Exact expressions for the renormalization constants in the MS-like schemes

Abstract

We briefly review how it is possible to derive some exact expressions for the renormalization constants for the MS-like renormalization prescriptions using the arguments based on the renormalization group. These expressions are obtained for a version of the dimensional technique in which the dimensionful parameter $\Lambda$ differs from the renormalization scale $\mu$. They encode the equations relating the coefficients at higher $\varepsilon$-poles, powers of $\ln \Lambda/\mu$, and mixed terms of the structure $\varepsilon^{-q} \ln^p \Lambda/\mu$ to the coefficients of the renormalization group functions (i.e. of the $\beta$-function and the anomalous dimension). The general results are verified by some multiloop calculations.