Strong Coupling Constant with Flavor Thresholds at Four Loops in the Modified Minimal-Subtraction Scheme

Abstract

We present in analytic form the matching conditions for the strong coupling constant ${\ensuremath{\alpha}}_{s}^{({n}_{f})}(\ensuremath{\mu})$ at the flavor thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient ${\ensuremath{\beta}}_{3}$ of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for ${\ensuremath{\alpha}}_{s}^{({n}_{f})}(\ensuremath{\mu})$ together with appropriate relationships between the asymptotic scale parameters ${\ensuremath{\Lambda}}^{({n}_{f})}$ for different numbers of flavors ${n}_{f}$.