Let Π∞∈V be the multilinear functional given by(30)Π∞({xi1,yi1}i=1q1,…,{xim,yim}i=1qm):=∏j=1m∏i=1qj〈xij,πj∞yij〉 where〈⋅,⋅〉 is the usual inner product onRn. The Hilbert-Schmidt norm of ρ(A1,…,Am,A)Π∞ is exactly equal to∏j=1m|||Ajπj∞A⁎|||qj, so by Lemma 1, it follows that(31)1=|||Π∞|||∏j=1nnj−qpjnj2=[BL({πj∞,pj}j=1m)]−q=infA1∈SLn1,…,Am∈SLnm,A∈SLn|||ρ(A1,…,Am,A)Π∞|||∏j=1mnj−qpjnj2. Combining this identity with Proposition 3 below, there must exist a nonconstant, homogeneous ρ-invariant polynomial P on V such that P(Π∞)≠0. Here homogeneous is meant in the usual sense of polynomials and specifically does not refer to (22). Suppose the degree of P is equal to d. Using the definition (30) of Π∞, we see that each entry of Π∞ is itself a product of entries of the πj∞. Thus we may regard P(Π∞) as a polynomial in the entries of the πj∞. To be explicit, regarding π1,…,πj as matrices of indeterminates, one can define Π exactly as was done in (30) by replacing π1∞,…,πm∞ with the π1,…,πm. The function P(Π) is now a polynomial in the indeterminate matrices π1,…,πm; to emphasize this dependence, define Φ({πj}j=1m):=P(Π). One must show that Φ satisfies (22), (23), and (24). In the former case, rescaling each πj by λj scales all entries of Π by λ1q1⋯λmqm, and since P is homogeneous of degree d, this means that Φ({λjπj}j=1m)=(λ1q1⋯λmqm)dΦ({πj}j=1m), which implies (22) with dj=dqj for each j (and because pjnj=qj/q for each j, the condition (24) holds for sΦ:=dq). Because P is ρ-invariant, Φ({πj}j=1m)=P(Π)=P(ρ(A1,…,Am,A)Π)=Φ({AjπjA⁎}j=1m), which is (23). Because P(Π∞)=1, the polynomial Φ just constructed contradicts (29) because (29) indicates that Φ({πj∞}j=1m) should equal zero rather than 1.The finite subset IP0 can be taken to be only those polynomials of the form P(Π) just described, for those P belonging to any finite generating set of the ρ-invariant algebra on V, since the contradiction just derived will still hold if P(Π∞)=0 for all such polynomials.□Proposition 3Let V be the real vector space of all maps(32)Π({xi1,yi1}i=1q1,…,{xim,yim}i=1qm)which are real and linear in eachxij∈Rnjand eachyij∈Rnfori=1,…,qjandj=1,…,m. The groupG:=SLn1×⋯×SLnm×SLnacts on the vector space V of all such Π by defining(33)ρ(A1,…,Am,A)Π({xi1,yi1}i=1q1,…,{xim,yim}i=1qm):=Π({A1⁎xi1,A⁎yi1}i=1q1,…,{Am⁎xim,A⁎yim}i=1qm).IfΠ∈Vhas the property that(34)infg∈G⁡|||ρgΠ|||=|||Π|||>0,where|||⋅|||is the Hilbert-Schmidt norm on V computed with respect to the standard basis, then there exists a homogeneous, nonconstant ρ-invariant polynomial P on V such thatP(Π)=1.ProofThis proposition is a special case of a fundamental and widely-known result in Geometric Invariant Theory as applied to real linearly reductive groups (to see that G is linearly reductive, express G as a subgroup of GLn1+⋯+nm+m; the Lie-Kolchin Theorem [25, Section 10.2] implies that the radical of G must consist of upper-triangular matrices, and consequently the unipotent radical must simply be the identity). Corollary 1.2 of Mumford [15] indicates that any two disjoint Zariski closed G-invariant subsets of V can be separated by an invariant, i.e., an invariant polynomial P exists which vanishes on one but not the other. Since {0} is certainly a Zariski closed, G-invariant subset of V, it suffices to establish that the G-orbit of Π is Zariski closed and does not contain zero. By (34), Π≠0. The criterion (34) indicates that Π is by definition a minimum vector in the sense of Richardson and Slodowy [20], and consequently Theorem 4.4 of [20] guarantees that the G-orbit of Π is indeed Zariski closed. But the G-orbit of Π cannot contain 0 (because 0=ρgΠ for some g∈G implies that 0=ρg−1(ρgΠ)=Π). Thus there must be a polynomial P which vanishes at 0 and is nonvanishing at Π. If P is not homogeneous, we can express P as a sum of homogeneous polynomials of distinct degrees, each of which must be G-invariant (which can be easily seen by an induction argument and comparing highest-degree terms before and after an application of ρg); at least one will vanish at 0 and not at Π, at which point it may be trivially rescaled to equal 1 at Π.□3.2Invariant polynomials and the Caley Ω processWhile Lemma 2 is the theoretical foundation upon which much of this paper rests, it is necessary to have a more concrete way of describing polynomials in the class IP. To that end, it is useful to appeal to the very old and well-known fact in invariant theory that invariants associated to the group SLn are generated by application of the “Cayley Ω process,” which is briefly described here as it applies to the more general situation of Brascamp-Lieb invariant polynomials satisfying (22), (23) and (24). As before, it will be assumed that the exponents pj are positive, rational, and satisfy the scaling condition (18).If Φ is any polynomial in {πj}j=1m satisfying (22), (23) and (24), then for any matrices A1,…,Am,A with strictly positive determinants, by homogeneity and ρ-invariance it must be the case that(35)Φ({AjπjA⁎}j=1m)=[(det⁡A)sΦ∏j=1m(det⁡Aj)pjsΦ]Φ({πj}j=1m). Since matrices with positive determinant form an open set in Rn×n for all n and since the left-hand side of the identity (35) must be a polynomial function in the entries of each Aj, this forces sΦ and pjsΦ to be positive integers and it further forces (35) to hold for all matrices A1,…,Am and A even if some of the determinants are zero or negative.Let ΩA be the Cayley Ω operator associated to A, i.e.,ΩA:=∑σ∈Sn(−1)σ∂∂A1σ1⋯∂∂Anσn. (Here and throughout the remainder of Section 3, σ will denote a permutation rather than referring to the measure (3).) The Cayley Ω operator associated to A satisfies the identityΩAs(det⁡A)s=cn,s>0 for all positive integers s and also satisfies ΩAf(BA)=(det⁡B)(ΩA˜f)(A˜)|A˜=BA for any n×n matrix B and any Cn function f of Rn×n (for both facts, see Theorem 4.3.4 of Sturmfels [24]). These facts together imply that(36)ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦΦ({AjπjA⁎}j=1m)=cΦ({πj}j=1m) for some nonzero constant c depending only on the exponents dj, pj, and nj when Φ satisfies (22), (23) and (24). They also imply that for any Φ satisfying (22) and (24) only, the function Φ˜ of {πj}j=1m given by(37)Φ˜({πj}j=1m):=ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦΦ({AjπjA⁎}j=1m) necessarily satisfies each of (22), (23), and (24) (note there is no dependence of Φ˜ on A or the Aj's because the orders of differentiation are chosen specifically to balance the degrees of dependence of Φ({AjπjA⁎}j=1m) on these matrices). To see why (23) holds, note that one must also have that ΩAf(AB)=(det⁡B)(ΩA˜f)(A˜)|A˜=AB because ΩA⁎=ΩA, so it follows thatΩAf(AB)=ΩA⁎f(AB)=ΩA⁎f((B⁎A⁎)⁎)=(det⁡B)ΩA˜⁎f((A˜⁎)⁎)|A˜⁎=B⁎A⁎=(det⁡B)ΩA˜f((A˜⁎)⁎)|A˜⁎=B⁎A⁎=(det⁡B)ΩA˜f(A˜)|A˜⁎=B⁎A⁎=(det⁡B)ΩA˜f(A˜)|A˜=AB. ConsequentlyΦ˜({BjπjB}j=1m)=ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦΦ({AjBjπjB⁎A⁎}j=1m)=[(det⁡B)sΦ∏j=1m(det⁡Bj)pjsΦ]⋅ΩA˜sΦΩA˜1p1sΦ⋯ΩA˜mpmsΦΦ({A˜jπjA˜⁎}j=1m)|A˜=AB,A˜j=AjBj=[(det⁡B)sΦ∏j=1m(det⁡Bj)pjsΦ]Φ˜({πj}j=1m) because the expression on the next to last line above is independent of the choice of A˜ and A˜j at which it is evaluated. Thus to understand the space of homogeneous invariant polynomials of a given multi-degree (d1,…,dm), then, it suffices to understand the image of the map Φ↦ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦΦ({AjπjA⁎}j=1m) for polynomials Φ satisfying (22) and (24) only.3.3Polynomial invariants of Brascamp-Lieb dataWe come now to the main result of this section, which gives a concrete characterization of the class IP in terms of polynomials which are expressible as determinants of block-form matrices. In light of Lemma 2, these determinants can be reasonably regarded as quantifying various sorts of transversality of the maps {πj}j=1m which allow for finiteness of the Brascamp-Lieb constant for any desired rational exponents {pj}j=1m∈(0,1]m. This approach to understanding the Brascamp-Lieb constant is complementary to the work of Bennett, Carbery, and Tao [4] and Bennett, Carbery, Christ, and Tao [3] in exactly the same way that direct computations with invariant polynomials complement characterizations of the nullcone in Geometric Invariant Theory. The strength of the finiteness criteria established in [3] is that one need only show that a single (cleverly-chosen) inequality is violated to deduce that the Brascamp-Lieb constant is infinite. Lemma 2, in contrast, allows one to deduce the finiteness of the constant by demonstrating the nonvanishing of a single (cleverly-chosen) invariant polynomial.Lemma 3Suppose{pj}j=1m∈(0,1]mare rational exponents satisfying the scaling condition(18). Let s be an integer such thatpjsis an integer for allj=1,…,m. LetVsbe the vector space of all polynomials Φ satisfying(22),(23), and(24)forsΦ=s. ThenVsis spanned by polynomials of the formdet⁡M({πj}j=1m), whereM({πj}j=1m)is anns×nsmatrix consisting of block elements of sizenj×nforj=1,…,marranged in the following way:•Each block entry is a constant multiple ofπjfor somej=1,…,m.•For eachj=1,…,m, there arepjsblock rows of heightnj(i.e., the block row is a group ofnjadjacent rows of M). In each such block row, all block entries are multiples ofπj. At mostnjof these block entries are nonzero.•There are s block columns of width n. In each block column, there are at most n nonzero block entries.Fig. 1illustrates the structure of all such matrices M.ProofThe proof proceeds by an analysis of the action of the Cayley Ω operator on general multilinear functionals. One could instead formulate this problem as a quiver representation and appeal to a number of general results concerning the structure of semi-invariants (see, for example Domokos and Zubikov [8]), but for the present purposes the Ω operator will yield a more elementary and transparent proof from the standpoint of analysis. Readers should also note the similarity of the matrices M({πj}j=1m) and the Brascamp-Lieb operator as defined in [9].Suppose that Π:(Rn)n→R is a multilinear functional on Rn. This Π is expressed in the standard basis by the formulaΠ({xi}i=1n):=∑j1,…,jn=1nΠj1⋯jnx1,j1⋯xn,jn where xi,j is the j-th coordinate of xi. For any n×n matrix A,Π({Axi}i=1n)=∑j1,…,jn=1nk1,…,kn=1Πj1⋯jnAj1k1⋯Ajnknx1,k1⋯xn,kn. If this sum is differentiated by ∂n/∂A1σ1⋯∂Anσn, the result will equal zero unless j1,…,jn are distinct and ki=σji for each i=1,…,n. Thus(38)∂n∂A1σ1⋯∂AnσnΠ({Axi}i=1n)=∑j1,…,jn=1ndistinctΠj1⋯jnx1,σj1⋯xn,σjn. Multiplying (38) by (−1)σ and summing over σ∈Sn gives thatΩAΠ({Axi}i=1n)=(∑τ∈Sn(−1)τΠτ1⋯τn)(∑σ∈Sn(−1)σx1,σ1⋯xn,σn)=(∑τ∈Sn(−1)τΠτ1⋯τn)[x1⋯xn]. The notation [x1⋯xn] is simply shorthand for the determinant of the n×n matrix whose columns are given by the vectors x1,…,xn. The quantity in parentheses on the last line above will be called the alternating contraction of Π in the indices (1,…,n) and will be denoted Π|(1,…,n). Suppose now that Π has some arbitrary degree of multilinearity, i.e., Π:(Rn)Λ→R for some ordered index set Λ. If #Λ<n, then ΩAΠ({Axi}i∈Λ)=0 trivially. If instead k>n, then by the product rule it must be the case thatΩAΠ({Axi}i∈Λ)=∑I⊂Λ#I=nΠ|I({Axi}i∈Λ∖I)[x]I where Π|I is the multilinear functional with index set Λ∖I obtained by performing an alternating contraction in the indices I (arranged in the usual order) and where [x]I:=[xi1⋯xin] with i1<⋯<in being the elements of I. By induction, for any s such that #Λ≥ns,(39)ΩAsΠ({Axi}i∈Λ)=∑#I1=n⋯∑#Is=nI1,…Is pairwise disjointΠ|I1⋯|Is({Axi}i∈Λ∖⋃j=1sIj))[x]I1⋯[x]Is. When #Λ=ns and Λ=I1∪⋯∪Is for pairwise disjoint Ij's, the quantity Π|I1⋯|Is is simply a scalar obtained by performing an alternating contraction in each of the index subsets I1,…,Is.Now consider the multilinear functional(40)Π({xi1,yi1}i=1q1,…,{xim,yim}i=1qm):=∏j=1m∏i=1qj〈xij,πjyij〉 where pjnj=qj/q and where the πj are as in the previous section; this is exactly the same construction as (30). If A∈SLn and Aj∈SLnj for each j=1,…,m, then we seek homogeneous polynomials of degree d in the entries of Π which are invariant under the action of these matrices given byΠ({A1xi1,Ayi1}i=1q1,…,{Amxim,Ayim}i=1qm). (Note that this action differs from ρ by replacing Aj⁎ and A⁎ by Aj and A; since the special linear group is closed under adjoints, this change is inconsequential and simplifies notation.) Any polynomial function of Π must belong to the span of d-fold products of the expressions (40), where in each term of the product, the xij's and yij's are regarded as fixed but may change from factor to factor (which is to say that evaluating Π on specific tuples of xij's and yij's gives a basis of functions from which the algebra of polynomial functions of Π can be generated). If this polynomial happens to be invariant under the action of the matrices (A1,…,Am,A), recall from (36) that polynomial must be preserved (up to multiplication by a nonzero constant) by the operator ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦ when sΦ:=dq. Moreover, as observed in (37), this compound Cayley operator maps all homogeneous polynomials of Π satisfying (22) and (24) into the space of invariant polynomials satisfying (22), (23), and (24). By virtue of the calculations above, the space of all such invariant homogeneous polynomials of a fixed degree is spanned by repeated alternating contractions of tensor powers of Π, where the contractions take place with respect to compatible entries. Specifically this means forming alternating contractions of the multilinear functional(41)Πd({xi1,yi1}i=1dq1,…,{xim,yim}i=1dqm):=∏j=1m∏i=1dqj〈xji,πjyji〉 in such a way that contractions are in n-tuples of indices corresponding to the variables yji for any values of i and j and in nj-tuples of indices corresponding to the variables xji for each j=1,…,m. After performing such an operation, the object that remains is a scalar quantity because dqj=njpjsΦ is an integer multiple of nj and d(q1+⋯+qm)=dq(p1n1+⋯+pmnm)=sΦn is an integer multiple of n.LetΛ:={(i,j)∈Z2|i∈{1,…,dqj},j∈{1,…,m}} and suppose Λ is given the lexicographic ordering. This is the index set associated to the product (41). For any λ∈Λ, let its coordinates be denoted iλ and jλ, i.e., λ:=(iλ,jλ). The structure of the expansion of(42)ΩAsΦΩA1p1sΦ⋯ΩAmpmsΦΠd({A1xi1,Ayi1}i=1dq1,…,{Amxim,Ayim}i=1dqm) will include a sum over all partitions J:={J1,…,JsΦ} of Λ into pairwise disjoint sets of cardinality n, where alternating contractions of length n are performed over the groups of variables yji indexed by each of the subsets J1,…,JsΦ. Summing over all such partitions will yield the expansion of the ΩAsΦ factor. The expansions of all the remaining factors of Ω can be expressed as a sum over a different type of partition I:={I1,…,IsΦ(p1+⋯+pm)} of Λ. In this case, the alternating contractions will involve nj indices and variables xji1,…,xjinj for values of j between 1 and m. In other words, each I1,…,IsΦ(p1+⋯+pm) must consist of indices of the form {(i1,j),…,(inj,j)} for some j. While it is perhaps clear what one means by applying the formula (39) to compute the alternating contraction of (41) with respect to these partitions I and J, carefully carrying out this computation explicitly and compactly requires some additional notation. First, for any λ∈Λ, let [λ]I denote the unique subset Iℓ∈I such that λ∈Iℓ. Likewise let [λ]J be the unique element of the partition J containing λ. Let SI be all permutations σ of Λ such that [σλ]I=[λ]I for all I (i.e., SI is restricted to permutations of Λ which preserve the partition I) and analogously for SJ. Lastly, let rI(λ) be the total number of indices λ′∈[λ]I such that λ′≤λ and similarly let cJ(ℓ) be the total number of indices λ′∈[λ]J such that λ′≤λ. It follows that the repeated alternating contraction of Πd associated to the partitions I and J is given exactly by(43)∑σ∈SI,τ∈SJ(−1)σ+τ∏λ∈Λ(πjλ)rI(σλ)cJ(τλ) where (πj)ℓℓ′ is the ℓℓ′-entry of the matrix of πj in the standard basis. The formula (43) can be seen to be an alternating contraction precisely because inside each Iℓ∈I, σ merely permutes elements of Iℓ, which means that the values of rI(σλ) for λ∈Iℓ are merely permutations of {1,…,njλ} and similarly for the partition J. The identity (39) guarantees that (42) is expressible of a linear combination of terms of the form (43) with coefficients which depend on the xji and the yji; moreover, it can be somewhat easily checked that each term of the form (43) is invariant under the action of (A1,…,Am,A) precisely because (43) is expressible in terms of alternating contractions and such contractions themselves have the desired invariance properties.Now for each λ∈Λ, let πλ be a #Λ×#Λ matrix with rows and columns indexed by Λ whose entries are(πλ)λ′λ″:={(πjλ)rI(λ′)cJ(λ″) if [λ]I=[λ′]I and [λ]J=[λ″]J0 otherwise. With this definition, it must be the case that (43) is equal to(44)∑σ,τ∈SΛ(−1)σ+τ∏λ∈Λ(πλ)σλτλ where the sums are now over all permutations σ and τ of Λ because the terms of the sum (44) simply vanish for all permutations σ∈SΛ∖SI and τ∈SΛ∖SJ (simply because there will necessarily be some λ such that [λ]I≠[σλ]I or [λ]J≠[τλ]J, which means that one of the entries of πλ in the product (πλ)σλτλ will necessarily be zero by definition of (πλ)λ′λ″). The expression (44) is itself exactly equal to the expression(∏λ∈Λ∂∂tλ)det⁡∑λ∈Λtλπλ for real parameters tλ, since by the product rule(∏λ∈Λ∂∂tλ)det⁡∑λ∈Λtλπλ=(∏λ∈Λ∂∂tλ)∑τ∈SΛ(−1)τ∏λ′∈Λ(∑λ∈Λtλπλ)λ′τλ′=∑σ,τ∈SΛ(−1)τ∏λ′∈Λ(πσλ′)λ′τλ′ (where the permutation σ comes from all orderings of the partial derivatives) which can be seen to equal (44) by replacing τ by τ∘σ, reordering the terms in the product, and then replacing σ by σ−1. Derivatives of polynomials can always be evaluated exactly as finite differences, which means that (43) itself be realized as a linear combination of determinants det⁡∑λ∈Λtλπλ for various values of the parameters tλ.To finish, observe that the matrices πλ have common block structure. To be precise, each row λ′ of the full matrix is uniquely associated with a unique element of I, namely, [λ′]I∈I, in the sense that πλ will be identically zero in row λ′ unless [λ]I=[λ′]I. The same goes for columns: πλ is zero in column λ′ unless [λ′]J=[λ]J. By reordering rows so that rows associated to the same set in I are adjacent and likewise bringing columns associated to the same set in J together to be adjacent, it follows that the alternating contraction (43) is expressible as a linear combination of determinants of #Λ×#Λ=nsΦ×nsΦ matrices of the exact form described in the statement of the lemma. To see that every block row associated to πj for fixed j contains no more than nj nonzero copies of πj, simply note that this block row is associated to exactly nj literal rows λ′ of the large matrix, and there are exactly nj values of λ such that πλ is not automatically zero in this row (namely, the values of λ such that [λ]I=[λ′]I). If each such λ belongs to a different element of the column partition J, then there can be at most nj nonzero block entries in this block row. The argument for block entries in block columns is similar.□4Radon-like operators: proof of Theorem 2This section contains the proof of Theorem 2. The general structure is to combine three elements: the characterization of the Brascamp-Lieb constant given by Lemma 2, the continuous Kakeya-Brascamp-Lieb inequality as it is formulated in Theorem 1, and key ideas from [11] formulated for the study of nonconcentration inequalities. The initial step is to observe that the quantity in the integrand on the left-hand side of (4) is an integral nonconcentration quantity and so may be directly estimated from below via a supremum: Lemma 4Suppose π is a continuous map from some ℓ-dimensional manifold M intoR(n−k)×n. For any Borel setF⊂Mand any finite nonnegative Borel measure σ on M, there is a Borel subsetF′⊂Fwithσ(F′)≥σ(F)/2such that∫Fm[W({π(tj)}j=1m)]1pdσ(t1)⋯dσ(tm)≳(σ(F))msupt1∈F′,…,tm∈F′[W({π(tj)}j=1m)]1pwith an implicit constant which depends only onn,k, and m.The proof of Lemma 4 is based on the following proposition, which is a mild extension of Lemma 1 from [11]: Proposition 4Let V be a normed vector space. For any positive integer d, any topological space X, any nonnegative finite Borel measure μ on X, any d-dimensional vector spaceFof continuous functionsf:X→V, and anyδ∈(0,1), there is a closed subsetXδ⊂Xwithμ(Xδ)≥(1−δ)μ(X)such that(45)μ({x∈X||f(x)|≥d−1supy∈Xδ⁡|f(y)|})≥δd−1μ(X)for allf∈F. The setXδhas the form(46)Xδ:={x∈X|fj(x)=0∀j<j0and|fj(x)|≤1,∀j≥j0}for some functionsf1,…,fd∈Fand somej0∈{0,…,d+1}.ProofInformally, the content of (45) is that there must always be a relatively large subset Xδ⊂X (large as a fraction of X with respect to the measure μ) such that each f∈F exceeds d−1supy∈Xδ⁡|f(y)| on some nontrivial fraction of X. In essence, it allows one to approximately reverse the usual inequalities of Lp-norms on X if one is allowed to compute the L∞ norm over a slightly smaller set than all of X. The main challenge is to show that the set Xδ can be defined independently of the particular choice of f∈F.By homogeneity of (45) and homogeneity of the inequality μ(Xδ)≥(1−δ)μ(X) with respect to the measure μ, it may be assumed that μ is a probability measure since (45) is clearly true for the zero measure. For any positive ϵ, letFϵ:={f∈F|μ({x∈X||f(x)|>1})≤ϵ}. The first task is to establish a number of elementary facts about the sets Fϵ. The most basic of such facts are that 0∈Fϵ and that Fϵ is star-shaped at the origin, i.e., f∈Fϵ implies tf∈Fϵ for all t∈[0,1]. This follows directly from the inequality |tf(x)|≤|f(x)| when t∈(0,1). Moreover, for any f∈F, tf∈Fϵ for all sufficiently small t>0, sincelimt→0+⁡μ({|tf|>1})=∫Xlimt→0+⁡χ|tf|>1dμ=0 by virtue of Dominated Convergence and the fact that tf(x)→0 for all x. A fourth important simple fact is that Fϵ is closed in the vector space topology on F. To see this, observe that for any sequence of functions fn→f as n→∞, at every point x∈X where |f(x)|>1, it will always be the case that |fn(x)|>1 for all n sufficiently large, simply by continuity of |⋅|. Thus by Dominated Convergence,μ({|fn|>1}∩{|f|>1})→μ({|f|>1}) as n→∞. In particular, if μ({|fn|>1})≤ϵ for all n, then necessarily μ({|f|>1})≤ϵ.Fix a norm ||⋅||F on F, and for all f on the unit sphere {||f||F=1}, letLϵ(f):=sup⁡{t>0|tf∈Fϵ}. This function Lϵ(f) is necessarily upper semicontinuous on the unit sphere because Fϵ is closed: if Lϵ(f)<a for some a>0 and some f with ||f||F=1, then (a−η)f∈Fϵc for all sufficiently small η>0. Because Fϵ is closed, (a−η)g∈Fϵc for all g sufficiently close to f, yielding Lϵ(g)<a. Because the unit sphere is compact, there is a dichotomy: either Lϵ is bounded on the unit sphere and Fϵ is a compact set (since in this case ||⋅||F is necessarily a bounded function on Fϵ), or Lϵ is unbounded and there exists a nonzero f∈F such that tf∈Fϵ for all t>0. By Dominated Convergence, any such f must satisfy(47)μ({f≠0})≤ϵ because limt→∞⁡χ|tf(x)|>1=1 at every point x where f(x)≠0.Now fix any δ∈(0,1). From here forward, fix ϵ:=d−1δ. Suppose there exists a nonzero f1∈Fϵ satisfying (47) when d=1. In this case, setting Xδ:={x∈X|f1(x)=0} will satisfy the hypotheses of the lemma because all functions f∈F will be identically zero on Xδ. This forces (45) to be vacuously true because the supremum over Xδ will always be zero. If d=1 and (47) does not hold for any nonzero f1∈Fϵ, one can instead let f1:=Lϵ(f)f for some nonzero f∈Fϵ and define Xδ:={x∈X||f1(x)|≤1}. Since f1∈Fϵ, it must be that μ(Xδ)≥1−ϵ=(1−δ)μ(X). Nowμ({x∈X||f1(x)|≥1})=lims→1−⁡μ({x∈X||f1(x)|>s}) by Dominated Convergence. If the value of the limit on the right-hand side were strictly less than ϵ, s−1f1 would belong to Fϵ for some s<1, which would mean that s−1Lϵf∈Fϵ, contradicting the maximality of the supremum Lϵ(f). Thusμ({x∈X||f1(x)|≥1})≥ϵ=d−1δμ(X), which implies (45) because 1≥supy∈Xδ⁡|f1(y)|. By homogeneity of (45) in f (and triviality of (45) when applied to the zero function), the lemma must hold when d=1.Thus it suffices to assume that d>1. If Fϵ is not compact, let f1 be taken to equal any nonzero f satisfying (47), let X˜:={x∈X|f1(x)=0}, and let F˜ be any maximal subspace of F which is linearly independent when restricted to X˜. Because f1=0 on X˜, the dimension d˜ of F˜ is at most d−1; if F˜ is trivial, then the lemma follows by fixing Xδ:=X˜. Thus it may be assumed that 1≤d˜≤d−1. By induction on dimension, setting δ˜:=d˜δ/(d−δ)∈(0,1) gives that there exists a set X˜δ˜⊂X˜ of the form (46) with measure at least (1−δ˜)(1−ϵ)≥(1−δ)μ(X) such thatμ({x∈X˜||f(x)|≥d˜−1supy∈X˜δ˜⁡|f(y)|})≥δ˜(1−ϵ)d˜=δdμ(X) for all f∈F˜; however, every function in F restricts to a function in F˜ on X˜, so without loss of generality, the inequality also holds for all f∈F with the same constants. Thus (45) must be true if one defines Xδ:=X˜∩X˜δ˜, which also has the form (46) because X˜ is merely equal to the set {x∈X|f1(x)=0} for some f.It now suffices to assume that Fϵ is compact. Let det be any nontrivial alternating d-linear functional on F (which is unique up to scalar multiples). By compactness of Fϵ, there exist f1,…,fd∈Fϵ such that|det⁡(f1,…,fd)|=suph1,…,hd∈Fϵ⁡|det⁡(h1,…,hd)|. The supremum must be strictly positive because |det⁡(h1,…,hd)|≠0 for any linearly independent set {h1,…,hd}⊂Fϵ and for any such set, there must exist a small positive constant t such that thi∈Fϵ for all i. Now by Cramer's rule, for any f⁎∈Fϵ,(48)f⁎=∑j=1d(−1)j−1det⁡(f⁎,f1,…,fjˆ,…,fd)det⁡(f1,…,fd)fi where ⋅ˆ denotes omission. By the choice of f1,…,fd, the coefficient of each fi in the sum on the right-hand side of (48) has magnitude at most 1. If one definesXδ:={x∈X||fj(x)|≤1∀j=1,…,d}, then Xδc is contained in the union of sets {x∈X||fj(x)|>1} for j=1,…,d; each of these sets has measure at most ϵ, so μ(Xδc)≤dϵ=δ. At any point x∈Xδ, each term in the sum (48) has magnitude at most 1. Thus(49)supy∈Xδ⁡|f(y)|≤d for all f∈Fϵ.Now suppose f∈F is any function which is not identically zero on Xδ and let α>0 be any number such that(50)α<d−1supy∈Xδ⁡|f(y)|, i.e.,supy∈X⁡|α−1f(y)|>d. By (49), α−1f∈F cannot belong to Fϵ. This means thatμ({|f|>α})=μ({|α−1f|>1})≥ϵ=d−1δ. Taking a supremum over all α satisfying (50) and applying Dominated Convergence a final time gives thatμ({|f|≥d−1supy∈Xδ⁡|f(y)|})≥d−1δ, which is exactly the desired inequality (45).□Proof of Lemma 4By Lemma 2, there is some finite collection {Φi}i=1N of polynomial functions of {πj}j=1m such that(51)[W({π(tj)}j=1m)]1p≈∑i=1N|Φi({π(tj)}j=1m)|n−kdi, where di is the degree of Φi as in (22). Apply Proposition 4 to the vector space F of polynomial functions of π of degree at most di, where the measure μ is σ restricted to F. It follows, fixing δ:=1/2, that there exists F′ with σ(F′)≥σ(F)/2 such that∫F|Φi({π(tj)}j=1m)|n−kdidσ(t1)≥∫F|Φi({π(tj)}j=1m)|n−kdiχ|Φi({π(tj)}j=1m)|≥supt1∈F′⁡|Φi({π(tj)}j=1m)|dim⁡Fdσ(t1)≥(supt1∈F′⁡|Φi({π(tj)}j=1m)|dim⁡F)n−kdi⋅σ({t1∈F||Φi({π(tj)}j=1m)|≥supt1∈F′⁡|Φi({π(tj)}j=1m)|dim⁡F})≥12(dim⁡F)−1−n−kdiσ(F)(|Φi({π(tj)}j=1m)|)n−kdiχF′(t1) for any values of t1,t2,…,tm. Note the slight abuse of notation in the inequality just derived: on the top line (which becomes the left-hand side), t1 denotes a variable of integration, while on the final line (the new right-hand side), t1 denotes a point which can be chosen arbitrarily (but yields a trivial inequality unless t1∈F′). We proceed inductively, integrating this inequality over t2 and deriving a new inequality, etc.; the final result of this process yields the inequality∫Fm|Φi({π(tj)}j=1m)|n−kdidσ(t1)⋯dσ(tm)≳(|Φi({π(tj)}j=1m)|)n−kdi(σ(F))m∏j=1mχF′(tj), where the implicit constant is a function of dim⁡F. Summing over i and taking a supremum of the right-hand side over all t1,…,tm∈F′ completes the lemma by virtue of (51).□With the proof of Lemma 4 in hand, the proof of Theorem 2 follows rather easily as well: Proof of Theorem 2Suppose that Σ⊂Ω⊂Rn×Rn is a left-algebraic incidence relation with defining function ρ:Ω→Rn−k. By Theorem 1, for any Borel measurable set E⊂Rn, the functionT˜mχE(x):=∫Σx∩E⋯∫Σx∩E[W({Dxρ(x,yj)}j=1m)]1pdσ(y1)⋯dσ(ym) belongs to Lp(Rn) with p:=n/(m(n−k)) and satisfies||T˜mχE||Lp(Rn)≲|E|m with implicit constant which is independent of E. Now apply Lemma 4 by fixing F to be any subset of Σx∩E on which σ is finite; this gives thatT˜mχF(x)≳(σ(F))msupy1,…,ym∈F′⁡[W({Dxρ(x,yj)}j=1m)]1p for some Borel set F′⊂F⊂E∩xΣ with σ(F′)≥σ(F)/2 and some implicit constant which is independent of E and x. The main hypothesis of Theorem 2 gives thatsupy1,…,ym∈F′⁡[W({Dxρ(x,yj)}j=1m)]1p≳(σ(F′))s≳(σ(F))s for some exponent s and an implicit constant independent of x and F′ and consequently independent of E. But σ(E∩xΣ)=TχE(x) for the Radon-like operator (6), and also σ is σ-finite on the manifold Σx since it has smooth density with respect to Lebesgue measure there, so by applying the newly-derived inequality T˜mχF(x)≳(σ(F))m+s to a sequence of choices of F selected so that σ(F)→σ(E∩xΣ) in the limit, it follows thatT˜mχE(x)≳(TχE(x))m+s with implicit constant that is independent of x and E. It follows that||(TχE)m+s||Lp(Rn)≲||T˜mχE||Lp(Rn)≲|E|m. Raising both sides to the power 1/(m+s) gives (7).□5Applications of Theorem 2This final main section looks at various applications of Theorem 2, which includes the proof of Theorem 3. It begins with some basic computations which show how to compute a suitable defining function and the measures (3) for a Radon-like operator whose incidence relation is given parametrically. Following that is an example application of Theorem 2 which yields an alternative to Christ's proof of the Lp-improving properties of the moment curve [7]. Then comes the proof of Theorem 3, followed by a few extensions and generalizations.5.1A preliminary observation about parametrized incidence relationsProposition 5Letx,y∈Rnbe regarded as ordered pairs(x′,x″),(y′,y″)∈Rk×Rn−kand letγ:Rk×Rn→Rn−kbe any polynomial function. Then the Radon-like operator given byTf(x):=∫Rkf(x′+t,x″+γ(t,x))dtis exactly the operator(6)fromTheorem 2for the defining function(52)ρ(x,y)=y″−x″−γ(y′−x′,x).In particular, the measure dσ defined by(3)equals Lebesgue measure dt.ProofLet B(t,x) be the (n−k)×k matrix given by[∂γ1∂t1(t,x)⋯∂γ1∂tk(t,x)⋮⋮⋮∂γn−k∂t1(t,x)⋯∂γn−k∂tk(t,x)], where γ1,…,γn−k are the coordinate functions of γ in the standard basis and t1,…,tk are the coordinates of t. Taking (52) as the definition of ρ, the right derivative matrix Dyρ (recall (2)) has the block structure[−B(y′−x′,x)In−k] where In−k is the (n−k)×(n−k) identity. The induced Riemannian metric 〈⋅,⋅〉 on the graph Σx satisfies(53)〈∂∂ti,∂∂tj〉=δi,j+∂γ∂ti⋅∂γ∂tj, where ⋅ is the usual dot product on Rn−k and δi,j is the Kronecker delta. When the right-hand side of (53) is regarded as a matrix, the square root of the determinant equals the density of Hausdorff measure with respect to coordinate measure, i.e.,dHk=det⁡(Ik+BTB)1/2dt. Similarly,det⁡(Dyρ(Dyρ)T)1/2=det⁡(In−k+BBT)1/2. ThereforedHkdet⁡(Dyρ(Dyρ)T)1/2=det⁡(Ik+BTB)1/2det⁡(In−k+BBT)1/2dt. Now both det⁡(Ik+BTB)1/2 and det⁡(In−k+BBT)1/2 are invariant under the transformation B↦On−kBOk where On−k and Ok are orthogonal matrices of size (n−k)×(n−k) and k×k, respectively. Thus by the Singular Value Decomposition, to compute the ratiodet⁡(Ik+BTB)1/2det⁡(In−k+BBT)1/2, it suffices to assume that the only nonzero entries of B appear on the diagonal and that Bii≥0 for all i, in which casedet⁡(Ik+BTB)1/2=det⁡(In−k+BBT)1/2=∏i=1min⁡{k,n−k}(1+Bii2)1/2. It follows that dσ=dt.□5.2Warm-up application: the moment curveAs a first example of how Theorem 2 can be applied in practice, consider the case of convolution with the standard measure on the so-called moment curve. In Rn this is exactly the Radon-like transform given by(54)Tf(x):=∫f(x1+t,x2+t2,…,xn+tn)dt. This operator was the titular case study of Christ's seminar work on the combinatorial approach to Lp-improving inequalities [7]. In particular, Christ established that this operator satisfies a restricted weak type (n+12,n(n+1)2(n−1)) and a corresponding dual inequality. Christ's method was later extended by Stovall to arrive at a full Lebesgue space bound for this and more general polynomial curves [22,23]. The arguments below show that Theorem 2 provides a rather direct route to an intermediate result, namely that (54) satisfies a restricted strong type (n+12,n(n+1)2(n−1)) inequality.As implied above, let x:=(x1,…,xn) and y:=(y1,…,yn). The incidence relation associated to (54) has an algebraic defining function which is given byρ(x,y):=(x2−y2+(y1−x1)2,…,xn−yn+(y1−x1)n).Proposition 5 guarantees that the operator (54) equals the operator (6) specified by Theorem 2. A simple computation gives that Dxρ(x,y)=π(y1−x1), whereπ(t):=[2t10⋯0−3t201⋱⋮⋮⋮⋱⋱0(−1)nntn−10⋯01] There is a centrally-important polynomial function Φ(t(1),…,t(n)) which depends only on π(t(1)),…,π(t(n)) and satisfies the invariance properties (22) and (23), given (as in Lemma 3) by a block-form determinant:Φ(t(1),…,t(n)):=det⁡[π(t(1))0⋯00π(t(2))⋱⋮⋮⋱⋱⋮0⋯0π(t(n−1))π(t(n))π(t(n))⋯π(t(n))]. Subtracting upper block rows from the bottom block row results in individual block entries which are zero in all but their first columns. Expanding the determinant in the columns which vanish in the last block row gives that Φ must equal ±(n!) timesdet⁡[t(1)−t(n)⋯t(n−1)−t(n)⋮⋱⋮(t(1))n−1−(t(n))n−1⋯(t(1))n−1−(t(n))n−1], which is equal to(−1)ndet⁡[1⋯1t(1)⋯t(n)⋮⋱⋮(t(1))n−1⋯(t(n))n−1]. This is simply the classical Vandermonde determinant. Now if F⊂R is any Borel measurable set with positive Lebesgue measure, it is always possible to find n distinct points t(1),…,t(n)∈F such that |t(i)−t(j)|≥|F|/(2n−1) whenever i≠j. This is because one can always partition R into nonoverlapping intervals of length |F|/(2n−1); the set F must intersect at least (2n−1) of these intervals in a set of positive measure, so one can always take t(1),…,t(n) from n such intervals which are not adjacent. Thussupt(1),…,t(n)∈F⁡|Φ(t(1),…,t(n))|n!=supt(1),…,t(n)∈F⁡∏1≤i<j≤n|t(i)−t(j)|≥|F|n(n−1)2(2n−1)n(n−1)2. Since Φ is a degree n−1 function of each π(t(j)) in the sense of (22), the inequality (25) gives thatsupt(1),…,t(n)∈F⁡[W({π(t(j))}j=1n)]1p≳|F|n(n−1)2. Thus Theorem 2 implies that (54) satisfies a restricted strong type (n+12,n(n+1)2(n−1)) inequality.5.3Results concerning nonconcentration inequalitiesBefore proceeding with the proof of Theorem 3, it is necessary to recall the main result from [11] concerning nonconcentration inequalities. The point of doing so is to give sufficient conditions of a quantitative nature which guarantee that the main hypothesis (5) of Theorem 2 is true. This will involve identifying certain invariant quantities which generalize the notion of rotational curvature, first introduced by Phong and Stein [18].From [11], recall that a multisystem ∂ of size N on an open set Ω⊂Rn−k is a collection of smooth vector fields {Xji}j=1,…,n−k,i=1,…,N such that for each i=1,…,N, the vector fields {Xji}j=1,…,n−k commute and are linearly independent at every point in Ω. The collection of all such multisystems is denoted M(N). For any fixed vectors X1,…,Xn−k at a point t∈Ω and any function α:{1,…,ℓ}→{1,…,n−k}, where ℓ≤N, the differential operator (X⋅∂)α is defined to equal Zαℓℓ⋯Zα11, where Zji is the unique constant-coefficient linear combination of X1i,…,Xn−ki which equals Xj at the point t. Such α will be called ordered multiindices in n variables and |α| will be used to denote the order of differentiation of (X⋅∂)α, i.e., |α|=ℓ. Matrices T∈GLn−k act on these differential operators by defining(T⁎X)i:=∑j=1n−kTjiXj and taking (T⁎X⋅∂)α:=((T⁎X)⋅∂)α. The main result from [11] that will be used here is the following: Theorem 4cf. Theorem 4 of [11]11Theorem 4 of [11], unlike the other main theorems of that paper, does not actually require one to assume that Φ vanishes to some positive order on the diagonal, but there is likewise no harm in doing so, since in the present case the Theorem will only be applied to polynomials which do indeed vanish to some positive order on the diagonal.SupposeΩ⊂Rn−kis an open set and thatΦ(t1,…,tm)is a polynomial function oft1,…,tm∈Rn−k. For anys>0, let(55)ω(t):=inf∂∈M(N)T∈GLn−kmax|α1|,…,|αm|≤N⁡|(T⁎e⋅∂)1α1⋯(T⁎e⋅∂)mαmΦ(t,…,t)|1s|det⁡T|wheree:={ej}j=1n−kis the collection of standard coordinate vectors at t and(T⁎e⋅∂)jαjdenotes the differential operator(T⁎e⋅∂)αjapplied in the variabletj. If σ is any nonnegative Borel measure which is absolutely continuous with respect to Lebesgue measure such thatdσdt(t)≤ω(t)at each pointt∈Ω, wheredσdtis the Radon-Nikodym derivative of σ with respect to Lebesgue measure, then for any Borel setF⊂Ω,(56)supt1,…,tm∈F⁡|Φ(t1,…,tm)|≳[σ(F)]swith implicit constant depending only on(n−k,m,s,deg⁡Φ,N).Suppose Φ(t1,…,tm) is a polynomial function of t1,…,tm∈Rn−k and that c1,…,cm are nonnegative integers such that(57)∂t1α1⋯∂tmαmΦ(t1,…,tm)≡0 identically on the diagonal t1=⋯=tm for all choices of α1,…,αm satisfying |αj|≤cj for each j and |αj|<cj for at least one j=1,…,m. By definition of (T⁎e⋅∂)α,(T⁎e⋅∂)α=Zαℓℓ⋯Zα11 where Zα11 is a linear combination of X11,…,Xn−k1 which equals ∑j=1n−kTjα1∂j at the point t and so on through Zαℓℓ, which is a linear combination of X1ℓ,…,Xn−kℓ that equals ∑j=1n−kTjαℓ∂j at the base point t. For convenience, let T⁎∂ denote the tuple(∑j=1n−kTj1∂j,…,∑j=1n−kTj(n−k)∂j) and let (T⁎∂)α be the composition(∑j=1n−kTjα1∂j)⋯(∑j=1n−kTjαℓ∂j). The difference (T⁎e⋅∂)α−(T⁎∂)α is a differential operator of order strictly less than ℓ at that distinguished point t where each Ziℓ is fixed to equal ∑j=1n−kTji∂j. By hypothesis on the vanishing of derivatives of Φ on the diagonal, then, it follows that(T⁎e⋅∂)1α1⋯(T⁎e⋅∂)mαmΦ(t,…,t)=(T⁎∂)1α1⋯(T⁎∂)mαmΦ(t,…,t) when |αj|=cj for each j=1,…,m. As before, the subscript j in the expression (T⁎∂)j refers to the partial derivative as it is applied in the variable tj. It follows that(58)ω(t)≥infT∈GLn−k⁡max|α1|=c1,…,|αm|=cm⁡|(T⁎∂)1α1⋯(T⁎∂)mαmΦ(t,…,t)|1s|det⁡T|.To further aid in the estimation of the right-hand side of (58), one may assume without loss of generality that the infimum over T is taken only over those T which are upper-triangular. The reason for this is that we may always write TE=U for some matrix E of determinant 1 with uniformly bounded entries (i.e., a bound independent of T) and some upper-triangular matrix U, which then implies that(59)|(U⁎∂)1α1⋯(U⁎∂)mαmΦ(t,…,t)|≲max|β1|=|α1|,…,|βm|=|αm|⁡|(T⁎∂)1α1⋯(T⁎∂)mαmΦ(t,…,t)| with universal implicit constants depending only on n. The proof of this fact is a direct application of the following proposition: Proposition 6For every positive integer d and everyT∈Rd×d, there existU,E∈Rd×dsuch thatU=TE, U is upper-triangular,det⁡E=1, and∑ℓ=1d∑i=1d|Eℓi|≤2d−1.ProofIf d=1, the proposition is trivially true simply by fixing E to be the 1×1 identity matrix. When d>1, suppose that the final row of T has at least one nonzero entry. Let i be an index which maximizes |Tdi|. Without loss of generality, it may be assumed that i=d, since otherwise we may permute columns of T to make it so, and compensate with a corresponding permutation of the rows of the matrix E to be constructed shortly (and if the permutation leaves det⁡E negative, simply multiply a single column of E by −1 to restore positivity). Under this assumption, let Tji′=Tji−TdiTdd−1Tjd for all i,j∈{1,…,d−1}. By induction, there exists E′∈R(d−1)×(d−1) with determinant 1 such that T′E′ is upper triangular. Now let E be defined so thatEℓi:={Eℓi′ℓ,i∈{1,…,d−1}−∑j=1d−1TdjTdd−1Eji′ℓ=d and i∈{1,…,d−1}0i=d and ℓ∈{1,…,d−1}1i=ℓ=d. Then for i∈{1,…,d−1}, we have∑ℓ=1dTjℓEℓi=(∑ℓ=1d−1TjℓEℓi′)−Tjd∑ℓ=1d−1TdℓTddEℓi′={∑ℓ=1d−1Tjℓ′Eℓi′j≠d0j=d, which ensures that TE is indeed an upper-triangular matrix. We have that det⁡E=det⁡E′ by expanding the determinant of E with respect to its d-th column. Lastly, if i<d, we have∑ℓ=1d|Eℓi|=∑ℓ=1d−1|Eℓi′|+|∑ℓ=1d−1TdℓTddEℓi′|≤2∑ℓ=1d−1|Eℓi′| because |Tdℓ/Tdd|≤1 for each ℓ∈{1,…,d−1}. Thus∑i,ℓ=1d|Eℓi|≤2∑i,ℓ=1d−1|Eℓi′|+∑ℓ=1d|Eℓd|≤2(2d−1−1)+1=2d−1.If Tdi=0 for all i, one can instead take E just as above with the exception that Edi:=0 for i∈{1,…,d−1}. The desired conclusion follows after a minor modification of the above argument.As a final remark, it may be of interest to note that a modification of this argument which involves a further step of multiplying both U and E on the right by a suitably optimized diagonal matrix yields the stronger inequality maxi⁡∑ℓ|Eℓi|≤2(d−1)/2. The extent to which this upper bound can be improved as a function of d is not immediately clear, but this will not be a concern under the present circumstances.□5.4Quadratic submanifolds: proof of Theorem 3Just as was done for the moment curve, the main idea behind the proof of Theorem 3 is to apply Theorem 2; to do so, one establishes the nonconcentration inequality (5) by studying a well-chosen invariant polynomial Φ and applying Lemma 2.To be more specific, the proof proceeds by applying Theorem 2 to the operator (8). The parameter s in Theorem 2 will be fixed to equal n−k and m will be taken equal to n. For an appropriate defining function ρ, the problem reduces to proving thatsupy1,…,ym∈F⁡|W({Dxρ(x,yj)}j=1n)|1p≳(σ(F∩xΣ))n−k uniformly for all x∈Rn and all Borel F⊂xΣ. To accomplish this, it suffices to identify a suitable invariant polynomial function Φ of the matrices {Dxρ(x,yj)}j=1n which satisfies the inequality|W({Dxρ(x,yj)}j=1n)|1p≳|Φ({Dxρ(x,yj)}j=1n)| uniformly in x and y1,…,yn. Since Dxρ(x,y) will depend only on the first k coordinates of y and since Proposition 5 guarantees that σ agrees with Lebesgue measure in these first k-coordinates, it will suffice by Theorem 4 and the inequalities (58) and (59) to show that (with c:=n−k here and throughout the rest of the section)(60)max|α1|=⋯=|αk|=c|(U⁎∂)1α1⋯(U⁎∂)kαkΦ(Dxρ(x,y),…,Dxρ(x,y))|≥|det⁡U|c|∏j=0k−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)]| for any upper-triangular matrix U∈Rk×k, where as before, the operator (U⁎∂)jαj is applied with respect to the variables of yj prior to restricting to the diagonal.To arrive at the final goal (60), one must first be precise about the defining function ρ and the polynomial Φ to be used. As for ρ, it is convenient to use (52) multiplied by a factor of −1 to simplify computation:ρj(u,v):=−vk+j+uk+j+12∑i=1kλji(vi−ui)2 for j=1,…,c. Here u:=(u1,…,un)∈Rn and v:=(v1,…,vn)∈Rn (where the symbols u and v are used to simply avoid the need to temporarily redefine the meaning of the subscripted variables y1,…,yn). Taking the unusual but harmless convention of ordering the entries of u as uk+1,…,un,u1,…,uk, the corresponding left derivative matrix of ρ is given by(61)Duρ=[10⋯0(u1−v1)λ11⋯(uk−vk)λ1k01⋱⋮⋮⋱⋮⋮⋱⋱0⋮⋱⋮0⋯01(u1−v1)λc1⋯(uk−vk)λck]. As already noted, the case m:=n of Theorem 2 is the one of interest here, and the quantity W({Dxρ(x,yj)}j=1n will be estimated from below in terms of well-chosen invariant polynomials (where once again it should be emphasized that each yj is still to be understood as an element of Rn for each j=1,…,n as opposed simply a coordinate entry of some single vector). In particular, by Lemma 1 and specifically using (26), it will be the case that(62)[W({Dxρ(x,yj)}j=1n]1p≳|Φ({Dxρ(x,yj)}j=1n)| whenever Φ satisfies (23) and (22) with d1=⋯=dn=d−k, as shall be the case for the specific Φ constructed below.As in earlier sections, suppose that π1,…,πn are real c×n matrices. To these matrices one may associate an nc×nc matrix M(π1,…,πn) as follows. First, regard each πj as possessing c×c block Aj and a c×k block Bj by fixing Aj to consist of the first c columns of πj and Bj to consist of the final k columns of πj. The matrix M will have a nested block structure:•an upper left block MUL of size kc×c2 which itself is divided into smaller blocks of size c×c which are denoted MijUL for i=1,…,k, j=1,…,c,•a lower left block MLL of size c2×c2 which is itself divided into smaller c×c blocks MijUR for i,j=1,…,c,•an upper right block MUR of size kc×kc consisting of smaller c×k blocks MijUR for i=1,…,k, j=1,…,c, and•a lower right block MLR of size c2×kc consisting of smaller c×k blocks MijLR for i,j=1,…,c. The various sub-blocks of M are derived from the matrices Aj and Bj as follows:•Let MijUR=Bi if the diagonal of MUR passes through MijUR and let MijUR=0 otherwise. (Here the diagonal is understood as the literal diagonal of the kc×kc matrix MUR.)•Let MijUL=Ai if (i,j) is a pair for which MijUR lies on the diagonal of MUR and MUL=0 otherwise. The layout of MUL matches the layout of MUR with the Bi blocks replaced by Ai blocks.•Let MiiLL=Ak+i and MijLL=0 when i≠j.•Let MiiLR=Bk+i and MijLR=0 when i≠j.Fig. 2 illustrates the structure of this matrix M(π1,…,πn). With the matrix M(π1,…,πn) defined, let(63)Φ(π1,…,πn):=det⁡M(π1,…,πn). (To apply Φ in the case of (60), one need only specify that πj=Dxρ(x,yj) for each j=1,…,n.) Permuting the columns of M(π1,…,πn) brings it exactly into the form identified in Section 3.3, so in particular (63) defines a polynomial Φ which has the invariance property (23) and is homogeneous of degree c in each of the matrices π1,…,πn (so m=n and d1,…,dn=c in (22)). In particular, this quantity (63) will satisfy (62) when πj:=Dxρ(x,yj) for each j=1,…,n.When (61) is used for the matrices π1,…,πn as described above, it will be the case that A1=⋯=An=Ic×c and Bj=B(x−yj) for each j=1,…,n withB(t):=[t1λ11⋯tkλ1k⋮⋱⋮t1λc1⋯tkλck], under the convention that t=(t1,…,tk)∈Rk and that each x−yj is understood to be projected down to Rk by retaining only the first k coordinates of each x−yj∈Rn. Restricting to the situation in which yk+1=⋯=yn, it will be the case that πk+1=⋯=πn. By elementary row operations and expanding the determinant of M, it follows thatΦ(π1,…,πk,πn,…,πn)=(−1)c3kdet⁡MUR(B1−Bn,…,Bk−Bn). For convenience, defineΦUR(B1,…,Bk,Bn):=det⁡MUR(B1−Bn,…,Bk−Bn), where by MUR(B1−Bn,…,Bk−Bn), we mean simply the matrix with the same structure as MUR but with each B1,…,Bk replaced by B1−Bn,…,Bk−Bn, respectively. On the full diagonal y1=⋯=yn, the matrix MUR will be identically zero, and so ΦUR will be zero as well.Since B(t) is some c×k real matrix which depends smoothly on the parameter t∈Rk, one can precisely understand the low-order derivatives of ΦUR on the diagonal. For each j=1,…,n, let t(j)∈Rk denote the first k coordinates of x−yj. The immediate goal is to compute ΦUR and its low-order derivatives at a point t(1)=⋯=t(n)=t(0) for some fixed value of t(0). Since Bk+1=⋯=Bn=B(t(0)) on the diagonal, for each index j∈{1,…,k}, there is a unique collection of c rows of the matrix MUR(B1−Bn,…,Bk−Bn) which vanish identically when t(j)=t(0); consequently∂t(1)α1⋯∂t(k)αkΦUR(B(t(1)),…,B(t(k)),B(t(0)))=0 when t(1)=⋯=t(k)=t(0) if |αj|<c for any j=1,…,k. This is precisely the situation anticipated by (57): taking c1=⋯=ck=c and ck+1=⋯=cn=0 in (58) establishes that the inequality (60) would in principle be sufficient to prove Theorem 3 by the application of Theorem 2.A precise analysis of higher derivatives of ΦUR on the diagonal is more delicate. By linearity of B as a function of t, it suffices to assume t(0)=0. To establish a lower bound for quantity ω(t) from (55), one may use (58) and (59). After these reductions, it suffices to compute or otherwise estimate the derivatives of ΦUR(B(t(1)),…,B(t(k)),0) with respect to constant-coefficient vector fields X1,…,Xk of the formXj(i)=∑ℓ=1jcjℓ∂tℓ(i). These are just the vector fields determined by U⁎∂ in (60). In particular, Xj(i) denotes the j-th operator among those defining U⁎∂, applied to the variable t(i). Note in particular that X1(i) points in the first coordinate direction in the variables t(i), X2(i) lies in the span of the first two coordinate directions, and so on. To simplify computations, it will be assumed for the moment that the diagonal entries cℓℓ=Uℓℓ are all equal to 1. It will also be useful to take the periodicity convention Xj+Nk(i):=Xj(i) for any positive integer N. When j and j′ are both integer subscripts of the vector fields just defined, the relation j<j′ will be said to hold when this inequality holds in the usual sense for the representatives of j,j′ taken from the interval {1,…,k} (i.e., the relation j<j′ will mean that the representative of j which belongs to {1,…,k} is less than the corresponding representative of j′).It will be shown by induction on ℓ that for any ℓ≤k, one has(64)Xℓc(ℓ)⋯X(ℓ−1)c+1(ℓ)⋯Xc(1)⋯X1(1)ΦUR(B(t(1)),…,B(t(k)),0)=det⁡MℓUR∏j=0ℓ−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)], where MℓUR is the (k−ℓ)c×(k−ℓ)c lower-right minor of the matrix MUR and where the columns of the matrix λ of coefficients associated to the operator (8) are regarded as periodic with period k just as was the case for the index j of the vectors Xj(i). There are two cases to consider: one case when the block Bℓ+1 appears exactly once in the matrix MUR (e.g., B1 or B4 in Fig. 2) and another case when the block appears twice in MUR with one copy appearing immediately to the right of the other (e.g., B2 or B3 in Fig. 2). In the first case, the truncated matrix MℓUR has the c×1 block[tℓc+1(ℓ+1)λ1(ℓc+1)⋯tℓc+1(ℓ+1)λc(ℓc+1)]T in its upper left-hand corner. As a function of t(ℓ+1), the determinant det⁡MℓUR does not depend on ti(ℓ+1) for any i<ℓc+1 (interpreted periodically), since all such columns of MUR that do depend on these variables lie outside the minor MℓUR. This means that the derivative of det⁡MℓUR with respect to Xℓc+1(ℓ+1) must simply equal the derivative with respect to tℓc+1(ℓ), the effect of which is to replace the upper left block in the first column with the new block[λ1(ℓc+1)⋯λc(ℓc+1)]T and to replace all other entries in the first column with zeros (if they do not vanish already) because they are constant with respect to t(ℓ+1). The argument then repeats for all the remaining derivatives Xℓc+1(ℓ+1) through X(ℓ+1)c(ℓ+1) by advancing to the second column and so on. At each stage, there is no dependence on t(ℓ) with respect to any “lower” coordinate directions. Once all derivatives of det⁡MℓUR with respect to t(ℓ+1) have been taken, the result is thatXℓc+c(ℓ+1)⋯Xℓc+1(ℓ+1)det⁡MℓUR may be expressed as the determinant of a matrix with a c×c minor in the upper-left corner equalling[λ1(ℓc+1)⋯λ1(ℓc+c)⋮⋱⋮λc(ℓc+1)⋯λc(ℓc+c)] and the matrix Mℓ+1UR in the lower right corner.On the other hand, if the block Bℓ+1 appears twice in MUR, then the argument above requires slight modification. First, there must be an index p in the range {ℓc+1,…,(ℓ+1)c} which is equivalent to 1 modulo periodicity. If any columns of the leftmost Bℓ+1 block appear in the minor MℓUR, they must appear alone on their own column since no block in MUR can have neighbors both on the right and below. This would mean that MℓUR has a block in the upper left hand corner with the form[tℓc+1(ℓ+1)λ1(ℓc+1)⋯tp−1(ℓ+1)λ1(p−1)⋮⋱⋮tℓc+1(ℓ+1)λc(ℓc+1)⋯tp−1(ℓ+1)λc(p−1)] and all other entries in these same columns must be zero. It follows when taking the determinant of MℓUR that factors of tℓc+1(ℓ+1),…,tp−1(ℓ+1) appearing on their own rows simply factor out by multilinearity of the determinant as a function of the columns. Furthermore, although these same columns of the leftmost Bℓ+1 appear again in the rightmost Bℓ+1 block, elementary column operations allow one to subtract the leftmost copy of these columns from the rightmost block without changing the determinant of MℓUR. Thus it may be assumed without loss of generality that det⁡MℓUR has no dependence on tℓc+1(ℓ+1),…,tp−1(ℓ+1) beyond the factors already obtained from the initial columns. By exactly the same argument as above, then, it follows thatX(ℓ+1)c(ℓ+1)⋯Xp(ℓ+1)det⁡MℓUR=tℓc+1(ℓ−1)⋯tp−1(ℓ+1)det⁡Mℓ+1URdet⁡[λ1(ℓc+1)⋯λ1(ℓc+c)⋮⋱⋮λc(ℓc+1)⋯λc(ℓc+c)], and from this identity the desired conclusion holds after differentiating once again with respect to the remaining derivatives Xℓc+1(ℓ+1),…,Xp−1(ℓ+1) in order just listed (Xℓc+1(ℓ+1) first, etc.), once again using the fact that at every step, there is no dependence on variables from the “lower” coordinate directions. Finally, because the X vector fields are constant-coefficient linear combinations of coordinate vector fields, we see that while the order of differentiation was extremely useful to exploit for computational purposes, it does not have an effect on the final result. Therefore in both cases we conclude thatX(ℓ+1)c(ℓ+1)⋯Xℓc+1(ℓ+1)det⁡MℓUR=det⁡Mℓ+1URdet⁡[λ1(ℓc+1)⋯λ1(ℓc+c)⋮⋱⋮λc(ℓc+1)⋯λc(ℓc+c)]. Now (64) with ℓ=k gives the final conclusion that(65)X1(1)⋯Xc(1)⋯X(k−1)c+1(k)⋯Xkc(k)ΦUR(B(t(1)),…,B(t(k)),0)=∏j=0k−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)].The inequality (65) gives exactly the desired inequality (60), i.e.,max|α1|=⋯=|αk|=c|(U⁎∂)1α1⋯(U⁎)kαkΦ(Dxρ(x,y),…,Dxρ(x,y))|≥|∏j=0k−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)]| under the assumption that Uii=1 for each i. When the diagonal elements of U are not all 1, one may instead apply (65) by choosingXj(i)=∑ℓ=1jUjj−1Ujℓ∂tℓ(i) for each i=1,…,k and j=1,…,k. Because each subscript index in the set {1,…,k} appears exactly c times among the derivatives on the left-hand side of (65), multiplying both sides of (65) by |det⁡U|c (which is simply the c-fold product of the absolute value of the diagonal elements of U) gives the more general inequalitymax|α1|=⋯=|αk|=c|(U⁎∂)1α1⋯(U⁎)kαkΦ(Dxρ(x,y),…,Dxρ(x,y))|≥|det⁡U|c|∏j=0k−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)]| for arbitrary invertible upper-triangular matrix U; if U is not invertible, the inequality just established is trivially true. This is exactly the desired inequality (60).By (58) and (59) (fixing s=c), it follows that the appropriate density ω(t) from (58) is at least bounded below by a fixed implicit constant (depending only on n) times K1/c, whereK:=|∏j=0k−1det⁡[λ1(jc+1)⋯λ1(jc+c)⋮⋱⋮λc(jc+1)⋯λc(jc+c)]|. By Theorem 4, the measure K1/cdt satisfies K1/cdt≤ωdt, so thatsupy1,…,ym∈F∩xΣ⁡|Φ({Dxρ(x,yj)})|≳[K1/cσ(F∩xΣ)]c for all Borel sets F⊂Rn. Assuming that K>0, the inequality (7) must hold by Theorem 2 after fixing m=n and s=n−k. This is exactly the desired conclusion of Theorem 3.5.5A generalizationThe nature of nonconcentration inequalities such as the main hypothesis (5) of Theorem 2 is that when (5) can be shown to for some model operator, this can often be used to show that it must hold for some generic class of operators and that there must exist some nontrivial polynomial functions of the data which govern the sort of nondegeneracy which (5) implicitly requires. The following result gives such an example: Theorem 5Let k and n be positive integers satisfying the inequalitiesk<n≤2kand let all vectorsx,y∈Rnbe regarded as pairs(x′,x″)∈Rk×Rn−kand(y′,y″)∈Rk×Rn−k, respectively. There exists a nonempty collection of nontrivial polynomials{P1,…,PN}on the space(Rk×k)n−k(i.e., on the space of(n−k)-tuples ofk×kreal matrices) such that the following holds: For any incidence relation ρ of the formρ(x,y):=y″−x″−Q(x′,y′)whereQ:Rk×Rk→Rn−kis a polynomial inx′andy′, ifΩ′⊂Rk×Rkis an open set such that∑i=1N|Pi(∂x′y′2Q(x′,y′)))|2≥cat every point(x′,y′)∈Ω′for some constantc>0, then for any Borel setE⊂Rn, the Radon-like operatorTf(x):=∫(x′,y′)∈Ω′f(y′,x″+Q(x′,y′))dy′satisfies||TχE||L2n−kn−k(Rn)≤C|E|n2n−kfor someC<∞independent of E (where|E|denotes Lebesgue measure of E).ProofAs noted above, let x=(x′,x″)∈Rk×Rn−k and similarly for y. Consider the Radon-like operator parametrized by y=x+(t,Q(x′,x′+t)) for t∈Rk, which has defining function ρ(x,y):=y″−x″−Q(x′,y′) as noted in the statement of the theorem. Using Theorem 4 and following the same initial derivation as in the proof of Theorem 3, to verify the main hypothesis of Theorem 2, it suffices to show thatΦ({Dxρ(x,yj)}j=1n)=det⁡MUR(Dx′Q(x′,y1′)−Dx′Q(x′,y′),…,Dx′Q(x′,x′+yk′−Dx′Q(x′,y′)) has the property that(66)maxT∈GLn−k⁡max|α1|=⋯=|αk|=n−k⁡|(T⁎∂′)1α1⋯(T⁎∂′)kαkΦ({Dxρ(x,y)}j=1n)|1n−k|det⁡T| is uniformly bounded below for all (x′,y′)∈Ω′⊂Rk×k, where ∂′ represents the partial derivatives with respect to the single-primed y′-variables. As before, note once again that |αj|<n−k for some j∈{1,…,k}, the matrix MUR will have a row which is identically zero when it is evaluated on the diagonal y1=⋯=yn=y; when |αj|=n−k for each j∈{1,…,n−k}, the resulting derivative (T⁎∂′)1α1⋯(T⁎∂′)kαkΦ({Dxρ(x,y)}j=1n) is expressible on the diagonal as a polynomial function of ∂x′y′2Q simply because each derivative must fall on a distinct row of MUR for the determinant to be nonzero, which means that no higher-order derivatives in y′ occur in nonzero terms. If R is any polynomial function of the quantities{∂1′α1⋯∂k′αkΦ({Dxρ(x,y)})}|α1|=⋯=|αk|=m which is invariant under the natural action of T∈SLn−k, then just as in the proof of Theorem 3, it must be the case thatmaxT∈GLn−kmax|α1|=⋯=|αk|=n−k⁡|(T⁎∂′)1α1⋯(T⁎∂′)kαkΦ({Dxρ(x,y)}j=1n)|1n−k|det⁡T|≳|R({∂1′α1⋯∂k′αkΦ({Dxρ(x,y)})}|α1|=⋯=|αk|=m)|1n−k for some implicit constant that depends only on n, k, and R. Because we know that the quantity (66) on the left-hand side is nonzero for some choice of ρ (namely, the case established by Theorem 3), this guarantees that it is possible to find a nontrivial invariant polynomial R because the null cone of the SLn−k representation associated to (66) does not trivially contain all vectors. Taking P(∂x′y′2Q(x′,y′))) to equal R({∂1′α1⋯∂k′αkΦ({Dxρ(x,y)})}|α1|=⋯=|αk|=m) for all possible nontrivial R establishes the conclusion of this theorem.□5.6Maximal codimensionThe final application of Theorem 2 is to establish boundedness of certain non-translation-invariant quadratic model operators which have the maximum possible codimension for the given dimension. When the dimension of the underlying submanifold is k, the codimension cannot exceed k2, which is simply equal to the number of mixed partial derivatives ∂x′y′2.Let x:=(x′,x″) for x′∈Rk and x″∈Rk2. For convenience, xi′ will denote the coordinates of x′ in the standard basis and xij″ will be the coordinates of x″, where i,j range over {1,…,k}. The operator which will be studied here is given by the definition(67)Tf(x):=∫Rkf(x′+t,{xij+xi′(tj+xj)}i,j=1n)dt for all measurable functions on Rk×Rk2. The associated defining function ρ(x,y) maps into Rk2 and hasρ(x,y):=−y″+x″+{xi′yj′}i,j=1k.Theorem 6The Radon-like operator given by(67)is of restricted strong type(2k+1k+1,2k+1k).ProofThe matrix Dxρ(x,y) consists of two blocks: one k2×k block on the left and a k2×k2 block on the right which simply equals the k2×k2 identity matrix. The block on the left can itself be understood as composed of k×1 sub-blocks which equal y′ (interpreted as a column matrix) along the block diagonal and 0 elsewhere, i.e., in row (i,j) and column ℓ, the entry of this matrix is yj′δi,ℓ with δ being the Kronecker δ. The simplest invariant polynomial which may be used to estimate the Brascamp-Lieb weight is the following:Φ({Dxρ(x,yj)}j=1k+1):=det⁡[Dxρ(x,y1)0⋯00Dxρ(x,y2)⋱⋮⋮⋱⋱00⋯0Dxρ(x,yk)Dxρ(x,yk+1)⋯⋯Dxρ(x,yk+1)]. To compute this determinant, subtract one copy of each of the upper block rows from the bottom block row and expand the determinant in those columns corresponding to the k2×k2 identity blocks of Dxρ(x,y1),…,Dxρ(x,yk); since there are now no nonzero entries in these columns in the final block row, the expansion is trivial and one concludes that, up to a possible factor of ±1, the determinant equalsdet⁡[yk+1−y10⋯0yk+1−yk0⋯00⋱⋱⋱⋱⋱⋱⋮⋮⋱⋱⋱⋱⋱⋱00⋯0yk+1−y10⋯0yk+1−yk] where each yk+1−yj is understood as a k×1 block, as is each 0. Rearranging columns, this matrix can itself be brought into block form, and consequently|Φ({Dxρ(x,yj)}j=1k+1)|=|det⁡[yk+1−y1⋯yk+1−yk]|k. This Φ corresponds to the case of a multilinear determinant functional, which has been studied in a variety of contexts [10]. In particular, it is known (see [11]) thatsupy1,…,yk+1∈F⁡|det⁡[yk+1−y1⋯yk+1−yk]|≳|F| for any Borel set F∈Rk, so it follows thatsupy1,…,yk+1∈F∩xΣ⁡|Φ({Dxρ(x,yj)})|≳|σ(Σx∩F)|k. By (26) with m=k+1, n=k(k+1) and d1=⋯=dk+1=k2 (one can see that the exponent is k2 by using multilinearity of the determinant defining Φ as a function of its rows), it follows that |W({Dxρ(x,yj)}j=1k+1|≳|Φ({Dxρ(x,yj)}j=1k+1)|, so Theorem 2 applies when s=k to give that||TχE||L2k+1k(Rk×Rk2)≲|E|k+12k+1 for all Borel E⊂Rk×Rk2.□6AppendixThis Appendix contains the proof of Lemma 5, which establishes the existence of a “normalized” defining function which satisfies a number of desirable properties. Lemma 5 was used in Section 2.3 to complete the proof of Theorem 1. The proof of Lemma 5 is essentially a consequence of a quantitative version of the Implicit Function Theorem.To simplify matters somewhat, it is useful to adopt some additional notation. For any x∈Rn and any r>0, let Qx,r:=x+(−r,r)n. Fix |⋅| to be the ℓ∞ norm Rn in the standard coordinates and further fix ||⋅|| to be the ℓ∞→ℓ∞ operator norm on matrices in Rn×n. There is no intrinsic reason why such a choice is required, but having norm balls equal to product boxes makes the application of these results somewhat simpler.Proposition 7Let Φ be an everywhere differentiable map from the ballQx0,rintoRn−k, where0≤k<n. LetDΦxbe the(n−k)×nderivative matrix of Φ at x and let R be ann×(n−k)matrix such thatsupx∈Qx0,r⁡||DΦxR−I||≤c<1.If|Φ(x0)|<r||R||−1(1−c), there exists someu∈Rnsuch that the pointx=x0+Rusatisfiesx∈Qx0,r,Φ(x)=0, and|x−x0|≤||R||(1−c)−1|Φ(x0)|.ProofThe point x will be the limit of the sequence given byxj+1:=xj−RΦ(xj) for all j≥0. By assumption, |Φ(x0)|<r||R||−1(1−c). Suppose that for some value of the index j, it is known that the following inequalities hold:|Φ(xj)|≤cj|Φ(x0)|,|xj−x0|≤||R||1−cj1−c|Φ(x0)|<r(1−cj). By definition of xj+1 and the above inequality for |Φ(xj)|,(68)|xj+1−xj|≤||R||cj|Φ(x0)| which gives that|xj+1−x0|≤|xj−x0|+||R||cj|Φ(x0)|≤||R||1−cj+11−c|Φ(x0)|<r(1−cj+1). One implication of this inequality is that the line segment joining xj and xj+1 belongs to Qx0,r. Consequently, the functiont↦Φ(xj−tRΦ(xj)) is well-defined and differentiable for all t in some open interval containing [0,1]. By the chain rule and the Mean Value Theorem, for any z∈Rn, there is some t∈[0,1] such that〈z,−DΦxj−tRΦ(xj)RΦ(xj)〉=〈z,Φ(xj+1)−Φ(xj)〉, where 〈⋅,⋅〉 is the usual inner product in standard coordinates. For convenience, let x⁎:=xj−tRΦ(xj). Rearranging terms in the above expression yields〈z,Φ(xj+1)〉=−〈z,(DΦx⁎R−I)Φ(xj)〉. Taking absolute values and a supremum over all z with coordinates whose magnitudes sum to 1 and applying the main hypothesis of this proposition gives that |Φ(xj+1)|≤c|Φ(xj)|, which implies that the induction hypotheses continue to hold when the index j is replaced by j+1. By (68), the sequence {xj} must be Cauchy; by continuity of Φ, defining x:=limj→∞⁡xj gives that Φ(x)=limj→∞⁡Φ(xj)=0. The definition of the sequence and continuity of matrix multiplication gives that x−x0=Ru for some u∈Rn, and the limit of the induction hypotheses gives that |x−x0|≤||R||(1−c)−1|Φ(x0)|<r.□Proposition 8Let Φ, R,x0, and r be as inProposition 7and supposek>0. Let V be the orthogonal complement of the image space of R and supposesupx∈Qx0,r|v|≤1,v∈V|DΦxv|≤C.If|Φ(x0)|<r3||R||−1(1−c), thenHk({x∈Qx0,r|Φ(x)=0})≥cnrk(min⁡{12,1−c6C||R||})kfor some constantcn>0that depends only on n.ProofThe new hypothesis guarantees that |Φ(x0+v)−Φ(x0)|≤C|v| whenever v∈V and |v|<r. The proof is by the Mean Value Theorem as it just appeared:|〈z,Φ(x0+v)−Φ(x0)〉|=|〈z,DΦx⁎v〉| for some x⁎∈Qx0,r; applying the new hypothesis of this proposition and taking a supremum over z gives |Φ(x0+v)−Φ(x0)|≤C|v|.Suppose now that |Φ(x0)|<r3||R||−1(1−c) as assumed in the statement of this proposition. For any v∈V such that |v|≤min⁡{r2,r6C||R||−1(1−c)},|Φ(x0+v)−Φ(x0)|≤C|v|≤r6||R||−1(1−c), which means that |Φ(x0+v)|≤|Φ(x0)|+r6||R||−1(1−c)<r2||R||−1(1−c). Moreover, Qx0+v,r/2⊂Qx0,r, so the previous proposition applies on the box with new center x0+v and new radius r/2. This implies that there exists u∈Rn such that Φ(x0+v+Ru)=0 and |Ru|≤r2. In other words, the zero set {x∈Qx0,r|Φ(x)=0} must contain a graph over the k-dimensional set {v∈V||v|≤min⁡{r2,r6C||R||−1(1−c)}}, which forces the graph to have k-dimensional Hausdorff measure at least as large as the k-dimensional Hausdorff measure of the parametrizing set. This establishes the conclusion of the proposition.□Lemma 5Suppose ρ is a smooth defining function on some open set Ω of an incidence relation Σ. There exists some open setΩ˜⊂Ωcontaining Σ and another smooth defining functionρ˜of Σ such that the following hold:1.At every point(x,y)∈Σ, the matrixDxρ˜(x,y)has rows which are orthonormal vectors inRn.2.At every point(x,y)∈Σ,1=det⁡Dxρ˜(Dxρ˜)Tanddet⁡Dyρ(Dyρ)Tdet⁡Dxρ(Dxρ)T=det⁡Dyρ˜(Dyρ˜)T.3.For every compact subsetK⊂Σ, there is an open setU⊂Ω˜containing K and a positiveδ0such that for any(x,y)∈U,|ρ˜(x,y)|<δκnfor anyδ≤δ0(whereκnis some fixed constant depending only on n) implies thatHk(Qx,δ∩Σy)≥cnδkfor some positivecndepending only on n.ProofFor any real symmetric positive-definite matrix A, let A−1/2 be the matrix such that every eigenvector e of A with eigenvector λ>0 of A is also an eigenvector with eigenvalue λ−1/2 of A−1/2. It is relatively easy to see that the mapping A↦A−1/2 is a smooth function of A; the standard way to see this is to use the identityA−1/2=12πi∫γz1/2(zI−A)−1dz where z1/2 is a branch of the square root on the right half space Rez>0 which equals the positive square root on the real axis and γ is, for example, a closed circular contour in the right half space which encloses all eigenvalues of A.Let Ω˜⊂Ω be the neighborhood of Σ on which det⁡Dxρ(Dxρ)T>0; the functionρ˜(x,y):=(Dxρ(Dxρ)T)−1/2ρ(x,y) is well-defined and smooth on Ω˜ provided that ρ is smooth. This mapping ρ˜ vanishes if and only if ρ vanishes (so that Σ is also the set of points (x,y) where ρ˜(x,y)=0), and by the product rule, Dxρ˜=(Dxρ(Dxρ)T)−1/2Dxρ at all points of Σ (since all terms in which derivatives fall on (Dxρ(Dxρ)T)−1/2 vanish because ρ vanishes). This implies that Dxρ˜(Dxρ˜)T is the identity matrix at all points (x,y)∈Σ, which means that the rows of Dxρ˜ are mutually orthogonal unit vectors when (x,y)∈Σ. The formula for det⁡Dyρ˜(Dyρ˜)T also follows directly from the definition of ρ˜.Now fix any compact subset K⊂Σ. Because K is compact, there must exist some r>0 such that Qx0,3r×Qy0,3r⊂Ω˜ for any (x0,y0)∈K. It may further be assumed (after possibly reducing the value of r) that||Dxρ˜(x′,y′)(Dxρ˜(x0,y0))T−I||<12 and|Dxρ˜(x′,y′)v|≤12 for all v∈ker⁡Dxρ˜(x0,y0) such that |v|≤1 whenever (x0,y0)∈K and (x′,y′)∈Ω˜ are any points that satisfy |x0−x′|<2r and |y0−y′|<2r (simply because the quantities on the left-hand sides of these inequalities will be identically zero when (x0,y0)=(x′,y′) and are continuous functions on compact sets, so are consequently uniformly continuous).Now suppose U is the open set of pairs (x,y) such that |x−x0|<r and |y−y0|<r for some (x0,y0)∈K. For any (x,y)∈U, fixing R:=(Dxρ˜(x0,y0))T gives thatsupx′∈Qx,δ⁡||Dxρ(x′,y)R−I||≤12 and supx′∈Qx,δ|v|≤1,v∈ker⁡RT|Dxρ(x′,y)v|≤12 for any δ<r. 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Available online at arXiv:1807.09604. diff --git a/tests/data/elsevier/j.aim.2021.107831_expected.yml b/tests/data/elsevier/j.aim.2021.107831_expected.yml new file mode 100644 index 0000000..3a139c0 --- /dev/null +++ b/tests/data/elsevier/j.aim.2021.107831_expected.yml @@ -0,0 +1,699 @@ +abstract: This paper considers the problem of establishing Lp-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension n of the ambient space and the dimension k of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang [28] and extended by Zorin-Kranich [29], and on recent results for geometric nonconcentration inequalities [11]. As an initial application of this new approach, this paper establishes sharp restricted strong type Lp-improving inequalities for certain model quadratic submanifolds in the range k<n≤2k. +copyright_holder: Elsevier Inc. +copyright_statement: © 2021 Elsevier Inc. All rights reserved. +copyright_year: 2021 +document_type: article +license_url: '' +license_statement: '' +keywords: ['Radon-like operators', 'Brascamp-Lieb inequalities'] +article_type: full-length article +journal_title: Advances in Mathematics +material: publication +publisher: Elsevier Inc. +year: 2021 +authors: +- full_name: Gressman, Philip T. + emails: + - gressman@math.upenn.edu +artid: '107831' +title: Lp-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality +dois: +- doi: 10.1016/j.aim.2021.107831 + material: publication +journal_volume: '387' +journal_issue: '' +is_conference_paper: false +publication_date: '2021-08-27' +collaborations: [] +documents: +- key: j.aim.2021.107831.xml + url: http://example.org/j.aim.2021.107831.xml + source: Elsevier Inc. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: J.-G.BakD.M.OberlinA.Seeger<maintitle>Restriction + of Fourier transforms to curves and related oscillatory integrals</maintitle><maintitle>Am. + J. 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Anal. + journal_volume: '17' + journal_issue: '5' + year: 2008 + page_start: '1343' + page_end: '1415' + label: '3' + authors: + - full_name: Bennett, J. + inspire_role: author + - full_name: Carbery, A. + inspire_role: author + - full_name: Christ, M. + inspire_role: author + - full_name: Tao, T. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: J.BennettA.CarberyT.Tao<maintitle>On + the multilinear restriction and Kakeya conjectures</maintitle><maintitle>Acta + Math.</maintitle>19622006261302J. Bennett, A. Carbery, T. Tao, On the multilinear restriction + and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261–302. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Acta Math. + journal_volume: '196' + journal_issue: '2' + year: 2006 + page_start: '261' + page_end: '302' + label: '4' + authors: + - full_name: Bennett, J. + inspire_role: author + - full_name: Carbery, A. + inspire_role: author + - full_name: Tao, T. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: J.Bourgain<maintitle>Averages + in the plane over convex curves and maximal operators</maintitle><maintitle>J. + Anal. Math.</maintitle>4719866985J. Bourgain, Averages in the plane over convex curves and maximal + operators, J. Analyse Math. 47 (1986) 69–85. + source: Elsevier Inc. + reference: + publication_info: + journal_title: J. Anal. Math. + journal_volume: '47' + year: 1986 + page_start: '69' + page_end: '85' + label: '5' + authors: + - full_name: Bourgain, J. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: J.Bourgain<maintitle>Besicovitch + type maximal operators and applications to Fourier analysis</maintitle><maintitle>Geom. + Funct. Anal.</maintitle>121991147187J. Bourgain, Besicovitch type maximal operators and applications + to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Geom. Funct. Anal. + journal_volume: '1' + journal_issue: '2' + year: 1991 + page_start: '147' + page_end: '187' + label: '6' + authors: + - full_name: Bourgain, J. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'M.Christ<maintitle>Convolution, + curvature, and combinatorics: a case study</maintitle><maintitle>Int. + Math. Res. Not.</maintitle>19199810331048M. Christ, Convolution, curvature, and combinatorics: a case study, + Internat. Math. Res. Notices, 19 (1998) 1033–1048.' + source: Elsevier Inc. + reference: + publication_info: + journal_title: Int. Math. Res. Not. + journal_volume: '19' + year: 1998 + page_start: '1033' + page_end: '1048' + label: '7' + authors: + - full_name: Christ, M. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: M.DomokosA.N.Zubkov<maintitle>Semi-invariants + of quivers as determinants</maintitle><maintitle>Transform. + Groups</maintitle>612001924M. Domokos, A.N. Zubkov, Semi-invariants of quivers as determinants, + Transform. Groups 6(2001), no. 1, 9–24. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Transform. Groups + journal_volume: '6' + journal_issue: '1' + year: 2001 + page_start: '9' + page_end: '24' + label: '8' + authors: + - full_name: Domokos, M. + inspire_role: author + - full_name: Zubkov, A.N. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'A.GargL.GurvitsR.OliveiraA.Wigderson<maintitle>Algorithmic + and optimization aspects of Brascamp-Lieb inequalities, via operator scaling</maintitle><maintitle>STOC''17—Proceedings + of the 49th Annual ACM SIGACT Symposium on Theory of Computing</maintitle>2017ACMNew + York397409MR3678197A. Garg, L. Gurvits, R. Oliveira, A. Wigderson, Algorithmic and + optimization aspects of Brascamp-Lieb inequalities, via operator scaling, in: + STOC''17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, + ACM, New York, 2017, pp. 397–409. MR3678197.' + source: Elsevier Inc. + reference: + publication_info: + parent_title: STOC'17—Proceedings of the 49th Annual ACM SIGACT Symposium on + Theory of Computing + year: 2017 + page_start: '397' + page_end: '409' + label: '9' + misc: + - MR3678197 + authors: + - full_name: Garg, A. + inspire_role: author + - full_name: Gurvits, L. + inspire_role: author + - full_name: Oliveira, R. + inspire_role: author + - full_name: Wigderson, A. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: P.T.Gressman<maintitle>On + multilinear determinant functionals</maintitle><maintitle>Proc. + Am. Math. Soc.</maintitle>1397201124732484P.T. Gressman, On multilinear determinant functionals, Proc. Amer. + Math. Soc. 139 (2011), no. 7, 2473–2484. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Proc. Am. Math. Soc. + journal_volume: '139' + journal_issue: '7' + year: 2011 + page_start: '2473' + page_end: '2484' + label: '10' + authors: + - full_name: Gressman, P.T. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: P.T.Gressman<maintitle>Geometric + averaging operators and nonconcentration inequalities</maintitle>Available + online atarXiv:1906.045992019P.T. Gressman, Geometric averaging operators and nonconcentration + inequalities, 2019. Available online at arXiv:1906.04599. + source: Elsevier Inc. + reference: + publication_info: + year: 2019 + arxiv_eprint: '1906.04599' + label: '11' + misc: + - Available online at + authors: + - full_name: Gressman, P.T. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: L.Guth<maintitle>The + endpoint case of the Bennett-Carbery-Tao multilinear Kakeya conjecture</maintitle><maintitle>Acta + Math.</maintitle>20522010263286L. Guth, The endpoint case of the Bennett-Carbery-Tao multilinear + Kakeya conjecture, Acta Math. 205 (2010), no. 2, 263–286. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Acta Math. + journal_volume: '205' + journal_issue: '2' + year: 2010 + page_start: '263' + page_end: '286' + label: '12' + authors: + - full_name: Guth, L. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'G.KempfL.Ness<maintitle>The + length of vectors in representation spaces</maintitle><maintitle>Algebraic + Geometry</maintitle>Proc. Summer Meeting, Univ. Copenhagen, + Copenhagen, 1978<maintitle>Lecture + Notes in Math.</maintitle>vol. 7321979SpringerBerlin233243G. Kempf, L. Ness, The length of vectors in representation spaces, + in: Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, + 1978), Lecture Notes in Math. vol. 732, Springer, Berlin, 1979, pp. 233–243.' + source: Elsevier Inc. + reference: + publication_info: + journal_title: Lecture Notes in Math. + parent_title: Algebraic Geometry + journal_volume: vol. 732 + year: 1979 + page_start: '233' + page_end: '243' + label: '13' + misc: + - Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978 + authors: + - full_name: Kempf, G. + inspire_role: author + - full_name: Ness, L. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: E.H.Lieb<maintitle>Gaussian + kernels have only Gaussian maximizers</maintitle><maintitle>Invent. + Math.</maintitle>10211990179208E.H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. + Math. 102 (1990), no. 1, 179–208. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Invent. 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Equ. + journal_volume: '264' + journal_issue: '3' + year: 2018 + page_start: '1841' + page_end: '1885' + label: '17' + authors: + - full_name: Patel, N. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: D.H.PhongE.M.Stein<maintitle>Singular + Radon transforms and oscillatory integrals</maintitle><maintitle>Duke + Math. J.</maintitle>5821989347369D.H. Phong, E.M. Stein, Singular Radon transforms and oscillatory + integrals, Duke Math. J. 58 (1989), no. 2, 347–369. + source: Elsevier Inc. + reference: + publication_info: + journal_title: Duke Math. 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Stovall, Endpoint bounds for a generalized Radon transform, + J. Lond. Math. Soc. (2) 80 (2009), no. 2, 357–374. + source: Elsevier Inc. + reference: + publication_info: + journal_title: J. Lond. Math. Soc. (2) + journal_volume: '80' + journal_issue: '2' + year: 2009 + page_start: '357' + page_end: '374' + label: '22' + authors: + - full_name: Stovall, B. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: B.Stovall<maintitle>Endpoint + <math altimg="si1100.svg"><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo + stretchy="false">→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup></math> + bounds for integration along certain polynomial curves</maintitle><maintitle>J. + Funct. Anal.</maintitle>25912201032053229MR2727644B. Stovall, Endpoint Lp→Lq bounds for integration along certain + polynomial curves, J. Funct. Anal. 259 (2010), no. 12, 3205–3229. MR2727644. + source: Elsevier Inc. + reference: + publication_info: + journal_title: J. Funct. 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Available + online at arXiv:1807.09604. + source: Elsevier Inc. + reference: + publication_info: + year: 2018 + arxiv_eprint: '1807.09604' + label: '29' + misc: + - Available online at + authors: + - full_name: Zorin-Kranich, P. + inspire_role: author diff --git a/tests/data/elsevier/j.cpc.2020.107740.xml b/tests/data/elsevier/j.cpc.2020.107740.xml new file mode 100644 index 0000000..63d7f9f --- /dev/null +++ b/tests/data/elsevier/j.cpc.2020.107740.xml @@ -0,0 +1 @@ +application/xmlFrom Feynman rules to conserved quantum numbers, IIIP. NogueiraParticle number conservation rulesFeynman diagramsGraphical correlation functionsSystems of linear Diophantine equationsComputer Physics Communications 260 (2021). doi:10.1016/j.cpc.2020.107740journalComputer Physics Communications© 2020 Elsevier B.V. All rights reserved.Elsevier B.V.0010-4655260March 202110.1016/j.cpc.2020.107740http://dx.doi.org/10.1016/j.cpc.2020.107740doi:10.1016/j.cpc.2020.107740107740JournalsS300.1COMPHY107740107740S0010-4655(20)30366-010.1016/j.cpc.2020.107740Elsevier B.V.Fig. 1Some 0-point topologies for the formal VP-relations v1=(p1)2 and (v1)2=p1p2p3.Fig. 2Two 0-point topologies for the formal VP-relation (v1)2=p1p2p3p4.☆The review of this paper was arranged by Prof. Z. Was.From Feynman rules to conserved quantum numbers, IIIP.Nogueirapaulo.nogueira@tecnico.ulisboa.ptCeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, PortugalCeFEMA, Instituto Superior Técnico, Universidade de LisboaAv. Rovisco Pais 1Lisboa1049-001PortugalCeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, PortugalAbstractThe present article complements the earlier ones in this series. The first part contains various results on the constituent system Cκ(M) of a graph model M, and on its feasibility system Iη(M) (which comprises a number of identities that define the number conservation rules). Those results include the general form of the (particle) number conservation rules in models without explicit propagator mixing.A few types of graph models (including self-conjugate and quasi-normal models) are defined. Oversimplifying, a model M is self-conjugate if it has a certain (weak) combinatorial type of C-symmetry, and M is quasi-normal if Iη(M) fully determines for which multisets of coloured, unlinked half-edges there is a graph in M with that exact multiset. It is shown not only that every self-conjugate model is quasi-normal, but also that some extensions of self-conjugate models are still quasi-normal. Most conveniently, self-conjugate models (and even some extended models) may be recognized in polynomial time.These results seem to lead to the following conclusion: for many (possibly ‘most’ of the) consistent, relevant QFT models, a complete correlation function 〈F〉 is graphical — i.e. there exist Feynman diagram(s) for 〈F〉 — if and only if 〈F〉 is allowed by the number conservation rules (i.e. by a complete system of such rules). Since these rules can be computed in polynomial time then, for many QFT models, deciding whether 〈F〉 is graphical also takes polynomial time.KeywordsParticle number conservation rulesFeynman diagramsGraphical correlation functionsSystems of linear Diophantine equations1IntroductionIn Quantum Field Theory (QFT), many models are described by a Lagrangian density that is invariant under certain continuous and/or discrete phase redefinitions of the quantum fields. Those symmetries lead to (particle) number conservation rules and to conserved charges, either additive or multiplicative. As shown in the first article of this series [1], there exist polynomial-time algorithms that are able to compute a complete system of number conservation rules for an arbitrary, finite graph model. Those rules, or identities, which are integer linear relations, — either modular11The term modular will be used in connection with modular arithmetic, i.e. congruences. or non-modular equalities, and in any case homogeneous — are essentially feasibility (or solvability) conditions for the system of constituent equations.Now that we have feasible algorithms to compute the number conservation rules, we might try to find other computable criteria that can be employed to assert the existence or non-existence of Feynman diagrams for the correlation functions, and that do not rely on the generation of those diagrams.This article complements the two previous papers in this series [1,2]. Section 2 introduces some terminology and notation. Sections 3 to 5 contain a number of basic results which will hopefully clarify some finer points not discussed in the previous papers. The generic form of the number conservation rules for models without explicit propagator mixing is discussed in Section 4. Sections 3 and 5 present a number of results on the system of constituent equations of a graph model and (mainly) on its feasibility system (i.e. system of identities). The latter section contains those results for which the existence of graphs is explicitly assumed, or required, while the former section contains results of a more general nature.The number conservation rules of a QFT model are necessary conditions for the existence of Feynman diagrams (for the correlation functions of that model). Section 6, which contains the main results of this paper, describes some classes of graph models for which the conjunction of those rules constitutes a sufficient condition as well (apart from a minor technicality). The graph models derived from many known QFT models belong to one of the classes described, which contains the self-conjugate models and some of their extensions. In addition, there exist efficient algorithms for deciding whether an arbitrarily chosen model is in that class.Several examples and counterexamples are included.2Basic terminology and notationA QFT model is usually described by a Lagrangian density ℒ, whose kinetic and interaction terms are essentially products of quantum fields and their spacetime derivatives. For each ℒ (with a finite number of terms) one may construct a graph model [1]M(ℒ) by ignoring derivatives, spins, dimensions, and so on. Let us suppose that a certain ℒ depends on q quantum fields Φi (1≤i≤q). Then, in M(ℒ), each field Φi is represented by an indeterminate (dubbed a colour) κi say, each kinetic term that (necessarily) depends on a pair of fields Φi and Φj (distinct or not) is represented by the monomial κiκj (called a p-rule), and each interaction term that depends on d (not necessarily distinct) fields Φi1, Φi2,…,Φid is represented by the monomial κi1κi2⋯κid (dubbed a v-rule). Those terms may be composite — for instance, a kinetic term may include two elementary terms such as 12∂μΦ∂μΦ−m22Φ2,but it may also consist of only one of those terms (either one yields the same p-rule). The p-rules and the v-rules represent the propagator and the vertex types of the quantum model. Moreover, the graphs that may be constructed in M(ℒ) are intrinsically the Feynman graphs for ℒ, with new (i.e. different) labels. Introducing a graph model should make it easier to concentrate on the combinatorial features of the quantum model, and to forget about the physical properties of the fields. Even though the reader may be tempted to mentally substitute κi→Φi there are situations where one has to distinguish between ℒ and M(ℒ) — note (eg) that the derivation ℒ→M(ℒ) is not injective. Further, instead of specifying the Lagrangian density ℒ which gives rise to a certain model M(ℒ), we will use nearly always the following strictly combinatorial definition.Given a set K of colours, a set RP of p-rules, and a set RV of v-rules, let M(K,RP,RV) denote the graph model defined by those (finite) sets. The original definition [1] requires that K and RV should be non-empty (ie K,RV≠∅), and that each colour should be a factor of at least one v-rule. From now on, one further requirement should be included in that definition, namely • every v-rule has nonzero (i.e. positive) degree. This simply asserts that the monomial ω=1 is not a valid v-rule (and that no graph has a vertex of degree zero, of course). For a (fixed) graph model M(K,RP,RV) let, by default, (1)nK=|K|,nP=|RP|,nV=|RV|,and (2)K={κ1,κ2,…,κnK},RP={p1,p2,…,pnP},RV={v1,v2,…,vnV}.The p-rules, the v-rules, and the colours themselves are special cases of K-monomials, i.e. monomials in the elements of K, and with coefficients equal to 1. The notation ω1|ω2 means that the monomial ω1 divides (or is a factor of) ω2, i.e. that ω2∕ω1 is also a monomial (for instance, ω1 may be a colour and ω2 some v-rule or p-rule). The degree of a monomial ω is noted deg(ω), and the degree of the indeterminate x in ω is noted deg(ω,x). If X is a non-empty set of monomials (more generally, if deg(x) is well defined for every element x of X), let (3)δ(X)=minx∈Xdeg(x),Δ(X)=maxx∈Xdeg(x).For example, the above additional constraint to be satisfied by any graph model may be expressed as δ(RV)≥1.A colour κi∈K is pairable if there is pk∈RP such that κi|pk, else it is solitary (or non-pairable). For any K-monomial ω let (4)ω=ϕP(ω)ϕS(ω),where ϕP(ω) and ϕS(ω) contain the pairable and the solitary colour factors of ω, respectively. For a v-rule vj one may have ϕP(vj)=1 or ϕS(vj)=1, but not simultaneously. A K-monomial ω≠1 is self-pairable if it can be expressed as a product of p-rules, viz (5)ω=∏k(pk)bk(pk∈RP,bk≥0).The graph sum G+G⋯+G (where G appears exactly k times) may also be denoted by k⋅G. This notation can be extended in the obvious way to a more general case involving distinct terms. Let Γ(vj) denote the 1-vertex graph that represents v-rule vj — i.e. Γ(vj) has deg(vj) unlinked half-edges attached to a single node, and each half-edge is labelled with one of the colour factors of vj (such that the product of all those labels equals vj). Then, each graph G in some fixed model M(K,RP,RV) may be built from a non-empty collection of vertices (6)k1⋅Γ(v1)+k2⋅Γ(v2)⋯+knV⋅Γ(vnV)by linking pairs of half-edges (where each edge thus created matches a p-rule in RP).The number of vertices (and the number of nodes) of a graph G is denoted by vˆ(G), the number of edges by eˆ(G), the number of unlinked (or unpaired) half-edges by hˆ(G), and the loop order (i.e. circuit rank) by γˆ(G). If hˆ(G)=n then G is an n-point graph; if γˆ(G)=ℓ then G is an ℓ-loop graph. It will be assumed that every graph has at least one vertex, and (as stated earlier) that every vertex has positive degree; thus, any graph G satisfies (7)vˆ(G)>0,eˆ(G)+hˆ(G)>0.The terms node and vertex are not completely interchangeable: node refers to the basic unit of a classic graph, whereas vertex consists of a node and a collection of attached, coloured half-edges. If colours are ignored, a 0-point graph becomes effectively a classic graph (i.e. a non-empty one, with no isolated nodes).The quantities η(G,κi), πV(G,vj) and πP(G,pk) denote the number of (respectively) unlinked half-edges of G with colour κi, vertices of G that match v-rule vj, and edges of G that match p-rule pk. It should be clear that (8)hˆ(G)=∑i=1nKη(G,κi),eˆ(G)=∑k=1nPπP(G,pk),vˆ(G)=∑j=1nVπV(G,vj).The u-product22The origin of this term lies in the fact that uˆ(G) may be defined as a product over the unlinked (or unpaired) half-edges of G. of G may be defined as (respectively, satisfies) (9)uˆ(G)=∏i=1nK(κi)η(G,κi),deg(uˆ(G))=hˆ(G).For example, uˆ(Γ(v))=v for any v-rule v. Given any K-monomial ω, let ηˆ(ω) be the column vector η∈(N0)nK with entries ηi=deg(ω,κi). The end-vector33One may think of the unlinked half-edges as the ends of a graph. of G, noted ηˆ(G), is the vector η with entries ηi=η(G,κi); equivalently, ηˆ(G)=ηˆ(uˆ(G)).The u-product uˆ(G) tells us which correlation function 〈F〉 the graph G contributes to (after converting the colours back into fields). For instance, if uˆ(G)=(κ1)3 then the Feynman graph derived from G contributes to 〈Φ1Φ1Φ1〉. By default, the graphs considered in this paper need not be connected. Accordingly, the relevant QFT correlation functions are the complete ones, which incorporate the contribution of every matching Feynman diagram, connected or not.In the context of QFT, the deletion of an edge is not necessarily a useful operation, as the resulting graph needs not be a graph in the same model. Instead, one may split an edge, which means breaking it into two half-edges which keep their own colours (this is the opposite of creating an edge from two coloured half-edges). Additionally, the deletion of a vertex ν should be implicitly preceded by the splitting of the edges incident at ν, so that other vertices remain intact.A bridge of a graph G is an edge whose splitting increases the number of connected components; equivalently, it is an edge that is not part of a circuit (or ‘loop’). Splitting an edge of an n-point graph produces an (n+2)-point graph, obviously. A bridge whose splitting generates at least one 1-point connected component is singular, otherwise it is regular.3Quasi-graphical monomialsThe system of constituent equations [1] for a graph model M — to be noted Cκ(M,x,η), or simply Cκ(M) — is an integer linear system of the form (10)Ax=η,where A is the constituent matrix (to be noted Aκ(M), in general), x∈ZnP+nV, and η∈ZnK. The entries of the column vectors x and η have different roles: while the xj are the unknowns, the ηi may be seen as parameters, given that each η yields a specific system of equations. For example, Cκ(M,x,0) denotes the homogeneous constituent system, which is related to 0-point graphs.The v-rules and p-rules of the model are encoded in the columns of the constituent matrix. If A∗,b is the column which encodes v-rule vj (respectively, if A∗,c encodes p-rule pk) then (11)A∗,b=ηˆ(vj),A∗,c=−ηˆ(pk).This matrix has neither null rows (since each colour must divide some v-rule) nor null columns (since no v-rule nor p-rule has degree zero). Moreover, each scalar equation in (10) (i.e. each constituent equation) effectively involves a single colour. Here we diverge from [1]. Instead of choosing explicit bijections b(j) and c(k), we will simply implement the following definitions: for each vj∈RV let xj[V]=xb(j) and, for each pk∈RP, let xk[P]=xc(k). Thus, (10) may be rewritten as (12)∑jxj[V]ηˆ(vj)−∑kxk[P]ηˆ(pk)=ηwhich, for a graph G, reduces to (13)∑jπV(G,vj)ηˆ(vj)−∑kπP(G,pk)ηˆ(pk)=ηˆ(G).This is just a graphical (de)composition of G. It tells us that if we construct G by starting from a collection of vertices (6), where kj=πV(G,vj), and then creating (for each pk∈RP) exactly πP(G,pk) edges matching p-rule pk, then the end-vector of G can be computed using (13). Obviously, creating an edge decreases the number of unlinked half-edges, hence the negative sign for the terms that depend on the p-rules.The relevance of the constituent system, either as (10) or as (12), lies in the fact that it is satisfied by every graph, as shown — actually, (10) was initially derived from the above mentioned decomposition. If we were interested only in graphs, we might consider treating (10) as a Diophantine system (in the stricter sense), imposing from the start the constraints44Inequalities such as these should be read entry-wise (e.g. x≥0 means xi≥0 for each entry xi of x).x≥0 and η≥0. Nevertheless, as temporarily disregarding those constraints is actually advantageous, in this section we shall not be primarily concerned about whether or not there exist graphs G such that η=ηˆ(G).For each Cκ(M) (and thus, for each model M), there is [3] a feasibility or solvability system, to be noted I(M,η) or simply Iη(M) — consisting of a (possibly empty) system of linear relations on the ηi — such that Cκ(M,x,z) has integer solutions if and only if z is a solution of (or z is not excluded by) I(M,η). Each of those relations (which were dubbed identities of the model) can have one of two possible forms, namely55Throughout this paper, even when considering m-identities, we will always be dealing with integers, not the elements of the finite rings Z∕kZ.(14a)cTη=0(14b)cTη=0(modk) where c∈ZnK, cT is the transpose of c, k>1, and ηi=η(κi); these identities are assumed to be nontrivial, and that places obvious constraints on c.Although there may exist many solvability systems for the same graph model, they are all equivalent, obviously. For exposition purposes, it is often convenient to pretend that Iη(M) is unique, though; whenever that is the case, Iη(M) will be a representative of that class of equivalent systems. The precise method for choosing the identities in Iη(M) is rather immaterial (at least for this paper), but it may be assumed that they are independent — hence Iη(M) is complete but not overcomplete (and thus finite).Definition 3.1A K-monomial ω is quasi-graphical (in a model M) if ηˆ(ω) is a solution of Iη(M).In physical terms, a quasi-graphical monomial represents a correlation function that is compatible with every particle number conservation rule of the model.3.1Generic modelsThe feasibility system has always integer solutions. For instance, η=0 is clearly a solution of every identity (14), and thus Iη(M) never rules out the existence of 0-point graphs (even for models in which no such graphs exist). The identities of a given model cannot exclude any v-rule vj of that model, since Γ(vj) is a valid graph. In fact, no p-rule may be excluded either, as shown by the next Proposition (that property may be used to derive the general form of the identities of type-SD models [2], see Section 4).Proposition 3.1LetM=M(K,RP,RV), and letωbe aK-monomial that can be expressed in the form(15)ω=∏j(vj)aj⋅∏k(pk)bk(aj,bk∈Z),wherevj∈RV,pk∈RP; then,ωis quasi-graphical.ProofLet us begin with two particular instances of (15), namely ω∈RP and ω∈RV. It is not difficult to find an explicit solution x of Cκ(M,x,η) for each relevant K-monomial. Let A=Aκ(M) and, for each pk∈RP, let xˆ(pk) denote the vector x∈ZnP+nV that satisfies (16)xi[P]=−δi,k,xj[V]=0,for every pi∈RP and every vj∈RV, respectively. This choice yields an integer solution of Cκ(M,x,ηˆ(pk)) since, owing to (12), Axˆ(pk)=ηˆ(pk); although every xˆ(pk) has a negative entry, that is irrelevant for this proof. In addition, for each vj∈RV, let xˆ(vj) be the vector x∈ZnP+nV such that (17)xi[V]=δi,j,xk[P]=0,for every vi∈RV and pk∈RP, respectively. Now, Axˆ(vj)=ηˆ(vj).By definition, Iη(M) never excludes integer solutions; thus, if ω∈RV∪RV, ω is quasi-graphical. The general case follows: for each x∈ZnP+nV, there is z∈ZnK (z=Ax, obviously) such that x is a solution of Cκ(M,x,z). Hence, for any aj,bk∈Z, the vector (18)η=A∑jajxˆ(vj)+∑kbkxˆ(pk)must satisfy every identity in Iη(M), as those identities are linear and homogeneous. If the aj and bk are such that the right-hand side of (15) is a monomial, the equality η=ηˆ(ω) will follow. □The basic reason for the validity of Proposition 3.1 is that the columns of Aκ(M) are precisely the vectors ηˆ(vj) and −ηˆ(pk), for each index j,k. The constraints imposed on Iη(M) by that Proposition may be checked for every example (in [1,2], and in this paper) that presents the identities of some model.In the following, a lattice is a point lattice in some Zm (or a Z-lattice), whose rank is allowed to be smaller than the dimension of the Euclidean space Rm where the lattice may be naturally embedded; a point is defined by a column vector. Let (19)XV=⋃j{xˆ(vj)},XP=⋃k{xˆ(pk)},(20)YV=⋃j{ηˆ(vj)},YP=⋃k{−ηˆ(pk)},(21)XA=XV∪XP,YA=YV∪YP.The elements of XA have been defined in (16) and (17). Each vector in XA has a single nonzero entry, equal to ±1; in addition, no two such vectors have a nonzero entry at the same position. The set XA is then a basis of the integer lattice ZnP+nV. The elements of YA coincide with the columns of Aκ(M).Clearly, the integer system (10) has a solution x if and only if η is an integer linear combination of the columns of Aκ(M). It is quite obvious too that the solutions of an integer, linear, underdetermined homogeneous system form a lattice (see e.g. [3] for an explicit formula). Proposition 3.2 is a straightforward consequence of those facts. A non-minimal proof is presented nonetheless, which includes some details intrinsic to graph models.Proposition 3.2For any (fixed) graph modelM=M(K,RP,RV), the integer solutions ofI(M,η)form a latticeΛI(M)⊆ZnKwhose rank is positive and equal to the rank of the matrixAκ(M); in addition, a pointz∈ZnKis inΛI=ΛI(M)if and only if it may be written as(22)z=∑j=1nVajηˆ(vj)+∑k=1nPbkηˆ(pk),whereaj,bk∈Z,vj∈RV,pk∈RP(i.e.ΛIis generated by the columns ofAκ(M)).ProofEven though Iη(M) may contain both modular and non-modular equations, neither type precludes the existence of a lattice. Given any finite collection of equations of the forms (14a) and (14b), and two solutions η1 and η2 of Iη(M), it is clear that any integer linear combination of those two vectors is also a solution — hence the lattice structure.The lattice ΛI(M) is unique even if the feasibility system is not, since equivalent systems must have the same solutions. The proof of Proposition 3.1 has shown that ΛI contains every point defined by (22). There are no other points: if z∈ZnK is a solution of Iη(M) that means that there is x∈ZnP+nV such that z=Ax; however, since XA is a basis of ZnP+nV, every solution of Iη(M) may be written as in (18), which reduces to (22). The set YA(21) generates ΛI but, as linear dependence is not excluded, YA needs not be a basis of that lattice.Since the rank and the column-rank of a given matrix are equal, (22) shows that the rank of ΛI and the rank of Aκ(M) are the same. The case where the null vector is the only solution of Iη(M) cannot occur: since every graph model has at least one v-rule v (and deg(v)>0), Aκ(M) must have at least one (non-null) column, namely ηˆ(v). Thus, ΛI never degenerates into the ‘null-rank lattice’ Λ0={0}. □Proposition 3.3GivenM=M(K,RP,RV)and aK-monomialω, the following statements are equivalent:(i)ωis quasi-graphical;(ii)ηˆ(ω)∈ΛI(M);(iii)ηˆ(ω)is an integer linear combination of the columns ofAκ(M).ProofThis is a direct consequence of Proposition 3.2. □There are at least two different ways to ascertain whether a monomial ω is quasi-graphical: in accordance with the definition, one may compute Iη(M) and then check if ηˆ(ω) satisfies every identity in Iη(M); or, one may check whether ηˆ(ω) is an integer linear combination of the columns of Aκ(M). There is a well known procedure that may be used in the latter case. One computes the (column) Hermite normal form of Aκ(M) and then deletes every null column (if any) of that normal form. The columns of the resulting matrix, noted A′, form a basis of ΛI(M). Let B denote the matrix that consists of the juxtaposition of the columns of A′ and the column vector ηˆ(ω). Then ηˆ(ω) is in ΛI(M) if and only if the (column) Hermite normal form of B equals A′, up to the addition (or deletion) of a null column.Every identity rules out an infinite number of points in ZnK. Acting on ZnK, a modular identity merely makes the lattice sparser but a diophantine identity reduces the lattice rank by one. If there are two or more identities, each (except the first) acting on the lattice produced by the previous identity, the result is similar provided each identity is independent from the previous ones. Nevertheless, after eliminating all those η incompatible with Iη(M), there still remains a lattice ΛI of positive rank. Therefore, there are also some limits to what Iη(M) can exclude.Proposition 3.4LetM=M(K,RP,RV)be any (fixed) graph model. Then, there exist quasi-graphicalK-monomials of degreenfor infinitely many integersn≥0; moreover, ifRP≠∅those integers include every non-negative even integer.ProofConsider the generic quasi-graphical K-monomial (15). Since every model has at least one v-rule (v1, say), the monomial ω=(v1)a1 is quasi-graphical for each a1≥0; in addition, deg(ω)=a1deg(v1), where deg(v1)>0.If RP≠∅ then the monomial ω=(p1)b1 is quasi-graphical for each b1≥0 and has degree 2b1. □Let M be a graph model such that ker(AT)≠{0}, where A=Aκ(M), and let c∈ZnK∖{0} such that ATc=0. Then there is a d-identity, since for each η∈ΛI(M) there exists an x such that Ax=η, and thus cTη=(cTA)x=0.Conversely, if (14a) is a d-identity then, given the decomposition for x implicit in (18), in the basis XA, it must follow that (23)ATc=0,c≠0.A similar analysis shows that each m-identity (14b) is associated with a pair (c,k) such that k>1 and (24)ATc=0(modk),c≠0(modk).An m-identity (14b) is primitive if c satisfies (24) but not (23), of course.Proposition 3.5LetMbe any (fixed) graph model, andA=Aκ(M)its constituent matrix; in addition, letASandAHdenote the Smith and the (row) Hermite normal forms [4,5]ofA. Then,(i)Mhas a d-identity if and only ifAhas linearly dependent row(s);(ii)Mhas a d-identity if and only ifAShas a null row;(iii)Mhas a d-identity if and only ifAHhas a null row;(iv)Mhas a primitive m-identity if and only ifAShas an entry not in{0,1};(v)Mhas no identities if and only if the columns ofAgenerate the full integer latticeZnK(ieΛI(M)=ZnK).ProofFor a proof of claims (i–iv) see [1]; in fact, (ii) and (iii) follow immediately from (i), which follows from (23). Regarding (v), if Iη(M) is empty, the constituent system has integer solutions for every η∈ZnK, and thus ΛI(M)=ZnK; if ΛI(M)=ZnK then Iη(M) must be empty, since any identity rules out infinitely many points in ZnK. □Let E=E(η) be a d-identity (14a), expressed as a linear homogeneous relation in the variables η(κi), no two terms depending of the same colour. Now let θ(E) denote the number of terms contributing to E, i.e. the number of (necessarily distinct) indices i such that the term ciη(κi) has non-null coefficient ci. Saying that E depends on η(κi) means that ci is non-null. Additionally, for any (fixed) K-monomial ω, let θ(E,ω) denote the number of non-null terms in E(η) immediately after substituting η→ηˆ(ω) (i.e. the number of indices i for which cideg(ω,κi) is non-null).Proposition 3.6LetM=M(K,RP,RV), and letE=E(η)be a d-identity forM. Then(i)θ(E)≥2, and there exist at least two termsciη(κi)andcjη(κj)inEsuch thatcicj<0;(ii)θ(E,ω)≠1for any quasi-graphicalK-monomialω.ProofIf E(η) had a single term, it could be written as η(κa)=0(κa∈K).By Proposition 3.1, this identity should be satisfied by every v-rule in RV. That means deg(vj,κa) should be null for every v-rule vj∈RV. Since colour κa necessarily divides some v-rule vj∈RV, deg(vj,κa)>0 (contradiction). Therefore, θ(E)≥2. Moreover, every d-identity must have (at least) one positive coefficient and one negative coefficient. If that were not the case then it would not be possible to achieve a cancellation between the terms of E(ηˆ(vj)). Alternatively, (23) would not hold, since the matrix Aκ(M) has neither null rows nor columns with both negative and positive entries.If exactly one nonzero term cideg(ω,κi) were to result from the substitution η→ηˆ(ω) in E(η) then no cancellation would be possible either. Nevertheless, one may have θ(E,ω)>1 or θ(E,ω)=0. □Example 3.1In Quantum Electrodynamics (QED), the fermion number conservation rule is a d-identity that involves the difference between the numbers of fermions and anti-fermions of the same species. Thus, θ(E)=2, and the two coefficients have opposite sign. ♢If a certain model has two or more linearly independent d-identities, E(η) can be any non-trivial linear combination of a basis of d-identities. Proposition 3.6 implies that no such linear combination may be reduced to a single nonzero term ciη(κi) (as pointed out before [2], that constraint is due to the definition of graph model), nor to a number of such terms whose coefficients all have the same sign. In particular, if there is a v-rule of the form vj=(κi)d then η(κi) does not appear in any d-identity.Example 3.2The 1-colour (type-S) model defined by the sets K={κ1},RP={(κ1)2},RV={(κ1)4},has a single primitive identity, namely the m-identity η(κ1)=0(mod2)which has a single term (and whose coefficient c1 equals 1). However, any quasi-graphical K-monomial must have the form ω=(κ1)a1 where a1 is even, in which case the term c1a1 cancels itself (in that identity). ♢3.2Type-SD, self-conjugate, and quasi-self-conjugate modelsFor many graph models derived from relevant QFT models (with no explicit propagator mixing), it is possible to define a colour-conjugation transformation cˆ which replaces colours by their ‘anti-colours’, so to say. In particular, every p-rule is invariant under cˆ, and cˆ2 is equal to the identity transformation. That transformation cˆ may be seen as a weak version of the QFT charge-conjugation C — a graph model M(ℒ) may be invariant under cˆ even if ℒ has no C-symmetry whatsoever.In a type-SD model [2], for each colour κi∈K there is a unique κj such that κiκj∈RP — we say that colour κj is the conjugate of κi (and vice versa) and write κj=cˆ(κi), κi=cˆ(κj). The conjugate of a K-monomial ω, to be noted cˆ(ω), is derived from ω by replacing each colour factor by its conjugate; also, given a set Ω of K-monomials, cˆ(Ω) denotes the set whose elements are the conjugates of the elements in Ω. A monomial (or a set) X is self-conjugate if cˆ(X)=X, else we say that X and cˆ(X) constitute a conjugate pair.Proposition 3.7LetM(K,RP,RV)be a type-SD model, andωaK-monomial; then,ωandcˆ(ω)are either both quasi-graphical or both non-quasi-graphical.ProofAs the case ω=cˆ(ω) is trivial, one may assume without loss of generality that ω≠cˆ(ω), and in particular that ω≠1. Let ω=∏i(κi)ai, ai≥0; for each κi∈K let κ[i]=cˆ(κi), i.e. let [i] denote the index of the conjugate of κi. Then, for each i≠[i], the exponents of κi and κ[i] in the product ωcˆ(ω)=∏i(κi)ai+a[i]are equal and non-negative; additionally, for each i=[i], the exponent of κi is even and non-negative. Thus, ωcˆ(ω) is self-pairable and (by Proposition 3.1) quasi-graphical. Moreover, given that ηˆ(ω)+ηˆ(cˆ(ω))=ηˆ(ωcˆ(ω)),if either ω or cˆ(ω) is quasi-graphical then both are. □Definition 3.2A self-conjugate model M(K,RP,RV) is a graph model for which there is a bijection cˆ:K→K such that (cˆ)−1=cˆ and (25)RP=⋃κa∈K{cˆ(κa)κa},RV=cˆ(RV).In this case cˆ will be dubbed a c-symmetry of the model (obviously, cˆ(RP)=RP). A self-conjugate model is necessarily a type-SD model — for instance, it has no solitary colours, |RP|≤|K|, and it is subject to Proposition 3.7.Definition 3.3A quasi-self-conjugate model M is a graph model that may be derived from a self-conjugate model by deleting one or more p-rules.Some quasi-self-conjugate models may be derived (in the manner just described) from more than one self-conjugate model. As a result, a quasi-self-conjugate model M may have more than one residualc-symmetry — i.e. a symmetry induced by the c-symmetry of a self-conjugate model from which M may be derived. A quasi-self-conjugate model has at least one non-pairable colour, and is necessarily a type-SD∗ model.Proposition 3.8LetM=M(K,RP,RV)be a quasi-self-conjugate model. Then, for each residual c-symmetrycˆ, and for eachvj∈RV, theK-monomials66See (4) for the definitions of ϕP and ϕS.(26)χj=cˆ(ϕP(vj))ϕP(vj)ωj=cˆ(ϕS(vj))ϕS(vj)are both quasi-graphical; additionally, if anyχjorωjdepends on a single colour (κa, say) then no d-identity depends onη(κa).ProofBy Proposition 3.1, the v-rules vj and cˆ(vj) are quasi-graphical, and so is χj because it is obviously a product of p-rules pa=cˆ(κa)κa. That same Proposition implies that ωj=vjcˆ(vj)∕χj must be quasi-graphical too. The cases where ϕP(vj)=1 or ϕS(vj)=1 are trivial, of course.The second claim follows at once from Proposition 3.6 (for ω=(κa)d). □3.3Simple model extensionsThe following property is likely to be commonly used in QFT model-building: if a Lagrangian density ℒ′ can be derived from another Lagrangian density ℒ by simply adding some new terms (but no new fields), and if the new terms do not break the symmetries of ℒ, then ℒ and ℒ′ will have exactly the same symmetries. Therefore, the next result presents yet more evidence that Iη(M) behaves as expected from a system of number conservation rules.Proposition 3.9GivenM=M(K,RP,RV), letωbe aK-monomial such thatdeg(ω)>0andω∉RV; let alsoR̄V=RV∪{ω}andM̄=M(K,RP,R̄V). Then,ΛI(M̄)=ΛI(M)if and only ifωis quasi-graphical inM.ProofBy Proposition 3.2, ηˆ(ω)∈ΛI(M̄). If ω is not quasi-graphical in M then ηˆ(ω)∉ΛI(M) and ΛI(M)≠ΛI(M̄). If ω is quasi-graphical in M, then ηˆ(ω)∈ΛI(M); since one cannot enlarge a lattice Λ by adding an element z∈Λ to a generating set of that lattice, it follows at once that ΛI(M)=ΛI(M̄) (in which case I(M,η) and I(M̄,η) are equivalent systems). □There is an analogous result for the case where M̄ is generated from M by adding a new p-rule, and the proof is similar (note, however, that not every model may be extended in this way).Proposition 3.10GivenM=M(K,RP,RV), letωbe aK-monomial such thatdeg(ω)=2andω∉RP; let alsoR̄P=RP∪{ω}andM̄=M(K,R̄P,RV). ThenΛI(M̄)=ΛI(M)if and only ifωis quasi-graphical inM.Proposition 3.11LetM=M(K,RP,RV)be a type-SD model. Then eitherMis self-conjugate, or there is a self-conjugate modelM̄=M(K,RP,R̄V), of whichMis a sub-model (ieRV⊂R̄V), such thatΛI(M̄)=ΛI(M).ProofSince M is a type-SD model, there is (as before) a map cˆ that transforms each colour κi∈K into its conjugate cˆ(κi). Let the set Ω consist of those v-rules vj∈RV such that cˆ(vj)∉RV, and let R̄V=RV∪cˆ(Ω). If Ω=∅ then M is self-conjugate, else M̄ is self-conjugate; in the latter case, Proposition 3.7 ensures that every element of cˆ(Ω) is quasi-graphical in M, and then Proposition 3.9 implies ΛI(M̄)=ΛI(M). □4Identities for type-SD and type-SD∗ modelsA QFT model features (explicit) propagator mixing if — besides the usual kinetic terms that depend on a field and on its anti-field — there is at least one kinetic term that depends on the ‘wrong’ pair of fields. In practice very few models are of that type — although that feature is not unusual in QFT model-building, it is nearly always eliminated by means of a unitary transformation of the fields involved (which simultaneously defines the actual physical fields). Thus, predominantly, common QFT models give rise to type-SD and to type-SD∗ models [2] (see also Section 7).If a graph model is either type-S or type-D, the form of the identities is considerably more restricted than that of a generic model. The general form of the identities of a type-SD model is presented below, as it was not included in [2].Proposition 4.1LetM(K,RP,RV)be a type-SD model withpconjugate colour pairs{yi,zi}(1≤i≤p)andqself-conjugate coloursxj(1≤j≤q); let alsoΔi=η(yi)−η(zi). Then, any d-identity is independent of the self-conjugate colours and can be reduced to the generic form(27)a1Δ1+a2Δ2⋯+apΔp=0.Furthermore, any m-identity can be reduced to the generic form(28)a1′Δ1+⋯+ap′Δp+b1′η(x1)+⋯+bq′η(xq)=0(modk),subject to the following additional constraints:(i)ifkis odd then everybj′may be set to zero;(ii)ifkis even then eachbj′may be set to one (and only one) of the values in{0,k∕2}.(iii)if every coefficientai′is null then one may choosek=2(i.e. if an m-identity depends only on the self-conjugate colours then it is a ‘parity-rule’).ProofWhatever the model, the expression Lˆ(E) on the left-hand side77By convention, the side that contains the non-constant terms. of any identity E may be written as a linear combination of the η(κi), i.e. (29)Lˆ(E)=∑κi∈Kc̄iη(κi).while on the right-hand side we have the constant zero, as in (14). A type-SD model has two types of p-rules and, by Proposition 3.1, every p-rule is quasi-graphical. That implies the following.For each p-rule yizi, every identity must be satisfied when η(yi)=η(zi)=1 and all the other entries of η are null. Thus, the coefficients of η(yi) and η(zi) must add up to zero (or to zero modulo k, depending on the type of identity), which means that the combined terms are proportional to Δi.For each p-rule (xj)2, every identity must be satisfied when η(xj)=2 is the only nonzero entry of η. Therefore, the coefficient of η(xj) must vanish in any d-identity, and 2bj′ must be null (modulo k) for any m-identity. The additional constraints follow at once. □Since, by definition, a solitary colour does not divide any p-rule, such colours do not constrain the form of the identities as pairable colours do. If (apart from the colour content described in Proposition 4.1) a type-SD∗ model has also r solitary colours sl (1≤l≤r) then (27) and (28) should be replaced by more general forms, viz (30)∑i=1paiΔi+∑l=1rclη(sl)=0,(31)∑i=1pai′Δi+∑j=1qbj′η(xj)+∑l=1rcl′η(sl)=0(modk),and constraint (iii) should be modified accordingly.Example 4.1The type-SD∗ model defined by the sets K={y1,z1,s1},RP={y1z1},RV={(y1)2(z1)2,(y1)2z1s1,(z1)2y1s1},has one solitary colour (i.e. s1) and one primitive identity, namely η(y1)−η(z1)+η(s1)=0(mod2).One may construct a self-conjugate model by adding the p-rule (s1)2, which means that cˆ(y1)=z1 and cˆ(s1)=s1. Proposition 3.8 may be cross-checked immediately: each v-rule vj that depends on s1 yields ωj=(s1)2, and ηˆ((s1)2) satisfies the above m-identity; no d-identity depends on η(s1), of course. ♢In general, the colour permutation symmetries of a graph model M should be reflected on Iη(M) even if some identities may (per se) lack the required invariance. Obviously, Proposition 4.1 applies to self-conjugate models and, if the analysis is kept as general as possible, without making further assumptions about the model, there will be no other constraints on the form of the identities. Indeed, apart from some obvious, minor rewriting, in which the additional constraints on the m-identities should not be neglected, (27) and (28) are separately invariant under the transformation that consists of the full set of exchanges yi↔zi (ie Δi↔−Δi). In the case of quasi-self-conjugate models, the residual c-symmetries may still impose the same type of conditions on the coefficients of the terms ciη(si).Example 4.2The model defined by the following sets is quite similar to the model given in the previous example. K={y1,z1,s1,s2},RP={y1z1},RV={(y1)2(z1)2,(y1)2z1s1,(z1)2y1s2}It also has one primitive identity (although a d-identity), e.g. η(y1)−η(z1)−η(s1)+η(s2)=0.Even though s1 and s2 are independent, this identity depends on the difference η(s1)−η(s2), as if s1 and s2 were a pair of conjugate colours. That dependence is not fortuitous: it is possible to construct a self-conjugate model by adding p-rule s1s2, and both models are invariant under the transformation cˆ defined by cˆ(y1)=z1 and cˆ(s1)=s2. ♢The restriction to self-conjugate models does not necessarily preclude primitive m-identities with intrinsic modulus of congruence k greater than 2, as the next example shows; nevertheless, such moduli are excluded in type-S models [2], which are obviously self-conjugate. Regarding QFT models see e.g. [6], and in particular the potential with Z3 symmetry, for which there is an m-identity with k=3. On the other hand, there is a model [7] with extended CP-symmetry for which one might expect to find an m-identity with k=4; however, it turns out that k=2 is sufficient.Example 4.3The (type-D) model defined by the sets RP={y1z1,y2z2,y3z3},RV={y1y2(z3)2,z1z2(y3)2,y2y3(z1)2,z2z3(y1)2,y3y1(z2)2,z3z1(y2)2},is clearly self-conjugate. It has one d-identity, namely Δ1+Δ2+Δ3=0where Δi=η(yi)−η(zi), and it has also one primitive m-identity, to be chosen among the following ones: Δi−Δj=0(mod3),(1≤i<j≤3).Those same identities also occur in each of the three sub-models obtained by deleting a conjugate pair of v-rules from RV. ♢5Graphical monomialsThe identities of a graph model M have been identified with the feasibility conditions for Cκ(M,x,η). For some models, however, not every non-negative solution η of I(M,η) (ie η∈ΛI(M), η≥0) is the end-vector of some graph. In the following definitions the graph model M=M(K,RP,RV) is assumed to be fixed (else one should e.g. replace ‘graphical’ by ‘graphical in M’).Definition 5.1A K-monomial ω is graphical if there is a graph G such that ω=uˆ(G).One of the differences between quasi-graphical and graphical monomials is as follows: given a set of quasi-graphical monomials ωi, the vector η defined by the integer linear combination (32)η=∑iaiηˆ(ωi)(ai∈Z)defines a quasi-graphical monomial ω (ie η=ηˆ(ω)) whenever η≥0 (Proposition 3.1); however, given a similar set of graphical monomials, the condition η≥0 does not always ensure that there is a graphical monomial ω such that η=ηˆ(ω), unless (eg) ai≥0 for each index i and ∑iai>0.Example 5.1Each monomial ωj defined in (26) is graphical. A graph G such that uˆ(G)=ωj may be obtained from the graph Γ(vj)+Γ(cˆ(vj)) by linking the half-edges with pairable colours (if any), so as to create valid edges. ♢Definition 5.2The pattern of a graph G, noted πˆ(G), is the column vector x∈ZnP+nV such that (a) xk[P]=πP(G,pk) for each p-rule pk∈RP, and (b) xj[V]=πV(G,vj) for each v-rule vj∈RV.If x is the pattern of some graph G then one may also say that x is a graphical solution of Cκ(M), and in that case (33)Ax=Aπˆ(G)=ηˆ(G)=η≥0,(34)∑ixi=∑jxj[V]+∑kxk[P]=vˆ(G)+eˆ(G).If a fixed η∈ZnK is the end-vector of a graph G then (obviously) there exists at least one pattern x such that (33) holds; however, not every x satisfying (33) is necessarily graphical.Although there may exist many graphs with the same pattern, generating a single graph with a given pattern is straightforward — see Algorithm 5.1, whose correctness follows from the proof of Proposition 5.1.Algorithm 5.1(0)input: a model M(K,RP,RV), and a vector x∈ZnP+nV;(1)make sure that x is a pattern (Proposition 5.1), else exit on error;(2)create the graph sum defined by (36);(3)(loop) for each pk∈RP, create xk[P] edges of that kind, by linking appropriate pairs of half-edges;(4)output: graph built in steps (2) and (3).Proposition 5.1LetM=M(K,RP,RV)be any (fixed) graph model, and letA=Aκ(M),x∈ZnP+nV; thenxis the pattern of some graph if and only if the following conditions hold:(35)Ax≥0,x≥0,x≠0.ProofThese conditions are obviously necessary, as they are satisfied by any graph G. The entries of Ax consist of the nK quantities η(G,κi), and the entries of x consist of the nV+nP quantities πV(G,vj) and πP(G,pk). Lastly, the pattern should not be null since G cannot have zero vertices.To prove sufficiency, let us first remark that the nonzero terms of the constituent equations have positive coefficients if the unknowns are of ‘vertex-type’, and negative coefficients if the unknowns are of ‘edge-type’ — see (12). If x has no negative entries, the graph (36)∑j=1nVxj[V]⋅Γ(vj)exists (it is just a collection of disjoint vertices) unless every coefficient xj[V] equals zero. However, that special case cannot really occur: since x≠0, there would have to exist a nonzero (positive) entry xk[P] and, as a result, at least one of the constituent equations (12) would fail (as there would be at least one negative term, but no positive terms).Additionally, if every entry of Ax is non-negative, there exist sufficient numbers of half-edges to create all the required coloured edges — this is simply what the constituent equations express. In other words, incrementally (say) and starting with (36), for each pk∈RP it is possible to create xk[P] edges matching p-rule pk by linking appropriate pairs of half-edges. The remaining half-edges define the end-vector Ax. □Example 5.2The (type-D) model M=M(K,RP,RV) defined by the sets K={y1,z1,y2,z2},RP={y1z1,y2z2},RV={y1(y2)2,z2(z1)2}, has a single primitive identity, namely η(y1)−η(z1)+η(y2)−η(z2)=0(mod3).Although this identity does not rule out 0-point graphs (as stated earlier, no identity does that) and every colour is pairable, no such graph may be built. Suppose that a 0-point graph G exists, and let v1=y1(y2)2, v2=z2(z1)2, m1=πV(G,v1), and m2=πV(G,v2). By splitting every edge of G one should obtain the graph Γ=m1⋅Γ(v1)+m2⋅Γ(v2). However, pairing all the half-edges of Γ according to the existing p-rules requires (a) m1=2m2, for p-rule y1z1, and (b) m2=2m1, for p-rule y2z2. Thus, no graph G satisfies uˆ(G)=1.There are no 1-point graphs either, but these are excluded by the above identity (Proposition 5.2). For 2-point graphs, the only graphical monomial is y2z1, although every monomial in the set {y1z1,y1z2,y2z1,y2z2} is quasi-graphical. That is consistent with the non-existence of 0-point graphs: had there been a graph G with either uˆ(G)=y1z1 or uˆ(G)=y2z2, or graphs G1 and G2 such that uˆ(G1)=y2z1 and uˆ(G2)=y1z2, we would have been able to construct a 0-point graph from those graphs (by linking one or two pairs of half-edges).The constituent matrix is square (4×4) and non-singular. Thus, for each K-monomial ω, there is at most one (possibly non-graphical) integer solution x of Cκ(M,x,ηˆ(ω)), and the number of graphs G with uˆ(G)=ω is necessarily finite, if not null. ♢If there are no 1-point graphs in some model then no graph may have a singular bridge; in addition, 0-point graphs (if any exist) must be bridgeless, since every bridge of a 0-point graph is singular. The next result yields two basic necessary conditions for the existence of 1-point graphs.Proposition 5.2LetGbe a 1-point graph in some modelM=M(K,RP,RV), and letκa=uˆ(G). ThenRVcontains at least one odd v-rule, andIη(M)does not depend onη(κa).ProofThe first claim is elementary, and is included for completeness only: splitting every edge of G would yield a collection of vertices with an odd number of half-edges (in total), and that number cannot be obtained by summing even numbers only.The entries of ηˆ(G) are η(G,κi)=δa,i for 1≤i≤nK. If M had an identity E(η) that depended on η(κa), after the substitution η→ηˆ(G) the term proportional to η(κa) would be non-null and there would be no other term to cancel it (even if that identity were modular, as every identity is homogeneous). Thus, ηˆ(G) would not be a solution of Iη(M), which is a contradiction. □Given n>0, we will say that n-point graphs are totally excluded by Iη(M) if no K-monomial of degree n is quasi-graphical, and that they are partially excluded by Iη(M) if there exist both quasi-graphical and non-quasi-graphical K-monomials of degree n. Unlike 0-point graphs, other n-point graphs may be partially or even totally excluded by Iη(M).The next example illustrates the basic difference between odd n and even n, regarding the existence of n-point graphs: for some graph models, and for some odd values of n, the complete non-existence of n-point graphs can be derived from the identities of the model — even if RP≠∅ and RV contains v-rules of odd degree (see also Proposition 3.4).Example 5.3Let d (≥3) be an odd integer, and consider the type-S model that has d colours κi(1≤i≤d), d p-rules pi=(κi)2, and 1 v-rule v1=κ1κ2⋯κd. Since each constituent equation has the form x1[V]−2xi[P]=η(κi)(1≤i≤d),each of the following relations η(κi)+η(κj)=0(mod2)(1≤i<j≤d),is an identity of the model (there are no d-identities [2]). Those m-identities may be seen as vectors in (F2)d, where F2={0,1} is the finite field with two elements. Any system comprising d−1 independent identities, for example the two-term identities for which (i,j)∈{(1,2),(2,3),…,(d−1,d)},defines a valid Iη(M). Proposition 5.2 rules out 1-point graphs: each η(κi) appears in at least one identity, and thus in at least one of the identities of any feasibility system. In fact, there are no n-point graphs for those odd n such that 0<n<d. To prove this, observe first that (for any n-point graph G) n=∑iη(G,κi) is the sum of d non-negative terms; thus, if n<d there must be a null term η(G,κa), otherwise that sum would be too large. If in addition n is odd, there must be at least one odd term η(G,κb). It is now clear that the two-term identity involving η(κa)+η(κb) cannot be satisfied by G. There are n-point graphs for any other value of n, though. ♢6Quasi-normal models, and regular modelsOne might wonder whether it is possible to find classes of graph models for which the conjunction of the identities in I(M,η) — and of the inequality η≥0, since ηˆ(G)≥0 for any graph G — is not only a necessary but also a sufficient condition for the existence of graphs G, connected or non-connected, such that ηˆ(G)=η. It is, actually, and at least one of those classes is strongly related to common QFT models.In this paper, that analysis is restricted to models in which every colour is pairable, which is the most interesting case (and for which the exception in the following definition is irrelevant). Graph models with non-pairable colours (which describe e.g. QFT models with external sources) will hopefully be addressed in a subsequent article.Definition 6.1A graph model M(K,RP,RV) is quasi-normal if every quasi-graphical K-monomial ω is also graphical — except possibly ω=1.Definition 6.2A VP-relation of a graph model M is an equality of the form (37)vj1vj2⋯vjr=pk1pk2⋯pks(r,s≥1),involving (not necessarily distinct) v-rules and p-rules of that model. A VP-relation is primitive if it cannot be obtained by combining (i.e. multiplying) other VP-relations (no substitutions are allowed, just the multiplications).One must have RP≠∅ in any model for which there is a VP-relation, of course. Moreover, for each relation (37), (38)δ(RV)⋅r≤2s≤Δ(RV)⋅r.Each VP-relation expresses a linear dependence between some points (i.e. vectors) in ΛI(M), but not conversely. Indirectly, each VP-relation (37) asserts that at least one 0-point graph can be constructed from the graph (39)Γ=∑i=1rΓ(vji),namely by pairing and then linking the half-edges in Γ so that the p-rules matched by the edges thus created fully reproduce the p-rules in that VP-relation (as in Algorithm 5.1). Conversely, a 0-point graph G defines implicitly a VP-relation. That construction will be used in the proof of Theorem 6.1. If the VP-relation is primitive, any graph thus constructed is always connected, otherwise each connected component would obviously define a VP-relation of lower order.Definition 6.3A formalVP-relation is a relation of the form (37) whose v-rules and p-rules are indeterminates. A formal VP-relation is realizable if those v-rules and p-rules can be assigned specific, valid monomials for which that relation holds.Example 6.1The connection between VP-relations and 0-point graphs may be illustrated with the help of Fig. 1. The formal VP-relation v1=(p1)2 is realizable, and the possible solutions are as follows: (a) v1=(κa)4, p1=(κa)2, and (b) v1=(κaκb)2, p1=κaκb (where κa and κb are distinct colours). The topology T1 (Fig. 1) fits both cases.The formal VP-relation (v1)2=p1p2p3 is also realizable (again, as a primitive relation only). The solutions are (a) v1=κaκbκc, p1=(κa)2, p2=(κb)2, p3=(κc)2, and (b) v1=κaκbκc, p1=κaκb, p2=κbκc, p3=κaκc. The topology T2 fits both solutions, but T3 fits only the second solution.The formal VP-relation (v1)2=p1p2p3p4 is realizable either as a non-primitive or as a primitive relation. For the non-connected topology T4 (Fig. 2) we could have (eg) v1=κ1κ2κ3κ4, p1=κ1κ3, p2=κ2κ4, p3=κ1κ2, p4=κ3κ4. In this case that VP-relation can be decomposed into a product of two primitive VP-relations (v1=p1p2 and v1=p3p4).Those same rules may also be used to colour the connected topology T5. However, T5 may also be coloured with other sets of rules, e.g. p1=κ2κ3, p2=κ1κ3, p3=κ1κ2, p4=(κ4)2, and v1 as before. Now the VP-relation (v1)2=p1p2p3p4 is primitive, as there is no way to express v1 as a product of two p-rules. ♢Example 6.2 Many formal VP-relations, such as (v1)3(v2)3=(p1)2p2p3,(v1)6=p1p2⋯p9,are not realizable. By computing the degree of the left- and right-hand sides of those equalities one may obtain (respectively) 3deg(v1v2)=8,6deg(v1)=18.The first relation may be excluded at once. The second relation involves at most 3 colours, given that v1 must be cubic; however, generating 9 distinct p-rules requires at least 4 colours. The relations (v1)4=(p1)5p2,(v1)4=p1p2p3p4(p5)2are not realizable either. We will deal with the first and leave the other as an exercise. Since v1 has to be cubic, it is sufficient to consider the cases v1=κaκbκc, v1=(κa)2κb, and v1=(κa)3, where κa, κb, κc are distinct colours (no other form is possible). The first form may be excluded since (v1)4 would be equal to (κa)4(κb)4(κc)4, which cannot be divided by (p1)5 whatever the exact form of p-rule p1. For the second form, (v1)4 would be equal to (κa)8(κb)4, which is not divisible by either (κa)10, (κb)10, or (κaκb)5. The last form depends on a single colour, which means that only a single p-rule may be involved (i.e. one could define p1=(κa)2 but not p2, as p2 must differ from p1). ♢6.1Regular modelsDefinition 6.4A v-rule (respectively, p-rule) is regular if there is at least one VP-relation that depends on that v-rule (respectively, p-rule). A colour is regular if it is a factor of at least one regular v-rule (equivalently, of a regular p-rule).Definition 6.5A graph model is regular if all the p-rules, v-rules, and thus colours, are regular (this implies that regular models cannot have solitary colours).Theorem 6.1A regular graph model is necessarily quasi-normal.ProofLet M=M(K,RP,RV) be a regular model, and A=Aκ(M). Then, for each vj∈RV there is a 0-point graph GjV such that πV(GjV,vj)>0. To build GjV, pick a VP-relation that depends on vj and then follow the construction based on (39). Likewise, for each pk∈RP there is a 0-point graph GkP such that πP(GkP,pk)>0. For each index j (respectively, k) one has πˆ(GjV)≥0 and πˆ(GkP)≥0, obviously.By definition, given a quasi-graphical K-monomial ω, there is at least one solution x∈ZnP+nV of the constituent system Ax=ηˆ(ω); obviously, ηˆ(ω)≥0. Let us suppose that one such solution x has been computed; although x may contain negative entries, the column vector (40)x′=x+∑jajπˆ(GjV)+∑kbkπˆ(GkP)will have only positive entries88This type of argument has been used before now (see [8], Theorem 8). for sufficiently large aj,bk∈N. In addition, x′ is a solution of the same system, given that for each index j (respectively, k) Aπˆ(GjV)=0 and Aπˆ(GkP)=0. Thus, ω is graphical (Proposition 5.1). □Example 6.3Each of the models considered in Examples 3.2 and 5.3 has one primitive VP-relation, respectively v1=(p1)2,(v1)2=p1p2⋯pd.Since these relations depend on every v-rule and p-rule, those models are regular and quasi-normal.The graph model described in Example 5.2 is neither. In fact, there is not a single VP-relation for that model (as there are no 0-point graphs). ♢Theorem 6.2LetMbe a quasi-normal graph model in which every colour is pairable. ThenMis regular.ProofLet M=M(K,RP,RV). Since K≠∅ and every colour is pairable, there is a function f:K→K such that f(κa)κa∈RP for each κa∈K (if more than one such function can be defined then choose one as f). It will be shown that every p-rule and v-rule is regular, using the correspondence between 0-point graphs and VP-relations (which is not bijective).If pk∈RP, there is at least one graph Gk such that uˆ(Gk)=pk (Proposition 3.1, quasi-normal model). If a new edge is then created by linking the two half-edges in Gk, the result is a 0-point graph GkP such that πP(GkP,pk)>0. Hence every p-rule is regular.If vj∈RV, one may generate a K-monomial χj with the same degree (which may or may not be a v-rule) by replacing each factor κa of vj by f(κa). By Proposition 3.1, χjvj must be quasi-graphical since it may be written as a product of p-rules f(κa)κa. Then χj is also quasi-graphical because ηˆ(χj)=ηˆ(χjvj)−ηˆ(vj) (Propositions 3.2 and 3.3).As M is quasi-normal there is a graph Ḡj such that uˆ(Ḡj)=χj. Creating deg(vj) edges in Γ(vj)+Ḡj by linking pairs of half-edges — where each edge consumes one half-edge in Γ(vj) with colour κa (say) and one half-edge in Ḡj with colour f(κa) — produces a 0-point graph GjV such that πV(GjV,vj)>0. Hence every v-rule is regular too. □Theorem 6.3A self-conjugate model is necessarily regular, and thus quasi-normal.ProofLet M=M(K,RP,RV) be self-conjugate. For each vj∈RV the monomial cˆ(vj)vj, which is a product of two v-rules, may obviously be expressed as a product of deg(vj) p-rules of the form cˆ(κi)κi. Moreover, for each pk∈RP (which may be written as pk=cˆ(κa)κa for some colour κa∈K) there is vs∈RV such that κa|vs. Thus, cˆ(vs)vs may be expressed as a product of p-rules that includes pk as a factor. Hence the necessary VP-relations exist. □It is a trivial matter to devise a polynomial-time algorithm — i.e. with a running time bounded by a polynomial in nK, nP, nV, and99This last parameter is relevant only if Δ(RV) is unbounded, or at least very large.log2Δ(RV) — that decides whether M(K,RP,RV) is self-conjugate or not. That algorithm has simply to check whether the conditions stated in Definition 3.2 are satisfied; nevertheless, the generic problem of deciding whether a graph model is regular is (almost certainly) harder. Hence, if M(ℒ) is self-conjugate, there is an efficient algorithm that can assert whether each complete correlation function is graphical or non-graphical, irrespective of the lowest loop order of such diagrams.For many QFT ‘physical’ models (i.e. containing no auxiliary fields) M(ℒ) is a self-conjugate graph model. However, in the case of gauge theories, that (induced) self-conjugacy may disappear after gauge-fixing ℒ (see Example 6.4). For that reason, one might be unable to apply Theorem 6.3 to some models derived from properly quantized gauge theories [9–12] (or to some of their correlation functions, at any rate). Nevertheless, at least some of those non-self-conjugate models are still quasi-normal.6.2Regular extensions of regular modelsTwo basic results on regular extensions of regular graph models are presented next. The procedures inherent to those results will be dubbed regular extensions. Note that, in either case, every p-rule in RP and every v-rule in RV is automatically regular in M̄.Proposition 6.1Given a regular modelM=M(K,RP,RV), letωbe a quasi-graphicalK-monomial such thatω∉RV∪{1}, and letR̄V=RV∪{ω},M̄=M(K,RP,R̄V). ThenΛI(M̄)=ΛI(M), andM̄is also regular.ProofThe equality ΛI(M̄)=ΛI(M) follows from Proposition 3.9. The monomial ω is graphical, even in M, since M is quasi-normal. As in the proof of Theorem 6.2, there is a K-monomial χ, graphical in M (and in M̄), such that χω is self-pairable. Thus, there is a 0-point graph in M̄ with a vertex matching v-rule ω, and ω is a regular v-rule (in M̄). □Proposition 6.2Given a regular modelM=M(K,RP,RV), letωbe a quasi-graphicalK-monomial such thatdeg(ω)=2andω∉RP, and letR̄P=RP∪{ω},M̄=M(K,R̄P,RV). ThenΛI(M̄)=ΛI(M), andM̄is also regular.ProofNow ΛI(M̄)=ΛI(M) follows from Proposition 3.10. Since M is quasi-normal, there is a graph G such that uˆ(G)=ω. Linking the two half-edges of G produces a 0-point graph (in M̄) with an edge matching p-rule ω. Hence ω is a regular p-rule in M̄. □Starting from a regular model, increasingly ‘larger’ regular models may be generated by performing consecutive regular extensions. There is a finite number of p-rules that may be added to any given model, though, if K is fixed. In practice, Proposition 6.2 might be useful only rarely. In any case, to prove that a certain model M̄ is regular, by making use of Propositions 6.1 and 6.2, one has to start from a sub-model M of M̄.Problem 6.1Given a type-SD, non-self-conjugate graph model M̄=M(K̄,R̄P,R̄V), determine whether M̄ has a self-conjugate sub-model M=M(K̄,RP,RV) such that ΛI(M̄)=ΛI(M), and if so compute M.Since M̄ is a type-SD model, no p-rule should be removed from R̄P, i.e. RP=R̄P (else M would not be self-conjugate). Thus, there is a unique colour-conjugation map cˆ, which may be trivially reconstructed from R̄P. Next, compute the largest subset RV⊆R̄V such that cˆ(RV)=RV (i.e. RV consists of the conjugate pairs of v-rules, and of those v-rules invariant under cˆ).Now, if M=M(K̄,R̄P,RV) is a valid graph model (in which case it is self-conjugate, and thus regular), and if every v-rule in R̄V∖RV is quasi-graphical in M, then M̄ is regular since it can be generated from a regular model by the repeated application of the type of regular extension implicit in Proposition 6.1. In that case, M is a solution to the above problem — else there is no solution. Moreover, computing M (or showing that no such M exists) requires polynomial time.Example 6.4The usual gauge-fixed Lagrangian density for the current Standard Model of particle physics (i.e. the one based on linear gauge-fixing conditions, in Rξ gauges) includes terms for the cubic vertices c̄W+cAϕW+ and c̄W−cAϕW− but not for their conjugates c̄AcW+ϕW− and c̄AcW−ϕW+. Here cA denotes the ghost field for the photon, and cW+, ϕW+ (respectively, cW−, ϕW−) denote the ghost field and the Goldstone boson for the W+ boson (respectively, W−); the conjugates of those ghost fields are noted c̄A, c̄W+, and c̄W−.Let M̄ be the corresponding graph model, and M the model obtained from M̄ by eliminating the v-rules which prevent M̄ from being a self-conjugate model (ie the v-rules for the two above mentioned vertices).A direct computation of Iη(M) will show that those two v-rules (in M̄ only) are quasi-graphical monomials in M. Since M is regular (Theorem 6.3), Proposition 6.1 ensures that M̄ is regular too (the existence, in both models, of 1-loop graphs for the correlation functions 〈c̄AcW+ϕW−〉 and 〈c̄AcW−ϕW+〉 confirms it). ♢7Concluding remarksLet us try to clarify the relation between QFT models and self-conjugate graph models by describing some predominant features of the former type of models. First of all, a ‘physical’ Lagrangian density ℒ usually involves two basic types of fields, neutral and charged (a field ϕ is neutral if it is its own anti-field, and charged if there is a distinct field ϕ̄ such that ϕ and ϕ̄ are each other’s anti-fields). If (in a given ℒ) each kinetic term involves only one field and its anti-field then that QFT model has no explicit propagator mixing, and M(ℒ) is a type-SD∗ graph model. If (in addition) each field contributes to a kinetic term then M(ℒ) is a type-SD model. Furthermore, if ℒ contains a certain term ℒa then (usually, at least if ℒa does not contain auxiliary fields) ℒ also contains its Hermitian conjugate ℒa† (which may or may not coincide with ℒa). If these conditions are satisfied (and they very often are) it follows that M(ℒ) is self-conjugate. In fact, the (combinatorial) self-conjugacy of M(ℒ) does not depend on the exact combination ℒa+ℒa† being present — it is sufficient that, for each nonzero term ℒa, there is either a nonzero term proportional to ℒa† or some other term that gives rise to the same p-rule or v-rule.Therefore, the results of Section 6, and Theorem 6.3 in particular, corroborate to a large extent the common expectation, based on experience, that there exist Feynman diagrams for any scattering process not forbidden by the number conservation rules of the model in hand. As mentioned earlier, for gauge-fixed QFT models (initially constructed as classic gauge theories, then quantized according to some gauge-fixing prescription) M(ℒ) is not necessarily self-conjugate. Nevertheless, that might not be sufficient to break quasi-normality. One could even conjecture that the constraints to which the quantization procedures [10–12] are subjected actually help preserving quasi-normality — if not in general, perhaps at least in standard usage. This seems to be an open question.Various topics will be postponed to some future paper(s). Those include quasi-normal graph models with non-pairable colours, the existence of connected graphs, and the computational complexity of some basic problems. For example, the graphs predicted by Theorem 6.1, with end-vector ηˆ(ω), need not be connected. However, if a regular graph model M satisfies some fairly minimal constraints then the existence of connected graphs follows. For instance, connected graphs exist if (in addition) M cannot be decomposed into two or more (disjoint) graph models and δ(RV)≥3.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsThis work was partially supported by Fundação para a Ciência e a Tecnologia (FCT, Portugal) through RD Units Strategic Plans UID/CTM/04540/2013 and UID/CTM/04540/2020.I also thank the reviewer for constructive suggestions, whose implementation will have improved (I hope) the readability of this paper.References[1]NogueiraP.Comput. Phys. Comm.21420178390P. Nogueira, From Feynman rules to conserved quantum numbers I, Comput. Phys. Commun. 214 (2017) 83–90.[2]NogueiraP.Comput. Phys. Comm.21520171319P. Nogueira, From Feynman rules to conserved quantum numbers II, Comput. Phys. 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D. 28 (1983) 2567–2582; diff --git a/tests/data/elsevier/j.cpc.2020.107740_expected.yml b/tests/data/elsevier/j.cpc.2020.107740_expected.yml new file mode 100644 index 0000000..2429228 --- /dev/null +++ b/tests/data/elsevier/j.cpc.2020.107740_expected.yml @@ -0,0 +1,302 @@ +abstract: 'The present article complements the earlier ones in this series. The first part contains various results on the constituent system Cκ(M) of a graph model M, and on its feasibility system Iη(M) (which comprises a number of identities that define the number conservation rules). Those results include the general form of the (particle) number conservation rules in models without explicit propagator mixing. A few types of graph models (including self-conjugate and quasi-normal models) are defined. Oversimplifying, a model M is self-conjugate if it has a certain (weak) combinatorial type of C-symmetry, and M is quasi-normal if Iη(M) fully determines for which multisets of coloured, unlinked half-edges there is a graph in M with that exact multiset. It is shown not only that every self-conjugate model is quasi-normal, but also that some extensions of self-conjugate models are still quasi-normal. Most conveniently, self-conjugate models (and even some extended models) may be recognized in polynomial time. These results seem to lead to the following conclusion: for many (possibly ‘most’ of the) consistent, relevant QFT models, a complete correlation function 〈F〉 is graphical — i.e. there exist Feynman diagram(s) for 〈F〉 — if and only if 〈F〉 is allowed by the number conservation rules (i.e. by a complete system of such rules). Since these rules can be computed in polynomial time then, for many QFT models, deciding whether 〈F〉 is graphical also takes polynomial time.' +copyright_holder: Elsevier B.V. +copyright_statement: © 2020 Elsevier B.V. All rights reserved. +copyright_year: 2020 +document_type: article +license_url: '' +license_statement: '' +keywords: ['Particle number conservation rules', 'Feynman diagrams', 'Graphical correlation functions', 'Systems of linear Diophantine equations'] +article_type: full-length article +journal_title: Computer Physics Communications +material: publication +publisher: Elsevier B.V. +year: 2021 +authors: +- full_name: Nogueira, P. + raw_affiliations: + - value: CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco + Pais 1, 1049-001 Lisboa, Portugal + source: Elsevier B.V. + emails: + - paulo.nogueira@tecnico.ulisboa.pt +artid: '107740' +title: From Feynman rules to conserved quantum numbers, III +dois: +- material: publication + doi: 10.1016/j.cpc.2020.107740 +journal_volume: '260' +journal_issue: '' +is_conference_paper: false +publication_date: '2021-03' +collaborations: [] +documents: +- key: j.cpc.2020.107740.xml + url: http://example.org/j.cpc.2020.107740.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: NogueiraP.<maintitle>Comput. + Phys. Comm.</maintitle>21420178390P. Nogueira, From Feynman rules to conserved quantum numbers I, Comput. + Phys. Commun. 214 (2017) 83–90. + source: Elsevier B.V. + reference: + publication_info: + journal_title: Comput. Phys. 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D + journal_volume: '30' + year: 1984 + page_start: '508' + label: '12' + misc: + - "(erratum)" + authors: + - full_name: Batalin, I.A. + inspire_role: author + - full_name: Vilkovisky, G.A. + inspire_role: author diff --git a/tests/data/elsevier/j.nima.2019.162728.xml b/tests/data/elsevier/j.nima.2019.162728.xml new file mode 100644 index 0000000..3801a79 --- /dev/null +++ b/tests/data/elsevier/j.nima.2019.162728.xml @@ -0,0 +1 @@ +application/xmlTransverse and longitudinal segmented forward hadron calorimeters with SiPMs light readout for future fixed target heavy ion experimentsF. GuberD. FinogeevM. GolubevaA. IvashkinA. IzvestnyyN. KarpushkinS. MorozovA. KuglerV. MikhaylovA. SengerNA61/SHINE CBM and BM@N collaborationsForward hadron calorimeterMicropixel photodetectorsHeavy ion collisionsNuclear Inst. and Methods in Physics Research, A 958 (2020). doi:10.1016/j.nima.2019.162728journalNuclear Inst. and Methods in Physics Research, A© 2019 Elsevier B.V. All rights reserved.Elsevier B.V.0168-90029581 April 20202020-04-01Proceedings of the Vienna Conference on Instrumentation 201910.1016/j.nima.2019.162728http://dx.doi.org/10.1016/j.nima.2019.162728doi:10.1016/j.nima.2019.162728162728JournalsS300.1NIMA162728162728S0168-9002(19)31193-310.1016/j.nima.2019.162728Elsevier B.V.Fig. 1Schematic view of the sampling lead/scintillator module, left. Photo of the assembled module without top cover, right.Fig. 2Photo of the supermodule at CERN test beams, left. Energy resolution of the supermodule in full proton energy range 1.5–150 GeV, right.Fig. 3Schematic view of the NA61/SHINE with proposed upgrades during LS2 (up). Schematic front view of the PSD at present NA61/SHINE, (down, left). Schematic front view of the FPSD and the MPSD, (down, right).Fig. 4Radial distributions of doses from ionizing particles in the calorimeters at a depth of 10 cm from the forward face of MPSD and FPSD, left. Radial neutron fluence distribution at the plane of MPPCs, right.Fig. 5Schematic view of ZDC position at BM@N, left. Schematic FHCal front view for upgraded BM@N, right.Fig. 6Radial distributions of doses from ionizing particles in FHCal at distance 10 cm from the forward face of calorimeters, left. Radial neutron fluence distribution at the plane of MPPCs, right.Fig. 7Schematic view of the PSD position at the CBM, left. The scheme of the PSD CBM, right.Fig. 8Radial distributions of doses from ionizing particles in calorimeter at distance 10 cm from the forward face of the PSD for Au+Au@10 AGeV, left. Radial neutron fluence distribution at the plane of MPPCs, right.Transverse and longitudinal segmented forward hadron calorimeters with SiPMs light readout for future fixed target heavy ion experimentsF.Gubera⁎guber@inr.ruD.FinogeevaM.GolubevaaA.IvashkinaA.IzvestnyyaN.KarpushkinabS.MorozovacA.KuglerdV.MikhaylovdeA.Sengerffor theNA61/SHINE, CBM and BM@N collaborationsaInstitute for Nuclear Research RAS, Moscow, RussiaInstitute for Nuclear Research RASMoscowRussiaInstitute for Nuclear Research RAS, Moscow, RussiabMoscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, RussiaMoscow Institute of Physics and Technology, DolgoprudnyMoscow RegionRussiaMoscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, RussiacNational Research Nuclear University MEPhI, Moscow, RussiaNational Research Nuclear University MEPhIMoscowRussiaNational Research Nuclear University MEPhI, Moscow, RussiadNuclear Physics Institute of CAS, R̆ez̆, Czech RepublicNuclear Physics Institute of CASR̆ez̆Czech RepublicNuclear Physics Institute of CAS, R̆ez̆, Czech RepubliceCzech Technical University, Prague, Czech RepublicCzech Technical UniversityPragueCzech RepublicCzech Technical University, Prague, Czech RepublicfGSI Helmholtz Centre for Heavy Ion Research, Darmstadt, GermanyGSI Helmholtz Centre for Heavy Ion ResearchDarmstadtGermanyGSI Helmholtz Centre for Heavy Ion Research, Darmstadt, Germany⁎Corresponding author.AbstractForward hadron calorimeters with transverse and longitudinal segmentation are developed for upgraded NA61/SHINE [BM@N] and future CBM experiments at CERN, JINR and FAIR respectively. The main purpose of these calorimeters is to provide an event-by-event measurements of centrality and reaction plane orientation in ion collisions. Hadron calorimeters in all these experiments are composed of sampling lead/scintillator modules. The light collection from longitudinal sections in modules is provided by Wave-Length Shifting (WLS) fibers embedded in scintillator plates. Micropixel photodetectors MPPCs are used for light detection. The light yield measured at muons beam for these modules is about 8 - 10 ph.el./MeV. Performance of the supermodule composed of these modules has been studied at proton beam energies 1.5 - 150 GeV at CERN. Radiation conditions at high heavy ion beam rates are studied by FLUKA simulations for calorimeters geometry optimization aimed to get acceptable radiation doses and neutron fluence both for the scintillator plates and for photodetectors in these experiments.KeywordsForward hadron calorimeterMicropixel photodetectorsHeavy ion collisions1IntroductionThe main goal of upgraded fixed target experiments- NA61/SHINE@SPS [1], BM@N [2] as well as new CBM experiment [3] at FAIR is to investigate properties of hot and dense nuclear matter created in the ion collision zone. Different observables measured in the relativistic ion collisions–particle yields, transverse momentum spectra, rapidity and angular distributions, as well as fluctuations and correlations of hadrons, are studied as a function of the collision energy and centrality. To study such observables as collective flows of the identified particles it is necessary to know the reaction plane angle. To measure the centrality and reaction plane on event-by-event base in ion collisions in these experiments the transverse and longitudinal segmented forward hadron calorimeters have been developed. The structure of sampling lead/scintillator module, as the unit of all hadron calorimeters at these experiments, with some details of light collection from scintillator plates and its detection by MPPCs is discussed in the first section. Results of performance study of the calorimeter supermodule assembled as 3 × 3 array of these modules are discussed in Section 2. Some features of operation of the hadron calorimeters assembled of these modules at high beam rates at NA61/SHINE, BM@N and CBM experiments are discussed in Section 3.2The design of sampling lead/scintillator module for forward hadron calorimetersForward hadron calorimeters in fixed target experiments- NA61/SHINE@SPS, BM@N and CBM@FAIR are assembled of sampling lead/scintillator modules with longitudinal segmentation. The similar modules will be used also for the FHCal calorimeters in the collider MPD experiment [4] at NICA. Schematic view of the module is shown in Fig. 1, left. It consists of 60 lead /scintillators samples with sampling ratio 4:1 (the thicknesses of the lead plates and scintillator tiles are 16 mm and 4 mm, correspondingly) that provides the compensation condition. Light readout is provided by WLS-fibers embedded in 1.2 mm grooves in the scintillator plates. Longitudinal sections in modules are forming by the WLS fibers light collection from each 6 consecutive scintillator tiles which are connected to a single photodetector at the end of the module. All 60 samples are tied together with a 0.5 mm stainless steel tape in one block and loaded into box made of 0.5 mm stainless steel sheet. The photo of assembled module is shown in Fig. 1, right. There are two types of such modules with different sizes. The modules with transverse size 20 × 20 cm2 and active length about 120 cm (5.6 nuclear interaction lengths) with 10 longitudinal sections are used for NA61/SHINE, CBM and outer part of the BM@M calorimeters. Total weight of such module is about 500 kg. The modules with transverse size 15 × 15 cm2 and active length about 85 cm (4 nuclear interaction lengths) with 7 longitudinal sections will be used for inner part of BM@N and MPD calorimeters. These modules have the weight about 200 kg.The longitudinal division into 10 (7) sections ensures the uniformity of the light collection along the module. Ten (seven) individual Hamamatsu MPPCs S12572-010P with active area 3 × 3 mm2 are used as photodetectors in each module. The light yield from minimum ionizing particles (MIPs) is about 50 ph.e./section for the modules with 10 sections. It was also shown [5] that high pixel density and fast recovery time of the MPPCs provide constant amplitude response for the beam rates up to 106 Pb ions per second.3Performance study of the calorimeter supermoduleThe performance was already studied for hadron calorimeter assembled from such modules at NA61/SHINE experiment with proton beam momenta in the range 13–150 GeV/c. However the BM@N, MPD and CBM experiments are interested in the lower proton energy range which is never studied before. To do this the 3 × 3 array assembled from 9 CBM (20 × 20 cm2) modules, Fig. 2, left, has been exposed at CERN T9, T10 proton beam lines in the energy range 1.5–9 GeV [6]. To readout the signals from the calorimeter sections the sampling ADC64s2 board [7] with 16 nsec sampling interval has been used. It was shown that at proton energy range 1.5–9 GeV stochastic term of the supermodule energy resolution is about 54%/E (GeV). The supermodule has been tested then at NA61/SHINE proton beam line with energy range of 10–150 GeV. Combining two measurements it was shown that stochastic term of energy resolution of the supermodule in full energy range of 1.5–150 GeV is about 58%/E (GeV) (Fig. 2, right). Constant term is about 3%.In addition, the response of the module with irradiated MPPCs has been measured. Module energy resolution shows only 10% deterioration for MPPCs irradiated with neutron fluence 2×1011 neutrons/cm2[8].4Forward hadron calorimeters at NA61/SHINE, BM@N and CBM heavy ions experimentsAll forward hadron calorimeters in the NA61/SHINE, BM@N and CBM heavy ions experiments have the same modular structure but due to different operation conditions they have some specific features and issues which are discussed below.4.1PSD at NA61/SHINE experimentNew physics program of NA61/SHINE includes experiment on open charm production in Pb+Pb collisions at 150 AGeV after LS2 [1]. This experiment requires increasing of the beam intensity up to 106 Pb ions per spill, and will operate at higher trigger rate (up to 1 kHz). Therefore, practically all NA61/SHINE subdetector systems and data acquisition will be significantly upgraded (Fig. 3, up). In particular, instead of present Projectile Spectator Detector (PSD), (Fig. 3, down, left) two calorimeters will be installed (Fig. 3, down, right). The first calorimeter — the main PSD (MPSD), is the present upgraded PSD. The present PSD [5] was used at NA61/SHINE experiments during 2010–2018 to measure centrality and reaction plane in Be+Be, Ar+Sc, Xe+La and Pb+Pb reactions in beam energy range of incident ions 13–150 AGeV. It consists of 16 central small modules with transverse sizes of 10 × 10 cm2 and 28 outer large modules with transverse sizes of 20 × 20 cm2. The beam rate in these experiments did not exceed 105 ions/spill (8 s spill in every 40 s) and due to rather low beam intensities it was possible to use the calorimeter without beam hole in a center. In this case the energy deposition in the calorimeter is a monotonic function of the centrality of ion interactions and thus it could be used for centrality determination of the ion collisions. The PSD has rather good energy resolution (stochastic term is about 66%/E (GeV) and constant term is about 4%) with linear response in given energy range. During the upgrade, instead of 16 small central modules, new 4 large modules with tapered edges will be used to have the beam hole with diameter 60 mm in the center of the MPSD to avoid irradiation of scintillators in MPSD. The second calorimeter — the forward PSD (FPSD), has no beam hole and consists of array of 3 × 3 modules (20 × 20 cm2). The FPSD will be placed at about 4 meters downstream the MPSD and will measure mainly the energy of heavy fragments, which are not detected in MPSD. To read the signals from the calorimeter sections the sampling readout board based on DRS4 chip will be used [9].FLUKA simulations of the corresponding doses from ionizing particles in MPSD and FPSD scintillators after 1 month of operation at beam rate 106 Pb ions per spill at a depth of 10 cm from the front face of the calorimeters, where maximum of dose is expected, are shown in Fig. 4, left. To avoid strong scintillators irradiation in central FPSD module the holes with diameter 40 mm are done in the centers of all 60 scintillators in this module. Corresponding neutron fluence at the place of MPPCs on back sides of calorimeters are shown in Fig. 4, right.These new forward hadron calorimeters will be able to measure centrality of ion interaction in NA61/SHINE experiments at high beam rate after 2020 with precision of about 10% for peripheral and semi peripheral events on event-by-event base.4.2New FHCal at BM@N experimentsUp to now, the ZDC calorimeter [10] has been used at BM@N setup (Fig. 5, left) in the experiments with C, Ar and Kr ions to measure centrality in reactions with C, Al, Cu and Pb targets at beam energies up to 4.5 AGeV. This calorimeter also has lead/scintillator sampling, the light readout is done here by WLS plates from all scintillator plates in the module and detected by one PMT placed behind the module, i.e. ZDC has no longitudinal segmentation. The ZDC also has no beam hole in the center. During BM@N upgrades it will be replaced by new FHCal calorimeter (Fig. 5, right). Due to the expected high beam rate of gold ions (up to 2×106 Au/sec) with energies up to 4.5 AGeV, the FHCal would have the beam hole to avoid strong scintillators irradiation in the central module of the calorimeter.Inner part of the FHCal will be assembled from 34 modules similar to that already constructed for the FHCal calorimeter for MPD experiment at NICA with beam hole 15 × 15cm2 in the center. The outer part of FHCal will be assembled from 20 modules constructed for the CBM experiment at FAIR. To measure fragments in the beam hole the quartz hodoscope is proposed to be installed behind the FHCal [11]. For signals readout from the calorimeter sections the sampling ADC64s2 board [7] with 16 nsec sampling interval will be used. It is expected that centrality of ion interaction can be measured with precision of about 10% for peripheral and semi peripheral events [11]. Corresponding doses from ionizing particles in FHCal scintillators after 1 month of operation with beam rate 2×106 Au ions were simulated by FLUKA and are shown in Fig. 6, left. The doses are shown as function of radius at a depth of 10 cm from the surface of the calorimeters, where maximum dose is expected. Corresponding neutron fluence at the place of MPPCs on back side of the calorimeter is shown in Fig. 6, right.4.3The PSD for the CBM experiment at FAIRThe main feature of the CBM installation (Fig. 7, left) is the operation at beam rate up to 108 Au ions per second and reaction rate up to 106 per second at free streaming DAQ and at beam energy range 2–11 AGeV. For signals readout from the calorimeter sections the adopted sampling ADC PANDA ECAL board [12] with 8 nsec sampling interval is planned to be used. The PSD CBM geometry (Fig. 7, right) takes into account irradiation of the scintillators in the calorimeter, photodetectors and readout electronics. Initially, in the PSD TDR [13], it was proposed to have diamond shape beam hole with inscribed circle diameter 60 mm. But, further FLUKA simulations with taking into account the multiple scattering of gold ions in the target have shown that PSD internal modules irradiation will be too high and it was decided to increase the beam hole size.FLUKA simulation results for dose distribution in the PSD and neutron fluence at the position of MPPCs placed behind the modules are shown in Fig. 8 for different shapes and sides of the beam hole and for Au beam energies 4 and 10 AGeV. Increasing of the beam hole significantly decreases the dose and neutron fluence in modules near beam axis. Final decision about the beam hole shape will be taken after detail simulation of physics performance of the PSD with different beam holes.5ConclusionForward hadron calorimeters based on lead/scintillator longitudinal segmented modules with MPPCs light readout are developed for upgraded NA61/SHINE and BM@N experiments and for future CBM experiment at FAIR. Results of the supermodule beam tests and performance of present PSD NA61/SHINE show that calorimeters composed of these modules satisfy the requirements of the heavy ions experiments in the proton energy range 1.5–150 GeV. Geometries of the hadron calorimeters in these experiments have been optimized to get acceptable doses and neutron fluences for scintillator plates and photodetectors, which was shown by corresponding FLUKA simulations.AcknowledgmentsThe authors thank the NA61/SHINE, BM@N and CBM collaborations for their support in the development of the corresponding forward hadron calorimeters. We thank CERN staffs M. Jeckel and L. Gatignon for their help in the supermodule tests at the T9 and T10 beamlines. This work was partially supported by RFBR grant 18-02-40081 and grant Czech MEYS - LM2015049 , OP VVV - CZ.02.1.01/0.0/0.0/16 013/0001677References[1]A. Aduszkiewicz, et al. CERN-APSC-2018-008 / SPSC-P330-ADD-10.[2]KapishinM.Nucl. Phys.A9822019967970M. Kapishin, Nucl.Phys. A982 (2019) 967-970[3]AblyazimovT.Eur. Phys. J.A533201760T. Ablyazimov et al., Eur.Phys.J. A53 (2017) no.3, 60[4]AbraamyanU.Nucl. Instrum. Meth.A628201199102U. Abraamyan et al., Nucl.Instrum.Meth. A628 (2011) 99-102[5]GolubevaM.J. Phys.: Conf. Ser.7982017012073M. Golubeva et al., Journal of Physics: Conf. Series798 (2017) 012073[6]N. Karpushkin, et al. Accepted at NIM A: https://doi.org/10.1016/j.nima.2018.10.054.[7]http://afi.jinr.ru.[8]MikhaylovV.Nucl. Instrum. Meth.A9122018241244V. Mikhaylov et al., Nucl.Instrum.Meth. A912 (2018) 241-244[9]http://dpnc.unige.ch/bravar/exp/NA61/Bravar-DRS.pdf.[10]O. Gavrishchuk, et al. arXiv:1406.1599v2[physics.ins-det], 2014.[11]GuberF.EPJ Web Conf.20420190700710.1051/epjconf/201920407007F. Guber et al., submitted to EPJ Web Conf. www.epj-conferences.org[12]https://ieeexplore.ieee.org/document/5402172.[13]F. Guber, et al. http://repository.gsi.de/record/109059/files/20150720-CBM-TDR-PSD.pdf. diff --git a/tests/data/elsevier/j.nima.2019.162728_expected.yml b/tests/data/elsevier/j.nima.2019.162728_expected.yml new file mode 100644 index 0000000..c7978e9 --- /dev/null +++ b/tests/data/elsevier/j.nima.2019.162728_expected.yml @@ -0,0 +1,262 @@ +abstract: Forward hadron calorimeters with transverse and longitudinal segmentation are developed for upgraded NA61/SHINE [BM@N] and future CBM experiments at CERN, JINR and FAIR respectively. The main purpose of these calorimeters is to provide an event-by-event measurements of centrality and reaction plane orientation in ion collisions. Hadron calorimeters in all these experiments are composed of sampling lead/scintillator modules. The light collection from longitudinal sections in modules is provided by Wave-Length Shifting (WLS) fibers embedded in scintillator plates. Micropixel photodetectors MPPCs are used for light detection. The light yield measured at muons beam for these modules is about 8 - 10 ph.el./MeV. Performance of the supermodule composed of these modules has been studied at proton beam energies 1.5 - 150 GeV at CERN. Radiation conditions at high heavy ion beam rates are studied by FLUKA simulations for calorimeters geometry optimization aimed to get acceptable radiation doses and neutron fluence both for the scintillator plates and for photodetectors in these experiments. +copyright_holder: Elsevier B.V. +copyright_statement: © 2019 Elsevier B.V. All rights reserved. +copyright_year: 2019 +document_type: conference paper +license_url: '' +license_statement: '' +keywords: ['Forward hadron calorimeter', + 'Micropixel photodetectors', + 'Heavy ion collisions'] +article_type: full-length article +journal_title: Nuclear Inst. and Methods in Physics Research A +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Guber, F. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. + emails: + - guber@inr.ru +- full_name: Finogeev, D. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. +- full_name: Golubeva, M. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. +- full_name: Ivashkin, A. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. +- full_name: Izvestnyy, A. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. +- full_name: Karpushkin, N. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. + - value: Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, + Russia + source: Elsevier B.V. +- full_name: Morozov, S. + raw_affiliations: + - value: Institute for Nuclear Research RAS, Moscow, Russia + source: Elsevier B.V. + - value: National Research Nuclear University MEPhI, Moscow, Russia + source: Elsevier B.V. +- full_name: Kugler, A. + raw_affiliations: + - value: Nuclear Physics Institute of CAS, R̆ez̆, Czech Republic + source: Elsevier B.V. +- full_name: Mikhaylov, V. + raw_affiliations: + - value: Nuclear Physics Institute of CAS, R̆ez̆, Czech Republic + source: Elsevier B.V. + - value: Czech Technical University, Prague, Czech Republic + source: Elsevier B.V. +- full_name: Senger, A. + raw_affiliations: + - value: GSI Helmholtz Centre for Heavy Ion Research, Darmstadt, Germany + source: Elsevier B.V. +artid: '162728' +title: Transverse and longitudinal segmented forward hadron calorimeters with SiPMs light readout for future fixed target heavy ion experiments +dois: +- material: publication + doi: 10.1016/j.nima.2019.162728 +journal_volume: '958' +journal_issue: '' +is_conference_paper: true +publication_date: '2020-04-01' +collaborations: ['NA61/SHINE, CBM and BM@N collaborations'] +documents: +- key: j.nima.2019.162728.xml + url: http://example.org/j.nima.2019.162728.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: A. + Aduszkiewicz, et al. CERN-APSC-2018-008 / SPSC-P330-ADD-10. + source: Elsevier B.V. + reference: + label: '1' + misc: + - A. Aduszkiewicz, et al. CERN-APSC-2018-008 / SPSC-P330-ADD-10 +- raw_refs: + - schema: Elsevier + value: KapishinM.<maintitle>Nucl. + Phys.</maintitle>A9822019967970M. Kapishin, Nucl.Phys. A982 (2019) 967-970 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Phys. + journal_volume: A982 + year: 2019 + page_start: '967' + page_end: '970' + label: '2' + authors: + - full_name: Kapishin, M. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: AblyazimovT.<maintitle>Eur. + Phys. J.</maintitle>A533201760T. Ablyazimov et al., Eur.Phys.J. A53 (2017) no.3, 60 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Eur. Phys. J. + journal_volume: A53 + journal_issue: '3' + year: 2017 + page_start: '60' + label: '3' + authors: + - full_name: Ablyazimov, T. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: AbraamyanU.<maintitle>Nucl. + Instrum. Meth.</maintitle>A628201199102U. Abraamyan et al., Nucl.Instrum.Meth. A628 (2011) 99-102 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Instrum. Meth. + journal_volume: A628 + year: 2011 + page_start: '99' + page_end: '102' + label: '4' + authors: + - full_name: Abraamyan, U. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'GolubevaM.<maintitle>J. + Phys.: Conf. Ser.</maintitle>7982017012073M. Golubeva et al., Journal of Physics: Conf. Series798 (2017) 012073' + source: Elsevier B.V. + reference: + publication_info: + journal_title: 'J. Phys.: Conf. Ser.' + journal_volume: '798' + year: 2017 + artid: "012073" + label: '5' + authors: + - full_name: Golubeva, M. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: 'N. + Karpushkin, et al. Accepted at NIM A: https://doi.org/10.1016/j.nima.2018.10.054.' + source: Elsevier B.V. + reference: + dois: + - 10.1016/j.nima.2018.10.054 + label: '6' + misc: + - 'N. Karpushkin, et al. Accepted at NIM A:' +- raw_refs: + - schema: Elsevier + value: http://afi.jinr.ru. + source: Elsevier B.V. + reference: + urls: + - value: http://afi.jinr.ru + label: '7' +- raw_refs: + - schema: Elsevier + value: MikhaylovV.<maintitle>Nucl. + Instrum. Meth.</maintitle>A9122018241244V. Mikhaylov et al., Nucl.Instrum.Meth. A912 (2018) 241-244 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Instrum. Meth. + journal_volume: A912 + year: 2018 + page_start: '241' + page_end: '244' + label: '8' + authors: + - full_name: Mikhaylov, V. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: http://dpnc.unige.ch/bravar/exp/NA61/Bravar-DRS.pdf. + source: Elsevier B.V. + reference: + urls: + - value: http://dpnc.unige.ch/bravar/exp/NA61/Bravar-DRS.pdf + label: '9' +- raw_refs: + - schema: Elsevier + value: O. + Gavrishchuk, et al. arXiv:1406.1599v2[physics.ins-det], + 2014. + source: Elsevier B.V. + reference: + arxiv_eprint: '1406.1599' + label: '10' + misc: + - O. Gavrishchuk, et al. , 2014 +- raw_refs: + - schema: Elsevier + value: GuberF.<maintitle>EPJ + Web Conf.</maintitle>20420190700710.1051/epjconf/201920407007F. Guber et al., submitted to EPJ Web Conf. www.epj-conferences.org + source: Elsevier B.V. + reference: + publication_info: + journal_title: EPJ Web Conf. + journal_volume: '204' + year: 2019 + page_start: '07007' + dois: + - 10.1051/epjconf/201920407007 + label: '11' + authors: + - full_name: Guber, F. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: https://ieeexplore.ieee.org/document/5402172. + source: Elsevier B.V. + reference: + urls: + - value: https://ieeexplore.ieee.org/document/5402172 + label: '12' +- raw_refs: + - schema: Elsevier + value: F. + Guber, et al. http://repository.gsi.de/record/109059/files/20150720-CBM-TDR-PSD.pdf. + source: Elsevier B.V. + reference: + urls: + - value: http://repository.gsi.de/record/109059/files/20150720-CBM-TDR-PSD.pdf + label: '13' + misc: + - F. Guber, et al diff --git a/tests/data/elsevier/j.nima.2019.162787.xml b/tests/data/elsevier/j.nima.2019.162787.xml new file mode 100644 index 0000000..7b7abf0 --- /dev/null +++ b/tests/data/elsevier/j.nima.2019.162787.xml @@ -0,0 +1 @@ +application/xmlLong-term and efficient operation of the MWPC muon detector at LHCbSofia Kotriakhovaon behalf of LHCb Muon groupMuon spectrometersWire chambers (MWPC)Malter effectNuclear Inst. and Methods in Physics Research, A 958 (2020). doi:10.1016/j.nima.2019.162787journalNuclear Inst. and Methods in Physics Research, A© 2019 Published by Elsevier B.V.Elsevier B.V.0168-90029581 April 20202020-04-01Proceedings of the Vienna Conference on Instrumentation 201910.1016/j.nima.2019.162787http://dx.doi.org/10.1016/j.nima.2019.162787doi:10.1016/j.nima.2019.162787162787JournalsS300.1NIMA162787162787S0168-9002(19)31233-110.1016/j.nima.2019.162787Fig. 1(a) Side view of the LHCb muon detector. (b) Station layout with the four regions R1–R4 indicated.Fig. 2A typical example of the appearing of a self-sustained current and the recovery procedure from ME during normal LHCb operation with beams, applied to one of the MWPC gaps in region M5R3. The plots on the left show the current (top) and HV setting (bottom) during a period of about 3 days around the first appearance of the HV trip and the subsequent start of the automatic HV training. The plots on the right show the current (top) and HV setting (bottom) during the full recovery procedure, which lasted about two weeks. The nominal HV setting for this gap is 2600 V; the average current in presence of colliding beams is about 0.6μA.Fig. 3Left: Average number of trips per day observed during each year of data taking, as a function of the effective run time (days); the total number of trips is shown in red (full line), the number of new trips in blue (dotted line). Right: Average number of recurrent trips per day observed during each year of data taking normalized to the total number of gaps tripped in the previous years; points are evaluated starting from 2011. Errors are statistical.Table 1The chamber gap recovery test with oxygen: chamber type, number of damaged zones, ignition voltage, and time needed for successful recovery. The 4 gaps of a MWPC are named A, B, C, D.Chamber typeNumber of detected ME zonesME ignition voltage, (kV)Time for recovery, (h)M2R42 (gap A)2.7532.81M4R41 (gap D)2.75M5R41 (gap A)2.641 (gap B)2.73M5R41 (gap A)2.741 (gap D)2.82Table 2Days of LHC beam and integrated luminosity per year since 2010, number of gaps tripping for the first time (new trips) and gaps which had already tripped in the previous years (recurrent trips).YearEffective run daysIntegrated lumin. (pb−1)New tripsRecurrent trips2010294090020115612208431201276221069672015393701115201686191076742017801990183220187224602732Total43810200375251Long-term and efficient operation of the MWPC muon detector at LHCbSofiaKotriakhovaskotriak@cern.chon behalf of LHCb Muon groupPetersburg Nuclear Physics Institute of National Research Centre ”Kurchatov Institute”, St.Petersburg, RussiaPetersburg Nuclear Physics Institute of National Research Centre ”Kurchatov Institute”St.PetersburgRussiaPetersburg Nuclear Physics Institute of National Research Centre ”Kurchatov Institute”, St.Petersburg, RussiaAbstractWith its ∼1650m2 of MWPCs, the muon detector of LHCb is one of the largest instrument of this kind worldwide, and one of the most irradiated. Currently we run at the relatively low instantaneous luminosity of 4⋅1032cm−2s−1, nevertheless the most irradiated MWPCs already integrated ∼0.7 C/cm of accumulated charge per wire. The statistics of gas gaps affected by high voltage trips in the proportional chambers is presented for the whole period of operation. Most of the problematic chambers were successfully recovered in situ during data taking, under the nominal LHC beam conditions, by means of a long-term HV training (with the working gas mixture). The appearing of self-sustained currents in one of the MWPC gaps and the effectiveness of the recovery procedures put in place, indicate that the large majority of the trips are due to Malter effect. The method has proven to be very effective, allowing to keep the muon detector efficiency very close 100%, as it was initially designed. In parallel, a test has been performed of a systematic addition of a small amount of oxygen to the nominal gas mixture: results of this test will be discussed.KeywordsMuon spectrometersWire chambers (MWPC)Malter effect1IntroductionWith its 1368 Multi-Wire-Proportional-Chamber (MWPCs), the muon detector of LHCb is one of the largest instrument of this kind worldwide, and one of the most irradiated. All of the MWPCs are supplied with a 40%Ar+55%CO2+5%CF4 gas mixture. Currently we run at the relatively low instantaneous luminosity of 4⋅1032cm−2s−1, nevertheless the most irradiated MWPCs already integrated ∼0.7 C/cm of accumulated charge per wire. In nine years of operation the chambers did not show any symptoms of ageing, but many gaps were affected by high currents, triggered by high radiation and self-sustained in absence of beam. To address this issue and avoid a loss in detector efficiency a dedicated HV training procedure in-situ has been developed. This paper will mainly concentrate on the description of the above training method, while a more detailed work on analysis and statistics of operations is under preparation and will be proposed for publication in the next months.2The LHCb muon detectorThe muon detector of the LHCb experiment [1,2], consists of five stations, M1–M5 placed along the beam axis. Each station is divided into four regions, R1–R4, with increasing distance from the beam axis, as shown in Fig. 1. The whole detector comprises 1380 chambers — 1368 MWPCs (20 types varying mainly in size) and 12 GEMs.3The recovery method to treat the high detector currentsWhile no typical ageing effects have been observed on the system, about 100 MWPC gaps suffered every year from the appearance of high currents and the concomitant HV trips.These currents, triggered by high radiation and self-sustained without presence of beam, suggests that most of the trips are due to Malter-like effects (ME) [3,4]. This can happen due to the presence of a thin (∼1 μm) dielectric polymer film on metal cathode surfaces. Depending on the film thickness and on the value of the positive electric charge deposited on the film, the resulting electric field in the dielectric may become sufficient to cause spontaneous secondary emission of electrons from the cathode. In the LHCb Muon MWPCs such thin insulator on the cathode could be caused by silicon-containing materials used during the construction process.These films can be etched from the cathode surface thanks to the presence of CF4 in the working gas mixture, which allows to increase the concentration of fluorine radicals reacting with silicon and polymers and leading to the creation of volatile products in the plasma. In proportional chambers, the most intense formation of free radicals takes place around anode wires where the electric field reaches 20–200 kV/cm. Dissociation of CF4 molecules resulting from the impact of electrons of about Ee∼3–5 eV occurs with the subsequent formation of chemically active radicals [5,6].However, in the vicinity of the cathode, which is several millimetres apart from anode wires and the plasma environment, the concentration of fluorine radicals is small. Thus, the recovery procedure often requires relatively long time and sometimes can be inefficient.4The HV training procedureTo recover problematic chambers in situ, under the nominal beam conditions, we are using the HV training procedure, as described below. Fig. 2 shows a typical example of the self-sustaining high current appearance and the following recovery procedure in one of the MWPC gaps in region M5R3, where the nominal operation voltage is 2.6 kV and the maximum current value, in presence of colliding beams is ∼0.6 μA. At some moment, the appearing of a high current causes the trip of the HV channel powering this gap. Right after that, the HV for that channel is automatically lowered to decrease the current below the nominal threshold. Notified expert can start the training then and choose optimal parameter of the procedure. The main task of the training procedure is to maintain a relatively high current (around 30 ± 4 μA) to establish the best conditions for polymer film etching. To keep current in this range and protect the gap from damage (in case of current values higher than a safe limit of ∼40 μA), the HV is varied automatically. As seen in Fig. 2, during the training period HV has been raised slowly until the maximum training value of 2.75 kV and decreased from time to time if the current exceeds a user-selectable threshold value.When the gap reaches nominal current at maximum HV in presence of colliding beams (and ∼zero current without beam) it is kept under training for another couple of days because sometime new areas with dielectric films can appear.During the periods when access to the detector is possible the training procedure with negative HV polarity can be applied. In this case the energy of electrons near the cathode is increased as well as the production of fluorine radicals capable of the chemical etching. This is of great help in removing the sources of ME.5Addition of oxygen in the gas mixture to accelerate recovery procedureTo optimize the duration and the efficiency of the training procedure a small amount of oxygen can be added to the gas mixture to increase the etching rate, as shown in [6,7]. This procedure has been tested during LS1 in 2013–2014. Four MWPCs with severe ME were examined with a collimated 90Sr β-source to localize affected cathode regions. These seven zones are listed in Table 1.In presence of 90Sr β-source for each zone the training with the standard gas mixture was applied for about 6 h with no observable current decrease. Then, the same procedure was repeated with a new gas mixture containing ∼2% of oxygen. In this case, the initial current of 25 μA (ignited at the voltage of 2.6–2.7 kV) drops sharply until it raises again by a corresponding increase of the high voltage in 50 V steps. After a few iterations the current finally drops to zero. The average recovery time is around 3.5 h, which is a significant improvement comparing to the procedure with standard mixture. This technique could be used in future during the year-end-technical-stops to recover chambers affected by ME.6Statistics of HV tripsDuring nine years of operation 375 MWPC gaps were affected by ME and passed the training procedure. All but 27 gaps were restored back to the normal operation. Table 2 shows the total number of affected gaps per year (excluding Long Shutdown in 2013–2014), together with integrated luminosity and the number of effective days with colliding beams. “New trips” here are the gaps tripped for the first time and “recurrent trips” are the gaps that had already tripped before. We can say that trip frequency is not correlated with integrated luminosity, since we observed more trips during LHC luminosity ramp-up phase when the integrated luminosity was quite low.The average number of affected gaps per day observed during each year of data taking is reported in Fig. 3 for all gaps and for the ones affected for the first time. In both cases a clear decreasing trend is observed. High number of recurrent trips, together with the decreasing trend of new trip confirms that ME is rather caused by initial chamber imperfections than ageing during operation. The spike around 250 effective run days (in 2016) is related to a period where an increase of the chamber gas gain was observed, is probably caused by an unwanted change of the gas mixture.In Fig. 3 (right) the number of recurrent trips per day is normalized to the total number of gaps tripping in the previous year. The recurrences observed during the last two years is around 0.1% per day, equivalent to less than 10% probability during the full run, which shows that adopted training procedure effectively solves ME problems in most of the cases.7ConclusionThe HV training procedure allows a non-invasive recovery of MWPCs in-situ with the standard gas mixture. The efficiency of the method had been proven during the years of LHCb operation and allowed to keep the detection efficiency close to 100% for almost a decade. As the standard procedure takes about two months on average, an accelerated method with the addition of the oxygen to the gas mixture has been investigated, providing encouraging results. This could be used in future operation of the Muon system at increased luminosity.References[1]AlvesA.A.Jr.LHCb CollaborationThe LHCb detector at the LHCJ. Instrum.32008S0800510.1088/1748-0221/3/08/S08005A. A. 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Flamm, Plasma etching of Si and SiO2 - The effect of oxygen additions to CF4 plasmas, Journal of Applied Physics 49 (1978) 3796–3803 doi:101063/1325382 diff --git a/tests/data/elsevier/j.nima.2019.162787_expected.yml b/tests/data/elsevier/j.nima.2019.162787_expected.yml new file mode 100644 index 0000000..8bc97e7 --- /dev/null +++ b/tests/data/elsevier/j.nima.2019.162787_expected.yml @@ -0,0 +1,207 @@ +abstract: 'With its ∼1650m2 of MWPCs, the muon detector of LHCb is one of the largest instrument of this kind worldwide, and one of the most irradiated. Currently we run at the relatively low instantaneous luminosity of 4⋅1032cm−2s−1, nevertheless the most irradiated MWPCs already integrated ∼0.7 C/cm of accumulated charge per wire. The statistics of gas gaps affected by high voltage trips in the proportional chambers is presented for the whole period of operation. Most of the problematic chambers were successfully recovered in situ during data taking, under the nominal LHC beam conditions, by means of a long-term HV training (with the working gas mixture). The appearing of self-sustained currents in one of the MWPC gaps and the effectiveness of the recovery procedures put in place, indicate that the large majority of the trips are due to Malter effect. The method has proven to be very effective, allowing to keep the muon detector efficiency very close 100%, as it was initially designed. In parallel, a test has been performed of a systematic addition of a small amount of oxygen to the nominal gas mixture: results of this test will be discussed.' +copyright_holder: +copyright_statement: © 2019 Published by Elsevier B.V. +copyright_year: 2019 +document_type: conference paper +license_url: '' +license_statement: '' +keywords: ['Muon spectrometers', 'Wire chambers (MWPC)', 'Malter effect'] +article_type: full-length article +journal_title: Nuclear Inst. and Methods in Physics Research A +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Kotriakhova, Sofia + raw_affiliations: + - value: Petersburg Nuclear Physics Institute of National Research Centre ”Kurchatov + Institute”, St.Petersburg, Russia + source: Elsevier B.V. + emails: + - skotriak@cern.ch +artid: '162787' +title: Long-term and efficient operation of the MWPC muon detector at LHCb +dois: +- material: publication + doi: 10.1016/j.nima.2019.162787 +journal_volume: '958' +journal_issue: '' +is_conference_paper: true +publication_date: '2020-04-01' +collaborations: ['on behalf of LHCb Muon group'] +documents: +- key: j.nima.2019.162787.xml + url: http://example.org/j.nima.2019.162787.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: AlvesA.A.Jr.LHCb + Collaboration<maintitle>The LHCb detector at + the LHC</maintitle><maintitle>J. + Instrum.</maintitle>32008S0800510.1088/1748-0221/3/08/S08005A. A. 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Aglieri RinellaALICE CollaborationMonolithic active pixel sensors (MAPS)Pixel detectorsStitchingCMOSTrackingLow radiation lengthBent sensorsNuclear Inst. and Methods in Physics Research, A 1049 (2023). doi:10.1016/j.nima.2023.168018journalNuclear Inst. and Methods in Physics Research, A© 2023 Published by Elsevier B.V.Elsevier B.V.0168-90021049April 202310.1016/j.nima.2023.168018http://dx.doi.org/10.1016/j.nima.2023.168018doi:10.1016/j.nima.2023.1680181680182023-01-02T00:00:00.000Z2023-01-17T00:00:00.000ZJournalsS250.1NIMA168018168018S0168-9002(23)00008-610.1016/j.nima.2023.168018Fig. 1CAD rendering of the ITS3 detector concept.Fig. 2Views of the DUT for beam tests with 6 ALPIDE chips bent at the radii of the ITS3.Fig. 3Mechanical integration prototype of a half barrel of ITS3.Fig. 4Principles of stitching. Left: design reticle with sub-frames. Right: circuits on the wafer (not to scale).Fig. 5Floor-plan of the wafers of the ER1 submission including six MOSS stitched sensors and six MOST stitched sensors. Insert: design reticle with sub-frames.Fig. 6Concept diagram of the MOSS chip.Fig. 7Architecture of one Bottom Half Unit.☆This report describes results of a large collaborative effort between the ALICE Collaboration and scientists contributing to the technology developments coordinated by CERN EP R&D Work Package 1.2. A list of participants to WP1.2 is available at: https://indico.cern.ch/event/1156197/contributions/4855158/subcontributions/385642/attachments/2465175/4257728/202207-Authors-List.html.Developments of stitched monolithic pixel sensors towards the ALICE ITS3G.Aglieri Rinellagianluca.aglieri.rinella@cern.chOn behalf of theALICE Collaboration1CERN European Organization for Nuclear Research, Esplanade des Particules 1, Meyrin, 1217, SwitzerlandCERN European Organization for Nuclear ResearchEsplanade des Particules 1Meyrin1217SwitzerlandCERN European Organization for Nuclear Research, Esplanade des Particules 1, Meyrin, 1217, Switzerland1https://alice-collaboration.web.cern.ch.AbstractThe ALICE collaboration is pursuing the development of a novel and considerably improved vertexing detector called ITS3, to replace the three innermost layers of the Inner Tracker System during the LHC Long Shutdown 3. The primary goals are to reduce the material budget to the unprecedented value of 0.05% X0 per layer, and to place the first layer at a radial distance of 18 mm from the interaction point. These features will boost the impact parameter resolution by a factor two over all momenta and drastically enhance the tracking efficiency at low transverse momentum.The new detector will consist of true cylindrical layers. Each half-cylinder is based on curved wafer-scale monolithic pixel sensors. The bending radii are 18, 24 and 30 mm, and the length of the sensors in the beam direction is 27 cm. The sensors will be produced using a commercial 65 nm CMOS Imaging technology and a recent technique called stitching. This allows to manufacture chips reaching the dimensions of 27 cm × 9 cm on silicon wafers of 300 mm diameter. The sensors will be thinned down to 50 µm or below. The ITS3 concept foresees cooling by air flow, ultra-light carbon foam support elements and no flexible printed circuits in the active area. This demands a power density limit of 20 mW/cm2 for the sensor, and the need to distribute supply and transfer data over the entire sensors towards circuits located at the short edges of the chip.This contribution summarises the status of the microelectronic developments and presents selected results from the characterisation of the first prototype chips. Furthermore, it describes the ongoing efforts on the design of a first wafer-scale stitched sensor prototype, the MOSS (Monolithic Stitched Sensor) chip.KeywordsMonolithic active pixel sensors (MAPS)Pixel detectorsStitchingCMOSTrackingLow radiation lengthBent sensors1IntroductionThe current Inner Tracker System (ITS2) of the ALICE experiment at the LHC is a 7 layers pixel silicon detector built with 24120 ALPIDE sensors [1]. The ALPIDE chip [2] is a monolithic active pixel sensor manufactured in a 180nm CMOS Imaging Technology. The ITS2 detector has a sensitive area larger than 10 m2 and more than 12.5 billion pixel channels.The ALICE ITS3 project [3–5] aims at replacing the three innermost layers of the ITS2 with a new detector in which flexible printed circuits, cooling and mechanical holding structures are removed or minimised and silicon sensors are the only components in the active area. This would provide a significant reduction of radiation length per layer from the current 0.35% X0 down to an unprecedented value of 0.05% X0. The radial distance of the innermost layer would also be reduced from 22 mm down to 18 mm in the new layout. This requires the replacement of the beam pipe.The tracking and vertexing performances will be significantly enhanced by these changes. The pointing resolution is expected to improve by a factor 2 over the full range of interest of transverse momentum. The tracking efficiency would increase significantly, especially in the low transverse momentum region of the spectrum.2Layout of the ITS3The new ITS3 detector will consist of three true cylindrical layers. These will be made with very large silicon dies curved and shaped into half cylinders (Fig. 1). The sensors will be thinned down to 50 µm or less, to allow the bending of the silicon. The bending radii of the three layers will be 18, 24 and 30 mm. The sensors need to reach a length of 27 cm along the z-axis and more than 9 cm on the azimuth direction. The required single hit position resolution is of the order of 5 µm, which demands a pixel pitch of the order of 20 µm.One of the primary goals is to remove or minimise external components from the active area to meet the target material budget. This requires that power supply and data transfer are implemented on the sensor, possibly entirely in the metal stack of the chip, over distances that can reach 27 cm.Cooling has to be by air flow to minimise the cooling components and material. Past developments [3] in the context of the ALICE ITS Upgrade have shown that cooling using a low-speed air flow is a viable option if the average power dissipation over the sensor surface is less than 20 mW/cm2.The ALPIDE chip dissipation is close to 40 mW/cm2, with the pixels consuming less than 7 mW/cm2 and the remainder being dissipated in the periphery and in the output drivers. The change of technology node from 180 nm node to 65 nm with the related voltage scaling is expected to bring a reduction of power by at least a factor 2 for similar sensor features.3Developments on bent sensors and mechanical integrationThe feasibility of operating monolithic sensors after bending has been demonstrated using ALPIDE sensor chips [6]. Several ALPIDE chips have now been operated after bending at 18 mm radius in various laboratory setups and beam tests. The tests have confirmed experimentally that a high detection efficiency is maintained over comfortable operating ranges when the sensors are bent.The ITS3 project has accumulated experience in operating bent ALPIDE chips in beam tests. Fig. 2 shows one of the more complex samples under test. Its layout mimics the configuration foreseen for ITS3. It contains 6 ALPIDE chips bent at the target radii and simultaneously functional.Another major development line relates to the questions on the feasibility of the mechanical integration of the ITS3 detector. A series of mechanical integration prototypes have been produced to develop the methods and address these questions. A mechanical prototype of a full half barrel integrating three half cylindrical layers has been manufactured (Fig. 3). This prototype is based on dummy silicon dies of the final dimensions, thinned down to 50 µm or less and bent at the nominal radii. The silicon dummies are held in place by very lightweight carbon foam support components.4Microelectronic developmentsThe primary objective for the microelectronic developments is to search and find a path to build single-die monolithic pixel sensors with the required dimensions of 27 cm on the long edge and 5.6 or 7.5 or 9.4 cm on the shorter edges, at the same time meeting the required resolution and readout rate capabilities. This has to be achieved with a single design corresponding to a single manufacturing masks set.The target technology is a 65nm CMOS Imaging Process by TPSCo [7]. This is implemented on wafers with a diameter of 300 mm and offers a stitching option which enables the building of very large devices.The developments are phased through a series of submissions in silicon planned over the years 2020–2026. The installation of the new detector in the cavern is planned in the course of LHC Long Shutdown 3 (2026–2028). A first submission was completed at the end of 2020. This employed a Multi-Layer per Reticle strategy and was called MLR1. At date, work is ongoing to complete the submission of an Engineering Run (ER1) using the full mask set and stitching. The microelectronics development activities have been framed within the CERN EP R&D Work Package focusing on the development of monolithic pixel sensors. This allowed the fruitful sharing and coordination of design effort among engineers of several institutes, including ones that are not members of the ALICE Collaboration.5Results of the first submissionThe MLR1 submission had as primary goals to develop know-how about the target technology, to verify that efficient particle detectors could be made with it and to characterise experimentally the performance of prototype circuits and pixel sensors.The submission included a large variety of prototype circuits, complete functional blocks and prototypes of pixel arrays. Most of the designs were implemented in small test chips with a common footprint of 1.5 mm × 1.5 mm. Tests on the prototype chips have been made or are currently being completed.Transistor Test Structure included in MLR1 allowed a detail characterisation of many basic devices. These work as expected and presented performances similar to other 65nm technologies that have been previously characterised for applications in particle detection.Prototypes of IP blocks implementing analog functions such as band-gaps, DACs, temperature sensing, VCOs, were produced and qualified. These designs are now silicon proven and are being re-used as components of circuits of the next submission.The MLR1 submission also contained prototypes of pixel sensors. A family of Analog Pixel Test Structures (APTS) chips implemented small (4 × 4) pixel arrays of pitches between 10 µm and 25 µm. The APTS chips feature direct analogue readout and explore variants of the design of the pixel sensor and of the analog front-end. The Digital Pixel Test Structure (DPTS) chip has a 32 × 32 array of pixels with 15 µm pitch with in-pixel fast discrimination, encoding of Time Over Threshold and data-driven asynchronous binary readout. The CE65 chips [8] have 32 × 32 pixels of 15 µm pitch with three variants of front-ends. The readout is analog and based on a rolling shutter architecture.Multiple beam tests with MLR1 pixel test chips have been performed. One key milestone was the measurement of particle detection efficiency larger than 99.5% with two DPTS chips simultaneously operated in combination with a beam telescope of six planes based on ALPIDE chips [4,5]. Results of the characterisation of the APTS and DPTS chips are expected to be published in 2022.The MLR1 submission also included process optimisation techniques that were pioneered in the previous generation of sensors built with the 180nm TowerJazz process [9,10]. Their effectiveness to increase the margins of sensing performance with the new 65 nm technology is being experimentally confirmed and results will be published.6Microelectronic developments in the next submissionThe main aims of the second silicon submission, Engineering Run 1 (ER1), are to learn and prove stitching. This is to be completed by mid 2022.The stitching technique [11] allows to manufacture devices that are much larger than the dimensions of the design reticle, normally limited to about 3 cm × 2 cm. Fig. 4 illustrates the principles. The design reticle (left side of the illustration) gets subdivided in sub-frames that correspond to sub-frames of the photomasks. During the photolithographic patterning of wafers, these are selectively exposed onto adjacent locations according to a pre-established pattern (right side of the illustration). This requires very accurate translations and alignment of the wafers between each exposure. The peripheral structures along the outer edges of the core array and at the four corners are designed in dedicated sub-frames of the reticle. The red lines in the figure indicate the dicing lanes. With a judicious design of the geometries in the reticle sub-frames, large chips with diagonals approaching the wafer diameter become feasible, provided their core area can be constructed as an array of sub-units regularly repeating in space. Interconnections across the sub-units are made with the joining of wiring geometries at the abutment boundaries.The floorplan of the ER1 submission is shown in Fig. 5. It contains two prototypes of stitched sensors, the Monolithic Stitched Sensor (MOSS) and Monolithic Stitched sensor with Timing (MOST) chips. These actually use stitching along only one axis (vertical in the figure), although the manufacturing process is a full 2D stitching process. This technique allows the dicing of sensors with different widths required for the different ITS3 layers (see Fig. 3) starting from a single design. A crucial unknown is the manufacturing yield achievable with such large designs. The two prototypes explore different approaches to mitigate the impact on the functional yield of possible, although rare, manufacturing defects.The submission will contain also additional small test chips. These include designs to further develop the pixel cells and prototypes of integrated functional blocks like data transmitters for fast serial links.7The monolithic stitched sensor prototypeThe primary goals of the MOSS chip are: to explore the viability of the stitching technique to make a particle detector, to learn how to make interconnects for the distribution of supplies and signalling on a wafer size chip, to learn about the yield that can be obtained with such a design and to study experimentally electrical performance parameters. These include IR drops, leakage currents, noise distributions, speed and spreads of characteristics.The MOSS chip, depicted in Fig. 6, is made abutting ten Repeated Sensor Units using stitching. The layout is completed on the two sides by two smaller end-cap regions. The overall dimensions are 25.9 cm × 14 mm. Referring to Figs. 4 and 5, the Repeated Sensor Unit is designed in the frame “M” of the reticle, while the end-cap regions are contained in the “T” and “B” frames.The Repeated Sensor Unit (RSU) is subdivided in two half-units with pixel arrays of different pitches. One contains four matrices of 256 × 256 pixels with a pitch of 22.5 µm. The other one contains four matrices of 320 × 320 pixels with a pitch of 18 µm. The two half units are characterised by different circuit densities and different widths and spacing of the interconnecting metal structures. The full sensor contains 6.72 million pixels. Each half unit is a standalone powering and functional unit including its own periphery and I/Os and it is independent from all the others.The analog front-end of the pixels is derived from the DPTS chip prototype. By design, the minimum and nominal power consumption values are 15 nW and 36 nW. The nominal value corresponds to analog power densities of 11 mW/cm2 and 7 mW/cm2 for the large and fine pixel pitch respectively.The architecture of one half unit is illustrated in Fig. 7. The four matrices have dedicated and independent analog biasing blocks and regional readout circuits. Each half unit has a top level peripheral control module and a readout module.Metal interconnects are the only structures crossing the stitching boundaries. The metal stripes of all power domains traverse all RSUs and extend from the left end-cap to the right end-cap. Wiring through the stitching boundaries is also used in the two Stitched Communication Backbones. These blocks prototype the wiring and the circuits needed to achieve signalling between the repeated units and the left end-cap. Each of them contains a control bus shared across all the RSUs and multiple point-to-point data readout busses between each half unit and the left end-cap. They are implemented in their own independent power domains and with conservative width and spacing layout rules.Thanks to the stitched backbone, control and readout of the full MOSS chip can be done via interfaces on the left end-cap. This feature complements the twenty control and readout ports distributed on the two long edges of the chip, each dedicated to one of the half units.The design of the MOSS chip supports different testing scenarios. The first tests will consist of powering and operating the twenty half units independently, studying the yield of the half units and its possible dependence on the density of circuits. In addition, the highly modular architecture is intended to quantify the yield at various levels of granularity below the half-unit, that is at region level, at pixel column or pixel row level, down to the single pixel cell level.8ConclusionsThis contribution described the aim of the ALICE ITS3 project which is to replace the three inner layers of the ALICE ITS2 detector with true cylindrical sensing layers, removing most of the electrical, mechanical and cooling components from the active volume.The sensors will consist of monolithic pixel chips implemented in a 65 nm CMOS Imaging technology with stitching. Their dimensions would reach full wafer scale, they will be thinned down to less than 50 µm.At the time of writing, the project has demonstrated the operation of thinned ALPIDE sensors, implemented in 180nm CMOS, curved at 18 mm bending radius. A full scale mechanical prototype based on dummy silicon dies has also been produced and it demonstrated the feasibility of the mechanical integration.The microelectronic developments towards the final sensor are ongoing. The technology has been validated for particle sensing and a detailed characterisation of prototype pixel chips from the first exploratory submission is on the way. The next silicon submission will focus on exploring the stitching technique.The MOSS chip is one of the stitched sensor prototypes under development. It is a full wafer scale chip design with dimensions of 25.9cm × 14mm. It is designed to investigate experimentally the achievable functional yield, to prototype the distribution of supply and signals over the full chip range and to prototype large arrays of pixels with different layout densities.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsThis work was supported by the ALICE Collaboration, the CERN EP R&D WP1.2 and collaborating institutes . Credits for the results go to members of the ALICE Collaboration and to several non-member contributors who worked on the electro-mechanical integration, characterisation and microelectronics developments. The following people contributed to the designs of MLR1 chips and stitched sensor prototypes: G. Aglieri Rinella, G. Borghello, S. Bugiel, L. Cecconi, J. De Melo, W. Deng, G. De Robertis, A. Dorda Martin, A. Dorokhov, P. Dorosz, K. Dort, X. 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DOI: 10.1016/j.nima.2017.07.046.[11]Tower Semiconductor LtdW.Stitching design rules for forming interconnect layers, US Patent 6225013B12001Tower Semiconductor Ltd, Stitching design rules for forming interconnect layers, US Patent 6225013B1 (2001). diff --git a/tests/data/elsevier/j.nimb.2019.04.063.xml b/tests/data/elsevier/j.nimb.2019.04.063.xml new file mode 100644 index 0000000..6f88583 --- /dev/null +++ b/tests/data/elsevier/j.nimb.2019.04.063.xml @@ -0,0 +1 @@ +application/xmlNew exotic beams from the SPIRAL 1 upgradeP. DelahayeM. DuboisL. MaunouryA. AnnaluruJ. AngotO. BajeatB. BlankJ.C. CamP. ChauveauR. FrigotS. HormigosB. JacquotP. JardinO. KamalouV. KuchiP. LecomteB. OsmondB.M. RetailleauA. SavalleT. StoraJ.C. ThomasV. ToivanenE. TraykovP. UjicR. VondrasekRadioactive ion beamsIon sourcesNuclear Inst. and Methods in Physics Research, B 463 (2020) 339-344. doi:10.1016/j.nimb.2019.04.063journalNuclear Inst. and Methods in Physics Research, B© 2019 The Authors. Published by Elsevier B.V.Elsevier B.V.0168-583X46315 January 20202020-01-15339-34433934410.1016/j.nimb.2019.04.063http://dx.doi.org/10.1016/j.nimb.2019.04.063doi:10.1016/j.nimb.2019.04.063http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2019-05-02T10:41:24Zhttp://vtw.elsevier.com/data/voc/oa/SponsorType#Authorhttp://creativecommons.org/licenses/by/4.0/This is an open access article under the CC BY license.JournalsS300.1NIMB63603S0168-583X(19)30247-210.1016/j.nimb.2019.04.063The AuthorsFig. 1Target–ion source system used at SPIRAL 1 since 2001. It couples the Nanogan III ECR ion source with a graphite target via a cold transfer tube.Fig. 2Elements produced by the different sources envisaged after the upgrade. The FEBIAD source is a universal technique to produce 1+ beam of all elements that exhibit a melting point below 2000 °C, and is therefore capable to ionize the same elements as the ECR ion sources and surface ionizers. Some of the elements that could be produced by fusion evaporation beyond Z = 41 are also shown. The lines delimit from left to right: alkali metals; metals of transition; post-transition metals; nonmetals; noble gases.Fig. 3FEBIAD TISS. The bottom insets is a zoom on the ion source critical parts. See text for details.Fig. 4Secondary intensities measured with the SPIRAL 1 FEBIAD TISS with 36Ar at nominal power (∼1200 W) in 2013.Fig. 5Photograph of the transfer tube and associated helical chicane.Fig. 6Secondary intensities measured with the SPIRAL 1 FEBIAD TISS with 36Ar at nominal power (800–1200 W) in 2018. A wrong steering of the primary beam is invoked as the reason for the lower yields as compared to 2013.Fig. 7Extrapolation of secondary beam intensities (before post-acceleration) for the fragmentation of different targets and fusion evaporation. See text for more explanations.Fig. 8Projections for secondary intensities for radioactive isotopes produced by the FEBIAD TISS for the “day 1” experiments.Fig. 9Extrapolation of secondary beam intensities (before post-acceleration) for the fragmentation of different stable beams available at GANIL [24].Fig. 10Ionization efficiencies measured at ISOLDE and GANIL with the different FEBIAD sources and in different ionization regimes. See text for more explanations.Fig. 11Energy profiles of the Nanogan and FEBIAD ion sources.Fig. 12Charge breeder efficiencies obtained at LPSC and SPIRAL 1 with the upgraded Phoenix charge breeder.New exotic beams from the SPIRAL 1 upgradeP.Delahayea⁎pierre.delahaye@ganil.frM.DuboisaL.MaunouryaA.AnnaluruaJ.AngotbO.BajeataB.BlankcJ.C.CamdP.ChauveauaeR.FrigotaS.HormigosaB.JacquotaP.JardinaO.KamalouaV.KuchiaP.LecomteaB.OsmondaB.M.RetailleauaA.SavalleaT.StorafJ.C.ThomasaV.ToivanenaE.TraykovgP.UjicaR.VondrasekhaGANIL, bd Henri Becquerel, 14000 Caen, FranceGANILbd Henri Becquerel14000 CaenFrancebLPSC, avenue des martyrs, 38026 Grenoble, FranceLPSCavenue des martyrs38026 GrenobleFrancecCENBG, 19 chemin du Solarium, 33175 Gradignan, FranceCENBG19 chemin du Solarium33175 GradignanFrancedLPC Caen, Bd Maréchal Juin, 14000 Caen, FranceLPC CaenBd Maréchal Juin14000 CaenFranceeCSNSM, bat. 101, Domaine de l'Université de Paris Sud, 91400 Orsay, FranceCSNSMbat. 101Domaine de l'Université de Paris Sud91400 OrsayFrancefISOLDE, CERN, route de Meyrin, 1211 Geneva, SwitzerlandISOLDECERNroute de Meyrin1211 GenevaSwitzerlandgIPHC, Batiment 27, 23 Rue du Loess, 67200 Strasbourg, FranceIPHCBatiment 27, 23 Rue du Loess67200 StrasbourgFrancehANL, 9700 Cass Avenue, Lemont, IL 60439, United StatesANL9700 Cass AvenueLemontIL60439United States⁎Corresponding author.AbstractSince 2001, the SPIRAL 1 facility has been one of the pioneering facilities in ISOL techniques for re-accelerating radioactive ion beams: the fragmentation of the heavy ion beams of GANIL on graphite targets and subsequent ionization in the Nanogan ECR ion source has permitted to deliver beams of gaseous elements (He, N, O, F, Ne, Ar, Kr) to numerous experiments. Thanks to the CIME cyclotron, energies up to 20 AMeV could be obtained. In 2014, the facility was stopped to undertake a major upgrade, with the aim to extend the production capabilities of SPIRAL 1 to a number of new elements. This upgrade, which is presently under commissioning, consists in the integration of an ECR booster in the SPIRAL 1 beam line to charge breed the beam of different 1+ sources. A FEBIAD source (the so-called VADIS from ISOLDE) was chosen to be the future workhorse for producing many metallic ion beams. This source was coupled to the SPIRAL 1 graphite targets and tested on-line with different beams at GANIL. The charge breeder is an upgraded version of the Phoenix booster which was previously tested in ISOLDE. It was recently commissioned at LPSC and lately in the SPIRAL 1 beam lines with stable beams. The upgrade additionally permits the use of other target material than graphite. In particular, the use of fragmentation targets will permit to produce higher intensities than from projectile fragmentation, and thin targets of high Z will be used for producing beams by fusion-evaporation. The performances of the aforementioned ingredients of the upgrade (targets, 1+ source and charge breeder) have been and are still being optimized in the frame of different European projects (EMILIE, ENSAR and ENSAR2). The upgraded SPIRAL 1 facility will provide soon its first new beams for physics and further beam development are undertaken to prepare for the next AGATA campaign. The results obtained during the on-line commissioning period permit to evaluate intensities for new beams from the upgraded facility. © 2001 Elsevier Science. All rights reserved.KeywordsRadioactive ion beamsIon sources1Scientific motivationsDuring the past decades, the SPIRAL 1 facility at GANIL has been delivering radioactive ion beams of unique intensity and purity for physics experiments. SPIRAL 1 makes use of the so-called “Isotope Separation On Line” (ISOL) technique [1]. Ionized in ion sources, ISOL beams have an optical quality which is comparable to this of the stable ion beams. In particular, the energies accessible at SPIRAL, from a few keV to 20 AMeV for the lightest nuclides accelerated by CIME, allow for a unique variety of studies using common techniques of decay spectroscopy, Coulomb excitation, fusion, transfer and direct reactions. During the past decade a number of interesting physics results were achieved at SPIRAL 1 using these techniques, leading to published highlights addressing questions that can only be handled with reaccelerated radioactive beams. Despite remarkable achievements [2] and its status of first world-class facility, SPIRAL 1 has been technically limited to the production of radioactive ion beams of gaseous elements, thus limiting the physics opportunities of the facility.A project of upgrade was started in 2011 to complement the radioactive ion beam production capabilities of the facility towards condensable elements (see Section 2). The scientific interest of the upgrade of SPIRAL 1 was sustained by different calls for letters of intent, which gathered a large response from the international nuclear physics community. In total, more than 100 beams were discussed during a workshop dedicated to SPIRAL 1, 2/3rd of which are using a new Target Ion Source System (TISS), which couples the SPIRAL target with a FEBIAD ion source [3]. First experiments using the new TISS should run during 2019. The upgrade opens on the one hand numerous opportunities with reaccelerated beams with innovative setups such as AGATA, MUGAST or ACTAR. On the other hand, it offers rich perspectives with low energy beams for weak interaction physics and beta/rare decay studies, in which DESIR [4] will play a major role in the upcoming years.2Upgrade layoutSince 2001, SPIRAL 1 makes use of thick graphite targets, on which the high power heavy ion beams of GANIL impinge. The radioactive fragments diffuse as atoms out of the hot target and effuse via a transfer passage to the ion source. Ionized to the required charge state by the Nanogan III Electron Cyclotron Resonance (ECR) ion source [5], the radioactive ions are then separated and accelerated in the Cyclotron d’Ions de Moyenne Energie (CIME) [1]. Fig. 1 presents the Target–Ion Source System (TISS) that was used so far at SPIRAL 1. The ionization technique prevents the ionization of condensable elements as the transfer tube and the ECR source is a cold assembly that stops the effusion of radioactive isotopes of condensable elements from the hot target to the ion source plasma. The SPIRAL upgrade project consisted in the development of a 1+ to n+ charge breeding system, permitting the use of versatile 1+ sources for extending the range of elements available for post-acceleration to condensable elements.2.1A FEBIAD source as workhorse for the upgradeCompared to rare gases, radioactive atoms of condensable elements produced by the ISOL method suffer from comparatively longer sticking times on surfaces that they encounter during their transport from the target to the ion source volume. As high temperature reduces the sticking time, the condensable elements are traditionally produced using hot target and ion source assemblies [6]. Typical ion sources used in this case are hot surface sources, such as surface ionization sources, Forced Electron Beam Induced Arc Discharge sources (FEBIAD sources also called “hot plasma” sources), and Resonant Ionization Laser Ion Source (RILIS). An alternative to this scenario was tested at GANIL in the past by coupling directly ECR sources to targets at the SIRa test bench [7,8]. Due to the diversity of the beams requested in the letters of intent, a FEBIAD source recently improved at ISOLDE, the so-called VADIS [9] was found to be the ion source of condensable elements to be coupled in priority with the SPIRAL targets. The FEBIAD source can deliver a number of beams in addition to those traditionally produced by surface ionization and ECR ion sources, as can be seen on Fig. 2. It is a first step towards a universal beam production that the more sophisticated, but cleaner RILIS could complement at a later stage.2.2New targetsThe upgrade was also aiming at extending the safety authorizations which were previously limited to the use of projectile fragmentation on graphite targets. In the frame of the upgrade, these authorizations were extended to other target materials for fragmentation, and to fusion evaporation reactions. The following primary beam – target combinations are now authorized:Reaction mechanismPrimary beamTargetAZXE (AMeV)I (pps)MaterialThicknessProjectile frag.12C → 238U≤95≤2∙1013GraphiteA few mmTarget frag.12C952∙1013Z ≤ 41 (Nb)A few mmFusion evap.12C → 238U∼8≤2∙1013Any Z∼1–2 mg/cm2A Nb target and fusion evaporation targets are presently being developed [10,11] and should be tested in the coming years. The fusion evaporation target benefits from a collaboration with IPN Orsay [11].2.3ECR charge breedingAt reacceleration facilities, charge breeding is a technique consisting in the injection of a 1+ beam into the plasma of an Electron Beam Ion Source (EBIS) or Electron Cyclotron Resonance Ion Source (ECRIS) and of its subsequent multi-ionization for reaching the charge state required by the post-accelerator [12]. The charge breeding is a competitive and cost effective solution compared to the stripping foil technique. It has benefited from multiple studies within the European research and technical development programs FP5 and FP6, and from the development of facilities using this technique at ISOLDE, TRIUMF or more recently ANL. Because of the continuous mode of operation and of the intrinsic resolving power of the cyclotron CIME, an ECR charge breeder has been found better suited than a pulsed EBIS. The ECR charge breeder of the SPIRAL upgrade is a Phoenix booster donated by the Daresbury Laboratory. It has been tested at ISOLDE from 2003 to 2008 [13,14] and upgraded since to reach the best performances with this device.3New beams from the FEBIAD TISS3.1TISS developmentThe FEBIAD TISS couples standard SPIRAL 1 1200 W C targets to the VADIS via an ohmic heated tantalum transfer tube (Fig. 3). The TISS was developed in a staged approach. The early versions were first tested online in 2011 using the SIRa test bench at modest primary beam power. They demonstrated the efficient ionization of radioactive isotopes of 8 new elements: the alkali Na, K, metallic Mg, Al, Fe, Cu, Mn and halogen Cl elements. The TISS lifetime was however limited due to various issues related to the dilatation of the transfer tube. In 2013, these issues were fixed by enabling a displacement of the ion source along the axis of the secondary ion beam extraction (Fig. 3) [15]. The modified TISS was then tested at nominal power (1200 W of 36Ar at 95 AMeV) in the SPIRAL 1 beam lines in December 2013, leading to the first scientific results obtained at SPIRAL 1 with a FEBIAD source [16,17]. These on-line tests at SPIRAL 1 gave generally even better secondary beam intensities than expected from the measurements at SIRa, showing that the yields do not scale linearly with the primary beam power. The intensities shown in Fig. 4 were published in [18]. At the very end of the tests, the anode entered in short circuit with the cathode. This failure was attributed to the slow dissociation and metallization during operation of BeO insulators supporting the anode of the FEBIAD (see Fig. 3), as Be+ ions were visible in high amount (1 µA), dominating the ion source spectrum. In order to increase the reliability of the target ion source, a program of test was rigorously followed on the SPIRAL 1 off-line test bench with the aim to select the best insulator, which could stand operation under high temperature. The results of these tests permitted to show that on the test bench boron nitride (BN) insulators permitted a stable operation over a period of about 3 weeks. These tests were realized with an important number of heating and cooling cycles, as the ion source was switched on in the morning and off in the evening for half of the tests period. The nominal and reproducible efficiencies for stable rare gases were determined (Section 3.3). At this stage, the target ion source was considered as ready for operation. 2016 and 2017 were used to provision the FEBIAD target ion sources for the startup of the upgrade and diverse characterizations detailed below. In April and May 2018, the SPIRAL 1 upgraded facility was commissioned for radioactive ion beam production. A beam of 95 AMeV of 20Ne first impinged on the SPIRAL 1 target for the production of a 17F beam for the E750 experiment. Unexpectedly, the anode entered again into a shortcut with the cathode very rapidly (6–8 h) after the beam was tuned on target. Again, the anode insulators were identified as the probable cause for the FEBIAD breakdown, contrasting with the observations done on the test bench without beam on target. During discussions with a panel of experts from ISOLDE and TRIUMF, the possible explanations mentioned for the different behavior of the source on-line and off-line were the different heating /thermal conditioning of the target ion source as well as the possible deposition of direct C vapors on different parts of the source, including the insulators. In order to fix these issues, a new FEBIAD ion source was conditioned with a different configuration of heat shields permitting to cool down the insulators. A helical chicane was inserted in the transfer tube to stop the direct C vapor from the irradiated target (Fig. 5). With these modifications, the FEBIAD target ion source could be run with first a low power (100–200 W) 40Ca beam and finally a high power (>800 W) 36Ar beam, both at 95 AMeV without failure. The secondary beam intensities measured for the latest test are shown in Fig. 6. As it clearly appeared later, the diagnostic used to center the primary beam on target was malfunctioning during the commissioning tests of 2018. The wrong steering caused lower yields than expected. As it was observed later with a Nanogan TISS delivering 14O to the E744 experiment, a factor of 50–100 improvement of the production yield could be obtained at the SPIRAL identification station [19] by tuning the primary beam steerers to correct for the faulty diagnostic. The wrong steering might even have been a possible cause for the early failure of the FEBIAD TISS with 20Ne.The FEBIAD target ion source is being presently further consolidated for final adjustment on the test bench, before being recommissioned online in the upcoming running period. In addition to the aforementioned modifications, a few adjustments are tested. In particular the material of the poles supporting the fragile insulators, originally in Mo, are replaced by Re (see inset of Fig. 3), which has a lower thermal conductivity. These Re poles will permit to further protect the insulators from the hottest parts of the ion source. With these adjustments, it is believed that the FEBIAD target ion source will have most chances to behave very well online.3.2Secondary beam intensitiesThe 1+ beam intensities measured in 2013 and 2018 from the FEBIAD TISS at high power with a beam of 36Ar at 95 AMeV are shown in Figs. 4 and 6. The yields measured in 2018 are considered as very conservative, as it was understood later that the primary beam positioning was not controlled during the on-line commissioning, resulting in degraded production conditions.An analysis of the data measured in 2013, briefly presented in [18], permitted to deduce from the measured intensities the typical release times and ionization efficiencies for the different elements observed with a Ge detector at the 1+ beam identification station. This analysis bases itself on the empirical release parametrization used in [20] for the ISOLDE yields. From the ionization, transport and release efficiencies deduced from this analysis, and using the EPAX V2 parametrization [21] for fragmentation cross sections, projections can be made for the isotopes of the different elements produced so far with the FEBIAD TISS. The results of these projections are shown in Fig. 7, making an optimized use of different primary beams available at GANIL. These projections were used as guideline for “day 1” beams, available for the first call for proposals of experiments with the SPIRAL 1 upgraded facility. For experiments using reaccelerated beams, these 1+ beam intensities have to be folded with charge breeding (∼5–10%, see Section 4) and acceleration efficiencies (∼20%), so that the resulting accelerated beam intensities are typically a factor 100 lower.More speculative extrapolations have been done for the production of radioactive isotopes for other elements typically produced by FEBIAD sources, as for instance observed at ISOLDE. For these elements, the release efficiencies have been estimated from the parameterization of on-line data obtained at ISOLDE [20], PARNNE [22], and if not available from diffusion – effusion coefficients found in the literature, using the parametrization of diffusion and effusion described by Kirchner in [23]. The FEBIAD ionization efficiencies are interpolated from measured data with noble gases, where a simple linear mass dependence was assumed, as described in [18]. The mass interpolation accounts for the time of exposure of atoms to the electrons in the anode: heavier atoms spend more time in the anode and consequently have more chances to get ionized. The dependence on the ionization potential was neglected here, although it will certainly play a role as one of the main ingredients defining the electron impact ionization cross section. Because the ionization potentials of noble gases are the highest of the table of elements, their ionization cross section will generally be weaker. Such approximation is therefore believed to be rather conservative. Sticking times were estimated considering a typical number of contacts from the target to the ion source of ∼10,000, which is also on the higher limit of what is usually considered in typical ISOL systems [23]. Even with these conservative assumptions, there is no guarantee that the results of such extrapolations are safe, and measurements are obviously needed. The results of the extrapolations are shown in Figs. 8 and 9 for the fragmentation of the different stable beams available at GANIL and using new target materials for fragmentation and fusion evaporation, respectively. The following fragmentation targets were considered: SiC, CaO, NiO, Nb using a 12C beam at 95 AMeV and at the highest power available at GANIL (3.6 kW). The fusion-evaporation cross sections have been estimated using the procedure described in [24] for a number of target material – stable beam combinations. The plot shows only the best combination. The same release efficiencies as for the fragmentation targets have been assumed, which is a priori conservative as for this reaction mechanism many tricks can be used in order to speed up the transport of radioactive atoms to the ion source [10]. A list of secondary beam intensities as obtained using these different estimates are available on the GANIL – SPIRAL 2 website [25].3.3Ongoing investigationsDuring the conditioning of the various versions of the TISS, the ionization efficiencies of the FEBIAD have been repeatedly monitored on the off-line test bench with stable rare gases. These ionization efficiencies are generally reproducing the efficiencies of the traditional MK5 of ISOLDE, which are a factor of ∼4 lower than the VADIS ones quoted in [9]. ISOLDE equally reports that similarly lower efficiencies are regularly obtained on-line with the VADIS. Nevertheless, it is remarkable that the efficiencies quoted in [9] could be sometimes obtained during the conditioning of the TISS at the SPIRAL 1 test bench, over periods of a few hours. The conditions for stabilizing this enhanced ionization regime have not yet been found, and are being investigated. One of the latest manifestation of this particular regime happened during the injection of a tiny leak of Xe (Fig. 10), whose flow rate enabled the control of the ionization efficiency, switching gradually from the standard low ionization regime to the enhanced ionization regime and vice-versa, for a couple of hours. As it was shown, only a hundred nA of Xe, compared to a total current of a few µA in the mass spectrum extracted from the FEBIAD, was sufficient to trigger the enhanced ionization regime.The energy profile of the 1+ beam from the FEBIAD source is of high interest for the charge breeding performances. The energy acceptance of the ECR charge breeder is defined by the width of its “ΔV” curve; the n+ extracted ion beam current as a function of the difference of voltage between the 1+ source and ECR charge breeder. The acceptance of ECR charge breeders is typically of 5–10 eV FWHM for condensable elements. The energy profiles of the FEBIAD and Nanogan III could lately be measured and compared thanks to an analyzer manufactured by LPC Caen (Fig. 11). The energy dispersion of the FEBIAD ion source is of the order of σEy1.5 eV, which is well within the acceptance range measured with the SPIRAL 1 charge breeder [26]. It is also remarkable that the plasma potentials of the Nanogan and FEBIAD source differ considerably. The origin for the voltage shown in Fig. 11 is somewhat arbitrary, because of the lack of calibration of our test bench HV power supplies. It corresponds to the 1+ source HV voltage within 20–30 V error bar. Latest simulations show that the FEBIAD can have plasma potentials as low as −70 V, with an electron beam current as high as 300 mA. This result was experimentally corroborated by the first charge breeding of a radioactive ion performed at SPIRAL 1 as described in Section 4.In order to identify the most important mechanisms that governs the ionization performances of the FEBIAD, simulations combining the SIMION software for calculating the trajectory of charged particles (electrons and ions) in fields, and FEMM 4.2 for accounting for space charge effects, are being undertaken. At present, these simulations reproduce the plasma potential reasonably well, as the latter defines the energy profile of the 1+ beam (mean energy and dispersion) of the extracted beam. Possible ionization regimes by pure electron impact or in a plasma will have then to be investigated to attempt to reproduce the order of magnitudes of the measured efficiencies. These efforts are carried out within a newly formed collaboration between GANIL, ISOLDE, TRIUMF, SPES, IPN Orsay and ISOL@MYRRHA, which are all interested in understanding the physical processes in play in the FEBIAD ion sources. The very recent comparison of our results with those of TRIUMF, which is using COMSOL for simulating the FEBIAD plasma potential, demonstrated an excellent agreement.4ECR charge breeder performancesBased on an early version of the Phoenix charge breeder used at ISOLDE for charge breeding studies [13,14], the SPIRAL 1 charge breeder has been upgraded in the frame of the EMILIE project [27] to benefit from the latest advances in the field [28,29]. The upgraded charge breeder is using ultra-high vacuum compliant components, including an Al plasma chamber. It is capable of 2 frequency heating, although only a single 14 GHz frequency heating was used so far. At injection, an iron plug originally truncated on one side for the RF injection has been symmetrized. The injection optics now include an electrostatic triplet and a mobile injection electrode. The puller is also mobile. The gas injection system is directly feeding the plasma chamber.Fig. 12 presents the charge breeding efficiencies obtained on the off-line LPSC test bench [26] and in 2018 at SPIRAL 1 [30] for Na, K and Rb alkali ions, with He as support gas. The efficiencies are a factor 2–5 higher than those recorded at ISOLDE [13,14]. The charge breeding performances for Na were lately benefiting from the work carried out at LPSC [31]: by fine tuning the magnetic field with iron rings and the extraction electrode position, the efficiency was improved by a factor about 2–3 compared to what was previously measured at SPIRAL 1 [30].During the radioactive ion beam commissioning in May 2018, the charge breeding of 37K1+ to 37K9+ was successfully demonstrated. The FEBIAD was run with an anode voltage of 150 V, similar to what is done for the plasma ionization of other elements. In practice, for optimizing the surface ionization of alkali ions, a much lower voltage could have been employed [32]. The secondary intensities of the 1+ FEBIAD TISS beams and charge bred beams were successively transported to the identification station. An efficiency of 5.3% was obtained after a very short period of beam tuning (couple of hours) by monitoring the activity of the rather short-lived (1.225 s) 37K isotopes at the SPIRAL identification station. For such short tuning time, the tuning only concerned the 1+ beam optics and voltage difference between the ECR charge breeder and FEBIAD anode (ΔV). With an unusually high electron beam current (300 mA) emitted from the FEBIAD cathode, the optimized ΔV was of the order of 80 V. Compared to usual optimized ΔV observed with surface ionization sources used for the off-line commissioning of the charge breeder (7–12 V) [26,30], this value pinpoints a FEBIAD plasma potential of the order of −70 V. Just prior to the injection of the radioactive ion beam, an efficiency of 10% was recorded for the charge breeding of a stable 40Ar1+ beam from the FEBIAD to 40Ar8+, which is a factor of ∼2 below the efficiencies measured with a properly tuned charge breeder. With optimized RF power, gas flow and magnetic confinement, it is quite likely that the 37K charge breeding efficiency would have attained a similar value as those measured with the off-line surface ionization source at LPSC and SPIRAL 1.5ConclusionsThe on-line commissioning of the SPIRAL 1 upgraded facility has started. As there was no charge breeding and subsequent acceleration of a radioactive ion beam delivered to an experiment, the full operation of the upgraded facility has not yet been fully demonstrated. Nevertheless, and despite early technical difficulties, the individual elements and combinations, which are required for the upgraded facility, are all operational. Beams have been successfully delivered from the FEBIAD TISS alone, from the FEBIAD TISS and charge breeder, from the charge breeder and CIME. The ultimate commissioning step, consisting in the successful delivery of beam to an on-line experiment, is being prepared. Final adjustments with the FEBIAD TISS are being undertaken on the off-line test bench. Charge breeding R&D is being pursued, with the aim of further improving the charge breeding efficiencies of light beams [31], and investigate the critical parameters on which the charge breeding time depends. The combination of FEBIAD TISS with an ECR charge breeder is a rather unusual one for reacceleration facilities. Experience will be gained over the years with this peculiar system. As seen with the operation of Nanogan for many years, the CIME cyclotron intrinsic resolving power permits to purify in many cases the light (A < 20) beams. For heavier masses, a case – by – case study has to be done. The test of different stripping foils behind CIME to separate masses around 56Ni delivered from the FEBIAD TISS and reaccelerated by CIME to energies of 10–12 AMeV will be undertaken in the coming months. For low energy beams, the High Resolution Separator [33] and the different beam purification techniques in ion traps such as PIPERADE [34,35], all present at DESIR, will be essential elements for the success of the associated experimental program.AcknowledgmentThis project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No 654002.References[1]A.C.C.VillariNuc. Phys. A7872007126c133cand references therein[2]A.NavinF.De Oliveira SantosP.Roussel-ChomazO.SorlinJ. Phys. G: Nucl. Part. Phys.38201124004[3]https://indico.in2p3.fr/event/12296/.[4]B.BlankB.PramanaJ. Phys.752010343[5]A.C.C.VillariNucl. Phys. A7012002476c479cand references therein[6]R.KirchnerNucl. Instrum. Meth. 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B3762016298 diff --git a/tests/data/elsevier/j.nimb.2019.04.063_expected.yml b/tests/data/elsevier/j.nimb.2019.04.063_expected.yml new file mode 100644 index 0000000..7937368 --- /dev/null +++ b/tests/data/elsevier/j.nimb.2019.04.063_expected.yml @@ -0,0 +1,697 @@ +abstract: "Since 2001, the SPIRAL 1 facility has been one of the pioneering facilities in ISOL techniques for re-accelerating radioactive ion beams: the fragmentation of the heavy ion beams of GANIL on graphite targets and subsequent ionization in the Nanogan ECR ion source has permitted to deliver beams of gaseous elements (He, N, O, F, Ne, Ar, Kr) to numerous experiments. Thanks to the CIME cyclotron, energies up to 20\u202fAMeV could be obtained. In 2014, the facility was stopped to undertake a major upgrade, with the aim to extend the production capabilities of SPIRAL 1 to a number of new elements. This upgrade, which is presently under commissioning, consists in the integration of an ECR booster in the SPIRAL 1 beam line to charge breed the beam of different 1+ sources. A FEBIAD source (the so-called VADIS from ISOLDE) was chosen to be the future workhorse for producing many metallic ion beams. This source was coupled to the SPIRAL 1 graphite targets and tested on-line with different beams at GANIL. The charge breeder is an upgraded version of the Phoenix booster which was previously tested in ISOLDE. It was recently commissioned at LPSC and lately in the SPIRAL 1 beam lines with stable beams. The upgrade additionally permits the use of other target material than graphite. In particular, the use of fragmentation targets will permit to produce higher intensities than from projectile fragmentation, and thin targets of high Z will be used for producing beams by fusion-evaporation. The performances of the aforementioned ingredients of the upgrade (targets, 1+ source and charge breeder) have been and are still being optimized in the frame of different European projects (EMILIE, ENSAR and ENSAR2). The upgraded SPIRAL 1 facility will provide soon its first new beams for physics and further beam development are undertaken to prepare for the next AGATA campaign. The results obtained during the on-line commissioning period permit to evaluate intensities for new beams from the upgraded facility. © 2001 Elsevier Science. All rights reserved." +copyright_holder: The Authors +copyright_statement: © 2019 The Authors. Published by Elsevier B.V. +copyright_year: 2019 +document_type: article +license_url: http://creativecommons.org/licenses/by/4.0/ +license_statement: This is an open access article under the CC BY license. +keywords: ['Radioactive ion beams', 'Ion sources'] +article_type: full-length article +journal_title: Nuclear Inst. and Methods in Physics Research B +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Delahaye, P. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. + emails: + - pierre.delahaye@ganil.fr +- full_name: Dubois, M. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Maunoury, L. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Annaluru, A. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Angot, J. + raw_affiliations: + - value: LPSC, avenue des martyrs, 38026 Grenoble, France + source: Elsevier B.V. +- full_name: Bajeat, O. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Blank, B. + raw_affiliations: + - value: CENBG, 19 chemin du Solarium, 33175 Gradignan, France + source: Elsevier B.V. +- full_name: Cam, J.C. + raw_affiliations: + - value: LPC Caen, Bd Maréchal Juin, 14000 Caen, France + source: Elsevier B.V. +- full_name: Chauveau, P. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. + - value: CSNSM, bat. 101, Domaine de l'Université de Paris Sud, 91400 Orsay, France + source: Elsevier B.V. +- full_name: Frigot, R. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Hormigos, S. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Jacquot, B. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Jardin, P. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Kamalou, O. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Kuchi, V. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Lecomte, P. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Osmond, B. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Retailleau, B.M. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Savalle, A. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Stora, T. + raw_affiliations: + - value: ISOLDE, CERN, route de Meyrin, 1211 Geneva, Switzerland + source: Elsevier B.V. +- full_name: Thomas, J.C. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Toivanen, V. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Traykov, E. + raw_affiliations: + - value: IPHC, Batiment 27, 23 Rue du Loess, 67200 Strasbourg, France + source: Elsevier B.V. +- full_name: Ujic, P. + raw_affiliations: + - value: GANIL, bd Henri Becquerel, 14000 Caen, France + source: Elsevier B.V. +- full_name: Vondrasek, R. + raw_affiliations: + - value: ANL, 9700 Cass Avenue, Lemont, IL 60439, United States + source: Elsevier B.V. +artid: '63603' +title: New exotic beams from the SPIRAL 1 upgrade +dois: +- material: publication + doi: 10.1016/j.nimb.2019.04.063 +journal_volume: '463' +journal_issue: '' +is_conference_paper: false +publication_date: '2020-01-15' +collaborations: [] +documents: +- key: j.nimb.2019.04.063.xml + url: http://example.org/j.nimb.2019.04.063.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: A.C.C.Villari<maintitle>Nuc. + Phys. 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Meth. Phys. Res. B</maintitle>43120185966 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Inst. Meth. Phys. Res. B + journal_volume: '431' + year: 2018 + page_start: '59' + page_end: '66' + label: '32' + authors: + - full_name: Palenzuela, Y. Martinez + inspire_role: author +- raw_refs: + - schema: Elsevier + value: T.Kurtukian + Nieto<maintitle>Nucl. + Instrum. Meth. B</maintitle>3172013284 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Instrum. Meth. B + journal_volume: '317' + year: 2013 + page_start: '284' + label: '33' + authors: + - full_name: Nieto, T. Kurtukian + inspire_role: author +- raw_refs: + - schema: Elsevier + value: P.Ascher<maintitle>EPJ + Web Conf.</maintitle>66201411029 + source: Elsevier B.V. + reference: + publication_info: + journal_title: EPJ Web Conf. + journal_volume: '66' + year: 2014 + page_start: '11029' + label: '34' + authors: + - full_name: Ascher, P. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: E.Minaya Ramirez<maintitle>Nucl. + Instrum. Meth. B</maintitle>3762016298 + source: Elsevier B.V. + reference: + publication_info: + journal_title: Nucl. Instrum. Meth. B + journal_volume: '376' + year: 2016 + page_start: '298' + label: '35' + authors: + - full_name: Ramirez, E. Minaya + inspire_role: author diff --git a/tests/data/elsevier/j.nuclphysa.2020.121991.xml b/tests/data/elsevier/j.nuclphysa.2020.121991.xml new file mode 100644 index 0000000..26a577b --- /dev/null +++ b/tests/data/elsevier/j.nuclphysa.2020.121991.xml @@ -0,0 +1 @@ +application/xmlDegeneracy effects and Bose condensation in warm nuclear matter with light and heavy clustersShun FurusawaIgor MishustinNuclear matterEquation of stateSupernovaeNuclear statistical equilibriumQuantum statistical effectBose–Einstein condensationNuclear Physics, Section A 1002 (2020). doi:10.1016/j.nuclphysa.2020.121991journalNuclear Physics, Section A© 2020 Elsevier B.V. All rights reserved.Elsevier B.V.0375-94741002October 202010.1016/j.nuclphysa.2020.121991http://dx.doi.org/10.1016/j.nuclphysa.2020.121991doi:10.1016/j.nuclphysa.2020.121991121991JournalsS250.1NUPHA121991121991S0375-9474(20)30301-810.1016/j.nuclphysa.2020.121991Elsevier B.V.Nuclear AstrophysicsFig. 1Critical temperatures below which Bose-Einstein condensation occurs as a function of the baryon density (mass number times number density) for deuterons (Aj = 2, blue dashed-dotted lines) and α-particles (Aj = 4, thick red solid lines). The symbols show the maximum values of the baryon densities, Ajnj, of symmetric nuclear matter (Yp = 0.5) for α-particles at T = 1 MeV (red triangle) and 3 MeV (red circle) and for deuterons at T = 1 MeV (blue diamond) and 3 MeV (blue square) in the full statistical ensemble including both light and heavy clusters over the whole density range (see Fig. 2). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)Fig. 1Fig. 2Number densities of dripped protons and neutrons (black dashed lines), deuterons (blue dashed dotted lines), tritons and helions (green dashed double-dotted lines), α-particles (thick red solid lines), and other nuclei (Zi > 2 or Ni > 2, magenta dotted lines) as functions of nB at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp = 0.2 (left column) and 0.5 (right column). The thin red solid lines represent the critical number density above which α-particles should form a condensate.Fig. 2Fig. 3Relative changes in the number densities of deuterons (blue dashed dotted lines), tritons (green dashed double-dotted lines), helions (cyan dashed lines), and α-particles (thick red solid lines) as calculated using quantum statistics vs. those obtained with classical (Boltzmann) statistics at T = 3.0 MeV and Yp = 0.2 (left panel) and 0.5 (right panel).Fig. 3Fig. 4Relative change in the number densities of deuterons (left panel) and α-particles (right panel) as calculated using quantum statistics vs. those obtained with classical (Boltzmann) statistics at Yp = 0.5 and T = 1.0 MeV (black solid lines), 2.0 MeV (blue dashed lines), 3.0 MeV (green dashed dotted lines), 4.0 MeV (red dashed double-dotted lines), and 5.0 MeV (magenta dotted lines).Fig. 4Fig. 5Mass fractions of α-particles (thick lines) and heavy clusters (thin lines) for a model solved for the full-ensemble of heavy clusters (black solid lines) and for a model that uses the single-nucleus approximation for heavy clusters (Zi > 5) (blue dashed line) at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp= 0.2 (left column) and 0.5 (right column). The red lines indicate the mass fraction of thermal α-particles (dashed dotted lines) and condensed α-particles (dashed double-dotted lines) for light-cluster matter in which we ignore the existence of heavy clusters other than d, t, h, and α.Fig. 5Fig. 6Number densities of dripped protons and neutrons (black dashed lines), deuterons (blue dashed dotted lines), tritons and helions (green dashed double-dotted lines), thermal α-particles (with kinetic energies ϵ > 0, thick red solid lines), and condensed α-particles (ϵ = 0, red dotted lines) as functions of nB for light-cluster matter at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp = 0.2 (left column) and 0.5 (right column).Fig. 6Fig. 7Proton chemical potential as a function of neutron chemical potential for light cluster matter at T = 1.0 MeV and Yp = 0.2 (thick dashed black line) and 0.5 (thin dashed black line) together with the critical lines at which the chemical potentials of the tritons and α-particles reach their rest masses without Coulomb-energy shifts, μp + 2μn = mt (green dashed double-dotted line) and 2μp + 2μn = mα (red solid line).Fig. 7Fig. 8Chemical potential of α-particles (with respect to the α-particle rest mass) per baryon (dashed lines) and the negative value of the α-particle binding energy per baryon (solid lines) as functions of nB for light-cluster matter at T = 1.0 MeV (left panel) and 3.0 MeV (right panel) and Yp = 0.5. The thin red lines display the results without the Coulomb energy shifts: ΔEjC=0.Fig. 8Degeneracy effects and Bose condensation in warm nuclear matter with light and heavy clustersShunFurusawaData curationInvestigationSoftwareWriting - original draftab⁎furusawa@rs.tus.ac.jpIgorMishustinConceptualizationMethodologySupervisionWriting - review & editingcdaDepartment of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, JapanDepartment of PhysicsTokyo University of ScienceShinjuku, Tokyo162-8601JapanDepartment of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, JapanbInterdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, JapanInterdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS)RIKENWakoSaitama351-0198JapanInterdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, JapancFrankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 Frankfurt am Main, GermanyFrankfurt Institute for Advanced StudiesJ.W. Goethe UniversityFrankfurt am Main60438GermanyFrankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 Frankfurt am Main, GermanydNational Research Center Kurchatov Institute, Moscow 123182, RussiaNational Research Center Kurchatov InstituteMoscow123182RussiaNational Research Center Kurchatov Institute, Moscow 123182, Russia⁎Corresponding author.AbstractWe have investigated the composition of nuclear matter under thermodynamic conditions relevant for the core-collapse of massive stars, with a focus on quantum-statistical effects for light clusters. We find that deviations in the number densities of light clusters between Boltzmann and quantum statistics are small at sub-nuclear densities and temperatures 1–3 MeV, ≈0.2% at most. The formation of heavy clusters in stellar matter leads to a reduction in the number densities of light clusters. As a result, quantum-statistical effects such as the Bose–Einstein condensation of deuterons and α-particles are suppressed. The condensation of α-particles in iso-symmetric nuclear matter is predicted under the assumptions that it is composed solely of nucleons and light clusters and that no heavy clusters are present. We also find that Coulomb screening hardly affects the critical baryon densities for alpha condensation, although it modifies the masses and chemical potentials of the α-particles.KeywordsNuclear matterEquation of stateSupernovaeNuclear statistical equilibriumQuantum statistical effectBose–Einstein condensation1IntroductionHot and dense stellar matter appears in such astrophysical phenomena as core-collapse supernovae and neutron-star mergers. Nuclear matter at sub-nuclear densities consists of a mixture of unbound nucleons, light clusters, and heavy clusters in chemical and thermal equilibrium [1]. The nuclear equation of state (EOS) determines the number densities of all these nuclear species as well as thermodynamic quantities as functions of temperature, baryon density, and charge fraction. The dynamical evolution and observable signatures of such compact-star phenomena are very sensitive to the EOS. In addition, weak interactions among the nuclear species play an important role, especially in core-collapse supernovae (see e.g., [2]). The roles of light clusters in core-collapse supernovae have not been completely clarified, although their potential influence on the dynamics, neutrino spectrum, and nucleosynthesis has been pointed out by many researchers [3–13].Modern EOSs for clustered nuclear matter at sub-nuclear densities are usually obtained for astrophysical simulations by using statistical models, e.g., [14–24]. Recently we have developed a self-consistent statistical approach where individual nuclear ground state properties and the full-ensemble distributions of nuclei are calculated consistently by minimizing the free energy density [25,26]. In most statistical models, the thermodynamic quantities for light clusters are calculated using Boltzmann statistics. This assumption is actually valid at high temperatures and low densities. However, it is a priori unclear whether this assumption is valid for conditions encountered in supernovae and neutron-star mergers. Certainly, at low temperatures and at high densities, they should be regarded as Fermi or Bose particles.The purpose of this work is to investigate such quantum-statistical effects for light clusters in stellar environments using realistic nuclear ensembles. The possibility of alpha condensation in nuclear matter has been analyzed previously by several authors. In ref. [27], the authors used a diagrammatic quantum-statistical approach to describe four-nucleon correlations in nuclear matter. They demonstrated the possibility of α condensation at sub-saturation densities, below a critical temperature of about 6 MeV. However, the calculations were done assuming a uniform nucleonic background; i.e., the presence of other light clusters in the system was completely ignored. Other previous studies [28–33] also did not consider the full nuclear ensemble of light and heavy clusters. In the present study, we generalize classical models of light clusters to a quantum-statistical description in the framework of our statistical approach, including the full-ensemble of nuclei.This paper is organized as follows. In Sec. 2, we formulate the modified statistical model, with a focus on the quantum-statistical effects of light clusters. The results of calculations with the full statistical ensemble, including both light and heavy clusters in some typical astrophysical conditions, are discussed in Sec. 3. For comparison, clustered nuclear matter consisting only of nucleons and light clusters is considered in Sec 4 in order to demonstrate the crucial role played by heavy clusters. Our conclusions are presented in Sec. 5.2Statistical model of clustered nuclear matter with quantum statisticsTo describe multi-component nuclear matter containing both light and heavy clusters, we use the same models as in our previous work [26]. The free energy density of uniformly distributed nucleons is evaluated with Skyrme-type interactions [34,35], using finite temperature expressions for the kinetic energies. The free energies of heavy clusters are given by expressions for Boltzmann gases, including excluded-volume effects and with mass free energies that are based on the compressible liquid-drop model. In previous works, Maxwell–Boltzmann statistics was employed to evaluate the free energies of both light and heavy clusters. In the present work, Bose–Einstein and Fermi–Dirac statistics are applied to light clusters with odd and even mass numbers, respectively.The total free energy density of the system is represented as(1)f=fnp+∑jflj+∑ini(Fit+Mi), where fnp is the free energy densities of dripped nucleons and flj represents the free energies of light clusters j with atomic numbers Zj≤5. The index i in the third term runs over all nuclear species of heavy clusters with 6≤Zi≤1000. In Eq. (1), ni, Fit, and Mi are the number densities, translational free energies, and mass free energies of nuclei i. The calculations of fnp, ni, Fit, and Mi are the same as in ref. [26], where one can also find all the details.For light clusters with atomic numbers Zj≤5 or neutron numbers Nj≤5, we include only those nuclides with available experimental mass data. The light clusters with even/odd mass numbers are assumed to obey Bose/Fermi statistics. The main thermodynamic quantities—the free energy density, particle density, and pressure of species j at temperature T—can then be expressed asaaBelow the units ħ=c=1 are used.(2)flj=njμj−pj,(3)nj=gjλj−32F1/2(ηj)π,(4)pj=gjλj−5F3/2(ηj)6πMj. Here μj is the chemical potential, gj is the ground state degeneracy factor, ηj=(μj−Mj)/T, λj=2π/(MjT) is the thermal wavelength, and(5)Fk(η)=∫0∞zk[exp(z−η)±1]−1dz are the Fermi (+) and Bose (−) integrals. In the low-density and high-temperature limit (nj≪λj−3), the free energy is equal to that of a Boltzmann gas, as was assumed in our previous works [25,26]. The masses of the light clusters are assumed to be the experimental masses, Mjex, with Coulomb energy shifts, ΔEjC; see details in refs. [25,26]:(6)Mj=Mjex+ΔEjC,(7)ΔEjC(np′,ne)=35(34π)−1/3e2n02Vj5/3×[(Zj−np′VjAj)2(1−32uj1/3+12uj)−(ZjAj)2], where Aj=Zj+Nj, Vj=Aj/n0, uj=(ne−np′)/(Zj/Vj−np′), ne is the electron density, and np′ is the local proton density in the vapor; see details in refs. [16,25]. The same formula for the Coulomb energy is also applied for heavy clusters in stellar environments. Excluded-volume effects are not implemented for light clusters in this work.Bose particles form a condensate under the condition, nj>njC, where the critical density, njC is given by Eq. (3) after substituting ηj=0 (or Mj=μj). The critical temperature below which Bose particles form a condensate can be expressed as(8)TjC=2πMj(njgjζ(3/2))2/3, where ζ(x) is the zeta function, and ζ(3/2)=2.612. Fig. 1 displays the critical lines in the (Ajnj, T) plane for deuterons (d) and α-particles (α). As one can see, at high densities and low temperatures, α-particle condensation can occur. Here we ignore the existence of electrons and dripped protons, which affect the nuclear Coulomb energies. The critical baryon number densities, AjnjC, are lower for deuterons than for α-particles at the same temperature, because the deuterons have a smaller value of Mj. However, the condensation of deuterons is less likely to occur, because their number density and binding energy are considerably smaller than those of α-particles, as shown later and as discussed in ref. [27].3Warm clustered matter with a realistic nuclear ensembleThe number densities of dripped protons and neutrons, deuterons, tritons (t), helions (h), α-particles, and other light and heavy clusters are shown in Fig. 2. We find that Bose-Einstein condensation of light bosonic clusters is unlikely because free nucleons and/or heavy clusters are the dominant components of nuclear matter under the considered conditions. As also shown in Fig. 1, the maximum values of the baryon densities for deuterons and α-particles over the whole density range are lower than the critical baryon densities, AjnjC, at which Bose–Einstein condensation may occur. In other words, almost all the nucleons are consumed, making the densities of light bosonic clusters insufficient to create a condensate. This result means that the inclusion of heavy clusters is crucial for a realistic description of stellar matter at sub-nuclear densities. Figs. 3 and 4 show the relative change (njQ−njC)/njC in the densities of light clusters calculated with Bose or Fermi statistics, njQ, as compared to those obtained with Boltzmann statistics, njC. In multi-component nuclear matter, we find little difference in the number densities of light clusters between the quantum and classical statistics; the deviations are less than ≈0.20%. In neutron-rich matter, Yp=0.2, and at high densities, nB≳10−3 fm−3, the chemical potentials of the neutrons are substantially larger than those of the protons, and as a result, the quantum-statistical effects for tritons are stronger than those for α-particles.In some previous works, the contributions of heavy clusters have been neglected or were represented by a single-nucleus such as Fe56[29,30] or by an optimized nucleus [23]. The single-nucleus approximation considerably underestimates the total mass fraction of heavy clusters and overestimates the average mass number of heavy clusters, even if the mass and proton numbers of the representative nucleus are optimized, as shown in ref. [25]. In the single-nucleus approximation, the free energy density is expressed as f=fnp+∑jflj+nrepFrep(Arep,Zrep) instead of Eq. (1), where nrep and Frep are, respectively, the number density and free energy of a representative nucleus. Its mass and proton numbers, Arep and Zrep, are optimized by the conditions ∂Frep/∂Arep|Zrep=μn and ∂Frep/∂Zrep|Arep=μp−μn. In addition, we assume chemical equilibrium among nucleons, light clusters, and the representative nuclei subject to the conservation of baryon number and charge (see details in ref. [25]). Here μp and μn are the chemical potentials of protons and neutrons, respectively.Fig. 5 shows the mass fractions of α-particles for a realistic nuclear ensemble, the single-nucleus approximation, and a model of light-cluster matter that is discussed in more detail in the next section, with no clusters having Z>2 or N>2. As one can see from the figure, the mass fraction of α-particles in the more realistic ensemble is considerably smaller than in the other more restrictive models, due to the larger population of heavy clusters. Indeed, the mass fractions of heavy clusters other than the representative nucleus in single-nucleus model, and the mass fractions of nuclei other than d, t, h, and α in light-cluster matter, are strictly zero. As a result, the mass fractions of α-particles in these two models are larger than in the full-ensemble model.4Warm light-cluster matter with a restricted nuclear ensembleIn this section, we consider clustered nuclear matter where heavy and some light elements with Z>2 or N>2 are artificially suppressed, e.g., only d, t, h, and α are allowed in the chemical equilibrium. Then we can discuss the possibility of alpha condensation under the same assumptions as in ref. [33]. Fig. 6 shows the number densities of nucleons and light clusters in the model where heavy clusters are suppressed. As one can see, in this more constrained ensemble, the α-particles may condense at nB≈0.02 fm−3 (T=1 MeV) and 0.09 fm−3 (T=3 MeV) in iso-symmetric matter (Yp=0.5). The free energy model for the light clusters in this work, however, is simple and lacks some in-medium effects such as interactions among nucleons and light and heavy clusters as will be noted in next section. Hence, at these high densities, α-condensation may be suppressed, as discussed in ref. [27].For neutron-rich matter (Yp=0.2), light clusters are less abundant, and α-particles do not condense even if the existence of heavy clusters is ignored. This is because the proton chemical potentials are much lower than those for Yp=0.5, especially at high densities, as also shown in Fig. 3. Fig. 7 displays the chemical potentials of nucleons for Yp=0.2 and 0.5 and the critical lines at which the chemical potentials of tritons and α-particles are equal to their rest masses (without Coulomb-energy shifts): μp+2μn=mt and 2μp+2μn=mα. Around and above these lines, the differences in number densities between classical and quantum statistics become large for each light cluster. For Yp=0.5, α-particles start to condense around this line, while for Yp=0.2, the quantum statistical effects for tritons become strong, μp+2μn≈mt at a lower density than the critical densities for α-condensation.To investigate the impact of the Coulomb energy, we also present the results obtained without the ΔEjC shift. Fig. 8 gives the chemical potential of α-particle relative to the α-particle rest mass, μα′=2(μn+μp−mn−mp) together with the negative value of its binding energy, −Bα=Mα−2(mn+mp). Obviously, condensation starts when the chemical potential becomes equal to the mass. We find that the Coulomb-energy shifts actually reduce both the chemical potential and the effective mass of the α-particles. Therefore, the critical densities at which α-particles may condense depend very weakly on the electron distribution.As shown in ref. [26], the ground state minimum in the bulk free energy disappears at temperatures around 14 MeV. At this point, heavy clusters disappear from the nuclear ensemble but the abundance of α-particles increases significantly. However, these temperatures are already too high for light-cluster condensation.5Concluding remarksWe have investigated the nuclear compositions of hot and dense stellar matter within a self-consistent model that includes a full nuclear ensemble. In contrast to previous works, here quantum statistics is used for bosonic clusters, while classical statistics is employed for heavier nuclei. We have demonstrated that the differences in the number densities of light clusters are small between quantum and classical statistics, at most 0.20%. The presence of heavy clusters leads to a reduction in the number densities of light clusters, and as a result, the condensation of bosonic clusters does not occur. On the other hand, if we assume that iso-symmetric nuclear matter is composed only of nucleons and light clusters that are treated as ideal Fermi or Bose particles, α-particles reach the threshold of Bose–Einstein condensation at rather high densities, nB≳ 0.01 fm−3 for T=1–3 MeV. It is difficult to expect such light clusters to exist in such dense stellar matter, where heavy clusters have already formed. For warm neutron-rich matter (T=1–3 MeV and Yp=0.2), the chemical potentials of the α-particles are too small for α-condensation, even when heavy clusters are suppressed.Our approach takes into account some in-medium modifications of the individual clusters: they include Coulomb-energy shifts for light clusters as well as excluded-volume corrections and shifts in the bulk, surface, and Coulomb energies for heavy clusters. In ref. [27], four-nucleon correlations were calculated more rigorously by solving the Schrödinger equation for four nucleons, and the possibility of α-condensation was predicted at sub-nuclear densities and T≲ 6 MeV. In the present work, strong-interaction effects are to some extent also taken into account by the Skyrme-like terms in the energy-density functionals used for dripped nucleons and for the bulk energies of heavy clusters. In a more realistic approach, one should introduce similar mean fields also for light clusters, as was done in refs. [31,32]. In addition, paring corrections [27] are not incorporated in our approach. However, they are considered to be washed out at T≳2–3 MeV [36], and therefore they may be relevant only to the late phase of core-collapse supernovae and the subsequent cooling of the neutron star.In the statistical models, the calculation of light-cluster energies e.g., the introduction of self-energy shifts [37], affects their number densities above nB≈10−5 fm−3 and at temperatures below T=5 MeV for Yp=0.3 and 0.5 [18]. Furthermore, calculations of dripped nucleons and heavy nuclei (e.g., nuclear interactions among dripped nucleons and nuclear excitations) also alter them due to baryon number conservation below T=3 MeV at any density [21]. In that sense, the number densities of light clusters obtained in Sec. 3 may change somewhat, say by the order of 10−3 fm−3, but the main conclusion of this paper—that light-cluster condensations are disturbed by heavy-cluster formation—would not be modified. As shown in ref. [37], the densities of light clusters in symmetric nuclear matter depend significantly on the interactions between nucleons and light clusters at nB≳10−3 fm−3. Hence, the results of Sec. 4, in which nuclear interactions among nucleons and light clusters are neglected, may be too crude, and the possibility of α-condensation in light-cluster matter should be considered within more realistic models.Reference [38] demonstrated for iso-symmetric nuclear matter that for strong enough α-N attractive interactions, an α-condensate appears even in the nuclear ground state. In relation to the present work, this means that an α-condensate may be present inside of heavy clusters. Another interesting possibility studied in ref. [32] is that pure nucleonic matter and α-matter are separated by a potential barrier. One may then expect the existence of nuclei made of α-particles. Such metastable α-nuclei also may be formed in dense stellar matter. In the future, we are planning to investigate these interesting possibilities for isospin-asymmetric nuclear matter.CRediT authorship contribution statementShun Furusawa: Data curation, Investigation, Software, Writing - original draft. Igor Mishustin: Conceptualization, Methodology, Supervision, Writing - review & editing.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsS.F. acknowledges H. Tajima and T. Hatsuda for useful discussion. This work was supported by JSPS KAKENHI (Grant Number JP17H06365, 19K14723) and HPCI Strategic Program of Japanese MEXT (Project ID: hp170304, 180111). A part of the numerical calculations was carried out on PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan. I. M. acknowledges fruitful discussions with L. M. 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C81 (1) (2010) 015803. doi:10.1103/PhysRevC.81.015803.[38]L.M.SatarovM.I.GorensteinI.N.MishustinH.StoeckerPossible Bose-Einstein condensate of alpha particles in the ground state of nuclear matter?Phys. Rev. C1012202002491310.1103/PhysRevC.101.024913L. M. Satarov, M. I. Gorenstein, I. N. Mishustin, H. Stoecker, Possible Bose-Einstein condensate of alpha particles in the ground state of nuclear matter? Phys. Rev. C101 (2) (2020) 024913. 10.1103/PhysRevC.101.024913. diff --git a/tests/data/elsevier/j.nuclphysa.2020.121991_expected.yml b/tests/data/elsevier/j.nuclphysa.2020.121991_expected.yml new file mode 100644 index 0000000..463948c --- /dev/null +++ b/tests/data/elsevier/j.nuclphysa.2020.121991_expected.yml @@ -0,0 +1,1180 @@ +abstract: We have investigated the composition of nuclear matter under thermodynamic conditions relevant for the core-collapse of massive stars, with a focus on quantum-statistical effects for light clusters. We find that deviations in the number densities of light clusters between Boltzmann and quantum statistics are small at sub-nuclear densities and temperatures 1–3 MeV, ≈0.2% at most. The formation of heavy clusters in stellar matter leads to a reduction in the number densities of light clusters. As a result, quantum-statistical effects such as the Bose–Einstein condensation of deuterons and α-particles are suppressed. The condensation of α-particles in iso-symmetric nuclear matter is predicted under the assumptions that it is composed solely of nucleons and light clusters and that no heavy clusters are present. We also find that Coulomb screening hardly affects the critical baryon densities for alpha condensation, although it modifies the masses and chemical potentials of the α-particles. +copyright_holder: Elsevier B.V. +copyright_statement: © 2020 Elsevier B.V. All rights reserved. +copyright_year: 2020 +document_type: article +license_url: '' +license_statement: '' +keywords: ['Nuclear matter', + 'Equation of state', + 'Supernovae', + 'Nuclear statistical equilibrium', + 'Quantum statistical effect', + 'Bose–Einstein condensation'] +article_type: full-length article +journal_title: Nuclear Physics A +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Furusawa, Shun + raw_affiliations: + - value: Department of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, + Japan + source: Elsevier B.V. + - value: Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), + RIKEN, Wako, Saitama 351-0198, Japan + source: Elsevier B.V. + emails: + - furusawa@rs.tus.ac.jp +- full_name: Mishustin, Igor + raw_affiliations: + - value: Frankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 + Frankfurt am Main, Germany + source: Elsevier B.V. + - value: National Research Center Kurchatov Institute, Moscow 123182, Russia + source: Elsevier B.V. +artid: '121991' +title: Degeneracy effects and Bose condensation in warm nuclear matter with light and heavy clusters +dois: +- material: publication + doi: 10.1016/j.nuclphysa.2020.121991 +journal_volume: '1002' +journal_issue: '' +is_conference_paper: false +publication_date: 2020-10 +collaborations: [] +documents: +- key: j.nuclphysa.2020.121991.xml + url: http://example.org/j.nuclphysa.2020.121991.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: S.Furusawa<maintitle>Nuclei + in central engine of core-collapse supernovae</maintitle><maintitle>Phys. + Scr.</maintitle>95202007400210.1088/1402-4896/ab8c15S. 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C + journal_volume: '101' + journal_issue: '2' + year: 2020 + artid: "024913" + dois: + - 10.1103/PhysRevC.101.024913 + label: '38' + authors: + - full_name: Satarov, L.M. + inspire_role: author + - full_name: Gorenstein, M.I. + inspire_role: author + - full_name: Mishustin, I.N. + inspire_role: author + - full_name: Stoecker, H. + inspire_role: author diff --git a/tests/data/elsevier/j.nuclphysa.2020.121992.xml b/tests/data/elsevier/j.nuclphysa.2020.121992.xml new file mode 100644 index 0000000..b4b6665 --- /dev/null +++ b/tests/data/elsevier/j.nuclphysa.2020.121992.xml @@ -0,0 +1,84 @@ + + + +application/xml +R2 +Shun FurusawaIgor Mishustin +Nuclear matterEquation of stateSupernovaeNuclear statistical equilibriumQuantum statistical effectBose–Einstein condensation +Nuclear Physics, Section A 1002 (2020). doi:10.1016/j.nuclphysa.2020.121992 +journal +Nuclear Physics, Section A +© 2020 Elsevier B.V. All rights reserved. +Elsevier B.V. +0375-9474 +1002 +October 2020 +10.1016/j.nuclphysa.2020.121992 +http://dx.doi.org/10.1016/j.nuclphysa.2020.121992 +doi:10.1016/j.nuclphysa.2020.121992 +121992 + +Journals +S250.1 + +NUPHA +121992 +121992 +S0375-9474(20)30301-8 +10.1016/j.nuclphysa.2020.121992 +Elsevier B.V. + +Nuclear Astrophysics + + + + +Fig. 1 +Critical temperatures below which Bose-Einstein condensation occurs as a function of the baryon density (mass number times number density) for deuterons (Aj = 2, blue dashed-dotted lines) and α-particles (Aj = 4, thick red solid lines). The symbols show the maximum values of the baryon densities, Ajnj, of symmetric nuclear matter (Yp = 0.5) for α-particles at T = 1 MeV (red triangle) and 3 MeV (red circle) and for deuterons at T = 1 MeV (blue diamond) and 3 MeV (blue square) in the full statistical ensemble including both light and heavy clusters over the whole density range (see Fig. 2). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) +Fig. 1Fig. 2Number densities of dripped protons and neutrons (black dashed lines), deuterons (blue dashed dotted lines), tritons and helions (green dashed double-dotted lines), α-particles (thick red solid lines), and other nuclei (Zi > 2 or Ni > 2, magenta dotted lines) as functions of nB at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp = 0.2 (left column) and 0.5 (right column). The thin red solid lines represent the critical number density above which α-particles should form a condensate.Fig. 2Fig. 3Relative changes in the number densities of deuterons (blue dashed dotted lines), tritons (green dashed double-dotted lines), helions (cyan dashed lines), and α-particles (thick red solid lines) as calculated using quantum statistics vs. those obtained with classical (Boltzmann) statistics at T = 3.0 MeV and Yp = 0.2 (left panel) and 0.5 (right panel).Fig. 3Fig. 4Relative change in the number densities of deuterons (left panel) and α-particles (right panel) as calculated using quantum statistics vs. those obtained with classical (Boltzmann) statistics at Yp = 0.5 and T = 1.0 MeV (black solid lines), 2.0 MeV (blue dashed lines), 3.0 MeV (green dashed dotted lines), 4.0 MeV (red dashed double-dotted lines), and 5.0 MeV (magenta dotted lines). +Fig. 4Fig. 5Mass fractions of α-particles (thick lines) and heavy clusters (thin lines) for a model solved for the full-ensemble of heavy clusters (black solid lines) and for a model that uses the single-nucleus approximation for heavy clusters (Zi > 5) (blue dashed line) at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp= 0.2 (left column) and 0.5 (right column). The red lines indicate the mass fraction of thermal α-particles (dashed dotted lines) and condensed α-particles (dashed double-dotted lines) for light-cluster matter in which we ignore the existence of heavy clusters other than d, t, h, and α.Fig. 5Fig. 6Number densities of dripped protons and neutrons (black dashed lines), deuterons (blue dashed dotted lines), tritons and helions (green dashed double-dotted lines), thermal α-particles (with kinetic energies ϵ > 0, thick red solid lines), and condensed α-particles (ϵ = 0, red dotted lines) as functions of nB for light-cluster matter at T = 1.0 MeV (top row) and 3.0 MeV (bottom row) and Yp = 0.2 (left column) and 0.5 (right column).Fig. 6Fig. 7Proton chemical potential as a function of neutron chemical potential for light cluster matter at T = 1.0 MeV and Yp = 0.2 (thick dashed black line) and 0.5 (thin dashed black line) together with the critical lines at which the chemical potentials of the tritons and α-particles reach their rest masses without Coulomb-energy shifts, μp + 2μn = mt (green dashed double-dotted line) and 2μp + 2μn = mα (red solid line). +Fig. 7Fig. 8Chemical potential of α-particles (with respect to the α-particle rest mass) per baryon (dashed lines) and the negative value of the α-particle binding energy per baryon (solid lines) as functions of nB for light-cluster matter at T = 1.0 MeV (left panel) and 3.0 MeV (right panel) and Yp = 0.5. The thin red lines display the results without the Coulomb energy shifts: ΔEjC=0.Fig. 8R2ShunFurusawaData curationInvestigationSoftwareWriting - original draftab⁎furusawa@rs.tus.ac.jpIgorMishustinConceptualizationMethodologySupervisionWriting - review & editingcda +Department of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, Japan +Department of Physics +Tokyo University of Science +Shinjuku, Tokyo162-8601 +Japan +Department of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, Japan + + +b +Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan +Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) +RIKEN +Wako +Saitama351-0198 +Japan +Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, Saitama 351-0198, Japan + +c +Frankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 Frankfurt am Main, Germany + +Frankfurt Institute for Advanced Studies +J.W. Goethe University +Frankfurt am Main +60438 +Germany + +Frankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 Frankfurt am Main, Germany + +d +National Research Center Kurchatov Institute, Moscow 123182, Russia + +National Research Center Kurchatov Institute +Moscow +123182 +Russia + +National Research Center Kurchatov Institute, Moscow 123182, Russia + +⁎ +Corresponding author. + +AbstractWe have investigated the composition of nuclear matter under thermodynamic conditions relevant for the core-collapse of massive stars, with a focus on quantum-statistical effects for light clusters. We find that deviations in the number densities of light clusters between Boltzmann and quantum statistics are small at sub-nuclear densities and temperatures 1–3 MeV, ≈0.2% at most. The formation of heavy clusters in stellar matter leads to a reduction in the number densities of light clusters. As a result, quantum-statistical effects such as the Bose–Einstein condensation of deuterons and α-particles are suppressed. The condensation of α-particles in iso-symmetric nuclear matter is predicted under the assumptions that it is composed solely of nucleons and light clusters and that no heavy clusters are present. We also find that Coulomb screening hardly affects the critical baryon densities for alpha condensation, although it modifies the masses and chemical potentials of the α-particles.KeywordsNuclear matterEquation of stateSupernovaeNuclear statistical equilibriumQuantum statistical effectBose–Einstein condensation1IntroductionHot and dense stellar matter appears in such astrophysical phenomena as core-collapse supernovae and neutron-star mergers. Nuclear matter at sub-nuclear densities consists of a mixture of unbound nucleons, light clusters, and heavy clusters in chemical and thermal equilibrium +[1]. The nuclear equation of state (EOS) determines the number densities of all these nuclear species as well as thermodynamic quantities as functions of temperature, baryon density, and charge fraction. The dynamical evolution and observable signatures of such compact-star phenomena are very sensitive to the EOS. In addition, weak interactions among the nuclear species play an important role, especially in core-collapse supernovae (see e.g., [2]). The roles of light clusters in core-collapse supernovae have not been completely clarified, although their potential influence on the dynamics, neutrino spectrum, and nucleosynthesis has been pointed out by many researchers [3–13].Modern EOSs for clustered nuclear matter at sub-nuclear densities are usually obtained for astrophysical simulations by using statistical models, e.g., [14–24]. Recently we have developed a self-consistent statistical approach where individual nuclear ground state properties and the full-ensemble distributions of nuclei are calculated consistently by minimizing the free energy density [25,26]. In most statistical models, the thermodynamic quantities for light clusters are calculated using Boltzmann statistics. This assumption is actually valid at high temperatures and low densities. However, it is a priori unclear whether this assumption is valid for conditions encountered in supernovae and neutron-star mergers. Certainly, at low temperatures and at high densities, they should be regarded as Fermi or Bose particles.The purpose of this work is to investigate such quantum-statistical effects for light clusters in stellar environments using realistic nuclear ensembles. The possibility of alpha condensation in nuclear matter has been analyzed previously by several authors. In ref. [27], the authors used a diagrammatic quantum-statistical approach to describe four-nucleon correlations in nuclear matter. They demonstrated the possibility of α condensation at sub-saturation densities, below a critical temperature of about 6 MeV. However, the calculations were done assuming a uniform nucleonic background; i.e., the presence of other light clusters in the system was completely ignored. Other previous studies [28–33] also did not consider the full nuclear ensemble of light and heavy clusters. In the present study, we generalize classical models of light clusters to a quantum-statistical description in the framework of our statistical approach, including the full-ensemble of nuclei. +This paper is organized as follows. In Sec. 2, we formulate the modified statistical model, with a focus on the quantum-statistical effects of light clusters. The results of calculations with the full statistical ensemble, including both light and heavy clusters in some typical astrophysical conditions, are discussed in Sec. 3. For comparison, clustered nuclear matter consisting only of nucleons and light clusters is considered in Sec 4 in order to demonstrate the crucial role played by heavy clusters. Our conclusions are presented in Sec. 5.2Statistical model of clustered nuclear matter with quantum statisticsTo describe multi-component nuclear matter containing both light and heavy clusters, we use the same models as in our previous work [26]. The free energy density of uniformly distributed nucleons is evaluated with Skyrme-type interactions [34,35], using finite temperature expressions for the kinetic energies. The free energies of heavy clusters are given by expressions for Boltzmann gases, including excluded-volume effects and with mass free energies that are based on the compressible liquid-drop model. In previous works, Maxwell–Boltzmann statistics was employed to evaluate the free energies of both light and heavy clusters. In the present work, Bose–Einstein and Fermi–Dirac statistics are applied to light clusters with odd and even mass numbers, respectively.The total free energy density of the system is represented as(1)f=fnp+∑jflj+∑ini(Fit+Mi), where fnp is the free energy densities of dripped nucleons and flj represents the free energies of light clusters j with atomic numbers Zj≤5. The index i in the third term runs over all nuclear species of heavy clusters with 6≤Zi≤1000. In Eq. (1), ni, Fit, and Mi are the number densities, translational free energies, and mass free energies of nuclei i. The calculations of fnp, ni, Fit, and Mi are the same as in ref. [26], where one can also find all the details.For light clusters with atomic numbers Zj≤5 or neutron numbers Nj≤5, we include only those nuclides with available experimental mass data. The light clusters with even/odd mass numbers are assumed to obey Bose/Fermi statistics. The main thermodynamic quantities—the free energy density, particle density, and pressure of species j at temperature T—can then be expressed asaaBelow the units ħ=c=1 are used.(2)flj=njμj−pj,(3)nj=gjλj−32F1/2(ηj)π,(4)pj=gjλj−5F3/2(ηj)6πMj. Here μj is the chemical potential, gj is the ground state degeneracy factor, ηj=(μj−Mj)/T, λj=2π/(MjT) is the thermal wavelength, and(5)Fk(η)=∫0∞zk[exp(z−η)±1]−1dz are the Fermi (+) and Bose (−) integrals. In the low-density and high-temperature limit (nj≪λj−3), the free energy is equal to that of a Boltzmann gas, as was assumed in our previous works [25,26]. The masses of the light clusters are assumed to be the experimental masses, Mjex, with Coulomb energy shifts, ΔEjC; see details in refs. [25,26]:(6)Mj=Mjex+ΔEjC,(7)ΔEjC(np′,ne)=35(34π)−1/3e2n02Vj5/3×[(Zj−np′VjAj)2(1−32uj1/3+12uj)−(ZjAj)2], where Aj=Zj+Nj, Vj=Aj/n0, uj=(ne−np′)/(Zj/Vj−np′), ne is the electron density, and np′ is the local proton density in the vapor; see details in refs. [16,25]. The same formula for the Coulomb energy is also applied for heavy clusters in stellar environments. Excluded-volume effects are not implemented for light clusters in this work.Bose particles form a condensate under the condition, nj>njC, where the critical density, njC is given by Eq. (3) after substituting ηj=0 (or Mj=μj). The critical temperature below which Bose particles form a condensate can be expressed as(8)TjC=2πMj(njgjζ(3/2))2/3, where ζ(x) is the zeta function, and ζ(3/2)=2.612. Fig. 1 displays the critical lines in the (Ajnj, T) plane for deuterons (d) and α-particles (α). As one can see, at high densities and low temperatures, α-particle condensation can occur. Here we ignore the existence of electrons and dripped protons, which affect the nuclear Coulomb energies. The critical baryon number densities, AjnjC, are lower for deuterons than for α-particles at the same temperature, because the deuterons have a smaller value of Mj. However, the condensation of deuterons is less likely to occur, because their number density and binding energy are considerably smaller than those of α-particles, as shown later and as discussed in ref. [27].3Warm clustered matter with a realistic nuclear ensembleThe number densities of dripped protons and neutrons, deuterons, tritons (t), helions (h), α-particles, and other light and heavy clusters are shown in Fig. 2. We find that Bose-Einstein condensation of light bosonic clusters is unlikely because free nucleons and/or heavy clusters are the dominant components of nuclear matter under the considered conditions. As also shown in Fig. 1, the maximum values of the baryon densities for deuterons and α-particles over the whole density range are lower than the critical baryon densities, AjnjC, at which Bose–Einstein condensation may occur. In other words, almost all the nucleons are consumed, making the densities of light bosonic clusters insufficient to create a condensate. This result means that the inclusion of heavy clusters is crucial for a realistic description of stellar matter at sub-nuclear densities. Figs. 3 and 4 show the relative change (njQ−njC)/njC in the densities of light clusters calculated with Bose or Fermi statistics, njQ, as compared to those obtained with Boltzmann statistics, njC. In multi-component nuclear matter, we find little difference in the number densities of light clusters between the quantum and classical statistics; the deviations are less than ≈0.20%. In neutron-rich matter, Yp=0.2, and at high densities, nB≳10−3 fm−3, the chemical potentials of the neutrons are substantially larger than those of the protons, and as a result, the quantum-statistical effects for tritons are stronger than those for α-particles.In some previous works, the contributions of heavy clusters have been neglected or were represented by a single-nucleus such as Fe56[29,30] or by an optimized nucleus [23]. The single-nucleus approximation considerably underestimates the total mass fraction of heavy clusters and overestimates the average mass number of heavy clusters, even if the mass and proton numbers of the representative nucleus are optimized, as shown in ref. [25]. In the single-nucleus approximation, the free energy density is expressed as f=fnp+∑jflj+nrepFrep(Arep,Zrep) instead of Eq. (1), where nrep and Frep are, respectively, the number density and free energy of a representative nucleus. Its mass and proton numbers, Arep and Zrep, are optimized by the conditions ∂Frep/∂Arep|Zrep=μn and ∂Frep/∂Zrep|Arep=μp−μn. In addition, we assume chemical equilibrium among nucleons, light clusters, and the representative nuclei subject to the conservation of baryon number and charge (see details in ref. [25]). Here μp and μn are the chemical potentials of protons and neutrons, respectively.Fig. 5 shows the mass fractions of α-particles for a realistic nuclear ensemble, the single-nucleus approximation, and a model of light-cluster matter that is discussed in more detail in the next section, with no clusters having Z>2 or N>2. As one can see from the figure, the mass fraction of α-particles in the more realistic ensemble is considerably smaller than in the other more restrictive models, due to the larger population of heavy clusters. Indeed, the mass fractions of heavy clusters other than the representative nucleus in single-nucleus model, and the mass fractions of nuclei other than d, t, h, and α in light-cluster matter, are strictly zero. As a result, the mass fractions of α-particles in these two models are larger than in the full-ensemble model.4Warm light-cluster matter with a restricted nuclear ensembleIn this section, we consider clustered nuclear matter where heavy and some light elements with Z>2 or N>2 are artificially suppressed, e.g., only d, t, h, and α are allowed in the chemical equilibrium. Then we can discuss the possibility of alpha condensation under the same assumptions as in ref. [33]. Fig. 6 shows the number densities of nucleons and light clusters in the model where heavy clusters are suppressed. As one can see, in this more constrained ensemble, the α-particles may condense at nB≈0.02 fm−3 (T=1 MeV) and 0.09 fm−3 (T=3 MeV) in iso-symmetric matter (Yp=0.5). The free energy model for the light clusters in this work, however, is simple and lacks some in-medium effects such as interactions among nucleons and light and heavy clusters as will be noted in next section. Hence, at these high densities, α-condensation may be suppressed, as discussed in ref. [27].For neutron-rich matter (Yp=0.2), light clusters are less abundant, and α-particles do not condense even if the existence of heavy clusters is ignored. This is because the proton chemical potentials are much lower than those for Yp=0.5, especially at high densities, as also shown in Fig. 3. Fig. 7 displays the chemical potentials of nucleons for Yp=0.2 and 0.5 and the critical lines at which the chemical potentials of tritons and α-particles are equal to their rest masses (without Coulomb-energy shifts): μp+2μn=mt and 2μp+2μn=mα. Around and above these lines, the differences in number densities between classical and quantum statistics become large for each light cluster. For Yp=0.5, α-particles start to condense around this line, while for Yp=0.2, the quantum statistical effects for tritons become strong, μp+2μn≈mt at a lower density than the critical densities for α-condensation.To investigate the impact of the Coulomb energy, we also present the results obtained without the ΔEjC shift. Fig. 8 gives the chemical potential of α-particle relative to the α-particle rest mass, μα′=2(μn+μp−mn−mp) together with the negative value of its binding energy, −Bα=Mα−2(mn+mp). Obviously, condensation starts when the chemical potential becomes equal to the mass. We find that the Coulomb-energy shifts actually reduce both the chemical potential and the effective mass of the α-particles. Therefore, the critical densities at which α-particles may condense depend very weakly on the electron distribution.As shown in ref. [26], the ground state minimum in the bulk free energy disappears at temperatures around 14 MeV. At this point, heavy clusters disappear from the nuclear ensemble but the abundance of α-particles increases significantly. However, these temperatures are already too high for light-cluster condensation.5Concluding remarksWe have investigated the nuclear compositions of hot and dense stellar matter within a self-consistent model that includes a full nuclear ensemble. In contrast to previous works, here quantum statistics is used for bosonic clusters, while classical statistics is employed for heavier nuclei. We have demonstrated that the differences in the number densities of light clusters are small between quantum and classical statistics, at most 0.20%. The presence of heavy clusters leads to a reduction in the number densities of light clusters, and as a result, the condensation of bosonic clusters does not occur. On the other hand, if we assume that iso-symmetric nuclear matter is composed only of nucleons and light clusters that are treated as ideal Fermi or Bose particles, α-particles reach the threshold of Bose–Einstein condensation at rather high densities, nB≳ 0.01 fm−3 for T=1–3 MeV. It is difficult to expect such light clusters to exist in such dense stellar matter, where heavy clusters have already formed. For warm neutron-rich matter (T=1–3 MeV and Yp=0.2), the chemical potentials of the α-particles are too small for α-condensation, even when heavy clusters are suppressed.Our approach takes into account some in-medium modifications of the individual clusters: they include Coulomb-energy shifts for light clusters as well as excluded-volume corrections and shifts in the bulk, surface, and Coulomb energies for heavy clusters. In ref. [27], four-nucleon correlations were calculated more rigorously by solving the Schrödinger equation for four nucleons, and the possibility of α-condensation was predicted at sub-nuclear densities and T≲ 6 MeV. In the present work, strong-interaction effects are to some extent also taken into account by the Skyrme-like terms in the energy-density functionals used for dripped nucleons and for the bulk energies of heavy clusters. In a more realistic approach, one should introduce similar mean fields also for light clusters, as was done in refs. [31,32]. In addition, paring corrections [27] are not incorporated in our approach. However, they are considered to be washed out at T≳2–3 MeV [36], and therefore they may be relevant only to the late phase of core-collapse supernovae and the subsequent cooling of the neutron star.In the statistical models, the calculation of light-cluster energies e.g., the introduction of self-energy shifts [37], affects their number densities above nB≈10−5 fm−3 and at temperatures below T=5 MeV for Yp=0.3 and 0.5 [18]. Furthermore, calculations of dripped nucleons and heavy nuclei (e.g., nuclear interactions among dripped nucleons and nuclear excitations) also alter them due to baryon number conservation below T=3 MeV at any density [21]. In that sense, the number densities of light clusters obtained in Sec. 3 may change somewhat, say by the order of 10−3 fm−3, but the main conclusion of this paper—that light-cluster condensations are disturbed by heavy-cluster formation—would not be modified. As shown in ref. [37], the densities of light clusters in symmetric nuclear matter depend significantly on the interactions between nucleons and light clusters at nB≳10−3 fm−3. Hence, the results of Sec. 4, in which nuclear interactions among nucleons and light clusters are neglected, may be too crude, and the possibility of α-condensation in light-cluster matter should be considered within more realistic models.Reference [38] demonstrated for iso-symmetric nuclear matter that for strong enough α-N attractive interactions, an α-condensate appears even in the nuclear ground state. In relation to the present work, this means that an α-condensate may be present inside of heavy clusters. Another interesting possibility studied in ref. [32] is that pure nucleonic matter and α-matter are separated by a potential barrier. One may then expect the existence of nuclei made of α-particles. Such metastable α-nuclei also may be formed in dense stellar matter. In the future, we are planning to investigate these interesting possibilities for isospin-asymmetric nuclear matter.CRediT authorship contribution statementShun Furusawa: Data curation, Investigation, Software, Writing - original draft. Igor Mishustin: Conceptualization, Methodology, Supervision, Writing - review & editing.Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsS.F. acknowledges H. Tajima and T. Hatsuda for useful discussion. This work was supported by JSPS KAKENHI (Grant Number JP17H06365, 19K14723) and HPCI Strategic Program of Japanese MEXT (Project ID: hp170304, 180111). A part of the numerical calculations was carried out on PC cluster at Center for Computational Astrophysics, National Astronomical Observatory of Japan. I. M. acknowledges fruitful discussions with L. M. 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We find that deviations in the number densities of light clusters between Boltzmann and quantum statistics are small at sub-nuclear densities and temperatures 1–3 MeV, ≈0.2% at most. The formation of heavy clusters in stellar matter leads to a reduction in the number densities of light clusters. As a result, quantum-statistical effects such as the Bose–Einstein condensation of deuterons and α-particles are suppressed. The condensation of α-particles in iso-symmetric nuclear matter is predicted under the assumptions that it is composed solely of nucleons and light clusters and that no heavy clusters are present. We also find that Coulomb screening hardly affects the critical baryon densities for alpha condensation, although it modifies the masses and chemical potentials of the α-particles. +copyright_holder: Elsevier B.V. +copyright_statement: © 2020 Elsevier B.V. All rights reserved. +copyright_year: 2020 +document_type: article +license_url: '' +license_statement: '' +keywords: ['Nuclear matter', + 'Equation of state', + 'Supernovae', + 'Nuclear statistical equilibrium', + 'Quantum statistical effect', + 'Bose–Einstein condensation'] +article_type: full-length article +journal_title: Nuclear Physics A +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Furusawa, Shun + raw_affiliations: + - value: Department of Physics, Tokyo University of Science, Shinjuku, Tokyo, 162-8601, + Japan + source: Elsevier B.V. + - value: Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), + RIKEN, Wako, Saitama 351-0198, Japan + source: Elsevier B.V. + emails: + - furusawa@rs.tus.ac.jp +- full_name: Mishustin, Igor + raw_affiliations: + - value: Frankfurt Institute for Advanced Studies, J.W. Goethe University, 60438 + Frankfurt am Main, Germany + source: Elsevier B.V. + - value: National Research Center Kurchatov Institute, Moscow 123182, Russia + source: Elsevier B.V. +artid: '121992' +title: R2 +dois: +- material: publication + doi: 10.1016/j.nuclphysa.2020.121992 +journal_volume: '1002' +journal_issue: '' +is_conference_paper: false +publication_date: 2020-10 +collaborations: [] +documents: +- key: j.nuclphysa.2020.121992.xml + url: http://example.org/j.nuclphysa.2020.121992.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: S.Furusawa<maintitle>Nuclei + in central engine of core-collapse supernovae</maintitle><maintitle>Phys. + Scr.</maintitle>95202007400210.1088/1402-4896/ab8c15S. Furusawa, Nuclei in central engine of core-collapse supernovae, + Physica Scripta 95 (2020) 074002. 10.1088/1402-4896/ab8c15. + source: Elsevier B.V. + reference: + publication_info: + journal_title: Phys. Scr. + journal_volume: '95' + year: 2020 + artid: "074002" + dois: + - 10.1088/1402-4896/ab8c15 + label: '1' + authors: + - full_name: Furusawa, S. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: S.FurusawaH.NagakuraK.SumiyoshiC.KatoS.Yamada<maintitle>Dependence + of weak interaction rates on the nuclear composition during stellar core collapse</maintitle><maintitle>Phys. + Rev. C</maintitle>95201702580910.1103/PhysRevC.95.025809S. Furusawa, H. Nagakura, K. Sumiyoshi, C. Kato, S. Yamada, Dependence + of weak interaction rates on the nuclear composition during stellar core collapse, + Phys. Rev. C 95 (2017) 025809. doi:10.1103/PhysRevC.95.025809. + source: Elsevier B.V. + reference: + publication_info: + journal_title: Phys. Rev. C + journal_volume: '95' + year: 2017 + artid: "025809" + dois: + - 10.1103/PhysRevC.95.025809 + label: '2' + authors: + - full_name: Furusawa, S. + inspire_role: author + - full_name: Nagakura, H. + inspire_role: author + - full_name: Sumiyoshi, K. + inspire_role: author + - full_name: Kato, C. + inspire_role: author + - full_name: Yamada, S. + inspire_role: author +- raw_refs: + - schema: Elsevier + value: W.C.Haxton<maintitle>Neutrino + heating in supernovae</maintitle><maintitle>Phys. + Rev. Lett.</maintitle>6019881999200210.1103/PhysRevLett.60.1999W. C. Haxton, Neutrino heating in supernovae, Physical Review + Letters 60 (1988) 1999–2002. doi:10.1103/PhysRevLett.60.1999. + source: Elsevier B.V. + reference: + publication_info: + journal_title: Phys. Rev. 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All rights reserved.Elsevier B.V.2095-927365715 April 20202020-04-15525-53052553010.1016/j.scib.2020.01.008http://dx.doi.org/10.1016/j.scib.2020.01.008doi:10.1016/j.scib.2020.01.008JournalsS300.1SCIB932S2095-9273(20)30020-710.1016/j.scib.2020.01.008Science China PressFig. 1(Color online) (a) Our best-quality VLBI image of J1430 + 4204 with natural weighting at 2.3 GHz (epoch April 9, 2001). The lowest contours are at ±3σ image noise level, the positive contour levels increase by a factor of 2. The peak brightness is 114.9 mJybeam-1. (b–f) Naturally weighted VLBI images of J1430 + 4204 at 8 GHz at the five epochs. Basic information of the maps can be found in Table B1 (online). For a consistent presentation of the images, we restored all of them with the largest beam corresponding to the 2012 epoch. The full width at half maximum (FWHM) of the restoring beam is shown at the bottom-left corner of each image. The lowest contours are at ±3σ level of individual images and the positive contour levels increase by a factor of 2. The peak intensities are 181.4, 69.7, 156.6, 182.3, and 168.3 mJybeam-1, respectively. In each image, the circles with a cross inside represent the locations and FWHM diameter of the fitted circular Gaussian model component; the cross symbols represent the positions of point models.Fig. 2(Color online) Component positions with respect to the core as a function of time and the fitted proper motions for J1 and J2. (a) The projected motions along the 2.3-GHz jet direction. (b) The motions perpendicular to the 2.3-GHz jet direction from southeast to northwest. The fitted parameter values can be found in Table B3 (online).Table 1Jet kinematic parameters of z>4 quasars derived from previous VLBI studies.NameRedshiftβaδbΓcθdRef.J0906 + 69305.472.5 ± 0.86.0 ± 0.83.6 ± 0.56.8 ± 2.2°[38]J1026 + 25425.2712.5  ±  3.9≳ 5≳ 12.54.6°[23]J1430 + 43044.722–4.28.6–124.6–6.83°e[19]J2134 − 04194.334.1  ±  2.73–82–73–20°[39]J1430 + 43044.7219.5 ± 1.922.2 ± 10.614.6 ± 3.82.2 ± 1.6°This papera: Fastest apparent transverse speed in unit of c;b: Doppler factor;c: Bulk Lorentz factor;d: Jet viewing angle with respect to our line of sight;e: This value is not estimated but fixed to an assumed value of 3° in their study [19].ArticleFast jet proper motion discovered in a blazar at z=4.72YingkangZhangabTaoAnac⁎antao@shao.ac.cnSándorFreydaShanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, ChinaShanghai Astronomical ObservatoryChinese Academy of SciencesShanghai200030ChinaShanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, ChinabUniversity of Chinese Academy of Sciences, Beijing 100049, ChinaUniversity of Chinese Academy of SciencesBeijing100049ChinaUniversity of Chinese Academy of Sciences, Beijing 100049, ChinacKey Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, ChinaKey Laboratory of Radio AstronomyChinese Academy of SciencesNanjing210008ChinaKey Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008, ChinadKonkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest, HungaryKonkoly ObservatoryResearch Centre for Astronomy and Earth SciencesKonkoly Thege Miklós út 15-17H-1121 BudapestHungaryKonkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary⁎Corresponding author.Graphical abstractJet component positions with respect to the core as a function of time and the fitted proper motions. The left and right panels show the projected motions along and perpendicular to the jet direction, respectively. AbstractHigh-resolution observations of high-redshift (z>4) radio quasars offer a unique insight into jet kinematics at early cosmological epochs, as well as constraints on cosmological model parameters. Due to the general weakness of extremely distant objects and the apparently slow structural changes caused by cosmological time dilation, only a couple of high-redshift quasars (HRQs) have been studied with parsec-scale resolutions, and with limited number of observing epochs. Here we report on very long baseline interferometry (VLBI) observations of a high-redshift blazar J1430 + 4204 (z=4.72) in the 8 GHz frequency band at five different epochs spanning 22 years. The source shows a compact core–jet structure with two jet components being identified within 3 milli-arcsecond (mas) scale. The long time span and multiple-epoch data allow for the kinematic studies of the jet components. That results in a jet proper motion of μ(J1) = 0.017 ± 0.002 mas a-1 and μ(J2) = 0.156 ± 0.015 mas a-1, respectively. For the fastest-moving outer jet component J2, the corresponding apparent transverse speed is (19.5±1.9)c. The inferred bulk jet Lorentz factor Γ=14.6±3.8 and viewing angle θ=2.2°±1.6° indicate highly relativistic beaming. The Lorentz factor and apparent proper motion are the highest measured to date among the z>4 jetted radio sources, while the jet kinematics is still consistent with the cosmological interpretation of quasar redshifts.KeywordsNucleiHigh-redshiftRadio continuumQuasarsIndividualJ1430+42041IntroductionActive galactic nuclei (AGNs) are among the most powerful and energetic objects in the Universe. It is believed that every AGN harbours a supermassive black hole (SMBH) in its central region. Because of their enormous energy production and persistent high luminosity, AGNs are excellent laboratories for studying black hole accretion and galaxy evolution across cosmic times. High-redshift quasars (HRQs) that represent powerful AGNs in the early Universe constitute a unique sample for evolutionary studies of AGNs as well as of the cosmic environment [1,2]. The most distant AGN known to date is the quasar J1342 + 0928 at z=7.54, corresponding to only 5% of the current age of the Universe [3]. The number of quasars known at z>5.6 already exceeds 100 (e.g., Refs. [4,5]). The discovery of extremely HRQs provided strong constraints on the formation history of the first generation of SMBHs [3,6].Among the HRQs, the radio-loud subsample is worth for exploring their relativistic jets and radio morphologies. Jets from radio-loud HRQs are useful probes of the intergalactic and interstellar environment in the very early stages (e.g., the epoch of reionization). Imaging of HRQ jets requires milli-arcsecond (mas) resolution or even higher. This can be achieved by using very long baseline interferometry (VLBI), which is the highest-resolution imaging technique. The facts that only ∼ 10% of the optically detected quasars are radio-loud, and that radio-loud quasars become too weak to be detected by radio telescopes at very large cosmological distances make high-redshift radio-loud quasars much more rare. The number of radio-detected AGNs with available spectroscopic redshift at z>4 is only about 170 [7], and the most distant radio-loud quasar to date is J1429 + 5447 at z=6.21[8,9].J1430 + 4204 was discovered as a radio-loud quasar at z=4.72[10]. Further X-ray and radio observations confirmed its blazar nature with broad-band variability and flat radio spectrum [11–13]. A distant (∼3.6″) jet component was detected in the northwestern direction of the nucleus from Chandra X-ray and Very Large Array (VLA) radio observations, making J1430 + 4204 the most distant quasar being detected with kiloparsec (kpc) scale radio/X-ray jet known so far [14,15]. The mas-scale radio structure of J1430 + 4204 has been studied with VLBI since 1996 [16]. With further more sensitive and higher-resolution VLBI observations, a compact core–jet structure was revealed, with a weak mas-scale jet ejecting towards the west-southwest seen at 2.3, 8.4 and 5 GHz frequencies (e.g., Refs. [15,17,18]). Veres et al. [19] conducted a two-epoch 15-GHz VLBI study of J1430 + 4204 before and after a major radio flare from 2005 to 2006. They did not find any evidence of newly ejected jet components. The jet emission in their 15-GHz VLBI images appears diffuse, hampering a direct measurement of jet proper motion.In this paper, we present a multi-epoch VLBI analysis of J1430 + 4204. Thanks to the long time span and multiple epochs, we are able to study the jet kinematics of J1430 + 4204. The five-epoch data were obtained in the 8-GHz frequency band, where the jet components are bright and compact enough for model-fitting. In Section 2, we provide the basic information about our VLBI data and describe their analysis. Section 3 presents the high-resolution images, the results of brightness distribution modeling, and the calculation of jet parameters. In Section 4, we discuss the properties of J1430 + 4204 and put them into context with other extremely distant quasars studied with VLBI.Throughout this paper, a flat ΛCDM cosmological model was adopted, with the parameters of H0=70 km s-1Mpc-1,Ωm=0.27, and ΩΛ=0.73. At the redshift of J1430 + 4204, 1 mas angular size corresponds to 6.68 pc projected linear size and 1 mas a-1 proper motion to 124.7c apparent transverse speed (c denotes the speed of light) [20].2MethodsData used for the jet kinematic study were obtained from VLBI observations in a total of five epochs from June 8, 1996 to May 1, 2018 (see the Supplementary materials). All these observations were made in geodetic/astrometric mode, in which a target source is observed during a few scans, usually for 5 min in each scan. The details of the observation logs are presented in Table B1 (online). Except for the fourth epoch observed only at 8.4 GHz, the other observations were carried out at dual 2/8 GHz frequency bands. The image noise depends on a combination of factors such as on-source integration time, number of telescopes and their diameters, data rate, (u,v) coverage, quality of calibration, etc. The final root-mean-square (rms) noise in the naturally-weighted clean images ranges from 0.25 to 1.27 mJy beam-1. The visibility data were calibrated using the US National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System (aips) software package [21]. The calibrated VLBI data were imported into the difmap software package [22] to carry out self-calibration, imaging and model fitting (see the Supplementary materials).3Results3.1Parsec-scale radio jetBecause of the limited sensitivity of geodetic-style “snapshot” VLBI experiments with short on-source integration times and poor (u,v) coverage (Table B1 online), not all epochs are sufficient to produce a reasonably detailed image of the core–jet structure. At 2.3 GHz band, we show only the best-quality image made from the epoch 2001 data (Fig. 1a) to indicate that the jet emission continues expanding out to an extent ∼20 mas (or a projected distance of ∼130 pc) from the core (the brightest component at the image center) in the southwest direction. Our 2.3-GHz VLBI image (Fig. 1a) is qualitatively consistent with that obtained in 1998 [15]. Both images reveal a compact core–jet structure within 10 mas and weaker knotty components beyond 10 mas.Fig. 1 shows the 8-GHz images at five available epochs. The 8-GHz imaging observations resolve out the diffuse emission structure at >10-mas scale and show jet components only within about 3 mas from the core (Fig. 1). The core showed a significant flux density variability from 182.3 mJy at epoch 1996 to 65.7 mJy at epoch 2001, then to 152.8 mJy at epoch 2003. The jet components are located to the west and southwest of the core, roughly in alignment with the 2.3-GHz jet structure (Fig. 1a). The VLA image at 1.4 GHz made by Cheung et al. [15] implies an extended jet/lobe at 3.6″ in the northwest. A large difference of position angles (∼50°) between the mas- and arcsec-scale jet indicates a possible large jet bending. Alternatively, the northwest lobe could be a relic from the past cycle of radio activity. To identify the connection between the VLBI and VLA jets, intermediate-resolution (∼100 mas) radio interferometric imaging would be required.3.2Jet proper motionThe typical one-sided apparent jet morphology of blazars provide a unique opportunity for their kinematic study, where the jet component proper motions, the viewing angles and the Lorentz factor of the relativistic jet can be constrained using high-resolution VLBI observations. Due to the cosmological time dilation effect, structural changes in the jet are seen (1+z) times slower in the observer’s frame than in the rest frame of the high redshift quasars [23]. Thus with an assumption of a continuous jet flow, we are able to link the jet components with each other at five different epochs over an observed period spanning ∼22 a that actually corresponds to just about 4 a in the rest frame of J1430 + 4204 at z=4.72.The 2.3 GHz observing frequency does not provide enough angular resolution to distinguish between the innermost jet components. Moreover, the outermost component seen at 2.3 GHz is too faint and diffuse, preventing us from making reliable proper motion estimates. Therefore the jet kinematic analysis is based on the 8-GHz observations. With model-fitting to our VLBI data, we were able to identify two jet components, named J1 and J2, across the different epochs. Component J1 is located well within 1 mas from the core and was detected at all 5 epochs. Component J2 shows an apparent separation of ∼2-3 mas from the core and was detected in the middle 3 epochs. The first and fifth epochs have relatively lower imaging sensitivity (see column 9 in Table B1 (online)) due to the smaller number of telescopes (in epoch 5) and short integration time (in epoch 1). These lead to a non-detection of the weaker jet component J2. The parameters of individual components derived from model-fitting are listed in Table B2 (online). The locations and sizes of the fitted model components are also indicated in Fig. 1.From the parameters in Table B2 (online), we found that both jet components are moving outwards from the core. The trajectory of J2 appears slightly curving towards the north (i.e., counter-clockwise direction in Fig. 1), with the jet position angle gradually changing from -135.2° at epoch 2001 to -104.4° at epoch 2012. The component J1 also shows a gradual change of position angle. Small changes in the direction of a relativistic jet flow beaming toward the observer could be amplified by projection effect. For example, apparent jet curvature could be attributed to a low-pitch helical trajectory projected on the sky [24,25], which is commonly seen in many blazars [26–28]. The northward rotation of the jet in our VLBI images might be related to the larger, kpc-scale radio structure of J1430 + 4204, where a distant jet component at 3.6″ from the core is seen in the VLA image [15]. An alternative possibility of a jet deflection off the dense interstellar medium cannot be ruled out [29–31]. Further polarisation-sensitive VLBI observations and long-term flux density monitoring to reveal periodicity could address the problem of the helical jet.To describe the apparent motions of J1 and J2 relative to the core, we estimated their linear proper motions along the 2.3-jet direction and perpendicular directions, respectively, using least-squares fitting. The component position angles from the best-resolved 2.3-GHz image (Fig. 1a) characterize a relativistic jet moving to southwest, where the mean position angle is -124.8°. This value is adopted to define the pc-scale jet direction here. Fig. 2 shows the fitted jet proper motions with respect to the core along and perpendicular to this direction. The values are μ|| = 0.100  ±  0.017 mas a-1 and μ⊥ = 0.120 ± 0.014 mas a-1 for J2, μ|| = 0.009 ± 0.002 mas a-1 and μ⊥ = 0.015  ±  0.002 mas a-1 for J1, respectively. Veres et al. [19] studied this source based on two-epoch 15-GHz VLBA observations. They estimated the jet parameters but did not detect a moving component due to their short observation period. Another reason is that the jet emission at 15 GHz (corresponding to 86 GHz in the source rest frame) is optically thin and becomes intrinsically weak.Veres et al. [19] estimated the Doppler factor using three independent approaches: from radio flux density variability, from radio and X-ray data and the inverse Compton process, and from the core brightness temperature measured with VLBI at 15 GHz. A moderate bulk Lorentz factor of 4.6–6.8 was derived from their VLBI study. We also applied the core brightness temperature to estimate the jet beaming parameters, while obtained a higher Lorentz factor Γ=14.6±3.8 and a viewing angle of 2.2°±1.6° (See the Supplementary materials). Our calculated jet parameters are in agreement with the blazar nature of J1430 + 4204 whose relativistic jet is inclined within a very small angle to the line of sight. The 15 GHz (correponding to a source-rest-frame frequency of about 86 GHz) data of Veres et al. [19] probe an inner jet section, while our 8 GHz data trace a relatively outer region. The difference in Lorentz factors derived from Ref. [19] and ours may imply an increase of the jet flow Lorentz factor from a distance ∼100 pc (deprojected distance of J1) to ∼500 pc (deprojected distance of J2).4Discussion and conclusionAccording to the unified scheme of radio-loud (jetted) AGNs, the appearance and luminosity of the objects depend on their jet orientation with respect to the viewing direction [32]. Blazars, whose relativistic jets point nearly towards us, are intriguing sources to probe the early Universe. Their jet emission is strongly enhanced due to the Doppler beaming effect which makes them more easily detectable compared to the coetaneous quasars with unbeamed jets, especially at high redshifts (e.g., Ref. [33]). Moreover, the powerful aligned jets are not affected by the obscuring torus that surrounds the central black hole and have more chances to break through the surrounding material [34], making the high-z sources prominent in multi-band studies [35,36].Apparent proper motion estimates of AGN jets provide a unique way of understanding their relativistic motions. This requires multi-epoch VLBI observations at a given frequency. The cosmological time dilation effect makes it more difficult to see apparent changes in high-redshift radio jets than in low-redshift ones: any change is observed (1+z) times slower compared to the rest frame of the quasar. Therefore determination of jet proper motion in HRQs has to possess long enough gaps from epoch to epoch. Among the known radio quasars at z>4.5, J1430 + 4204 belongs to the most luminous ones [37], owing to its Doppler-boosted jet emission. Until now, direct estimates of AGN jet proper motion at z>4 are rare, available only for 3 objects: J0906 + 6930 (z=5.47) at 15 GHz [38], J1026 + 2542 (z=5.26) at 5 GHz [23], and J2134 − 0419 (z=4.33) at 5 GHz [39]. Their derived jet parameters are collected in Table 1, along with our new results obtained for J1430 + 4204. Compared with previous proper motion estimates for other HRQs, our results are derived from more epochs over a longer time range, making the structural changes more evident and the estimates more reliable. Our estimated vector proper motion of ∼0.156 mas a-1 (∼19.5c) for the component J2 stands as the highest among all the measured values for other z>4 quasars. In addition, a γ-ray flare was also reported to originate from this source [40], indicating that J1430 + 4204 is an exceptionally powerful blazar in the early Universe and deserves comprehensive multi-band studies.HRQ jets open a window to explore the earliest activity of the accreting SMBHs [15]. In addition, the jet parameters (e.g., bulk Lorentz factor, viewing angle) yield useful constraints on the X-ray jet emission mechanism of HRQs [41]. The resulting μ and Γ of J1430 + 4204 are at the higher end compared to the known high-z jets, but sit in the middle of the large low-redshift blazar samples [42–44]. The reason why (the most powerful) HRQ jets do not have extremely high Lorentz factors (as some of the low-redshift blazars) is still an open question and needs more investigations on larger samples.As the kinematic analysis of VLBI jets in the large sample of the 15-GHz MOJAVE (Monitoring of Jets in AGNs with VLBA Experiments) survey indicates, accelerations as well as non-ballistic component motions are common in powerful AGN jets [45,46]. An increase in the Lorentz factor dominates the parallel accelerations which usually take place from the origin until a distance of 102 pc away from the core [45], as may also account for the observed increasing jet speed from J1 to J2 in J1430 + 4204. Considering that the jet knot may trace a curved path (see Section 3.1), an alternative possibility of the apparent jet acceleration might be due to the jet bending; for a highly beamed jet pointing toward us, any small change in the jet inclination angle may cause a large change of the apparent jet speed, even if the intrinsic jet bulk speed remains constant.Our current study helps accumulating kinematic data on high-redshift radio jets. Eventually, with a sufficiently large sample, the possible evolution of jetted radio sources on cosmological time scales, as well as cosmological model parameters could be constrained with such data [42,47]. The number of blazars at high redshifts can also help constrain the space density of radio-loud high-z AGNs. That yields the information on the space distribution of radio sources and cosmological evolution of the number density. If one assumes that AGN jets point in any direction with equal probability, and define blazars with a viewing angle θ<1/Γ, where Γ is the bulk Lorentz factor of the emitting jet plasma, the total number of radio-loud AGNs could be obtained as Ntotal = Nblazars×2Γ2[48,49]. Taking the highest Lorentz factor of J1430 + 4204 into account, that implies a total number of ∼450 radio-loud AGNs at the redshift of 4.72. This number is much higher than that currently detected. High-sensitivity all-sky surveys using the next generation radio telescopes, such as the Square Kilometre Array (SKA) and the next-generation VLA (ngVLA), may boost the detection of radio-emitting HRQs [50,51], allowing for more sophisticated study of the cosmological evolution of radio sources.Conflict of interestThe authors declare that they have no conflict of interest.AcknowledgmentsThis work was supported by the National Key R&D Programme of China (2018YFA0404603), the Chinese Academy of Sciences (114231KYSB20170003), and the Hungarian National Research, Development and Innovation Office (2018-2.1.14-TÉT-CN-2018-00001). The authors acknowledge the use of Astrogeo Center database of brightness distributions, correlated flux densities, and images of compact radio sources produced with VLBI. YZ thanks Shu Fengchun, Alexey Melnikov, Jamie McCallum, and Bo Xia for providing the Asia-Oceania VLBI (AOV) data and auxiliary telescope system temperature files.Author contributionsYingkang Zhang analysed the data and drafted the paper. Tao An initiated the project. Sandor Frey contributed to the data analysis. All contributed to the modification of the manuscript and interpretation of the data.Data availabilityAll data used in this study are public and can be accessed through the different data archives of the various instruments. NRAO VLBA archive: https://archive.nrao.edu/archive/advquery.jsp. The authors can provide data supporting this study upon request.Code availabilityUpon reasonable request the authors will provide all code supporting this study. Astronomical Image Processing System (aips) software can be found at http://www.aips.nrao.edu/index.shtml. 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He is leading the China SKA Regional Centre prototype construction. His research fields are astrophysics, radio astronomy, and very long baseline interferometry (VLBI). diff --git a/tests/data/elsevier/j.scib.2020.01.008_expected.yml b/tests/data/elsevier/j.scib.2020.01.008_expected.yml new file mode 100644 index 0000000..e051d5a --- /dev/null +++ b/tests/data/elsevier/j.scib.2020.01.008_expected.yml @@ -0,0 +1,1298 @@ +abstract: Jet component positions with respect to the core as a function of time and + the fitted proper motions. The left and right panels show the projected motions + along and perpendicular to the jet direction, respectively. High-resolution observations + of high-redshift (z>4) + radio quasars offer a unique insight into jet kinematics at early cosmological + epochs, as well as constraints on cosmological model parameters. Due to the general + weakness of extremely distant objects and the apparently slow structural changes + caused by cosmological time dilation, only a couple of high-redshift quasars (HRQs) + have been studied with parsec-scale resolutions, and with limited number of observing + epochs. Here we report on very long baseline interferometry (VLBI) observations + of a high-redshift blazar J1430 + 4204 (z=4.72) + in the 8 GHz frequency band at five different epochs spanning 22 years. The source + shows a compact core–jet structure with two jet components being identified within + 3 milli-arcsecond (mas) scale. The long time span and multiple-epoch data allow + for the kinematic studies of the jet components. That results in a jet proper + motion of μ(J1) = 0.017 ± 0.002 + mas a-1 + and μ(J2) = 0.156 ± 0.015 mas a-1, + respectively. For the fastest-moving outer jet component J2, the corresponding + apparent transverse speed is (19.5±1.9)c. The inferred bulk jet Lorentz factor + Γ=14.6±3.8 + and viewing angle θ=2.2°±1.6° + indicate highly relativistic beaming. The Lorentz factor and apparent proper motion + are the highest measured to date among the z>4 + jetted radio sources, while the jet kinematics is still consistent with the cosmological + interpretation of quasar redshifts. +copyright_holder: Science China Press +copyright_statement: © 2020 Science China Press. Published by Elsevier B.V. and Science China Press. All rights reserved. +copyright_year: 2020 +document_type: article +license_url: '' +license_statement: '' +keywords: ['Nuclei', 'High-redshift', 'Radio continuum', 'Quasars', 'Individual', 'J1430+4204'] +article_type: full-length article +journal_title: Science Bulletin +material: publication +publisher: Elsevier B.V. +year: 2020 +authors: +- full_name: Zhang, Yingkang + raw_affiliations: + - value: Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai + 200030, China + source: Elsevier B.V. + - value: University of Chinese Academy of Sciences, Beijing 100049, China + source: Elsevier B.V. +- full_name: An, Tao + raw_affiliations: + - value: Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai + 200030, China + source: Elsevier B.V. + - value: Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing + 210008, China + source: Elsevier B.V. + emails: + - antao@shao.ac.cn +- full_name: Frey, Sándor + raw_affiliations: + - value: Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, + Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary + source: Elsevier B.V. +artid: '932' +title: Fast jet proper motion discovered in a blazar at z=4.72 +dois: +- material: publication + doi: 10.1016/j.scib.2020.01.008 +journal_volume: '65' +journal_issue: '' +is_conference_paper: false +publication_date: '2020-04-15' +collaborations: [] +documents: +- key: j.scib.2020.01.008.xml + url: http://example.org/j.scib.2020.01.008.xml + source: Elsevier B.V. + fulltext: true + hidden: true +references: +- raw_refs: + - schema: Elsevier + value: K.I.Kellermann<maintitle>The + cosmological deceleration parameter estimated from the angular-size/redshift + relation for compact radio sources</maintitle><maintitle>Nature</maintitle>3611993134136Kellermann KI. 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All rights reserved.Elsevier B.V.0375-94741002October 202010.1016/j.nuclphysa.2020.121991http://dx.doi.org/10.1016/j.nuclphysa.2020.121991doi:10.1016/j.nuclphysa.2020.121991121991JournalsS250.1NUPHA121991121991S0375-9474(20)30301-810.1016/j.nuclphysa.2020.121991Elsevier B.V. diff --git a/tests/data/elsevier/sample_consyn_record.xml b/tests/data/elsevier/sample_consyn_record.xml new file mode 100644 index 0000000..09fdd6b --- /dev/null +++ b/tests/data/elsevier/sample_consyn_record.xml @@ -0,0 +1,493 @@ + + + + application/xml + Toward classification of conformal theories + Cumrun Vafa + Physics Letters B 206 (1988) 421-426. doi:10.1016/0370-2693(88)91603-6 + journal + Physics Letters B + Copyright unknown. Published by Elsevier B.V. + Elsevier B.V. + 0370-2693 + 206 + 3 + 26 May 1988 + 1988-05-26 + 421-426 + 421 + 426 + 10.1016/0370-2693(88)91603-6 + http://dx.doi.org/10.1016/0370-2693(88)91603-6 + doi:10.1016/0370-2693(88)91603-6 + + + + Journals + S350.2 + + + + PLB + 91603 + 0370-2693(88)91603-6 + 10.1016/0370-2693(88)91603-6 + + + + Toward classification of conformal theories + + + Cumrun + Vafa + + + Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA + + + + + Abstract + + By studying the representations of the mapping class groups which arise in 2D conformal theories we derive some restrictions on the value of the conformal dimension hi of operators and the central charge c of the Virasoro algebra. As a simple application we show that when there are a finite number of operators in the conformal algebra, the hi and c are all rational. + + + + Copyright 2014 Elsevier B.V. All rights reserved. + + Keywords + + Heavy quarkonia + + + Quark gluon plasma + + + Mott effect + + + X(3872) + + + + + + References + + + [1] + + + + + Belavin + A.A. + + + Polyakov + A.M. + + + Zamolodchikov + A.B. + + + + + + + + Nucl. Phys. B + + 241 + + 1984 + + + 333 + + + + + + [2] + + + + + Friedan + D. + + + Qiu + Z. + + + Shenker + S.H. + + + + + + + + Phys. Rev. Lett. + + 52 + + 1984 + + + 1575 + + + + + + [3] + + + + + Cardy + J.L. + + + + + + + + Nucl. Phys. B + + 270 + + 1986 + + + 186 + + + [FS16] + + + + + + Capelli + A. + + + Itzykson + C. + + + Zuber + J.-B. + + + + + + + + Nucl. Phys. B + + 280 + + 1987 + + + 445 + + + [FS 18] + + + + + + Capelli + A. + + + Itzykson + C. + + + Zuber + J.-B. + + + + + + + + Commun. Math. Phys. + + 113 + + 1987 + + + 1 + + + + + + + + Gepner + D. + + + + + + + + Nucl. Phys. B + + 287 + + 1987 + + + 111 + + + + + + [4] + + G. Anderson and G. 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Moore and N. Seiberg, unpublished. + + + + + + + diff --git a/tests/fixtures.py b/tests/fixtures.py new file mode 100644 index 0000000..8e341e3 --- /dev/null +++ b/tests/fixtures.py @@ -0,0 +1,38 @@ +from __future__ import ( + absolute_import, + division, + print_function, +) + +import os + + +def get_test_suite_path(*path_chunks): + """ + Args: + *path_chunks: Optional extra path element (strings) to suffix the responses directory with. + **kwargs: The test type folder name, default is the ``unit`` test suite, + e.g. ``test_suite='unit'``, ``test_suite='functional'``. + + Returns: + str: The absolute path to the test folder, if ``path_chuncks`` and ``kwargs`` + provided the absolute path to path chunks. + + Examples: + Default:: + + >>> get_test_suite_path() + '/home/myuser/inspire_utils/tests' + + Using ``path_chunks`` and ``kwargs``:: + + >>> get_test_suite_path('one', 'two', test_suite='functional') + '/home/myuser/inspire_utils/tests/functional/one/two' + """ + project_root_dir = os.path.abspath( + os.path.join( + os.path.dirname(os.path.abspath(__file__)), + '..', + ) + ) + return os.path.join(project_root_dir, 'tests', 'data', *path_chunks) diff --git a/tests/test_date.py b/tests/test_date.py index 25f6ea3..1086f02 100644 --- a/tests/test_date.py +++ b/tests/test_date.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/tests/test_dedupers.py b/tests/test_dedupers.py index d24a93a..0981619 100644 --- a/tests/test_dedupers.py +++ b/tests/test_dedupers.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/tests/test_helpers.py b/tests/test_helpers.py index f0e9478..c0f0672 100644 --- a/tests/test_helpers.py +++ b/tests/test_helpers.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/tests/test_name.py b/tests/test_name.py index 7f2956f..2044ff1 100644 --- a/tests/test_name.py +++ b/tests/test_name.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by diff --git a/tests/test_parsers_arxiv.py b/tests/test_parsers_arxiv.py new file mode 100644 index 0000000..5e17a5e --- /dev/null +++ b/tests/test_parsers_arxiv.py @@ -0,0 +1,85 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +from __future__ import ( + absolute_import, + division, + print_function, +) + +from inspire_utils.parsers.arxiv import ArxivParser + + +def test_latex_to_unicode_handles_arxiv_escape_sequences(): + expected = u"Kähler" + result = ArxivParser.latex_to_unicode(u'K\\"{a}hler') + + assert result == expected + + +def test_latex_to_unicode_handles_non_arXiv_escape_sequences(): + expected = u"\u03bd\u03bd\u0305process" + result = ArxivParser.latex_to_unicode(u"\\nu\\bar\\nu process") + + assert result == expected + + +def test_latex_to_unicode_preserves_math(): + expected = u'$H_{\\text{Schr\\"{o}dinger}}$' + result = ArxivParser.latex_to_unicode(u'$H_{\\text{Schr\\"{o}dinger}}$') + + assert result == expected + + +def test_latex_to_unicode_preserves_braces_containing_more_than_one_char(): + expected = u"On the origin of the Type~{\\sc ii} spicules - dynamic 3D MHD simulations" + result = ArxivParser.latex_to_unicode(u"On the origin of the Type~{\\sc ii} spicules - dynamic 3D MHD simulations") + + assert result == expected + + +def test_latex_to_unicode_preserves_comments(): + expected = u"A 4% measurement of $H_0$ using the cumulative distribution of strong-lensing time delays in doubly-imaged quasars" + result = ArxivParser.latex_to_unicode(u"A 4% measurement of $H_0$ using the cumulative distribution of strong-lensing time delays in doubly-imaged quasars") + + assert result == expected + + +def test_latex_to_unicode_handles_parens_after_sqrt(): + expected = u"at \u221a(s) =192-202 GeV" + result = ArxivParser.latex_to_unicode(u"at \\sqrt(s) =192-202 GeV") + + assert result == expected + + +def test_latex_to_unicode_handles_sqrt_without_parens(): + expected = u"\u221a(s)" + result = ArxivParser.latex_to_unicode(r"\sqrt s") + + assert result == expected + + +def test_latex_to_unicode_preserves_spacing_after_macros(): + expected = u"and DØ Experiments" + result = ArxivParser.latex_to_unicode(u"and D\\O Experiments") + + assert result == expected diff --git a/tests/test_parsers_crossref.py b/tests/test_parsers_crossref.py new file mode 100644 index 0000000..f6b32c5 --- /dev/null +++ b/tests/test_parsers_crossref.py @@ -0,0 +1,130 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +from __future__ import ( + absolute_import, + division, + print_function, +) + +import json +import pytest +import yaml + +from inspire_schemas.utils import validate +from inspire_utils.parsers.crossref import CrossrefParser +from fixtures import get_test_suite_path + + +def get_parsed_from_file(filename): + """A dictionary holding the parsed elements of the record.""" + path = get_test_suite_path('crossref', filename) + with open(path) as f: + aps_dict = yaml.load(f) + + return aps_dict + + +def get_parser_by_file(filename): + """A CrossrefParser instanciated on an crossref API response.""" + path = get_test_suite_path('crossref', filename) + with open(path) as f: + aps_crossref = json.load(f) + + return CrossrefParser(aps_crossref) + + +@pytest.fixture(scope='module', params=[ + ('2018.3804742.json', '2018.3804742_expected.yml'), + ('tasc.2017.2776938.json', 'tasc.2017.2776938_expected.yml'), + ('9781316535783.011.json', '9781316535783.011_expected.yml'), + ('PhysRevB.33.3547.2.json', 'PhysRevB.33.3547.2_expected.yml'), + ('s1463-4988(99)00060-3.json', 's1463-4988(99)00060-3_expected.yml'), +]) +def records(request): + return { + 'crossref': get_parser_by_file(request.param[0]), + 'expected': get_parsed_from_file(request.param[1]), + 'file_name': request.param[0], + } + + +UNREQUIRED_FIELDS = [ + 'abstract', + 'license', + 'journal_title', + 'authors', + 'artid', + 'references', + 'journal_volume', + 'journal_issue', + 'page_start', + 'page_end', + 'imprints', + 'parent_isbn', +] + +REQUIRED_FIELDS = [ + 'title', + 'dois', + 'document_type', + 'year', + 'material', +] + + +def test_data_completeness(records): + all_fields = REQUIRED_FIELDS + UNREQUIRED_FIELDS + for field in records['expected'].keys(): + assert field in all_fields + + +@pytest.mark.parametrize( + 'field_name', + REQUIRED_FIELDS +) +def test_required_fields(field_name, records): + '''Check every field in this list since all of them are required in a Crossref record''' + result = getattr(records['crossref'], field_name) + expected = records['expected'][field_name] + + assert result == expected + + +@pytest.mark.parametrize( + 'field_name', + UNREQUIRED_FIELDS +) +def test_unrequired_fields(field_name, records): + '''Check if the field was parsed correctly only if the field exists in this record''' + if field_name in records['expected']: + result = getattr(records['crossref'], field_name) + expected = records['expected'][field_name] + + assert result == expected + else: + assert True + + +def test_parse(records): + record = records['crossref'].parse() + assert validate(record, 'hep') is None diff --git a/tests/test_parsers_elsevier.py b/tests/test_parsers_elsevier.py new file mode 100644 index 0000000..0800d7e --- /dev/null +++ b/tests/test_parsers_elsevier.py @@ -0,0 +1,176 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +from __future__ import ( + absolute_import, + division, + print_function, +) + +import pytest +import yaml +import sys + +from deepdiff import DeepDiff +from inspire_schemas.utils import validate +from inspire_utils.parsers.elsevier import ElsevierParser +from fixtures import get_test_suite_path + + +def get_parsed_from_file(filename): + """A dictionary holding the parsed elements of the record.""" + path = get_test_suite_path('elsevier', filename) + with open(path) as f: + elsevier_expected_dict = yaml.load(f) + + return elsevier_expected_dict + + +def get_parser_by_file(filename): + """A ElsevierParser instanciated on an APS article.""" + path = get_test_suite_path('elsevier', filename) + with open(path) as f: + aps_elsevier = f.read() + + return ElsevierParser(aps_elsevier) + + +@pytest.fixture(scope='module', params=[ + ('j.nima.2019.162787.xml', 'j.nima.2019.162787_expected.yml'), + ('j.nuclphysa.2020.121991.xml', 'j.nuclphysa.2020.121991_expected.yml'), + ('j.nima.2019.162728.xml', 'j.nima.2019.162728_expected.yml'), + ('j.nimb.2019.04.063.xml', 'j.nimb.2019.04.063_expected.yml'), + ('j.cpc.2020.107740.xml', 'j.cpc.2020.107740_expected.yml'), + ('j.scib.2020.01.008.xml', 'j.scib.2020.01.008_expected.yml'), + ('aphy.2001.6176.xml', 'aphy.2001.6176_expected.yml'), + ('j.aim.2021.107831.xml', 'j.aim.2021.107831_expected.yml'), + ('j.nuclphysa.2020.121992.xml', 'j.nuclphysa.2020.121992_expected.yml'), +]) +def records(request): + return { + 'elsevier': get_parser_by_file(request.param[0]), + 'expected': get_parsed_from_file(request.param[1]), + 'file_name': request.param[0], + } + + +FIELDS_TO_CHECK = [ + 'abstract', + 'copyright_holder', + 'copyright_statement', + 'copyright_year', + 'document_type', + 'license_url', + 'license_statement', + 'keywords', + 'article_type', + 'journal_title', + 'material', + 'publisher', + 'year', + 'authors', + 'artid', + 'title', + 'dois', + 'references', + 'journal_volume', + 'journal_issue', + 'is_conference_paper', +] +FIELDS_TO_CHECK_SEPARATELY = [ + 'publication_date', + 'documents', + 'collaborations' +] + + +def test_data_completeness(records): + tested_fields = FIELDS_TO_CHECK + FIELDS_TO_CHECK_SEPARATELY + for field in records['expected'].keys(): + assert field in tested_fields + + +@pytest.mark.parametrize( + 'field_name', + FIELDS_TO_CHECK +) +def test_field(field_name, records): + result = getattr(records['elsevier'], field_name) + expected = records['expected'][field_name] + if field_name == 'authors': + diffs = DeepDiff(result, expected, ignore_order=True) + if sys.version_info[0] < 3 and 'type_changes' in diffs: + del diffs['type_changes'] + assert diffs == {} + else: + assert result == expected + + +def test_publication_date(records): + result = records['elsevier'].publication_date.dumps() + expected = records['expected']['publication_date'] + + assert result == expected + + +def test_collaborations(records): + result = records['elsevier'].collaborations + expected = records['expected']['collaborations'] + + assert result == expected + + +def test_parse(records): + record = records['elsevier'].parse() + assert validate(record, 'hep') is None + + +def test_attach_fulltext_document(records): + parser = records['elsevier'] + parser.attach_fulltext_document( + records['file_name'], + 'http://example.org/{}'.format(records['file_name']) + ) + result = parser.parse() + assert result['documents'] == records['expected']['documents'] + + +def test_get_identifier(records): + parser = records['elsevier'] + result_doi = parser.get_identifier() + assert result_doi == records['expected']['dois'][0]['doi'] + + +def test_record_should_be_harvested(records): + parser = records['elsevier'] + assert parser.should_record_be_harvested() + + +def test_record_shouldnt_be_harvested(): + parser = get_parser_by_file("record-that-shouldnt-be-harvested.xml") + assert not parser.should_record_be_harvested() + + +def test_imprints_date_should_be_taken_from_avaliable_online(): + parser = get_parser_by_file("j.nima.2023.168018.xml") + result = parser.parse() + assert result['imprints'] == [{'date': '2023-01-02'}] diff --git a/tests/test_parsers_jats.py b/tests/test_parsers_jats.py new file mode 100644 index 0000000..3f842d7 --- /dev/null +++ b/tests/test_parsers_jats.py @@ -0,0 +1,158 @@ +# -*- coding: utf-8 -*- +# +# This file is part of INSPIRE. +# Copyright (C) 2014-2024 CERN. +# +# INSPIRE is free software: you can redistribute it and/or modify +# it under the terms of the GNU General Public License as published by +# the Free Software Foundation, either version 3 of the License, or +# (at your option) any later version. +# +# INSPIRE is distributed in the hope that it will be useful, +# but WITHOUT ANY WARRANTY; without even the implied warranty of +# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +# GNU General Public License for more details. +# +# You should have received a copy of the GNU General Public License +# along with INSPIRE. If not, see . +# +# In applying this license, CERN does not waive the privileges and immunities +# granted to it by virtue of its status as an Intergovernmental Organization +# or submit itself to any jurisdiction. + +from __future__ import ( + absolute_import, + division, + print_function, +) + +import pytest +import yaml +import sys + +from deepdiff import DeepDiff +from inspire_schemas.utils import validate +from inspire_utils.parsers.jats import JatsParser +from fixtures import get_test_suite_path + + +def get_parsed_from_file(filename): + """A dictionary holding the parsed elements of the record.""" + path = get_test_suite_path('aps', filename) + with open(path) as f: + aps_expected_dict = yaml.load(f) + + return aps_expected_dict + + +def get_parser_by_file(filename): + """A JatsParser instanciated on an APS article.""" + path = get_test_suite_path('aps', filename) + with open(path) as f: + aps_jats = f.read() + + return JatsParser(aps_jats) + + +@pytest.fixture(scope='module', params=[ + ('PhysRevD.102.014505.xml', 'PhysRevD.102.014505_expected.yml'), + ('PhysRevX.7.021022.xml', 'PhysRevX.7.021022_expected.yml'), + ('PhysRevX.4.021018.xml', 'PhysRevX.4.021018_expected.yml'), + ('PhysRevD.96.095036.xml', 'PhysRevD.96.095036_expected.yml'), + ('PhysRevX.7.021021.xml', 'PhysRevX.7.021021_expected.yml'), +]) +def records(request): + return { + 'jats': get_parser_by_file(request.param[0]), + 'expected': get_parsed_from_file(request.param[1]), + 'file_name': request.param[0], + } + + +FIELDS_TO_CHECK = [ + 'abstract', + 'copyright_holder', + 'copyright_statement', + 'copyright_year', + 'document_type', + 'license_url', + 'license_statement', + 'article_type', + 'journal_title', + 'material', + 'publisher', + 'year', + 'authors', + 'artid', + 'title', + 'number_of_pages', + 'dois', + 'references', + 'journal_volume', + 'journal_issue', + 'is_conference_paper', +] +FIELDS_TO_CHECK_SEPARATELY = [ + 'publication_date', + 'documents', +] + + +def test_data_completeness(records): + tested_fields = FIELDS_TO_CHECK + FIELDS_TO_CHECK_SEPARATELY + for field in records['expected'].keys(): + assert field in tested_fields + + +@pytest.mark.parametrize( + 'field_name', + FIELDS_TO_CHECK +) +def test_field(field_name, records): + result = getattr(records['jats'], field_name) + expected = records['expected'][field_name] + + if field_name == 'authors': + diffs = DeepDiff(result, expected, ignore_order=True) + if sys.version_info[0] < 3 and 'type_changes' in diffs: + del diffs['type_changes'] + assert diffs == {} + else: + assert result == expected + + +def test_publication_date(records): + result = records['jats'].publication_date.dumps() + expected = records['expected']['publication_date'].isoformat() + + assert result == expected + + +@pytest.mark.skip(reason='No collaboration in input') +def test_collaborations(records): + result = records['jats'].collaborations + expected = records['expected']['collaborations'] + + assert result == expected + + +def test_parse(records): + record = records['jats'].parse() + assert validate(record, 'hep') is None + + +def test_attach_fulltext_document(records): + parser = records['jats'] + parser.attach_fulltext_document( + records['file_name'], + 'http://example.org/{}'.format(records['file_name']) + ) + result = parser.parse() + + assert result['documents'] == records['expected']['documents'] + + +def test_journal_title_physcis_is_converted_to_aps_physics(): + parser = get_parser_by_file("Physics.15.168.xml") + result = parser.parse() + assert result['publication_info'][0]['journal_title'] == "APS Physics" diff --git a/tests/test_record.py b/tests/test_record.py index 2ebeb2f..31e65c3 100644 --- a/tests/test_record.py +++ b/tests/test_record.py @@ -1,7 +1,7 @@ # -*- coding: utf-8 -*- # # This file is part of INSPIRE. -# Copyright (C) 2014-2017 CERN. +# Copyright (C) 2014-2024 CERN. # # INSPIRE is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by