DIS: TMC for FL

[felixhekhorn]

• We believe TMC for FL is wrong

• apfel/src/DIS/FLlight.f

  Lines 56 to 58 in a6bd3c1

  	FLlight = FLlight + w_int_gen(n,alpha,xi)
  	1 * ( FL(3,0,alpha)
  	2 + c1 * F2(3,0,alpha) + c2 * I2(3,0,alpha) )

• APFEL Implementation corresponds to https://arxiv.org/pdf/0808.1231.pdf sec. 3.8
• we believe Eq. 26 of http://iopscience.iop.org/0954-3899/35/5/053101 to be correct
• APFEL is not producing physical results for this - see log below

log

FLlight with theory=109 using CT14llo_NF6
           x          Q2         APFEL        yadism  yadism_error    rel_err[%]
0   0.000317   90.000000 -1.096388e-09  6.453450e-09           0.0   -688.610175
1   0.001007   90.000000 -7.971752e-09  4.682564e-08           0.0   -687.394585
2   0.003195   90.000000 -5.319054e-08  3.368383e-07           0.0   -733.267388
3   0.010138   90.000000 -3.578034e-07  2.480502e-06           0.0   -793.258370
4   0.032169   90.000000 -3.500474e-06  1.894879e-05           0.0   -641.320747
5   0.102077   90.000000 -4.737209e-05  1.355193e-04           0.0   -386.074071
6   0.304545   90.000000 -5.363571e-04  5.690540e-04           0.0   -206.096106
7   0.536364   90.000000 -6.862291e-04  3.825738e-04           0.0   -155.750147
8   0.768182   90.000000 -1.603016e-04  4.102670e-05           0.0   -125.593442
9   1.000000   90.000000 -1.977082e-07  5.276186e-09           0.0   -102.668674
10  0.010000    2.000000 -4.734504e-07  7.321240e-05           0.0 -15563.583255
11  0.010000    3.088904 -1.401287e-06  5.034189e-05           0.0  -3692.547885
12  0.010000    4.770665 -1.653054e-06  3.449513e-05           0.0  -2186.751469
13  0.010000    7.368063 -1.562449e-06  2.353910e-05           0.0  -1606.551240
14  0.010000   11.379620 -1.330316e-06  1.599546e-05           0.0  -1302.380041
15  0.010000   17.575279 -1.065192e-06  1.082591e-05           0.0  -1116.334043
16  0.010000   27.144176 -8.190154e-07  7.299937e-06           0.0   -991.306470
17  0.010000   41.922880 -6.119055e-07  4.905680e-06           0.0   -901.705498
18  0.010000   64.747880 -4.474956e-07  3.286587e-06           0.0   -834.440182
19  0.010000  100.000000 -3.219011e-07  2.195792e-06           0.0   -782.132479

corresponding theory

{'ID': 22,
 'PTO': 0,
 'FNS': 'FFNS',
 'DAMP': 0,
 'IC': 0,
 'ModEv': 'EXA',
 'XIR': 1.0,
 'XIF': 1.0,
 'NfFF': 3,
 'MaxNfAs': 6,
 'MaxNfPdf': 6,
 'Q0': 1,
 'alphas': 0.11800000000000001,
 'Qref': 91.2,
 'QED': 0,
 'alphaqed': 0.007496251999999999,
 'Qedref': 1.777,
 'SxRes': 0,
 'SxOrd': 'LL',
 'HQ': 'POLE',
 'mc': 2,
 'Qmc': 2,
 'kcThr': 1.0,
 'mb': 4,
 'Qmb': 4,
 'kbThr': 1.0,
 'mt': 173.07,
 'Qmt': 173.07,
 'ktThr': 1.0,
 'CKM': '0.97428 0.22530 0.003470 0.22520 0.97345 0.041000 0.00862 0.04030 0.999152',
 'MZ': 91.1876,
 'MW': 80.398,
 'GF': 1.1663787e-05,
 'SIN2TW': 0.23126,
 'TMC': 2,
 'MP': 0.938,
 'Comments': 'LO baseline for small-x res',
 'global_nx': 0,
 'EScaleVar': 1,
 '_modify_time': '2020-05-28 16:55:02.232461'}

[juanrojochacon]

Hi @felixhekhorn just to check, the problem is in the combination of FL with TMCs right? What happens if one runs the corresponding comparison without TMCs?

[alecandido]

We believe that the problem is the TMC expression of FL, as we already advertised in the mailing list.
The benchmark of FL itself, with all the other features, has already been performed, and apart from the known issues (for example this one and #22) they are passing.

As soon as we regenerate them (I will do it tomorrow) we can also post them here in the usual way.

[alecandido]

Sorry, I got confused: of course the FL is trivial at LO, so there is nothing to put. While when TMC are activated FL gets contributions from F2, so the TMC one is non-trivial also at LO (and it is the one that @felixhekhorn posted in the first place, and we have not detected in the first place because I put the wrong query when updating the one of non-TMC, since in that one FLlight was excluded because trivial).

If you are interested in FLlight NLO non-TMC I'm attaching it here, but in some sense it's not a fair comparison (even if you can notice that is perfectly good)

FLlight NLO non-TMC FFNS3

FLlight with theory=53 using CT14llo_NF6
           x          Q2         APFEL        yadism  yadism_error  rel_err[%]
0   0.001000   90.000000  2.498185e-01  2.499128e-01  3.836100e-14    0.037761
1   0.001520   90.000000  2.059528e-01  2.060371e-01  2.562909e-14    0.040900
2   0.002310   90.000000  1.683839e-01  1.684586e-01  8.900248e-15    0.044413
3   0.003511   90.000000  1.364486e-01  1.365152e-01  7.048782e-15    0.048852
4   0.005337   90.000000  1.095561e-01  1.096167e-01  3.501491e-15    0.055257
5   0.008111   90.000000  8.716633e-02  8.722395e-02  3.162206e-15    0.066099
6   0.012328   90.000000  6.877557e-02  6.882533e-02  1.748575e-15    0.072347
7   0.018738   90.000000  5.387528e-02  5.391338e-02  9.610212e-16    0.070713
8   0.028480   90.000000  4.196447e-02  4.199269e-02  7.263558e-16    0.067252
9   0.043288   90.000000  3.254091e-02  3.256148e-02  5.487533e-16    0.063218
10  0.065793   90.000000  2.507387e-02  2.508867e-02  4.213667e-16    0.059030
11  0.100000   90.000000  1.896229e-02  1.897291e-02  3.180218e-16    0.056018
12  0.150000   90.000000  1.373134e-02  1.373798e-02  2.253912e-16    0.048357
13  0.218182   90.000000  9.146310e-03  9.147783e-03  1.292445e-16    0.016099
14  0.286364   90.000000  6.002795e-03  6.003463e-03  8.380196e-17    0.011118
15  0.354545   90.000000  3.795886e-03  3.796400e-03  5.419578e-17    0.013520
16  0.422727   90.000000  2.279551e-03  2.279966e-03  3.376780e-17    0.018201
17  0.490909   90.000000  1.280753e-03  1.281045e-03  1.987831e-17    0.022830
18  0.559091   90.000000  6.602595e-04  6.604181e-04  1.085105e-17    0.024032
19  0.627273   90.000000  3.036095e-04  3.036681e-04  5.354479e-18    0.019324
20  0.695455   90.000000  1.190374e-04  1.190583e-04  2.306522e-18    0.017565
21  0.763636   90.000000  3.673050e-05  3.677278e-05  8.432469e-19    0.115107
22  0.831818   90.000000  7.620628e-06  7.639085e-06  2.950815e-19    0.242198
23  0.900000   90.000000  6.937581e-07  7.610920e-07  1.428958e-19    9.705684
24  0.800000   31.622777  2.627876e-05  2.627817e-05  8.006001e-19   -0.002246
25  0.800000   50.118723  2.151534e-05  2.151392e-05  6.648787e-19   -0.006622
26  0.800000   79.432823  1.782991e-05  1.782785e-05  5.585406e-19   -0.011569
27  0.800000  125.892541  1.493506e-05  1.493233e-05  4.739900e-19   -0.018292
28  0.800000  199.526231  1.262958e-05  1.262630e-05  4.058612e-19   -0.025995
29  0.800000  316.227766  1.077149e-05  1.076789e-05  3.503334e-19   -0.033485

corresponding theory

{'ID': 22,
 'PTO': 1,
 'FNS': 'FFNS',
 'DAMP': 0,
 'IC': 0,
 'ModEv': 'EXA',
 'XIR': 1.0,
 'XIF': 1.0,
 'NfFF': 3,
 'MaxNfAs': 3,
 'MaxNfPdf': 3,
 'Q0': 1.275,
 'alphas': 0.11800000000000001,
 'Qref': 91.2,
 'QED': 0,
 'alphaqed': 0.007496251999999999,
 'Qedref': 1.777,
 'SxRes': 0,
 'SxOrd': 'LL',
 'HQ': 'POLE',
 'mc': 1.275,
 'Qmc': 1.275,
 'kcThr': 1.0,
 'mb': 4.18,
 'Qmb': 4.18,
 'kbThr': 1.0,
 'mt': 173.07,
 'Qmt': 173.07,
 'ktThr': 1.0,
 'CKM': '0.97428 0.22530 0.003470 0.22520 0.97345 0.041000 0.00862 0.04030 0.999152',
 'MZ': 91.1876,
 'MW': 80.398,
 'GF': 1.1663787e-05,
 'SIN2TW': 0.23126,
 'TMC': 0,
 'MP': 0.938,
 'Comments': 'LO baseline for small-x res',
 'global_nx': 0,
 'EScaleVar': 1,
 '_modify_time': '2020-06-11 12:14:39.286465'}

[felixhekhorn]

Conclusion: it is a bug in APFEL https://github.com/NNPDF/papers/blob/master/minutes/minutes-2020-06-12.txt

[felixhekhorn]

maybe someone should fix this in APFEL as well?

[juanrojochacon]

I was thinking about that. What is the policy concerning possible bug fixes in APFEL? Do we just document them or we do actually fit them? Ideally we would like to keep APFEL as bug-less as possible to facilitate the comparison with EKO/YADISM

[vbertone]

Dear @felixhekhorn, @alecandido, @juanrojochacon, I would like to point out that the expressions in https://arxiv.org/pdf/0808.1231.pdf and http://iopscience.iop.org/0954-3899/35/5/053101 are not incompatible. More specifically, while the expressions of the latter are accurate up to next-to-next-to-leading power (NNLP) in M2/Q2, i.e. they include corrections up to O((M2/Q2)^2), those in the former are accurate up to NLP, i.e. they only include corrections of O(M2/Q2). You should easily be able to check that this is the case also for FL when using the definition FL = F2 - 2xF1 (note that this is the definition of FL used in https://arxiv.org/pdf/0808.1231.pdf that differs by NNLP corrections w.r.t. that of http://iopscience.iop.org/0954-3899/35/5/053101). Therefore, I believe that it cannot be stated that one is wrong and the other is right but rather that one is accurate to NLP and the other to NNLP. Probably the numerical discrepancy that you observe can be traced back to this difference. In conclusion, I believe that this cannot be considered a bug but just an implementation choice. I hope this helps.

[juanrojochacon]

Hi @vbertone many thanks for the tips and suggestions, these are really helpful in the context of this benchmarking exercise!!

[alecandido]

Hi @vbertone, thank you for your answer. We were thinking about the extra term and as you said we figured out that it was related to the different order of expansion.
What we were now claiming as a possible source of bug is only related to the Callan-Gross relation itself, but as you pointed out this is also related to the different expansion (i.e. different order).

However the bug itself was the negativity of the FLlight, as reported in the OP under the log section. Do you have any idea/opinion about this?

[vbertone]

Hi @alecandido, as far as I understand you're working at LO where FL without TMCs is identically zero. It follows that the predictions you get when you turn on TMCs are by definition NLP. Therefore, it's hard to give them any physical meaning (pretty much like NLO corrections could be either negative or positive). As a consequence, I think that these predictions are not bound to be positive definite. Reassuringly, as confirmed by your numbers, they are small in absolute value and tend to get smaller and smaller as Q increases, and this seems to happen consistently in both YADISM and APFEL.

Of course, this does not exclude the presence of bugs. However, I suppose that if you want to cross-check the two codes you need to set them in the same conditions which in this case probably means removing NNLP corrections from YADISM to match what APFEL does.

[alecandido]

What we were now claiming as a possible source of bug is only related to the Callan-Gross relation itself, but as you pointed out this is also related to the different expansion (i.e. different order).

Sorry @vbertone I should take back what I said before: the modified Callan-Gross relation is affecting already at the level of NLP, because the actual expression is:

and is exactly , i.e.:

as you can read in arXiv:0709.1775, equations 26 and 2 respectively

[vbertone]

Hi @alecandido, I suppose that the authors of 0808.1231 are more qualified than me to explain this but you should be able to see that r^2F2 = F2 + O((M2/Q2)^2) [hint: you need to use the expansion 1/sqrt(1+x) + sqrt(1+x) = 2 + x^2/4 +O(x^3)].

[alecandido]

I think you mean the first term of equation 75 and then equation 76, whose tau dependent prefactor multiplied by tau is:

and in your calculation.

[vbertone]

Correct! With x --> 4x^2M^2/Q^2. That prefactor is 1+O((M2/Q2)^2).

[alecandido]

Sorry to insist but I thought on it a little bit more, and it seems to me that that one is the prefactor of F2 once multiplied by but:

because considering only the first term (the correction on the second is for sure at order x^2):

that starts at order x.

Maybe I'm still making some mistakes, and of course the authors of 0808.1231 are reading (at least in principle) and they can reply too ;)

[vbertone]

Correct again! And that corresponds exactly to the second term in the r.h.s. of Eq. (85) of 0808.1231 proportional to .

[felixhekhorn]

mmm I still can not see the two things matching ... I wrote a little Mathematica snippet comparing the two equations:

expanding the Schienbein expression we get:

but expanding the APFEL expression we get

so they do not match at order ρ=M2/Q2

snippet

(* definitions *)
r[x_, \[Rho]_] := Sqrt[1 + 4 x^2 \[Rho]]
\[Xi][x_, \[Rho]_] := 2 x/(1 + r[x, \[Rho]])
\[Tau][x_, \[Rho]_] := r[x, \[Rho]]^2

(* Schienbein: Eq. 26 of arXiv:0709.1775 *)

fLs[x_, \[Rho]_] := 
 x^2/\[Xi][x, \[Rho]]^2/r[x, \[Rho]]*fL0[\[Xi][x, \[Rho]]] + 
  4 \[Rho] x^3/r[x, \[Rho]]^2 h2[\[Xi][x, \[Rho]]]

(* NNPDF/APFEL: (corrected) Eq. 85 of arXiv:0808.1231 *)

fLa[x_, \[Rho]_] := 
 fL0[\[Xi][x, \[Rho]]] + 
  x^2 (1 - \[Tau][x, \[Rho]])/\[Tau][x, \[Rho]]^(3/2)/\[Xi][
      x, \[Rho]]^2 f20[\[Xi][x, \[Rho]]] + 
  2 \[Rho] x^3 (3 - \[Tau][x, \[Rho]])/\[Tau][x, \[Rho]]^2 h2[\[Xi][
     x, \[Rho]]]

Series[fLs[x, \[Rho]], {\[Rho], 0, 1}]
Series[fLa[x, \[Rho]], {\[Rho], 0, 1}]

[vbertone]

Dear @felixhekhorn, I would just like to stress that what you are comparing is not Schienbein et al. vs. APFEL but rather Schienbein et al. vs. Forte et al.. So perhaps you can first clarify the differences that you find with the authors of the second paper (that I just used as a reference for the implementation of the TMCs in APFEL and whose expressions were used in all the previous NNPDF analyses, even when APFEL was not used). Should you ascertain any mistake in any of the expressions of the papers above I can surely help correct them also in the APFEL implementation if needed.