Coefficients at powers of logarithms in the HD+MSL renormalization scheme

TL;DR

This paper derives explicit formulas for coefficients of logarithmic terms in renormalization constants within the HD+MSL scheme, relating them to the beta function and anomalous dimensions, verified against four-loop calculations in supersymmetric QED.

Contribution

It provides general explicit expressions for logarithmic coefficients in renormalization constants in the HD+MSL scheme, connecting them to fundamental renormalization group functions.

Findings

Derived formulas relate logarithmic coefficients to beta function and anomalous dimensions.

Confirmed formulas match four-loop calculations in ${ m N}=1$ SQED.

Rederived NSVZ relation within the HD+MSL scheme.

Abstract

For renormalizable theories with a single coupling constant regularized by higher derivatives we investigate the coefficients at powers of logarithms present in the renormalization constants assuming that divergences are removed by minimal subtractions of logarithms. According to this (HD+MSL) renormalization prescription the renormalization constants include only powers of $\ln\Lambda/\mu$, where $\Lambda$ and $\mu$ are the dimensionful regularization parameter and the renormalization point, respectively. We construct general explicit expressions for arbitrary coefficients at powers of this logarithm present in the coupling constant renormalization and in the field renormalization constant which relate them to the $\beta$-function and (in the latter case) to the corresponding anomalous dimension. To check the correctness, we compare the results with the explicit four-loop calculation made earlier for ${\cal N}=1$ SQED and (for the supersymmetric case) rederive a relation between the renormalization constants following from the NSVZ equation.