We present in analytic form the matching conditions for the strong coupling constant αₛ^(nf)(μ) at the flavor thresholds to three loops in the modified minimal-subtraction scheme. Taking into account the recently calculated coefficient β₃ of the Callan-Symanzik beta function of quantum chromodynamics, we thus derive a four-loop formula for αₛ^(nf)(μ) together with appropriate relationships between the asymptotic scale parameters Λ^(nf) for different numbers of flavors nf.
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Chetyrkin et al. (1997) studied this question.
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