Update N3LO (2512.10783)

[giacomomagni]

This PR aims to update the N3LO splitting function to include the results from: https://arxiv.org/abs/2512.10783

[giacomomagni]

Hi @felixhekhorn, this should be ready to go.
I haven't checked the impact of the changes on $P_{gq}$ at pheno level yet.

[felixhekhorn]

thanks for also taking care of Rust!

[felixhekhorn]

I haven't checked the impact of the changes on P g q at pheno level yet.

I think running the N3LO benchmark tables might be sufficient

[giacomomagni]

I tried to rerun the lh23 benchmarks (there were some minimal updates needed 4bb672b), but then I get this error:

(eko_dev) giacomomagni@Giacomos-MacBook-Pro lh_bench_23 % python run-n3lo.py FFNS central 0 0 0 0 0 0 0  --rerun --use_fhmruvv
(Re)running eko ...
Traceback (most recent call last):
  File "/Volumes/Git_Workspace/physicstools/NN3PDF/eko/extras/lh_bench_23/run-n3lo.py", line 120, in <module>
    pdf = apply.apply_pdf_flavor(eko_, toy.mkPDF("ToyLH", 0), xgrid, rot, lab)
  File "/Volumes/Git_Workspace/physicstools/NN3PDF/eko/src/ekobox/apply.py", line 113, in apply_pdf_flavor
    new_grids = rotate_result(eko, grids, flavor_labels, targetgrid, flavor_rotation)
  File "/Volumes/Git_Workspace/physicstools/NN3PDF/eko/src/ekobox/apply.py", line 158, in rotate_result
    new_grids[ep] = np.einsum(
  File "<__array_function__ internals>", line 200, in einsum
  File "/Users/giacomomagni/.conda/envs/eko_dev/lib/python3.10/site-packages/numpy/core/einsumfunc.py", line 1383, in einsum
    operands, contraction_list = einsum_path(*operands, optimize=optimize,
  File "<__array_function__ internals>", line 200, in einsum_path
  File "/Users/giacomomagni/.conda/envs/eko_dev/lib/python3.10/site-packages/numpy/core/einsumfunc.py", line 862, in einsum_path
    raise ValueError("Einstein sum subscript %s does not contain the "
ValueError: Einstein sum subscript a does not contain the correct number of indices for operand 0.

apply_pdf_flavor seems not working anymore... Do you have idea how to fix this ?

Another way to solve this is address #397 by adding the paper table to the yaml file.

[felixhekhorn]

apply_pdf_flavor seems not working anymore... Do you have idea how to fix this ?

I haven't tried the fix in 8d5f706 (since I'm too lazy to run the script, i.e. I'm worried of the runtime 🙈 ), but can you please try that? (It's most likely an unaccounted side effect of 4ff9381)

Another way to solve this is address #397 by adding the paper table to the yaml file.

agreed

[giacomomagni]

Okay this should be the 100% difference for the central value and no SV (FFNS evolution):

            x           u_v           d_v           L_m       L_p           s_v       s_p       c_p         g
0          1e-7  8.664728e-07  5.014256e-07  4.211370e-06 -3.653628  1.699765e-06 -3.705042 -3.740130 -3.809492
1          1e-6 -8.243997e-08 -4.793707e-08 -3.841519e-07 -2.185996 -6.995284e-07 -2.237829 -2.273772 -2.310250
2          1e-5  2.299838e-08 -6.135612e-08  2.135208e-06 -0.966026  9.997723e-05 -1.005775 -1.034155 -1.068547
3          1e-4 -2.681288e-07 -1.607748e-07 -2.393038e-06 -0.204168  1.073067e-05 -0.219579 -0.231237 -0.261837
4          1e-3 -4.828829e-09 -2.749653e-09 -8.200111e-08  0.055102  3.883568e-07  0.063413  0.070542  0.066460
5          1e-2 -1.065837e-08 -1.420698e-08 -5.736981e-07  0.025154 -2.459712e-06  0.033456  0.042962  0.057076
6          1e-1 -2.312421e-08 -2.626855e-08  1.589755e-07 -0.000791  8.361596e-09 -0.001354 -0.002577 -0.002879
7          3e-1 3.249139e-08  2.051314e-07  3.988460e-07  0.000192  9.622578e-09  0.000375  0.001016  0.000927
8          5e-1  4.171960e-07  4.628890e-07 -4.215597e-06 -0.000154  3.943884e-09 -0.000314 -0.001014 -0.000196
9          7e-1  3.150741e-06 -9.116919e-06 -7.634300e-06  0.000405  5.117496e-08  0.000703  0.001846 -0.001342
10         9e-1  3.011481e-04 -2.757193e-03  3.643656e-02  0.134844  4.421074e-04  0.156290  0.178091  0.044351

[felixhekhorn]

do I understand correctly this is up to a ~4% correction? this would be quite significant ... i.e. PDF change by 4%? actually this can not be, N3LO should be of that order, no?

[giacomomagni]

do I understand correctly this is up to a ~4% correction? this would be quite significant ... i.e. PDF change by 4%? actually this can not be, N3LO should be of that order, no?

So this is the current version of eko vs the paper table.
To check the impact of this specific update I need to run against master, but I'd prefer to do it after #489.
If I recall correctly #414, #380 were merged after the benchmark paper #354, and this could explain the magnitude of the change. Also this 3.5% difference is at $x=10^{-7}$, while for $x > 10^{-4}$ everything seems in good order.