Fermion Traces Without Evanescence

TL;DR

This paper develops a method for evaluating fermion traces in arbitrary dimensions suitable for dimensional regularization, addressing evanescent structures crucial for accurate anomaly and higher-loop calculations in quantum field theory.

Contribution

It introduces a framework for handling fermion traces in dimensionally continued regularization without lifting algebra to non-integer dimensions, ensuring correct treatment of evanescent structures.

Findings

Proper treatment of evanescent structures is essential for anomaly calculations.

The method enables three-loop Standard Model Z-factor evaluations.

Addresses differences between DREG and DCREG approaches.

Abstract

We outline the evaluation of $n$-dimensional fermion traces ($n\in\mathbb{N}$) built by products of Dirac-$\gamma$ matrices suitable for a uniform dimensional continuation. Such a continuation is needed for calculations employing a dimensional regulator whenever intrinsically integer dimensional tensors yield non-vanishing contributions. A prime example for such a tensor is given by $\gamma_5$ for $n=4$. The main difference between Dimensional Regularization (DREG) and a Dimensionally Continued Regularization (DCREG) is that DCREG does not attempt to lift the algebra to continuous $d$ dimensions ($d\in\mathbb{R}$). As a consequence one has to properly deal with evanescent structures in order to ensure the uniform application of the regulator. In basic steps we identify evanescent structures in fermion traces and show that their proper treatment is crucial for example when calculating the $VVA$ anomaly in four dimensions. We checked that the performed considerations enable the evaluation of Standard Model (SM) $Z$-factors within DCREG up to including three loops.