Most exact FONLL implementation about yadism

[alecandido]

More details about alternatives will appear later on :)

Alternatives:

• current APFEL (inconsistent approximation of two masses)
• current yadism (consistent approximation, hard cut of charm mass effects above bottom matching)
  • possibly with a different bottom matching scale than its mass, to mitigate the effects of the cut (but introducing a trade-off with bottom PDF resummation)
• this proposal
• ACOT (requires encoding perturbative matching, i.e. an expanded EKO in x-space)
• doubly analytical matched FONLL (requires encoding many and lengthy expressions, still to be computed, and approximations)
  • outlined in sec. 4.1 of BBR

[alecandido]

More about ACOT

As spelled out in BBR sec. 2.3 (summarized in eq. 2.25) ACOT boils down to using PDFs in the $n+1$ scheme, together with coefficient functions obtained by applying the inverse evolution (inverse evolved, matched down, evolved to final scale) to the $n$ flavors scheme ones.
In this way, mass effects are kept in the coefficient functions, while collinear logs are resummed in the PDFs at the same time.

However, if the inverse evolution would unwind exactly the evolution, this would be perfectly equivalent to compute everything in the $n$ flavors scheme.
Thus, to apply ACOT, the inverse evolution has to be computed at fixed order, in order to preserve the resummed result from PDF evolution. Thus, the two options are:

• compute the fully truncated evolution operator (not even LO exact)
• inline the evolution in the coeff funcs

Usually (personal poor experience from FONLL implementation) the second approach is more convenient, since a large part of $n+1$ inverse evolution is canceled by $n$ forward evolution.

[alecandido]

More about doubly matched

Implementing eqs. 4.16-17 in BBR is full of troubles:

1. it is declared to be an approximation, since no intrinsic is taken into account
2. it is even more an approximation than declared, since terms like $C_g^{(3)} \otimes K_{gb} \otimes f_b^{(4)}$ (that should appear in the second branch of eq. 4.17) are neglected, and definitely not intrinsic
3. it is sketchy, since only the subtraction of charm logs is taken into account, but $B_i^{(5)}(m_c^2, m_b^2)$ contains also bottom mass logs, that are redundant with $C_i^{(5)}(0,0)$ and thus should be subtracted as well, not to be double counted
4. two mass effects are still neglected, otherwise the whole thing would complicate even more
5. computing the double match analytically is feasible, but challenging, since terms starts growing
   • even more in the case of FONLL-B and intrinsic
   • doing it at N3LO starts being prohibitive

Taking into account the first three points, it is manifest that either this strategy outlines a further approximation, without any clear improvement, or it requires quite some work to be implemented. Taking into account the last two, it is manifest that it does not scale, while the main PR proposal does it much better.

[felixhekhorn]

Implementing eqs. 4.16-17 in BBR is full of troubles

while I agree on the statement that this is highly non-trivial, I believe the FONLL expressions (that we would use of course) are given by Eqs. (4.10 - 4.13). Also to say explicitly Eq. (4.3) is already a complicated expression since it already contains a matching (i.e. this C is under the hood already an B).