Fix NNLO polarized

[felixhekhorn]

thanks @igborsa!

anomalous dimensions

While comparing with the results from EKO and APFEL, I noticed that there were some differences between the EKO and polPEGASUS for the distributions related to the singlet combination (for the valence distributions, on the other hand, there was complete agreement). I traced that back to a couple of bugs in the "\Delta P_{gq}^{(2)}" and "\Delta P_{PS}^{(2)}" implementations in EKO. Specifically:

*) In lines 180-182, the harmonic sums S_2, S_3 and S_4 are being called using the value of S_1.
*) There is a similar bug in lines 263 and 264.

I have checked that correcting those calls leads to similar results as those of polPEGASUS (with relative differences that are typically between 10^{-6}-10^{-5} for most of the range in x). 

these are obvious typos and fixed in 5538c2e

matching conditions

We have now also implemented the matching coefficients from 2211.15337. I turned those factors off to produce the tables in the repo, but, in case you want, I can easily upload similar results including the OMEs. Just in case, I compared numerically the values of the OMEs against those in EKO. For A_{Hq}^{PS,(2)}", "A_{gq,H}^{S,(2)}", "A_{gg,H}^{S,(2)}" and "A^{S,(2)}(N)" I obtained the exact same results. However, there was a difference for "A_{Hg}^{S,(2)}". I think the difference is originating in the following lines:

*) Line 176: The factor should be a 6.
*) Line 191: The correct term should be (2 + 3 * n + 3 * n**2).
*) Line 210: The correct power of the factor (1+n) should be 0.
*) Lines 242-252: It seems that the Mellin transform of [Li_2(x)/(1+x)] is being evaluated in N, instead of N+1. 

@giacomomagni can you please have a look? I remember you fixed a (significant) number of typos in the original

[giacomomagni]

The first 3 comments on A_Hg are correct and are typos, due to copy and paste the expression form the paper. About the last I'm not 100% sure I need to double check.

[giacomomagni]

So regarding the last remark on the matching condition my understanding is:

• we use equation 113 and equation 131 to simplify the expression.
• in the rhs of eq 131, $S_{-2,1}$ should be called as $S_{-2,1}(N)$. This is consistent with the notation of the paper where all the harmonics are evaluated at $N$ and with this numerical check:

Ntest = 5;
beta[n_] = 1/2 (PolyGamma[0, (n+1)/2] - PolyGamma[0, n/2]);
lhs = NIntegrate[ PolyLog[2,x]/(1+x) x^(n-1) /. n-> (Ntest + 1),{x,0,1}] - Zeta[2] beta[n+1];
rhs = (-1)^(n+1)(S[-2,1, n] + 5/8 Zeta[3]);
rhs - lhs /. n -> Ntest //N

[igborsa]

@giacomomagni, you are right. That last remark was a mistake from my side. Sorry for the confusion on that one.

[giacomomagni]

No problem, thanks again for spotting all the typos.

[felixhekhorn]

just for reference - no idea why NS is not digit-by-digit the same 🤔

before

$ python diff.py 19 1 eko pol_pegasus
rel. difference eko to pol_pegasus
             u_v           d_v       L_m        L_p        s_p        c_p  b_p         g
0  -8.631063e-07  9.166424e-06 -0.000003  -0.000766  -0.000947  -0.001242  NaN -0.001341
1   1.388581e-06  2.155961e-06 -0.000004  -0.000425  -0.000524  -0.000690  NaN -0.000789
2  -8.866478e-07  4.729081e-06 -0.000005  -0.000036  -0.000043  -0.000055  NaN -0.000255
3  -2.225760e-06  7.676201e-06 -0.000009   0.000406   0.000560   0.000903  NaN  0.000286
4  -1.772993e-07  2.880613e-06 -0.000017   0.001005   0.001700   0.005948  NaN  0.000795
5   5.597354e-07 -9.802332e-07 -0.000062   0.001766   0.017271  -0.002171  NaN  0.000783
6   2.391394e-06  4.780345e-06 -0.000714  -0.008951   0.011503   0.003468  NaN -0.004039
7  -2.598061e-06 -8.837742e-06 -0.006484  -0.077518   0.154859   0.038424  NaN -0.021812
8  -2.560658e-05 -1.216003e-05 -0.030537  -0.421411   0.670087   0.183437  NaN -0.047814
9  -1.252425e-04  3.310350e-04 -0.196365  54.798657   1.888001   0.924593  NaN -0.076559
10 -9.557881e-03  1.040812e-02 -3.679376  13.549184  13.267819  12.357667  NaN -0.071871

after

$ python diff.py 19 1 eko pol_pegasus
rel. difference eko to pol_pegasus
             u_v           d_v       L_m        L_p        s_p        c_p  b_p         g
0  -8.524011e-07  9.182548e-06 -0.000003  -0.000113  -0.000140  -0.000184  NaN -0.000309
1   1.390565e-06  2.158463e-06 -0.000004  -0.000119  -0.000144  -0.000185  NaN -0.000297
2  -8.870562e-07  4.728602e-06 -0.000005  -0.000104  -0.000129  -0.000175  NaN -0.000272
3  -2.230360e-06  7.668618e-06 -0.000009  -0.000087  -0.000111  -0.000169  NaN -0.000245
4  -1.755593e-07  2.880701e-06 -0.000017  -0.000014  -0.000013  -0.000010  NaN -0.000188
5   5.595451e-07 -9.804523e-07 -0.000062   0.000184   0.001875  -0.000250  NaN -0.000097
6   2.391708e-06  4.781068e-06 -0.000714   0.000287  -0.000374  -0.000117  NaN  0.000078
7  -2.597151e-06 -8.838582e-06 -0.006484  -0.003992   0.008187   0.002059  NaN  0.000179
8  -2.561067e-05 -1.217211e-05 -0.030537  -0.059125   0.094584   0.025895  NaN  0.000201
9  -1.252990e-04  3.310898e-04 -0.196387  19.787450   0.678971   0.330185  NaN -0.000270
10 -9.562666e-03  1.043962e-02 -3.679905  11.457027  11.254180  10.511372  NaN  0.009149