Coefficients at powers of logarithms in the higher-derivatives and minimal-subtractions-of-logarithms renormalization 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 higher-derivatives and minimal-subtractions-of-logarithms ($\mathrm{HD}+\mathrm{MSL}$) renormalization prescription, the renormalization constants include only powers of $\mathrm{ln}\mathrm{\ensuremath{\Lambda}}/\ensuremath{\mu}$, where $\mathrm{\ensuremath{\Lambda}}$ and $\ensuremath{\mu}$ 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 $\ensuremath{\beta}$-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 $\mathcal{N}=1$ supersymmetric quantum electrodynamics and (for the supersymmetric case) rederive a relation between the renormalization constants following from the Novikov, Shifman, Vainshtein, and Zakharov equation.