Renormalization scheme factorization of one-loop Fierz identities

TL;DR

This paper proves that renormalization scheme factors can be separated in one-loop Fierz identities, simplifying basis and scheme transformations in quantum field theory calculations.

Contribution

It introduces a method for scheme factorization in one-loop Fierz identities, enabling independent basis and scheme transformations using physical operator relations.

Findings

Validated scheme factorization with a two-loop anomalous dimension matrix

Transformed results between different schemes without explicit evanescent operator calculations

Reproduced known basis transformation results efficiently

Abstract

We present a proof of the factorization of renormalization scheme in one-loop-corrected Fierz identities. This scheme factorization facilitates the simultaneous transformation of operator basis and renormalization scheme using only relations between physical operators; the evanescent operators in the respective bases may be chosen entirely independently of each other. The relations between evanescent operators in the two bases is automatically accounted for by the corrected Fierz identities. We illustrate the utility of this result with a two-loop anomalous dimension matrix computation using the Naive-Dimensional Regularization scheme, which is then transformed via one-loop Fierz identities to the known result in the literature given in a different basis and calculated in the Larin scheme. Additionally, we reproduce results from the literature of basis transformations involving the rotation of evanescent operators into the physical basis using our method, without the need to explicitly compute one-loop matrix elements of evanescent operators.