Check the GLS sum rule

[Radonirinaunimi]

Adding a module that checks the GLS sum rules [Gross, Smith] a posteriori.

[juanrojochacon]

Very curious to see what is the outcome of this check. We should study the GLS both in the "data" region and in the "yadism" region. Probably we want to quote results for different values of the lower integration cut xmin to check that the neutrino structure functions are indeed integrable as they should

[Radonirinaunimi]

@juanrojochacon Here is the first result for $F^\nu_3 (A=1)$ with fixed $nf = 3$ and $x{\rm min}=10^{-2}$. The value of $x_{\rm min}$ was chosen to correspond to the smallest $x$ value included in the fit. Also, since the following result is for $A=1$, we are "basically" looking at Yadism through the fit.

The blue points essentially represent what is on the right-hand side of the equation while the red points represent the integral of $xF^\nu_3$ NN functions.

I only have two minor questions:

• In our paper, the equation is defined in terms of the neutrino part while in this paper (https://arxiv.org/pdf/hep-ph/9405254.pdf) it is defined in terms of the average. Which definition should we adopt?
• What should be done about non-isoscalar targets?

[juanrojochacon]

Hi @Radonirinaunimi thanks for the update! Looks good, we agree with the GLS sum rule within uncertainties. I am a bit surprised that the errors are so large from the parametrisation, but we can look at this later - at least the basic test is succesful.

Some further questions, and answering your comments:

• How do results change if xmin is varied? Are results stable wrt to the lower integration limit? They should, since xF3 should be an integrable structure functions (maybe we should impose this, as we do in NNPDF4.0)?
• I think the GLS should be the same for other values of A, since it is related to the valence sum rules, and valence sum rules are unaffected by nuclear corrections. We can check in the literature, but my guess is that if you try for other values of A you will find also agreement.
• The definition should be that in terms of the average, we need to correct this in our paper. But there are other options, for example you can average neutrino and antineutroino on a proton target or the average of neutrino in proton and nuclear targets, I think results would be the same
• Why we need to look at non-isoscalar targets? I don't think this is relevant for the discussion here

[Radonirinaunimi]

Hi @Radonirinaunimi thanks for the update! Looks good, we agree with the GLS sum rule within uncertainties. I am a bit surprised that the errors are so large from the parametrisation, but we can look at this later - at least the basic test is succesful.

Some further questions, and answering your comments:

* How do results change if xmin is varied? Are results stable wrt to the lower integration limit? They should, since xF3 should be an integrable structure functions (maybe we should impose this, as we do in NNPDF4.0)?

* I think the GLS should be the same for other values of A, since it is related to the valence sum rules, and valence sum rules are unaffected by nuclear corrections. We can check in the literature, but my guess is that if you try for other values of A you will find also agreement.

* The definition should be that in terms of the average, we need to correct this in our paper. But there are other options, for example you can average neutrino and antineutroino on a proton target or the average of neutrino in proton and nuclear targets, I think results would be the same

* Why we need to look at non-isoscalar targets? I don't think this is relevant for the discussion here

There is indeed something not fully correct about the uncertainties. I'm fixing this asap.

[Radonirinaunimi]

In addition to the bug in the uncertainties, I was also using an outdated model to generate $x F_3$ predictions. Now, the NN results look as expected. There are some slight differences between the proton and iron results, although both agree within the uncertainties. The only doubt now is the value of GLS for $Q^2 = 1, 2 \rm{GeV}^2$. For the time bein $\alpha_s$ is computed at 2-loop but I doubt that this would be the reason.

[juanrojochacon]

Very nice @Radonirinaunimi ! I am happy with the results. At some point at very low Q the perturbation expansion stops working, so I would not worry too much. One can also have HT effects and all sorts of crap. Provided we agree with GLS in the region where Q2 is of a few GeV2 or higher, we are in good shape, since this is an independent constraint that the ML parametrisation makes sense

I guess the only thing to check is the dependence with xmin - what happens for example if xmin goes down to 10^-3? Do results change a lot or they are instead stable?

[Radonirinaunimi]

One can indeed add HT contributions such as presented in this paper published last year which also computes the GLS up to four-loop. But since the expressions are lengthy I'd not bother to implement this for the time being (surely later one even just as a crosscheck).

Below are some results by varying $x_{\rm min}$. Note that these $x$ values belong to the extrapolation region where we are not doing anything yet (such as the idea proposed before, #53), nevertheless the results are more or less stable.

$x_{\rm min} = 10^{-3}$	$x_{\rm min} = 10^{-4}$

[juanrojochacon]

Hi @Radonirinaunimi I think this is sufficient as cross-check: results are stable as xmin is decreased as they should, and the fact that at small-Q there is some disagreement can be explained away easily. So I would consider this test as succesful and just let's remember to add this discussions to the results section of the paper once we have the final parametrisation

[Radonirinaunimi]

Merging this as it also contains the changes to have the experimental $\chi^2$ of the real data quoted in the report, as per #51.