diff --git a/doc/pdfs/CC_observables_def.pdf b/doc/pdfs/CC_observables_def.pdf index c282aeac..4d9cae8b 100644 Binary files a/doc/pdfs/CC_observables_def.pdf and b/doc/pdfs/CC_observables_def.pdf differ diff --git a/doc/pdfs/Complex_DGLAP.pdf b/doc/pdfs/Complex_DGLAP.pdf index 4c529874..b46fa277 100644 Binary files a/doc/pdfs/Complex_DGLAP.pdf and b/doc/pdfs/Complex_DGLAP.pdf differ diff --git a/doc/pdfs/DIS.pdf b/doc/pdfs/DIS.pdf index f32a9607..904c867d 100644 Binary files a/doc/pdfs/DIS.pdf and b/doc/pdfs/DIS.pdf differ diff --git a/doc/pdfs/Evolution_code.pdf b/doc/pdfs/Evolution_code.pdf index d3b6b323..cda13482 100644 Binary files a/doc/pdfs/Evolution_code.pdf and b/doc/pdfs/Evolution_code.pdf differ diff --git a/doc/pdfs/Lagrange_derivative.pdf b/doc/pdfs/Lagrange_derivative.pdf index d96ee6a6..0debbbbb 100644 Binary files a/doc/pdfs/Lagrange_derivative.pdf and b/doc/pdfs/Lagrange_derivative.pdf differ diff --git a/doc/pdfs/Luminosities.pdf b/doc/pdfs/Luminosities.pdf index 6c2184f1..de1818b0 100644 Binary files a/doc/pdfs/Luminosities.pdf and b/doc/pdfs/Luminosities.pdf differ diff --git a/doc/pdfs/QCD_QED_common_basis.pdf b/doc/pdfs/QCD_QED_common_basis.pdf index 3c388fde..05b615eb 100644 Binary files a/doc/pdfs/QCD_QED_common_basis.pdf and b/doc/pdfs/QCD_QED_common_basis.pdf differ diff --git a/doc/pdfs/TruncatedSolution.pdf b/doc/pdfs/TruncatedSolution.pdf index c5ad3c8e..becc6509 100644 Binary files a/doc/pdfs/TruncatedSolution.pdf and b/doc/pdfs/TruncatedSolution.pdf differ diff --git a/doc/pdfs/intrinsic_charm.pdf b/doc/pdfs/intrinsic_charm.pdf index 62a92005..ce108c89 100644 Binary files a/doc/pdfs/intrinsic_charm.pdf and b/doc/pdfs/intrinsic_charm.pdf differ diff --git a/doc/pdfs/manual.pdf b/doc/pdfs/manual.pdf index 4762263e..e335720b 100644 Binary files a/doc/pdfs/manual.pdf and b/doc/pdfs/manual.pdf differ diff --git a/doc/pdfs/running_mass.pdf b/doc/pdfs/running_mass.pdf index 3dbfc0bb..daeb2d35 100644 Binary files a/doc/pdfs/running_mass.pdf and b/doc/pdfs/running_mass.pdf differ diff --git a/doc/sources/QCD_QED_common_basis.tex b/doc/sources/QCD_QED_common_basis.tex index 20a2a0c7..9b719f76 100644 --- a/doc/sources/QCD_QED_common_basis.tex +++ b/doc/sources/QCD_QED_common_basis.tex @@ -57,7 +57,7 @@ \newpage{} -\section{The Structure of the DGLAP Equation} +\section{The structure of the DGLAP equation} The DGLAP equation that governs the PDF evolution has a general structure that in QCD holds at any perturbative order. Suppose one @@ -637,7 +637,7 @@ \section{The Structure of the DGLAP Equation} \end{equation} -\section{Evolution Basis} +\section{Evolution basis} In order to diagonlize as much as possible the evolution matrix in the presence of QED corrections avoiding unnecessery couplings between parton distributions, we propose @@ -908,7 +908,7 @@ \section{QED corrections at LO} $2e_\Sigma^2=\theta^-$ and $2\delta_e^2=\theta^+$, is consistent with eq.~(9) of the APFEL paper. -\section{Including the Lepton PDFs} +\section{Including the lepton PDFs} In order to include the lepton PDFs in the coupled QCD$\times$QED DGLAP evolution we first need to make some preliminary @@ -1408,7 +1408,7 @@ \section{Including the Lepton PDFs} \end{equation} -\subsection{Evolution Equations at LO} +\subsection{Evolution equations at LO} At LO in QED, considering that $\mathcal{P}_{q\ell}^S = 0$ and that $\mathcal{P}_{ij}=\widetilde{P}_{ij}$, the evolution equations above reduce to: @@ -1529,8 +1529,79 @@ \subsection{Evolution Equations at LO} that we need to introduce a new threshold between the charm and the bottom thresholds and this will complicate the structure of the code. -\bibliographystyle{unsrt} -\phantomsection\addcontentsline{toc}{section}{\refname}\nocite{*} -\bibliography{bibliography} +\section{QED corrections at NLO} + +In this section we discuss the details of the implementation of the +NLO QED corrections. While the inclusion of the LO corrections +presents many simplifications, $e.g.$ QED and QCD corrections do not +mix and thus the DGLAP equations as well as the $\alpha_s$ and +$\alpha$ evolution equantions are decoupled, when including NLO +corrections QED and QCD corrections mix both in the DGLAP and in the +coupling evolution equations. In addition, as far as DIS structure +functions are concerned, such corrections induce photon initiated +diagrams that have to be included in order to have the full set of +corrections. In the following we will first discuss how to generalize +the coupling evolution equations, we will then turn to the DGLAP, and +finally we will consider the DIS structure functions. + +\subsection{Evolution equations for the couplings} + +As already mentioned, NLO QED corrections induce a mixing with QCD. +At the level of the couplings, this essentially means that the QCD +$\beta$-function will get corrections proportional to $\alpha$ and +vice-versa, the QED $\beta$-function will get corrections proportional +to $\alpha_s$, that is: +\begin{equation} +\begin{array}{rcl} +\displaystyle \mu^2\frac{\partial \alpha_s}{\partial \mu^2} &=& \displaystyle + \beta^{\rm QCD}(\alpha_s,\alpha)\,.\\ +\\ +\displaystyle \mu^2\frac{\partial \alpha}{\partial \mu^2} &=& \displaystyle \beta^{\rm QED}(\alpha_s,\alpha)\,. +\end{array} +\end{equation} +As a consequence, these evolution equations form a set of coupled +differential equations. + +Before discussing the numerical solution of these equations, we first +need to know explicitly the new contributions to the +$\beta$-functions. In particular, up to NLO one has: +\begin{equation} +\beta^{\rm QCD}(\alpha_s,\alpha) = -\alpha_s\left[\beta_0^{(\alpha_s)}\left(\frac{\alpha_s}{4\pi}\right)+\beta_1^{(\alpha_s\alpha)}\left(\frac{\alpha_s}{4\pi}\right) \left(\frac{\alpha}{4\pi}\right)+\beta_1^{(\alpha_s^2)}\left(\frac{\alpha_s}{4\pi}\right)^2+\dots\right]\,, +\end{equation} +and: +\begin{equation} +\beta^{\rm QED}(\alpha_s,\alpha) = -\alpha\left[\beta_0^{(\alpha)}\left(\frac{\alpha}{4\pi}\right)+\beta_1^{(\alpha\alpha_s)}\left(\frac{\alpha}{4\pi}\right) \left(\frac{\alpha_s}{4\pi}\right)+\beta_1^{(\alpha^2)}\left(\frac{\alpha}{4\pi}\right)^2+\dots\right]\,, +\end{equation} +where the new terms $\beta_1^{(\alpha_s\alpha)}$ and +$\beta_1^{(\alpha\alpha_s)}$ can be taken from +Ref.~\cite{Surguladze:1996hx} and, taking into account the additional +factor four in the definition of the expansion parameter and writing +explicitly the color factors one finds: +\begin{equation} +\beta_1^{(\alpha_s\alpha)} = -2\sum_{i=1}^{n_f} +q_i^2\quad\mbox{and}\quad \beta_1^{(\alpha\alpha_s)} = -\frac{16}{3}N_c\sum_{i=1}^{n_f} q_i^2\,. +\end{equation} +where $n_f$ is the number of active quark flavors and $N_c=3$ is the +number of colors. We also need the term $\beta_1^{(\alpha^2)}$ which +can again be taken from the same paper: +\begin{equation} +\beta_1^{(\alpha^2)} = -4\left(n_l+N_c\sum_{i=1}^{n_f} q_i^2\right)\,. +\end{equation} +where $n_l$ is the number of active lepton flavors. + + + +\begin{thebibliography}{alp} + +%\cite{Surguladze:1996hx} +\bibitem{Surguladze:1996hx} + L.~R.~Surguladze, + %``O(alpha**n alpha-s**m) corrections in e+ e- annihilation and tau decay,'' + hep-ph/9803211. + %%CITATION = HEP-PH/9803211;%% + %3 citations counted in INSPIRE as of 29 Apr 2016 + +\end{thebibliography} + \end{document} diff --git a/src/Evolution/MatchPDFs.f b/src/Evolution/MatchPDFs.f index bd6d7431..a30b5b9e 100644 --- a/src/Evolution/MatchPDFs.f +++ b/src/Evolution/MatchPDFs.f @@ -33,10 +33,10 @@ subroutine MatchPDFs(nf,sgn,fevQCD) * double precision fevQCD(0:13,0:nint_max) * -* Get alphas value at the heavy quark threshold (with nf active flavours) +* Get alphas value at the heavy quark threshold (with nf+1 active flavours) * -c coup = asthUp(nf) - coup = asthDown(nf) + coup = asthUp(nf) +c coup = asthDown(nf) * * Singlet map * diff --git a/src/Evolution/a_QCD.f b/src/Evolution/a_QCD.f index 1be37b50..67403786 100644 --- a/src/Evolution/a_QCD.f +++ b/src/Evolution/a_QCD.f @@ -339,7 +339,7 @@ function fbeta(a,nf,ipt) elseif(ipt.eq.2)then fbeta = - a**2 * ( beta0apf(nf) 1 + a * ( beta1apf(nf) + a * beta2apf(nf) ) ) - elseif(ipt.eq.3)then + elseif(ipt.ge.3)then fbeta = - a**2 * ( beta0apf(nf) 1 + a * ( beta1apf(nf) 2 + a * ( beta2apf(nf) + a * beta3apf(nf) ) ) ) diff --git a/src/Evolution/a_QED.f b/src/Evolution/a_QED.f index 9301e3f1..312993a3 100644 --- a/src/Evolution/a_QED.f +++ b/src/Evolution/a_QED.f @@ -206,7 +206,7 @@ FUNCTION ALPHAQEDEV(NF,NL,Q2F,Q2I,ALPHAREF) BETA0 = BETA0QED(NF,NL) L = DLOG( Q2F / Q2I ) * - ALPHAQEDEV = ALPHAREF / ( 1D0 - ALPHAREF * BETA0 * L ) + ALPHAQEDEV = ALPHAREF / ( 1D0 + ALPHAREF * BETA0 * L ) * RETURN END @@ -240,7 +240,105 @@ function beta0qed(nf,nl) sumch2(5) = 11d0 / 9d0 sumch2(6) = 5d0 / 3d0 * - beta0qed = 8d0 / 3d0 * ( nc * sumch2(nf) + nl ) + beta0qed = - 8d0 / 3d0 * ( nc * sumch2(nf) + nl ) +c beta0qed = - 4d0 / 3d0 * ( nc * sumch2(nf) + nl ) +* + return + end +* +**************************************************************** + function beta1qed(nf,nl) +* + implicit none +** +* Input Variables +* + integer nf,nl +** +* Internal Variables +* + integer nc + double precision sumch4(3:6) +** +* Output Variables +* + double precision beta1qed +* +* Number of colours +* + nc = 3 +* +* Sum of the first 3, 4, 5 and 6 electric charges to the fouth +* + sumch4(3) = 18d0 / 81d0 + sumch4(4) = 34d0 / 81d0 + sumch4(5) = 35d0 / 81d0 + sumch4(6) = 51d0 / 81d0 +* + beta1qed = - 16d0 / 4d0 * ( nc * sumch4(nf) + nl ) +* + return + end +* +**************************************************************** + function beta1qcdqed(nf) +* + implicit none +** +* Input Variables +* + integer nf +** +* Internal Variables +* + double precision sumch2(3:6) +** +* Output Variables +* + double precision beta1qcdqed +* +* Sum of the first 3, 4, 5 and 6 squared electric charges +* + sumch2(3) = 2d0 / 3d0 + sumch2(4) = 10d0 / 9d0 + sumch2(5) = 11d0 / 9d0 + sumch2(6) = 5d0 / 3d0 +* + beta1qcdqed = - 2d0 * sumch2(nf) +* + return + end +* +**************************************************************** + function beta1qedqcd(nf) +* + implicit none +** +* Input Variables +* + integer nf +** +* Internal Variables +* + integer nc + double precision sumch2(3:6) +** +* Output Variables +* + double precision beta1qedqcd +* +* Number of colours +* + nc = 3 +* +* Sum of the first 3, 4, 5 and 6 squared electric charges +* + sumch2(3) = 2d0 / 3d0 + sumch2(4) = 10d0 / 9d0 + sumch2(5) = 11d0 / 9d0 + sumch2(6) = 5d0 / 3d0 +* + beta1qedqcd = - 16d0 / 3d0 * nc * sumch2(nf) * return end @@ -276,10 +374,10 @@ function a_as_exact(nf,nl,as0,a0,as,ipt) * Fourth-order runge-kutta beyond the leading order * do n=1,nstep - k1 = h * fbetaQED(a,nf,nl) / fbeta(as,nf,ipt) - k2 = h * fbetaQED(a+k1/2d0,nf,nl) / fbeta(as+h/2d0,nf,ipt) - k3 = h * fbetaQED(a+k2/2d0,nf,nl) / fbeta(as+h/2d0,nf,ipt) - k4 = h * fbetaQED(a+k3,nf,nl) / fbeta(as+h,nf,ipt) + k1 = h * fbetaQED(a,nf,nl,ipt) / fbeta(as,nf,ipt) + k2 = h * fbetaQED(a+k1/2d0,nf,nl,ipt) / fbeta(as+h/2d0,nf,ipt) + k3 = h * fbetaQED(a+k2/2d0,nf,nl,ipt) / fbeta(as+h/2d0,nf,ipt) + k4 = h * fbetaQED(a+k3,nf,nl,ipt) / fbeta(as+h,nf,ipt) * a = a + ( k1 + 2d0 * k2 + 2d0 * k3 + k4 ) / 6d0 as = as + h @@ -295,24 +393,29 @@ function a_as_exact(nf,nl,as0,a0,as,ipt) * QED beta function. * **************************************************************************** - function fbetaQED(a,nf,nl) + function fbetaQED(a,nf,nl,ipt) * implicit none ** * Input Variables * double precision a - integer nf,nl + integer nf,nl,ipt ** * Internal Variables * - double precision beta0qed + double precision beta0qed!,beta1qed ** * Output Variables * double precision fbetaQED * - fbetaQED = a**2 * beta0qed(nf,nl) + fbetaQED = - a**2 * beta0qed(nf,nl) +c if(ipt.eq.0)then +c fbetaQED = - a**2 * beta0qed(nf,nl) +c elseif(ipt.ge.1)then +c fbetaQED = - a**2 * ( beta0qed(nf,nl) + a * beta1qed(nf,nl) ) +c endif * return end diff --git a/src/Evolution/integralsQED.f b/src/Evolution/integralsQED.f index 7bb7dbbd..6a95be9d 100644 --- a/src/Evolution/integralsQED.f +++ b/src/Evolution/integralsQED.f @@ -51,7 +51,7 @@ function integralsQED(alpha,beta,coup,kk) if(PDFEvol.eq."exactmu".or.Th.eq."QUniD")then integralsQED = coup * SQ(igrid,pnf,wnl,kk,alpha,beta) else - integralsQED = SQ(igrid,pnf,wnl,kk,alpha,beta) + integralsQED = - SQ(igrid,pnf,wnl,kk,alpha,beta) 1 / beta0qed(bnf,wnl) / coup endif * diff --git a/src/commons/CheckAPFEL.h b/src/commons/CheckAPFEL.h index 854ebbda..afc0897c 100644 --- a/src/commons/CheckAPFEL.h +++ b/src/commons/CheckAPFEL.h @@ -11,295 +11,295 @@ 1 3.000E-01, 5.000E-01, 7.000E-01, 9.000E-01/ data Ref / 1 2.4423490E-01, - 1 1.1939223E-03, 7.4858485E-04, 6.8684503E+00, 1.8321320E+00, - 1 2.6560823E+01, - 1 6.0824859E-03, 3.6740014E-03, 4.3111983E+00, 9.8276287E-01, - 1 1.7089551E+01, - 1 3.1542453E-02, 1.8628394E-02, 2.6356330E+00, 4.8396049E-01, - 1 9.8707682E+00, - 1 1.6006304E-01, 9.2860680E-02, 1.5065607E+00, 2.0301457E-01, - 1 4.7524677E+00, - 1 5.9241427E-01, 3.0960038E-01, 4.6979232E-01, 3.8074420E-02, - 1 1.2535894E+00, - 1 5.5326046E-01, 2.2278383E-01, 6.6416151E-02, 4.0412765E-03, - 1 2.1485760E-01, - 1 2.6098665E-01, 7.4586190E-02, 6.2465416E-03, 3.5570529E-04, - 1 2.9829282E-02, - 1 6.2066608E-02, 1.0588029E-02, 2.0331899E-04, 1.2618583E-05, - 1 1.8974303E-03, - 1 2.0655154E-03, 1.1692735E-04, 1.7884365E-07, 1.8676692E-08, - 1 9.0467836E-06, - 1 2.0592027E+00, 6.9916472E-01, 2.0931788E-02, 2.7792992E+00, - 1 3.8760245E-01, 1.0081136E-01, 6.6853977E-04, 4.8908235E-01, - 1 1.2569250E-06, 3.4806578E-08, 0.0000000E+00, 1.2917316E-06, - 1 1.2613241E+00, 3.4853859E-01, 8.8587907E-03, 1.6187214E+00, - 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