Non-perturbative renormalization in lattice QCD+QED

Flavour physics is a very powerful tool to test the Standard Model and quantify the effects of New Physics by exploring possible departures from it. However, it is necessary to increase the level of precision of experimental measurements and theoretical predictions in order to be sensitive to possible deviations. In this talk we address this issue by providing theoretical improvements for both non-perturbative and perturbative calculations of flavour observables. On the one hand, we present a new strategy to renormalize lattice operators in QCD+QED in the RI-MOM scheme, fully including the non perturbative dynamics of QCD, and QED at $\mathcal{O}(\alpha_{em})$. We show how to keep systematically into account all contributions not separable between the two interactions, thus overcoming the factorization approximation in which the mixed effects are neglected. This strategy is applied to quark bilinear operators and to the matrix elements relevant for pion and kaon leptonic decays. On the other hand, we discuss the calculation of the anomalous dimension matrices of quark bilinear and four-fermions semileptonic operators at $\mathcal{O}(\alpha_{s} \alpha_{em})$, necessary to improve the evolution and matching of the Wilson coefficients related to such operators in Effective Field Theories.

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