The Single-Mass Variable Flavor Number Scheme at Three-Loop Order

TL;DR

This work derives the three-loop single-mass VFNS matching relations in both unpolarized and polarized QCD, enabling a consistent transition of heavy quarks to light partons at $Q^2 \gg m_Q^2$ via massive operator matrix elements $A_{ij}$. It provides the full set of single-mass OMEs at three-loop order and delivers fast $x$-space numerical representations and public codes (Fortran OMEUNP3, OMEPOL3 and C++ libome) for phenomenology. The results confirm that the FFNS and VFNS descriptions of inclusive observables such as $F_2(x,Q^2)$ and $g_1(x,Q^2)$ agree up to $O(a_s^3)$, while highlighting that heavy-quark distributions should be viewed as a convention rather than a threshold at $Q^2=m_Q^2$ and that large scales are necessary for meaningful heavy-quark PDFs. The work lays groundwork for future two-mass VFNS extensions and broad phenomenological use in global analyses and simulations of inclusive DIS data.

Abstract

The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $\overline{\rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided.

Figures (13)

• Figure 1: The distribution $x\Delta^{\rm NS,+}(x,Q^2)$ for $Q^2 = 30~\mathrm{GeV}^2$ (dotted line), $Q^2 = 100~\mathrm{GeV}^2$ (dashed line), $Q^2 = 10000~\mathrm{GeV}^2$ (full line) in the unpolarized case.
• Figure 2: Left panel: The distribution $x\Sigma(x,Q^2)$ for $Q^2 = 30~\mathrm{GeV}^2$ (dotted line), $Q^2 = 100~\mathrm{GeV}^2$ (dashed line), $Q^2 = 10000~\mathrm{GeV}^2$ (full line) in the unpolarized case. Right panel: the same for the distribution $xG(x,Q^2)$.
• Figure 3: The distribution $x\Delta_8(x,Q^2)$ for $Q^2 = 30~\mathrm{GeV}^2$ (dotted line), $Q^2 = 100~\mathrm{GeV}^2$ (dashed line), $Q^2 = 10000~\mathrm{GeV}^2$ (full line) in the polarized case in the Larin scheme Blumlein:2024euz.
• Figure 4: Left panel: the distribution $x\Delta\Sigma(x,Q^2)$ for $Q^2 = 30~\mathrm{GeV}^2$ (dotted line), $Q^2 = 100~\mathrm{GeV}^2$ (dashed line), $Q^2 = 10000~\mathrm{GeV}^2$ (full line) in the polarized case in the Larin scheme Blumlein:2024euz. Right panel: the same for the distribution $x\Delta G(x,Q^2)$.
• Figure 5: The ratio $\Delta^{\rm c,b}(x,Q^2)/\Delta^{\rm N_F =3}(x,Q^2)$ as a function of $x$, referring to the matching scales $Q^2 = 30~\mathrm{GeV}^2$ (dotted line), $Q^2 = 100~\mathrm{GeV}^2$ (dashed line), $Q^2 = 10000~\mathrm{GeV}^2$ (full line) in the unpolarized case.