Approximate massive N3LO for F2 and FL

[giacomomagni]

This PR can introduce the approximate massive N3LO contribution for F2,FL from @niclaurenti thesis and the relative code: https://github.com/niclaurenti/adani

For the time being:

• Asymptotic coefficient functions can be computed directly using adani.
• Massive coefficient functions have to be interpolated from a precomputed grid, since their computation is too slow.
• A new theory key is needed to select the possible coefficient function variation

[giacomomagni]

@niclaurenti to implement the asymptotic limit needed in FONLL,
following the yadism strategy we need to provide the high-scale (high-scale = Q^2 >> m^2 in adani convention) coefficients split by logs $L=log(Q^2/m^2)$ with $L_\mu = L_Q$.
This requires some further manipulation of the expression DX_yn_highscale given in adani right?
Any suggestion how to proceed ?

[niclaurenti]

• Massive coefficient functions have to be interpolated from a precomputed grid, since their computation is too slow.

Hi @giacomomagni , for the approximation are you using the argument method_flag=1? (if not provided it will use 0).
Since method_flag=1 uses monte carlo integration and it is much faster than method_flag=0.

[niclaurenti]

@niclaurenti to implement the asymptotic limit needed in FONLL,
following the yadism strategy we need to provide the high-scale (high-scale = Q^2 >> m^2 in adani convention) coefficients split by logs with .
This requires some further manipulation of the expression DX_yn_highscale given in adani right?
Any suggestion how to proceed ?

Inside adani it is not possible to get only the contribution proportional to $\log Q^2/m^2$.
The only thing that I can think of is to reimplement the expressions splitting the terms. It will not be difficult but only very long (given that the expressions are huge)

[niclaurenti]

@niclaurenti to implement the asymptotic limit needed in FONLL,
following the yadism strategy we need to provide the high-scale (high-scale = Q^2 >> m^2 in adani convention) coefficients split by logs with .
This requires some further manipulation of the expression DX_yn_highscale given in adani right?
Any suggestion how to proceed ?

Inside adani it is not possible to get only the contribution proportional to $\log Q^2/m^2$.
The only thing that I can think of is to reimplement the expressions splitting the terms. It will not be difficult but only very long (given that the expressions are huge)

[giacomomagni]

Hi @niclaurenti, yes also using method_flag=1 was quite slow, but I hope we can do a good job also using the grids.

Regarding the high-scale limit, that is mandatory to implement FONLL, so it has to be separated...

[niclaurenti]

Regarding the high-scale limit, that is mandatory to implement FONLL, so it has to be separated...

Ok I can take care of it.
Two questions:

1. I have to set $\mu=Q$ and split the terms proportional to $\log Q^2/m^2$ right? Or I have to keep the $\mu$ dependence?
2. Given that it will require me almost the same effort I did the first time I implemented those functions, should I implement it directly in yadism or you prefer to keep the functions in adani ( @felixhekhorn @alecandido )

[alecandido]

It would be helpful to know what is $\mu$.

Concerning point 2.: we're (or I'm) planning to move yadism coefficient functions to a compiled extension (sooner or later). Making the effort of coding them with Numba, to possibly undo it seems not worth it.
On the other hand, yadism has no compiled extension, for the time being. I would keep them in adani for a while, or in case move them to LeProHQ (to have a single other provider, but it should not change a lot), and migrate whenever we'll start implementing #219 (if ever).

[niclaurenti]

It would be helpful to know what is μ.

It's the factorization scale: those functions have a log dependence on both $Q$ and $\mu$ (see for example arXiv:1403.6356v1 from eq. 597)

Concerning point 2.: we're (or I'm) planning to move yadism coefficient functions to a compiled extension (sooner or later). Making the effort of coding them with Numba, to possibly undo it seems not worth it. On the other hand, yadism has no compiled extension, for the time being. I would keep them in adani for a while, or in case move them to LeProHQ (to have a single other provider, but it should not change a lot), and migrate whenever we'll start implementing #219 (if ever).

Ok perfect. I will try to do it as soon as possible

[giacomomagni]

It's the factorization scale: those functions have a log dependence on both Q and μ (see for example arXiv:1403.6356v1 from eq. 597)

I'd agree to set $\mu_f = Q$ and then split massive logs.

Regarding point 2) let's keep them in adani is possible also because here we might not have all the polylogarithms needed..

[alecandido]

The renormalization/factorization scale contributions are managed separately, you never need to take care of them in the coefficient functions (for yadism, in adani to whatever you wish).
Sooner or later they might also fully migrate to Pineko, cf. #220

[felixhekhorn]

1. I have to set µ=Q

yes! you have to do something - and that is setting $\mu=Q$ since you're doing the same as me in my thesis, i.e. to follow previous papers where the scale variations are written in terms of $\ln(\mu^2/m^2)$ e.g. for LeProHQ this leads to

yadism/src/yadism/coefficient_functions/heavy/f2_nc.py

Lines 37 to 39 in 337758c

	LeProHQ.cg1("F2", "VV", self._xi, self._eta(z))
	+ LeProHQ.cgBar1("F2", "VV", self._xi, self._eta(z))
	* np.log(self._xi)

and the same will happen for you - because the central scale in yadism is $Q$ (and that you need to provide)

[giacomomagni]

This should be now ready for review.