Two-loop anomalous dimensions in the LEFT: dimension-six four-fermion operators in NDR

A bstract 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 γ 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 $$ O\left({\alpha}_s^2\right) $$ O α s 2 , O ( α s α ) and O ( α 2 ) corrections. The method employed is an extension of a common approach to deal with traces containing γ 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.