Reproducing PDF4LHC21_40

[Francesco-Montagno]

Dear authors,
I am familiarizing myself with EKO, so I started reading this tutorial. In particular, I was able to reproduce the results for CT14llo.

Now, I would like to apply the same idea to PDF4LHC21_40 (Index: 93100). I am following the same steps (with some adjustments that I will point out), but I now obtain a 5–10% difference. I would like to understand if I am doing something wrong.

# library
from math import nan
import lhapdf
import eko
from pathlib import Path

import lhapdf
import numpy as np

from eko.interpolation import make_grid
from eko.quantities.heavy_quarks import HeavyQuarks, QuarkMassRef
from ekobox.cards import example

# get the PDF object
pdf4lhc21 = lhapdf.mkPDF("PDF4LHC21_40", 0)

# setup the operator card
op_card = example.operator()
op_card.xgrid = eko.interpolation.XGrid(make_grid(30, 30))  # x grid
op_card.mugrid = [(float(q), 5) for q in np.geomspace(5.0, 100, 5)]  # Q2 grid
op_card.init = (1.4001, 4)

Here I changed op_card.init = (1.4001, 4) with respect to the tutorial, since the .info file reports QMin: 0.140010E+01.

# setup the theory card - this can be mostly inferred from the PDF's .info file
th_card = example.theory()
th_card.orders = (3, 0)  
th_card.heavy.masses = HeavyQuarks(
    [QuarkMassRef([1.4, nan]), QuarkMassRef([4.75, nan]), QuarkMassRef([1.0E8, nan])]
)  # quark mass
th_card.couplings.alphas = 0.118000  # reference value of alpha_s
th_card.couplings.ref = (
    0.911870E+02,
    5,
)  # the reference scale together with the number of flavors at which alpha_s is provided
th_card.matching_order = (2,0)

I also changed th_card.orders = (3, 0) since the .info file says OrderQCD: 2 (if I understand correctly, there is a +1 shift with respect to the LHAPDF definition). I have also updated the masses and alpha_s accordingly.

from ekobox.evol_pdf import evolve_pdfs

path = Path("./myeko_pdf4lhc21_40.tar")
path.unlink(missing_ok=True)
evolve_pdfs(
    [pdf4lhc21], th_card, op_card, install=True, name="my_pdf4lhc21_40", store_path=path
)

import pandas as pd
import lhapdf
import numpy as np

# load evolved pdf
my_pdf4lhc = lhapdf.mkPDF("my_pdf4lhc21_40", 0)
pdf4lhc21 = lhapdf.mkPDF("PDF4LHC21_40", 0)

pid = 21  # gluon pid

# collect data
log = {"x": [], "Q2": [], "pdf4lhc": [], "my_pdf4lhc": [], "relative_diff %": []}
for q in np.geomspace(5, 100, 10):
    q2 = q**2.0
    for x in np.geomspace(1e-3, 0.9, 5):
        value = pdf4lhc21.xfxQ2(pid, x, q2)
        my_value = my_pdf4lhc.xfxQ2(pid, x, q2)
        log["x"].append(x)
        log["Q2"].append(q2)
        log["pdf4lhc"].append(value)
        log["my_pdf4lhc"].append(my_value)
        log["relative_diff %"].append((value - my_value) / value * 100)
        

print(pd.DataFrame(log))

The results are:

           x            Q2    pdf4lhc  my_pdf4lhc  relative_diff %
0   0.001000     25.000000  11.174477   10.107407         9.549175
1   0.005477     25.000000   6.886143    6.239664         9.388117
2   0.030000     25.000000   3.320114    3.152412         5.051105
3   0.164317     25.000000   0.647208    0.666043        -2.910148
4   0.900000     25.000000   0.000166    0.000256       -54.555759
5   0.001000     48.647193  13.563415   12.346124         8.974809
6   0.005477     48.647193   7.716495    7.027922         8.923389
7   0.030000     48.647193   3.403969    3.245856         4.644965
8   0.164317     48.647193   0.602479    0.625690        -3.852592
9   0.900000     48.647193   0.000108    0.000184       -69.418313
10  0.001000     94.661975  15.887784   14.555285         8.386945
11  0.005477     94.661975   8.456656    7.790664         7.875368
12  0.030000     94.661975   3.458768    3.331861         3.669148
13  0.164317     94.661975   0.562992    0.586625        -4.197796
14  0.900000     94.661975   0.000074    0.000118       -58.043478
15  0.001000    184.201575  18.114820   16.650442         8.083867
16  0.005477    184.201575   9.110798    8.450994         7.242005
17  0.030000    184.201575   3.490734    3.387222         2.965333
18  0.164317    184.201575   0.528008    0.552761        -4.687849
19  0.900000    184.201575   0.000053    0.000079       -47.366530
20  0.001000    358.435582  20.238202   18.654366         7.825971
21  0.005477    358.435582   9.689877    9.033304         6.775874
22  0.030000    358.435582   3.505765    3.419979         2.447003
23  0.164317    358.435582   0.496844    0.522832        -5.230575
24  0.900000    358.435582   0.000040    0.000059       -48.602376
25  0.001000    697.475397  22.257244   20.565692         7.600005
26  0.005477    697.475397  10.203359    9.559972         6.305634
27  0.030000    697.475397   3.507994    3.441315         1.900773
28  0.164317    697.475397   0.468947    0.495552        -5.673291
29  0.900000    697.475397   0.000030    0.000043       -42.848983
30  0.001000   1357.208808  24.173450   22.384158         7.401890
31  0.005477   1357.208808  10.659549   10.028269         5.922204
32  0.030000   1357.208808   3.500602    3.449985         1.445964
33  0.164317   1357.208808   0.443849    0.471072        -6.133444
34  0.900000   1357.208808   0.000024    0.000034       -40.349605
35  0.001000   2640.975950  25.990155   24.114522         7.216707
36  0.005477   2640.975950  11.065589   10.451091         5.553229
37  0.030000   2640.975950   3.485941    3.450920         1.004622
38  0.164317   2640.975950   0.421165    0.448708        -6.539755
39  0.900000   2640.975950   0.000019    0.000026       -36.554783
40  0.001000   5139.042664  27.711698   25.735549         7.131100
41  0.005477   5139.042664  11.427577   10.819982         5.316920
42  0.030000   5139.042664   3.465777    3.443362         0.646741
43  0.164317   5139.042664   0.400576    0.428831        -7.053818
44  0.900000   5139.042664   0.000016    0.000021       -35.027042
45  0.001000  10000.000000  29.342834   27.328070         6.866288
46  0.005477  10000.000000  11.750766   11.174280         4.905946
47  0.030000  10000.000000   3.441460    3.433356         0.235478
48  0.164317  10000.000000   0.381813    0.409632        -7.285994
49  0.900000  10000.000000   0.000013    0.000017       -30.189670

Naively, I was expecting permille-level precision. Am I doing something wrong?

I also increased the number of x and Q² points to (60, 60) and 10, but this did not improve the results. Moreover, I applied the same procedure to a LO set (NNPDF31_lo_as_0130), obtaining 0.3–1% differences — better than for PDF4LHC21_40, but still about one order of magnitude worse than for CT14llo.

Thanks in advance,
Francesco

[felixhekhorn]

Ciao Francesco,
thanks for reaching out! I haven't had yet time to look into this more closely, but I have a potential candidate on which setting you might still missing: evolution_method, i.e. with which strategy you are solving the differential equation. This is a peculiarity with PDF4LHC: recall that it is a plain glueing from three different groups and if I look to page 21 of https://arxiv.org/pdf/2203.05506 I don't see that they also harmonized on the evolution strategy. This indicates that the groups use their inherent evolution strategies, which are indeed different: while CT and MSHT use what EKO calls iterated-exact, NNPDF uses truncated (until very recently but surely in NNPDF3.1 which is going into PDF4LHC21).

So my first guess is that you need to evolve the "CT and MSHT component" with iterated-exact and the "NNPDF component" with truncated. Can you please try to do so and tell if that solves the problem?

I guess this is the main problem, but in order to check my conjectured solution you are facing (at least) two more problems:

1. As said the different groups are just glued together (as replicas) and then together they define PDF4LHC21 - this means that the central PDF, which you are using above, does not obey neither of the two solution strategies, because it is an average over different solutions. So when I said "CT component" above, I more specifically meant the CT replicas that went into PDF4LHC21 (and likewise for the others)
2. Let me repeat: 'when I said "CT component" above, I more specifically meant the CT replicas' - which is another problem with your choice of PDF4LHC21_40. If I understood Section 4.3 of the paper correctly by just glimpsing over it, this specific PDF set is a hessian version of the true PDF4LHC21, which originally is a replica set. Now, when doing this rewriting you are again loosing all information about evolution strategies as you are transforming the set as a whole. Thus you need to use the original replica version PDF4LHC21

Conclusion: please try using PDF4LHC21 instead; use a definite replica (i.e. not the central PDF); adjust the evolution_method to that replica

I couldn't quickly find the order in which the groups are glued together, but I'm sure it is written somewhere.

Does this help you?

---

Further reading:

• you can find more about different evolution methods in the documentation here or some brief comments in Sec. 3.2 of our paper
• also NNPDF is moving for the next release to iterate-exact as a consequence of our QED paper, where you can also read more formal aspects of iterate-exact vs. truncated

[Radonirinaunimi]

Conclusion: please try using PDF4LHC21 instead; use a definite replica (i.e. not the central PDF); adjust the evolution_method to that replica

I don't think this is feasible, at least not with the released PDF4LHC21 sets. Note that PDF4LHC21 was actually a combination of three different 300 replicas sets and then compressed to get to the released sets. With the compression in-between, it is nearly impossible to track which replicas from which group is a given member.

[felixhekhorn]

I see - thanks for pointing out! and yes I agree: the compression is a operation on the set, so this difference is lost.

@Francesco-Montagno I'm afraid this means that this particular PDF set is out of our scope as it is a mix of things. Note that this is only true for eko, meaning reproducing the evolution - from a physics point of view it is just fine as the difference is always higher order with respect to a given determination (i.e. in principle out of control). Also remember that this discussion is going all under the assumption that my conjecture (about evolution_method being misaligned) is correct, which I'm not sure about. However, we do know that at NNLO one can feel the difference between the two methods - see the discussion in our QED paper cited above and also this recent paper on a related topic.

[Francesco-Montagno]

Thanks for the detailed explanation. I now understand the issue with PDF4LHC21. In the meantime, I tried evolving CT14nlo and it worked well, so it seems the problem was specific to that set rather than to NLO sets in general