Analytic extension of the modified minimal subtraction renormalization scheme

The conventional definition of the running coupling (Formula presented) 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 (Formula presented) 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 (Formula presented) to the physical (Formula presented) scheme at specific scales, thus naturally including finite quark masses. The analytic extension inherits the exact analyticity of the (Formula presented) scheme and matches the conventional (Formula presented) scheme far above and below mass thresholds. Furthermore just as in the (Formula presented) scheme, there is no renormalization scale ambiguity, since the position of the physical mass thresholds is unambiguous.