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

We present in analytic form the matching conditions for the strong coupling constant ${\alpha }_s^{\left(n_f\right)}\left(\mu \right)$ at the flavor thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient ${\beta }_3$ of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for ${\alpha }_s^{\left(n_f\right)}\left(\mu \right)$ together with appropriate relationships between the asymptotic scale parameters ${\Lambda }^{\left(n_f\right)}$ for different numbers of flavors $n_f$.