Two-loop scale dependence of the static QCD potential including quark masses

The interaction potential ${V(Q}^2)$ between static test charges can be used to define an effective charge ${\alpha }_V{(Q}^2)$ and a physically based renormalization scheme for quantum chromodynamics and other gauge theories. In this paper we use recent results for finite-mass fermionic corrections to the heavy-quark potential at two loops to derive the next-to-leading order term for the Gell-Mann–Low function of the V scheme. The resulting effective number of flavors $N_F{(Q}^2{/m}^2)$ in the ${\alpha }_V$ scheme is determined as a gauge-independent and analytic function of the ratio of the momentum transfer to the quark pole mass. The results give an automatic decoupling of heavy quarks and are independent of the renormalization procedure. Commensurate scale relations then provide the next-to-leading order connection between all perturbatively calculable observables to the analytic and gauge-invariant ${\alpha }_V$ scheme without any scale ambiguity and a well-defined number of active flavors. The inclusion of finite quark mass effects in the running of the coupling is compared with the standard treatment of finite quark mass effects in the minimal subtraction $\left(\overline{\mathrm{MS}}\right)$ scheme.