General non-leptonic ∆F = 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 Nf 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.

BSM master formula for ε′/ε in the WET basis at NLO in QCD

“…With this goal in mind we have recently calculated, including NLO QCD corrections, the RG evolution matrix ÛBMU (µ had , µ ew ) [23] that allows to express the WCs C BMU (µ had ) at the hadronic scale in the BMU basis in terms of the C JMS (µ ew ) in the JMS basis at the electroweak scale as follows C BMU (µ had ) = ÛBMU (µ had , µ ew ) MJMS (µ ew ) C JMS (µ ew ).…”

“…• It includes NLO QCD RG evolution below the electroweak scale. Further, also the LO QED and NLO QED×QCD evolution is known, and can be applied if needed [23].…”

“…As the two items above have been accomplished in [23], the main goal of the present paper is the evaluation of all hadronic matrix elements for I = 0 and I = 2 in the BMU basis at µ had using the results of [15] obtained in the DQCD approach. Combining them with (1.5), we will be able to present a new master formula for ε /ε with the virtues listed above.…”

“…with the operators Q b and Q b given in the JMS basis as defined in appendix A and the ordering of operators in VLL, VLR and SRR sectors as given in [23]. The chirality-flipped operators Q b in the sectors VRR, VRL and SLL are obtained from Q b by interchanging the chirality-projectors…”

“…The list of all contributing operators in the JMS basis is given in appendix A and the one for the BMU basis in appendix B. We mention further that the choice of the JMS basis in [23] contains hermitian conjugated operators…”