General non-leptonic $ΔF=1$ WET at the NLO in QCD

Abstract

We reconsider the complete set of four-quark operators in the Weak Effective Theory (WET) for non-leptonic ∆F = 1 decays that govern s → d and b → 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 → ππ 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.

Simple Rules for Evanescent Operators in One-Loop Basis Transformations

“…The presented method has been already used successfully in several contexts such as NLO basis transformations [11,18,19], one-loop matching calculations [20] as well as in several two-loop calculations [10]. But it has not been presented in any detail and in particular in this generality in the literature so far.…”

“…Note that we choose the operators Q 26 and Q 28 with an opposite sign, compared to the basis in [11]. For the SRR sector with four different quarks we define the structures…”

“…where the tensor structures have again opposite sign compared to the convention adopted in [11]. With these definitions we define the operators…”

“…It has been used in all two-loop calculations performed by the second author and will be used in the following but this time in the context of basis change. This procedure is much simpler than the formal method presented in [10] and used recently in [11].…”

“…In the process of performing a NLO analysis, it is often necessary to perform one-loop transformations between different operator bases, since for instance anomalous dimension matrices (ADMs) are known only in a particular basis, whereas the matching conditions are given in a different one. This is for example the case for the WET, where the two-loop ADMs are known in the BMU basis [10] as elaborated on recently in [11]. On the other hand the one-loop matching from SMEFT onto WET is given in the JMS basis defined in [2].…”