The quantity ε ′ /ε measures direct CP violation in Kaon decays. Recent SM predictions show a 2.9σ tension with data, with the theoretical uncertainty dominating. As rapid progress on the lattice is bringing nonperturbative long-distance effects under control, a more precise knowledge of short-distance contributions is needed. We describe the first NNLO results for ε ′ /ε and discuss future prospects, as well as issues of scheme dependence and the separation of perturbative and nonperturbative effects. Finally we also comment on the solution of the renormalisation-group evolution in one of the talks at this conference and present the correct solution.

Recent progress in Lattice QCD and subsequent Next-to-Leading Order analysis of the ratio ε /ε resulted in a 2.9σ discrepancy between the Standard Model predictions and experimental data. This inconsistency could have several sources, one of which could be the missing contribution of new particles in the theory predictions. However, a reliable Standard Model prediction is essential to disentangle possible new physics effects from the Standard Model background. Indeed, possible higher order corrections could significantly alter the theory prediction. This is particularly true for ε /ε where the Next-to-Leading Order corrections have been found to be large. To close this gap, we aim to calculate the relevant matching corrections at Next-to-Next-to-Leading Order for this physical quantity and present a more accurate theoretical prediction within the Standard Model.

We consider scenarios with a heavy Z gauge boson with flavour non-universal quark and lepton couplings with the goal to illustrate how the cancellation of gauge anomalies generated by the presence of an additional U(1) gauge symmetry would imply correlations between FCNC processes within the quark sector, within the lepton sector and most interestingly between quark flavour and lepton flavour violating processes. To this end we present simple scenarios with only left-handed flavour-violating Z couplings and those in which also right-handed flavour-violating couplings are present. The considered scenarios are characterized by a small number of free parameters but in contrast to gauge anomaly cancellation in the Standard Model, in which it takes place separately within each generation, in our scenarios anomaly cancellation involves simultaneously quarks and leptons of all three generations. Our models involve, beyond the ordinary quarks and leptons, three heavy right-handed neutrinos. The models with only left-handed FCNCs of Z involve beyond g Z and M Z two real parameters characterizing the charges of all fermions under the U(1) gauge symmetry and the CKM and PMNS ones in the quark and lepton sectors, respectively. The models with the right-handed FCNCs of Z involve few additional parameters. Imposing constraints from well measured ∆F = 2 observables we identify a number of interesting correlations that involve e.g. ε /ε, B s,d → µ + µ − , B → K(K * ) + − , K + → π + νν, K L → π 0 νν and purely lepton flavour violating decays like µ → eγ, µ → 3e, τ → 3µ and µ − e conversion among others. Also (g − 2) µ,e are considered. The impact of the experimental µ → eγ, µ → 3e and in particular µ − e conversion bounds on rare K and B decays is emphasized.

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