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.
Highlights
The main obstacle to fully profiting from the potential of non-leptonic decays in tests of the Standard Model (SM) and searches for New Physics (NP) is our imprecise knowledge of nonperturbative hadronic bound-state effects
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
In the SM, the left-handed nature of weak interactions leads to a specific form of the corresponding Weak Effective Theory (WET)1 that allows us to develop specially tailored strategies for elimination of uncertainties that stem from such bound-state effects in the aforementioned non-leptonic decays
Summary
For the purpose of our transformation of the Wilson coefficients between the JMS and BMU bases, we introduce the following reference order: VLL : VLR : SRR : Q1, Q2, Q3, Q4, Q9, Q10, Q11, Q14 , Q5, Q6, Q7, Q8, Q12, Q13, Q15, . As we already have mentioned, non-vanishing entries in ∆rs arise whenever the tree-level transformation between the JMS and BMU bases involves Dirac algebra manipulations that are valid in D = 4 dimensions only In such cases, evanescent operators need to be taken into account when working out the basis transformation in D = 4 dimensions. The transformations described by ∆Band ∆Ccorrespond to the SRR-sector operators with two or three different flavours, respectively To determine these matrices, we had to evaluate one-loop diagrams with insertions of various Fierz-evanescent operators. The simple structure of ∆Ain eq (3.9) stems from the fact that only penguin diagrams with Fierz-evanescent operator insertions contribute to this matrix
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