Most exact FONLL implementation

[alecandido]

We should implement the exact statement of FONLL:

$$ F^{\text{FONLL}} = F^{(n-1)}(m) - F^{(n-1, 0)}(m) + F^{(n)} $$

I.e. implement the prescription at the observable level, computing the observables in the most trivial way.
In particular, the most trivial way involves PDF sets in two different schemes, $n-1$ flavors ( $f^{(n-1)}$ ) and $n$ flavors ( $f^{(n)}$ ). This yields that we can't store the whole FONLL observable in a single grid, but compositions of different grids is required, since they need to be convoluted separately with different PDFs/evolution operators.

This would not be possible without external support for many flavors, but since the people implementing the external support are the same working to this project, it is definitely simple.

Moreover, this avoids encoding the perturbative matching in two places (evolution and DIS), since matching will disappear from DIS expressions. Only asymptotics will be required to implement FONLL, that are considerably simpler than the current convolution with the matching. This will make FONLL easily scalable to higher orders, and more options will be allowed:

• when two thresholds are involved, at the moment, we need to use only one FONLL prescription at a time, to avoid interferences
• while in this way we can do both at the same time, including different options, in an extremely simple way; e.g. if charm and bottom are involved, we can implement both in the 3 flavor schemes (more consistent with two mass effects)

$$ F^{\text{FONLL}} = F^{(3)}(m_c) - F^{(3, 0)}(m_c) + F^{(3)}(m_b) - F^{(3, 0)}(m_b) + F^{(5)} $$

or resumming the charm contributions, while accounting for bottom mass (possibly more accurate, because of the resummation of additional collinear logs)

$$ F^{\text{FONLL}} = F^{(3)}(m_c) - F^{(3, 0)}(m_c) + F^{(4)}(m_b) - F^{(4, 0)}(m_b) + F^{(5)} $$

or full two-masses effects

$$ F^{\text{FONLL}} = F^{(3)}(m_c, m_b) - F^{(3, 0)}(m_c, m_b) + F^{(5)} $$

In particular, in order to implement these, we only to provide a generic $F^{(n)}(m)$ (structure function in $n$ flavors, depending on mass $m$, only its numerical value is relevant, and not the associated quark), $F^{(n)}$ (fully light structure function in $n$ flavors), and the asymptotic correspondent of the first one, i.e. $F^{(n, 0)}(m)$.
The first two are already available, and components of the last one are implemented as well, so the implementation of this prescription mostly correspond in the collection of different elements.
This has also the effect that yadism won't depend any longer on matching scales, so we will drop all the kDISqThr parameters.

Another relevant implementation detail would be to support (either in yadism or yadbox) the concept of observable bundle, i.e. an observable made of multiple grids.

[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).