X space conversion of the N3LO matching

[giacomomagni]

Just to keep track, not sure if we really want to merge.

[felixhekhorn]

let's keep this - it's not much of a burden and it might be useful in the future (or for MSHT to check we're doing the right thing)

[giacomomagni]

Hi @felixhekhorn,
so I managed to extract the large-N limit $\mathcal{O}(1/N)$ of our $A_{qq}^{(3)}$ and $A_{gg}^{(3)}$ expressions,
i.e. the coefficients of $S_1(N)$ and the constant bit.

The method used is visible in the notebook.
I basically took the original expressions and did the expansion for $N \to \infty$. For $A_{gg}^{(3)}$ I had to add also the terms computed in https://arxiv.org/pdf/2211.05462 using directly eq 4.6 and 4.7.

The idea is to explicitly subtract the limit before doing the $N \to x$ conversion and give to the MSHT people also the two expansions (converted in x-space) so the they can add the terms as they like.
Do you have any comment ?
Robert has confirmed me that the .txt format I'm using is fine for them.

To have a cleaner x-space asymptotic expression should I keep in the asymptotic also the $1/N$ terms ?
PS : If we want to improve the accuracy this might be a way to go.
From what I see, this might have an effect for $A_{Hg}$, which contains large-x divergencies as $\ln^5(1-x)$ and might be the reason why the integration of the interpolation is not fully able to reproduce the moment.

[felixhekhorn]

please double check that the bug I claim actually is one