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

TL;DR

This paper addresses divergences in quark anomalous dimension matrices at three loops in the Standard Model and proposes a new prescription that yields finite, consistent results without the need for non-Hermitian square-root matrices.

Contribution

It introduces an alternative method to compute anomalous dimensions directly from counterterms, avoiding divergences and simplifying calculations in the Standard Model.

Findings

The new prescription produces finite anomalous dimensions in the SM.

It reproduces previous results obtained with the non-Hermitian square-root method.

The approach simplifies higher-loop calculations in quantum field theory.

Abstract

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 $\sqrt{Z}$.