Symmetry of anomalous dimension matrices explained

TL;DR

This paper proves that the anomalous dimension matrices in quantum field theory are complex symmetric in a natural basis, explaining a previously observed pattern across various physical processes.

Contribution

It provides a rigorous proof that these matrices are complex symmetric in a specific natural basis used for physical calculations.

Findings

Anomalous dimension matrices are complex symmetric in a natural basis.

The symmetry property holds only in a particular subset of orthonormal bases.

This basis choice simplifies the analysis of physical processes.

Abstract

In a previous paper, one of us pointed out that the anomalous dimension matrices for all physical processes that have been calculated to date are complex symmetric, if stated in an orthonormal basis. In this paper we prove this fact and show that it is only true in a subset of all possible orthonormal bases, but that this subset is the natural one to use for physical calculations.