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