The leading electromagnetic (e.m.) and strong isospin-breaking corrections to the π þ → μ þ ν½γ and K þ → μ þ ν½γ leptonic decay rates are evaluated for the first time on the lattice. The results are obtained using gauge ensembles produced by the European Twisted Mass Collaboration with N f ¼ 2 þ 1 þ 1 dynamical quarks. The relative leading-order e.m. and strong isospin-breaking corrections to the decay rates are 1.53(19)% for π μ2 decays and 0.24(10)% for K μ2 decays. Using the experimental values of the π μ2 and K μ2 decay rates and updated lattice QCD results for the pion and kaon decay constants in isosymmetric QCD, we find that the Cabibbo-Kobayashi-Maskawa matrix element jV us j ¼ 0.22538ð46Þ, reducing by a factor of about 1.8 the corresponding uncertainty in the particle data group review. Our calculation of jV us j allows also an accurate determination of the first-row Cabibbo-Kobayashi-Maskawa unitarity relation jV ud j 2 þ jV us j 2 þ jV ub j 2 ¼ 0.99988ð46Þ. Theoretical developments in this paper include a detailed discussion of how QCD can be defined in the full QCD þ QED theory and an improved renormalization procedure in which the bare lattice operators are renormalized nonperturbatively into the regularization independent momentum subtraction (RI'-MOM) scheme and subsequently matched perturbatively at Oðα em α s ðM W ÞÞ into the W-regularization scheme appropriate for these calculations.

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We present a model-independent and relativistic approach to analytically derive electromagnetic finite-size effects beyond the point-like approximation. The key element is the use of electromagnetic Ward identities to constrain vertex functions, and structure-dependence appears via physical form-factors and their derivatives. We apply our general method to study the leading finitesize structure-dependence in the pseudoscalar mass (at order 1/L3) as well as in the leptonic decay amplitudes of pions and kaons (at order 1/L2). Knowledge of the latter is essential for Standard Model precision tests in the flavour physics sector from lattice simulations.