Flavor Number Schemes
FNS or Heavy Quark Matching Schemes are dealing with the ambiguity of including massive quark contributions to physical cross sections. There is not a unique prescription on how to do this and thus we implement several strategies. Unfortunately there is no consistent implementation of the different schemes in the commonly used tools and a comparison of the different outputs has to judged on a case by case basis.
In general we can consider two different kinematic regimes that require a different handling of the massive contributions: For \(Q^2 \lesssim m^2\) the heavy quark should be treated with the full mass dependence. For \(Q^2 \gg m^2\) however the quark should be considered massless, because otherwise a resummation of the occurring terms \(\ln(m^2/Q^2)\) would be required.
We define Heavyness as the split up of the physical total structure functions into several subparts that represent the heavy quark contributions. Again this is not a unique prescription and there are lots of ways to define physical observables, e.g. tagging the outgoing state and imposing kinematics cuts. However, we will not use any of these definitions as they are prone to be theoretically unsafe, if not properly designed. Instead we are defining new observables by considering new theories, derived from the SM by just setting to \(0\) some of its bare couplings.
We are thus providing the observables Flight, Fheavy and Ftotal (for all the unpolarized kinds).
FFNS
As the name FFNS suggests we are considering a fixed number of flavors \(n_f=n_l+1\) with \(n_l\) light flavors and 1 (and only 1) heavy flavor with a finite mass \(m\). The number of light quarks \(n_l\) is arbitrary but fixed and can range between 3 and 5. Except for intrinsic contributions we are NOT allowing the heavy (and the other non-existent) PDF to contribute.
Although this is the most naïve scheme, it is NOT consistently implement in some of the commonly used tools. This scheme is adequate for \(Q^2\sim m^2\).
Flight corresponds to the interaction of the purely light partons, i.e. the coefficient functions may only be a function of \(z,Q^2\) and eventually unphysical scales; in especially they may NOT depend on any quark mass. This may be consistently obtained computing contributions for a Lagrangian with all masses set to \(0\).
This definition is consistent with [MRV08, MV00, MVV05, MVV09, VVM05], QCDNUM
but is not consistent with APFEL, which instead it’s calling Flight the sum of contributions in which a light quark is coupled to the EW boson (but this definition would contain massive corrections, but not consistently, and so it’s theoretically unsafe)
Ftotal is NOT the sum of Flight and the single Fheavy, but contains additional terms Fmissing such as the Compton diagrams in [Hek19]. This is the proper physical object, accounting for all contributions coming from the full Lagrangian.
Fheavy is defined by having in the Lagrangian only the EW charges that are associated to the specific quark active (the only massive one). In NC this corresponds to the electric and weak charges of the quarks but in CC the situation is bit more involved: we divide the CKM-matrix into several parts:
\[\begin{split}V_{CKM} = \begin{pmatrix} {\color{red}V_{ud}} & {\color{red}V_{us}} & {\color{green}V_{ub}}\\ {\color{blue}V_{cd}} & {\color{blue}V_{cs}} & {\color{green}V_{cb}}\\ {\color{purple}V_{td}} & {\color{purple}V_{ts}} & {\color{purple}V_{tb}} \end{pmatrix}\end{split}\]and associate the blue couplings to the charm structure functions, green to bottom and purple to top. For \({\color{blue} F_{2,c}^{\color{black} \nu,p}}\) this in effect amounts to
\[\begin{split}{\color{blue} F_{2,c}^{\color{black} \nu,p}} &=& 2x\Big\{C_{2,q}\otimes\Big[|{\color{blue}V_{cd}}|^2(d+\overline{c}) + |{\color{blue}V_{cs}}|^2 (s+\overline{c})\Big]\\ &+& 2\left(|{\color{blue}V_{cd}}|^2+|{\color{blue}V_{cs}}|^2\right)C_{2,g}\otimes g\Big\}\\\end{split}\]Note that even heavier contributions are NOT available. E.g.:
there is no contributions coming from either bottom or top to \(F_{2,c}\)
while charm would contribute to \(F_{2,b}\), but only as a massless flavor.
ZM-VFNS
As the name ZM-VFNS suggests we are considering a variable number of light flavors \(n_f\) with \(n_f = n_f(Q^2)\). We associate an activation scale \(Q_{thr, i}^2\) to each “heavy” quark and whenever \(Q^2 \ge Q_{thr, i}^2\) we consider this quark massless, otherwise infinitely massive.
Note
\(Q_{thr,i}^2\) are not necessarily, but are usually chosen, to be the quarks’ masses.
This scheme is adequate for \(Q^2\gg m^2\).
Fheavy is NOT defined, as quark masses are either \(0\) or \(\infty\) (so no massive correction is available at all)
Ftotal thus is equal to Flight
Flight corresponds to the interaction of the purely light partons, i.e. the coefficient functions may only be a function of \(z,Q^2\) and eventually unphysical scales; in especially they may NOT depend on any quark mass.
ZM-VFNS dependence on thresholds is simple, they just define the \(Q^2\) patches in which \(n_f\) is constant (and they are of course different from the quark masses, that are always considered to be zero or infinite).
FFN0
This is the high-virtuality limit \(Q^2 \gg m^2\) of the FFNS, where only the collinear logs \(\log(Q^2/m^2)\) are retained, but all power-like heavy quark mass corrections are neglected. These collinear logs are the finite expansion of the DGLAP evolution if the heavy quark would be considered massless. Thus these terms are the overlap between the FFNS using \(n_l\) light quarks and one heavy quark and FFNS using \(n_l+1\) light quarks. These terms are needed in the construction of the FONLL scheme [FLNR10].
FONLL
FONLL [FLNR10] is a GM-VFNS that includes parts of the DGLAP equations into the matching conditions, i.e., two different schemes are considered, and they are matched at a given scale, accounting for the resummation of collinear logarithms. Hence, FONLL relies on both coefficient functions and evolution kernels, so yadism is not solely responsible for the scheme, but also eko and pineko.
In the original paper the prescription is only presented for the charm contributions, but we extend it here to an arbitrary quark.
The prescription defines two separate regimes, below and above the next heavy quark threshold: \(Q_{thr,n_f+2}\).
Note
As in the case of ZM-VFNS, the thresholds are not necessarily, but usually chosen, to be the quarks’ masses.
for \(Q^2 < Q_{thr,n_f+2}^2\):
The general expression, 14-15 of [FLNR10], is:
\[\begin{split}F^{\textrm{FONLL}}(x, Q^2) = F^{(d)}(x, Q^2) + F^{(n_f)}(x, Q^2)\\ F^{(d)}(x, Q^2) = F^{(n_f + 1)}(x, Q^2) - F^{(n_f, 0)}(x, Q^2)\end{split}\]Here we include explicitly the scheme change between the schemes with \(n_f\) (i.e. the FFNS scheme in which the active flavor is the only one considered to be massive) and \((n_f + 1)\) flavors (i.e. the FFNS scheme with only massless quarks, including the formerly active one).
This scheme change is related to the DGLAP matching conditions: in particular the massive corrections are only coming from the \(n_f\) scheme, but the collinear contribution is present in both:
- the \(n_f\) scheme includes the logarithms of the active mass,
while the PDF of the massive object are scale-independent by definition (since the factorization terms are kept in the matrix element)
- the \((n_f + 1)\) scheme does not account for them in them in the coefficient
function, but instead they are resummed in the PDF evolution through the DGLAP equation
By matching the two schemes a GM-VFNS is obtained, accounting for both the massive corrections and the resummation of collinear logarithms.
The matching is obtained subtracting the asymptotic massless limit of the massive expression, namely \(F^{(n_f, 0)}(x, Q^2)\), while adding the \((n_f + 1)\) expression, such that for large \(Q^2\) the massive \(n_f\) contribution cancels with the asymptotic one, and only the truly light contribution survives.
Actually below the former threshold, so \(Q^2 < Q_{thr,n_f+1}^2\), FNS with \(n_f\) flavors is employed, i.e. a \(\theta(Q^2 - Q_{thr,n_f+1}^2)\) is prepended to \(F^{(d)}\).
above this threshold:
The ZM-VFNS is employed and this leads to an inconsistency at this \(Q_{thr,n_f+2}\) threshold, but a good approximation nevertheless.
This amounts to simply make an hard cut to the original smooth decay of massive contributions, and to add the subsequent thresholds for the following massive quarks.
Damping
Continuity
Up to NLO the scheme change (from \(n_f - 1\) flavors to \(n_f\)) is continuous, but in general it is not.
In order to recover the continuous transition a damping procedure may be adopted, turning the scheme in the so called damp FONLL.
Continuity on its own is not an issue, but it is one symptom of a feature of \(F^{(d)}\): while it improves the behavior at large \(Q^2\) it is unreliable for \(Q^2 \sim Q_{thr,n_f+1}^2\).
For this reason might be a good idea to suppress \(F^{(d)}\) near threshold, and then this restore continuity.
The generic shape of this suppression is written in 17 of [FLNR10], and it is:
In particular the following conditions are needed for \(f_{\textrm{thr}} (x, Q^2)\) to fit the task:
be such that \(F^{(d, th)} (x, Q^2)\) and \(F^{(d)} (x, Q^2)\) is power suppressed for large \(Q^2\)
enforce the vanishing of \(F^{(d, th)} (x, Q^2)\) at and below threshold
A common shape for \(f_{\textrm{thr}} (x, Q^2)\) is then:
Note
The power used here is \(2\), but in general this is arbitrary, and thus
it is a user choice in yadism.
Threshold different from heavy quark mass
The matching scale \(\mu^2\) seems to play a relevant role in FONLL, since it is deciding where (in \(Q^2\)) the conversion between the schemes should happen. A typical choice is to put the matching scale on top of the relevant quark mass (also in ZM-VFNS, mimicking the opening of a new channel). This is not mandatory, as the matching scale is just an FNS parameter it can be freely chosen.
However, in practice, choosing the matching scale different from the quark masses has no effect in FONLL since the matching conditions follow the same evolution as PDF which are inlined in either case. We demonstrate this explicitly in the following. We follow the notation of [FLNR10] and denote the coefficient functions in the massive scheme (with \(n_f\) light and 1 heavy flavor) by \(C^{(n_f)}\) and the coefficient functions in the mass-less scheme (with \((n_f+1)\) light flavors) by \(B\). For the sake of readability we suppress in the following any dependence on parton momenta (i.e. \(x\) or \(z\)).
We used the (raw) eko \(E\) for DGLAP evolution
and the matching conditions \(K\) to match PDF between different number of light flavors
where we can assume that the matching scale \(\mu^2\) might be different from the quark mass \(m^2\). This latter equation we can, however, relate back to the case of \(\mu^2=m^2\) using eko again:
Inserting this last equation back into the definition of \(B\) we find
by using the transitive relation of eko. Now, since any eko \(E(Q_1^2\leftarrow Q_0^2)\) may only depend on \(\log(Q_1^2/Q_0^2)\) (it is exactly resumming that log!) \(B\) can not depend on the matching scale \(\mu^2\).
Note, that while nor \(C^{(n_f)}\) nor \(B\) may depend on the matching scale, yadism still has a dependency on the
matching scale: this scales decides which quark to actually treat in the FONLL prescription.
Moreover, the physical observable (i.e. the FK table) still depends on the matching scale as it simply
inherits the dependency from the evolution (which has an explicit, higher-order dependency on the matching scale).