General non-leptonic $\Delta F=1$ WET at the NLO in QCD

TL;DR

This paper provides a comprehensive NLO QCD analysis of four-quark operators in the Weak Effective Theory for non-leptonic $ riangle F=1$ decays, crucial for understanding flavor-changing processes in the Standard Model and beyond.

Contribution

It calculates WET Wilson coefficients at NLO, including two-loop RG running, and discusses basis transformations for matching to SMEFT.

Findings

Computed two-loop RG evolution of Wilson coefficients.

Analyzed operator basis transformations beyond leading order.

Facilitated matching between WET and SMEFT at NLO.

Abstract

We reconsider the complete set of four-quark operators in the Weak Effective Theory (WET) for non-leptonic $\Delta F=1$ decays that govern $s\to d$ and $b\to d, s$ transitions in the Standard Model (SM) and beyond, at the Next-to-Leading Order (NLO) in QCD. We discuss cases with different numbers $N_f$ of active flavours, intermediate threshold corrections, as well as the issue of transformations between operator bases beyond leading order to facilitate the matching to high-energy completions or the Standard Model Effective Field Theory (SMEFT) at the electroweak scale. As a first step towards a SMEFT NLO analysis of $K\to\pi\pi$ and non-leptonic $B$-meson decays, we calculate the relevant WET Wilson coefficients including two-loop contributions to their renormalization group running, and express them in terms of the Wilson coefficients in a particular operator basis for which the one-loop matching to SMEFT is already known.