Following the updated measurement of the lepton flavour universality (LFU) ratio R K in B → K decays by LHCb, as well as a number of further measurements, e.g. R K * by Belle and B s → µµ by ATLAS, we analyse the global status of new physics in b → s transitions in the weak effective theory at the b-quark scale, in the Standard Model effective theory above the electroweak scale, and in simplified models of new physics. We find that the data continues to strongly prefer a solution with new physics in semi-leptonic Wilson coefficients. A purely muonic contribution to the combination C 9 = −C 10 , well suited to UV-complete interpretations, is now favoured with respect to a muonic contribution to C 9 only. An even better fit is obtained by allowing an additional LFU shift in C 9 . Such a shift can be renormalization-group induced from four-fermion operators above the electroweak scale, in particular from semi-tauonic operators, able to account for the potential discrepancies in b → c transitions. This scenario is naturally realized in the simplified U 1 leptoquark model. We also analyse simplified models where a LFU effect in b → s is induced radiatively from four-quark operators and show that such a setup is on the brink of exclusion by LHC di-jet resonance searches.

New physics realized above the electroweak scale can be encoded in a model independent way in the Wilson coefficients of higher dimensional operators which are invariant under the Standard Model gauge group. In this article, we study the matching of the SU(3) C × SU(2) L × U(1) Y gauge invariant dimension-six operators on the standard B physics Hamiltonian relevant for b → s and b → c transitions. The matching is performed at the electroweak scale (after spontaneous symmetry breaking) by integrating out the top quark, W , Z and the Higgs particle. We first carry out the matching of the dimension-six operators that give a contribution at tree level to the low energy Hamiltonian. In a second step, we identify those gauge invariant operators that do not enter b → s transitions already at tree level, but can give relevant one-loop matching effects.

We present a global likelihood function in the space of dimension-six Wilson coefficients in the Standard Model Effective Field Theory (SMEFT). The likelihood includes contributions from flavour-changing neutral current B decays, lepton flavour universality tests in charged-and neutral-current B and K decays, mesonantimeson mixing observables in the K, B, and D systems, direct CP violation in K → ππ, charged lepton flavour violating B, tau, and muon decays, electroweak precision tests on the Z and W poles, the anomalous magnetic moments of the electron, muon, and tau, and several other precision observables, 265 in total. The Wilson coefficients can be specified at any scale, with the one-loop running above and below the electroweak scale automatically taken care of. The implementation of the likelihood function is based on the open source tools flavio and wilson as well as the open Wilson coefficient exchange format (WCxf) and can be installed as a Python package. It can serve as a basis either for model-independent fits or for testing dynamical models, in particular models built to address the anomalies in B physics. We discuss a number of example applications, reproducing results from the EFT and model building literature.

wilson is a Python library for matching and running Wilson coefficients of higher-dimensional operators beyond the Standard Model. Provided with the numerical values of the Wilson coefficients at a high new physics scale, it automatically performs the renormalization group evolution within the Standard Model effective field theory (SMEFT), matching onto the weak effective theory (WET) at the electroweak scale, and QCD/QED renormalization group evolution below the electroweak scale down to hadronic scales relevant for low-energy precision tests. The matching and running encompasses the complete set of dimension-six operators in both SMEFT and WET. The program builds on the Wilson coefficient exchange format (WCxf) and can thus be easily combined with a number of existing public codes.a

General analyses of B-physics processes beyond the Standard Model require accounting for operator mixing in the renormalization-group evolution from the matching scale down to the typical scale of B physics. For this purpose the anomalous dimensions of the full set of local dimension-six operators beyond the Standard Model are needed. We present here for the first time a complete and non-redundant set of dimension-six operators relevant for B-meson mixing and decay, together with the complete one-loop anomalous dimensions in QCD and QED. These results are an important step towards the automation of general New Physics analyses.

We define a data exchange format for numerical values of Wilson coefficients of local operators parameterising low-energy effects of physics beyond the Standard Model. The format facilitates interfacing model-specific Wilson coefficient calculators, renormalisation group (RG) runners, and observable calculators. It is designed to be unambiguous (defining a non-redundant set of operators with fixed normalisation in each basis), extensible (allowing the addition of new EFTs or bases by the user), and robust (being based on industry standard file formats with parsers implemented in many programming languages). We have implemented the format for the Standard Model EFT (SMEFT) and for the weak effective theory (WET) below the electroweak scale and have added interfaces to a number of public codes dealing with SMEFT or WET. We also provide commandline utilities and a Python module for convenient manipulation of WCxf files, including translation between different bases and matching from SMEFT to WET.

We present for the first time a master formula for ε ′ /ε, the ratio probing direct CP violation in K → ππ decays, valid in any theory beyond the Standard Model (BSM). The formula makes use of hadronic matrix elements of BSM operators calculated recently in the Dual QCD approach and the ones of the SM operators from lattice QCD. We emphasize the large impact of several scalar and tensor BSM operators in the context of the emerging ε ′ /ε anomaly. We have implemented the results in the open source code flavio.The non-conservation of the product of parity (P) and charge-conjugation (C) symmetries in nature, known under the name of CP violation, was established experimentally for the first time in 1964 via K → ππ decays [1]. Since then, this fundamental phenomenon has been confirmed also in other processes in the quark sector and is rather consistently described by the so-called Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [2, 3] within the Standard Model (SM) of elementary particle physics. Currently there are experimental efforts to establish analogous CP violation in the lepton sector.CP violation proves to be a prerequisite [4] for our present understanding of matter dominance over antimatter in the universe. However, the CP-violating contribution from the CKM matrix in the SM fails to account for this observation and it remains to be seen whether the CP-violating contributions in the lepton sector will be able to do so. As direct collider searches have not yet revealed any presence of new physics, rare processes in the quark sector remain a good territory to search for new sources of CP violation. This is especially the case for the kaon physics observables ε and ε ′ , which measure indirect and direct CP violation in K 0 -K 0 mixing and K 0 decay into ππ, respectively.Recently, there has been a renewed interest in the ratio ε ′ /ε [5-25], due to hints for a significant tension between measurements and the SM prediction from the RBC-UKQCD lattice collaboration [26,27] and the Dual QCD approach (DQCD) [28,29]. While on the experimental side the world average from the NA48 [30] and KTeV [31,32] collaborations reads

Following the 1999 analysis of Gambino, Haisch and one of us, we stress that all the recent NLO analyses of ε /ε in the Standard Model (SM) suffer from the renormalization scheme dependence present in the electroweak penguin contributions as well as from scale uncertainties in them related to the matching scale µ W and in particular to µ t in m t (µ t ). We also reemphasize the important role of isospin-breaking and QED effects in the evaluation of ε /ε. Omitting all these effects, as done in the 2015 analysis by RBC-UKQCD collaboration, and choosing as an example the QCD penguin (Q 6 ) and electroweak penguin (Q 8 ) parameters B(1/2) 6 and B(3/2) 8 to be B(1/2) 6 = 0.80 ± 0.08 and B(3/2) 8 2. Use the result for isospin-breaking and QED corrections from [18], which are compatible with the ones obtained already 30 years ago in [36].3. Use the NNLO QCD contributions to EWP in [34] in order to reduce the unphysical renormalization scheme and scale dependences.4. Include NNLO QCD contributions to QCDP from [19,20] in order to reduce left-over renormalization scale uncertainties.In view of the fact that meson evolution and the remaining three effects tend to suppress ε /ε, the expectation based on DQCD approach in [14] that ε /ε ≈ (5 ± 2) × 10 −4 in the SM could indeed be confirmed one day by LQCD.