High precision flavour physics

Flavour Physics is a very powerful tool to test the Standard Model and quantify the effects of New Physics by exploring possible departures from the model. It is necessary, however, to increase the level of precision of the experimental measurements and of the theoretical predictions in order to be sensitive to possible deviations. In this thesis we address this issue by providing theoretical improvements for both non-perturbative and perturbative calculations of some relevant 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 O(αem). 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. A numerical analysis in the electro-quenched approximation is carried out for quark bilinear operators and for the case of the matrix elements relevant for Kl2 and πl2 processes and for semileptonic decays like Kl3. A detailed discussion of the calculation of the leading isospin breaking corrections to the leptonic decay rates Γ(Kμ2) and Γ(πμ2) is presented. On the other hand, we compute for the first time the anomalous dimension matrices of quark bilinear and weak four-fermions semileptonic operators at O(αs αem), thus improving the evolution and matching of the matrix elements related to such operators in Effective Field Theories.