Higher $\varepsilon$-poles and logarithms in the MS-like schemes from the algebraic structure of the renormalization group

We investigate the structure of renormalization constants within the MS-like renormalization prescriptions for a version of dimensional regularization in which the dimensionful regularization parameter $\Lambda$ differs from the renormalization point $\mu$. Namely, we rewrite the all-loop equations relating coefficients at higher $\varepsilon$-poles and higher powers of $\ln\Lambda/\mu$ to the coefficients of the renormalization group functions in a simple unified form. It is argued that this form follows from the algebraic structure of the renormalization group.