Strong-Coupling Constant at Three Loops in Momentum Subtraction Scheme

TL;DR

This paper calculates the three-loop corrections to the strong coupling constant in a momentum subtraction scheme with a massive quark, clarifying its relation to the MSbar scheme and the decoupling phenomenon.

Contribution

It provides the first detailed three-loop calculation of the MOM scheme beta function with a massive quark and compares it to the MSbar scheme, demonstrating their equivalence outside the threshold region.

Findings

The MOM scheme can be related to the MSbar scheme across different energy regions.

Decoupling is effectively described in both schemes.

The MSbar scheme can be used to connect high- and low-energy coupling constants.

Abstract

In this paper we compute the three-loop corrections to the $\beta$ function in a momentum subtraction (MOM) scheme with a massive quark. The calculation is performed in the background field formalism applying asymptotic expansions for small and large momenta. Special emphasis is devoted to the relation between the coupling constant in the MOM and $\overline{\rm{MS}}$ schemes as well as their ability to describe the phenomenon of decoupling. It is demonstrated by an explicit comparison that the $\overline{\rm{MS}}$ scheme can be consistently used to relate the values of the MOM-scheme strong-coupling constant in the energy regions higher and lower than the massive-quark production threshold. This procedure obviates the necessity to know the full mass dependence of the MOM $\beta$ function and clearly demonstrates the equivalence of both schemes for the description of physics outside the threshold region.