Renormalization-group equations of the LEFT at two loops: dimension-six operators

TL;DR

This work delivers the complete two-loop renormalization-group equations for the LEFT arising from dimension-six operators that conserve baryon number, within the algebraically consistent HV scheme for $\gamma_5$. Finite one-loop renormalizations remove evanescent pieces and restore chiral spurion symmetry, enabling a clean, scheme-dependent calculation of the physical two-loop RGEs, including down-mixing into lower-dimension operators such as dimension-five dipoles, masses, gauge couplings, and $\theta$ terms. The results show a rich mixing structure, including nontrivial holomorphy properties for three-gluon operators and explicit scheme translations to the NDR framework in cases where $\gamma_5$ is not encountered. Supplementary material provides the full two-loop RGE notebook; these results are essential for precision, low-energy tests of physics beyond the Standard Model and connect to earlier dimension-five and baryon-number-violating analyses within the same HV scheme.

Abstract

We present the third part of a systematic calculation of the two-loop anomalous dimensions for the low-energy effective field theory below the electroweak scale (LEFT): insertions of dimension-six operators that conserve baryon number. In line with our previous publications, we obtain the results in the algebraically consistent 't Hooft-Veltman scheme for $γ_5$, corrected for evanescent as well as chiral-symmetry-breaking effects through finite renormalizations. We compute the renormalization of the dimension-six four-fermion and three-gluon operators, as well as the power corrections to lower-dimension operators in the presence of masses, i.e., the down-mixing into dimension-five dipole operators, masses, gauge couplings, and theta terms. Our results are of interest for a broad range of low-energy precision searches for physics beyond the Standard Model.