Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR

TL;DR

This paper computes the complete two-loop anomalous dimensions for four-fermion operators in the Low Energy Effective Field Theory using NDR, extending previous methods and providing results in multiple flavor schemes.

Contribution

It presents the full two-loop anomalous dimension matrix for four-fermion operators in LEFT within the NDR scheme, including implementation in public code DsixTools.

Findings

Derived the two-loop anomalous dimensions for four-fermion operators

Provided results for different numbers of active quark flavors

Implemented the results in the DsixTools software

Abstract

We derive the complete set of two-loop anomalous dimensions describing the mixing of four-fermion operators in the Low Energy Effective Field Theory (LEFT). The calculation is performed in Naive Dimensional Regularization with anticommuting $\gamma_5$ (the NDR scheme), and the results are given in the "JMS basis" of dimension-six operators. The derivation relies on known results for UV poles in two-loop diagrams in QCD, which are then used to derive the two-loop Anomalous Dimension Matrix (ADM) for the full set of four-fermion operators including $O(\alpha_s^2)$, $O(\alpha_s\alpha)$ and $O(\alpha^2)$ corrections. The method employed is an extension of a common approach to deal with traces containing $\gamma_5$ in NDR. Our results have been implemented in the public code DsixTools. We also discuss and provide the results in the LEFT with 5, 4 and 3 active quark flavors.