A Hybrid Scheme for Heavy Flavors: Merging the FFNS and VFNS

TL;DR

The work introduces the Hybrid Variable Flavor Number Scheme (H-VFNS), which embeds explicit N_F dependence into both PDFs and α_s to generate coexisting NF-specific sets that are linked by MSbar matching. This framework offers maximal flexibility to select the optimal NF for each observable, allowing low-scale FFNS-style fits (e.g., F2^charm at HERA) alongside high-scale VFNS analyses (e.g., LHC processes) without backward evolution. By enabling separate matching and switching scales and retaining multiple NF grids, H-VFNS achieves resummed heavy-quark logs where needed while preserving threshold accuracy, improving the description of heavy-flavor data across kinematics. The approach is demonstrated conceptually with an example involving HERA and LHC data and is supported by detailed discussion of NF-dependent PDFs, α_s running, and their interplay in physical observables.

Abstract

We introduce a Hybrid Variable Flavor Number Scheme for heavy flavors, denoted H-VFNS, which incorporates the advantages of both the traditional Variable Flavor Number Scheme (VFNS) as well as the Fixed Flavor Number Scheme (FFNS). By including an explicit $N_F$-dependence in both the Parton Distribution Functions (PDFs) and the strong coupling constant $α_S$, we generate coexisting sets of PDFs and $α_S$ for $N_F=\{3,4,5,6\}$ at any scale $μ$, that are related analytically by the $\overline{\text{MS}}$ matching conditions. The H-VFNS resums the heavy quark contributions and provides the freedom to choose the optimal $N_F$ for each particular data set. Thus, we can fit selected HERA data in a FFNS framework, while retaining the benefits of the VFNS to analyze LHC data at high scales. We illustrate how such a fit can be implemented for the case of both HERA and LHC data.

Figures (11)

• Figure 1: Schematic of a H-VFNS: PDF (left) and $\alpha_{S}$ (right) vs. $\mu$ for a selection of ${N_F}$ values. The preferred range of each ${N_F}$ branch is indicated by the thicker line. Thus, $f_{i}(x,\mu,{N_F}=3)$ can be used slightly above the $m_{c}$ transition, but for very large $\mu$ scales the ${N_F}=4,5,6$ branches are preferred as these resum the $m_{c}$ mass singularities.
• Figure 2: (a) Gluon momentum fraction; (b) Momentum fraction for $c+\bar{{c}}$, $b+\bar{{b}}$ and $t+\bar{{t}}$ quarks. The results have been obtained using NLO PDFs ($\overline{\rm MS}$) with a 2-loop $\alpha_{S}$.
• Figure 3: Momentum fraction carried by the (a) $u$-quark and (b) $\bar{u}$-quark in the 3, 4, 5, and 6 flavor schemes.
• Figure 4: (a) 2-loop $\alpha_{S}$ for different number of flavors; (b) ratio of 3, 4, 5, and 6-flavor $\alpha_{S}$ to the 3-flavor one.
• Figure 5: ${\alpha_{S}}^{(N_{R})}xg^{(N_{F})}(x)$ as a function of $x$, for different values of $\mu$. The curves are labeled ${\alpha_{S}}^{(N_{R})}xg^{(N_{F})}(x)=\{N_{R},N_{F}\}$ where the first term in braces indicates the $N_{R}$ for the $\alpha_{s}$ and the second indicates the $N_{F}$ for $g$.