An Analytic Extension of the $\bar{MS}$ Renormalizaton Scheme

TL;DR

This work addresses the nonanalytic behavior of the conventional $\alpha_{\overline{\mathrm{MS}}}(\mu)$ running coupling at quark thresholds by constructing an analytic extension $\tilde{\alpha}_{\overline{\mathrm{MS}}}$ tied to the physical charge $\alpha_V$ via a commensurate scale relation. By computing the one-loop mass-dependent running in the $\alpha_V$ scheme and mapping to $\alpha_{\overline{\mathrm{MS}}}$ through the CSR, the authors obtain an analytic, threshold-smooth coupling with no renormalization-scale ambiguity. The extension absorbs mass effects into the running, yielding percent-level differences from the standard scheme and reproducing conventional results to sub-percent accuracy across many scales; applications to hadronic Z width and related observables demonstrate its practicality. This framework provides a physically motivated, analytically well-behaved alternative to massless flavor counting, with potential implications for precision QCD and grand-unified theory studies.

Abstract

The conventional definition of the running coupling $α_{\bar{MS}}(μ)$ in quantum chromodynamics is based on a solution to the renormalization group equations which treats quarks as either completely massless at a renormalization scale $μ$ above their thresholds or infinitely massive at a scale below them. The coupling is thus nonanalytic at these thresholds. In this paper we present an analytic extension of $α_{\bar{MS}}(μ)$ which incorporates the finite-mass quark threshold effects into the running of the coupling. This is achieved by using a commensurate scale relation to connect $α_{\bar{MS}}(μ)$ to the physical $α_V$ scheme at specific scales, thus naturally including finite quark masses. The analytic-extension inherits the exact analyticity of the $α_V$ scheme and matches the conventional $\bar {MS}$ scheme far above and below mass thresholds. Furthermore just as in $α_V$ scheme, there is no renormalization scale ambiguity, since the position of the physical mass thresholds is unambiguous.

Figures (7)

• Figure 1: Single insertion of massive quark-antiquark loop into a gluon propagator, giving the quark part of the one-loop gluon vacuum polarization.
• Figure 2: The curve shows the contribution to the continuous $\hbox{$N_{F,V}^{ (0) }$}$ for just one quark as a function of $\rho=Q^2/m^2$ where $m$ is the mass of the quark. $\hbox{$N_{F,V}^{ (0) }$}$ is found by using the massive quark part of the one-loop gluon propagator instead of using the theta function thresholds conventionally used in dimensional regularization schemes.
• Figure 3: Continuous $N_{F,V}^{ (0) }$ for various quarks, lightest to heaviest goes top to bottom (d, c, b, t as one proceed downwards; the u and s plots are virtually identical at this scale to the d). Q runs from 1 to $M_Z$ GeV (for reference, the quark masses (in GeV) used are: $m_u=.004$; $m_d=.008$; $m_s=.200$; $m_c=1.5$; $m_b=4.5$; $m_t=175$.)
• Figure 4: The continuous $\widetilde{N}_{F,\overline{\hbox{\tiny MS}}}^{(0)}$ in the analytic extension of the $\overline{\hbox{MS}}$ scheme as a function of the physical scale $Q$. (For reference the continuous $N_F$ is also compared with the conventional procedure of taking $N_F$ to be a step-function at the quark-mass thresholds.)
• Figure 5: The solid curve shows the relative difference between the solutions to the 1-loop renormalization group equation using continuous $N_F$, $\widetilde{\alpha}_{\overline{\hbox{\tiny MS}}}(Q)$, and conventional discrete theta-function thresholds, $\alpha_{\overline{\hbox{\tiny MS}}}(Q)$. The dashed (dotted) curves shows the same quantity but using the scale $2Q$ ($Q/2$) in $\widetilde{N}_{F,\overline{\hbox{\tiny MS}}}^{(0)}$. The solutions have been obtained numerically starting from the world average Burrows$\alpha_{\overline{\hbox{\tiny MS}}}(M_Z) = 0.118$.