Abstract: The Standard Model Effective Field Theory (SMEFT) provides a powerful framework for parameterizing potential new physics in a model-independent way. To analyze data from experiments across different energy scales and to reliably extrapolate ultraviolet (UV) physics effects, it is crucial to know the renormalization group evolution (RGE) of SMEFT operators.

In this talk, I will present the complete set of two-loop SMEFT $\beta$-functions up to dimension-six in the baryon number-conserving sector, calculated with an anti-commuting $\gamma_5$, and discuss the methods used to compute them as well as the challenges encountered in this process. In particular, $\gamma_5$-odd traces involving six or more ordinary $\gamma$-matrices require special care, since cyclicity is lost. This subtlety can lead to potentially inconsistent results, which I will discuss in detail.