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

The conventional definition of the running coupling $\alpha_{\bar{MS}}(\mu)$ in quantum chromodynamics is based on a solution to the renormalization group equations which treats quarks as either completely massless at a renormalization scale $\mu$ 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 $\alpha_{\bar{MS}}(\mu)$ 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 $\alpha_{\bar{MS}}(\mu)$ to the physical $\alpha_V$ scheme at specific scales, thus naturally including finite quark masses. The analytic-extension inherits the exact analyticity of the $\alpha_V$ scheme and matches the conventional $\bar {MS}$ scheme far above and below mass thresholds. Furthermore just as in $\alpha_V$ scheme, there is no renormalization scale ambiguity, since the position of the physical mass thresholds is unambiguous.