Higher logarithms and $\varepsilon$-poles for the MS-like renormalization prescriptions

TL;DR

This paper analyzes a variant of dimensional regularization with distinct regularization and renormalization scales, providing explicit formulas for the structure of divergences and their relation to renormalization group functions.

Contribution

It derives explicit expressions for the coefficients of $ ext{log}$ and $ ext{epsilon}$-pole structures in MS-like schemes, relating them to RG functions and solving 't Hooft pole equations.

Findings

Explicit formulas for divergence coefficients in MS-like schemes.

All-loop expressions for renormalization constants with divergence structures.

Relations established between divergence coefficients and RG functions.

Abstract

We consider a version of dimensional regularization (reduction) in which the dimensionful regularization parameter $\Lambda$ is in general different from the renormalization scale $\mu$. Then in the scheme analogous to the minimal subtraction the renormalization constants contain $\varepsilon$-poles, powers of $\ln\Lambda/\mu$, and mixed terms of the structure $\varepsilon^{-q}\ln^{p}\Lambda/\mu$. For the MS-like schemes we present explicit expressions for the coefficients at all these structures which relate them to the coefficients in the renormalization group functions, namely in the $\beta$-function and in the anomalous dimension. In particular, for the pure $\varepsilon$-poles we present explicit solutions of the 't~Hooft pole equations. Also we construct simple all-loop expressions for the renormalization constants (also written in terms of the renormalization group functions) which produce all $\varepsilon$-poles and logarithms and establish a number of relations between various coefficients at $\varepsilon$-poles and logarithms. The results are illustrated by some examples.