Divergences in anomalous dimension matrices of quarks at three loops: Explanation and simple solution

Three-loop counterterms for the Standard Model (SM) revealed that the matrix of anomalous dimensions ($gamma$) of quarks is divergent in the $d to 4$ limit unless a carefully chosen non-Hermitian square-root of $Z$ matrix is used in the textbook formula for $gamma$. Here, an alternative prescription is given, which expresses $gamma$ and $beta$ functions directly in terms of counterterms (instead of $sqrt{Z}$ and conventional `bare couplings) and produces finite results. In the SM, this prescription $automatically$ reproduces results obtained previously by adjusting $sqrt{Z}$.