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.

We discuss the theoretical framework required for the computation of radiative corrections to semileptonic decay rates in lattice simulations, and in particular to those for K 3 decays. This is an extension of the framework we have developed and successfully implemented for leptonic decays. New issues which arise for semileptonic decays, include the presence of unphysical terms which grow exponentially with the time separation between the insertion of the weak Hamiltonian and the sink for the final-state meson-lepton pair. Such terms must be identified and subtracted. We discuss the cancellation of infrared divergences and show that, with the QED L treatment of the zero mode in the photon propagator, the O(1/L) finite-volume corrections are "universal". These corrections however, depend not only on the semileptonic form factors f ± (q 2 ) but also on their derivatives d f ± /dq 2 . (Here q is the momentum transfer between the initial and final state mesons.) We explain the perturbative calculation which would need to be performed to subtract the O(1/L) finite-volume effects.

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Abstract.A new experiment is proposed to measure the running of the electromagnetic coupling constant in the space-like region by scattering high-energy muons on atomic electrons of a low-Z target. The differential cross section of the elastic process μe → μe provides direct sensitivity to the leading-order hadronic contribution to the muon anomaly a HLO μ . It is argued that by using the 150-GeV muon beam available at the CERN North Area, with an average rate of ∼ 1.3 × 10 7 muon/s, a statistical uncertainty of ∼ 0.3% can be achieved on a HLO μ after two years of data taking. The direct measurement of a HLO μ via μe scattering will provide an independent determination and consolidate the theoretical prediction for the muon g-2 in the Standard Model. It will allow therefore a firmer interpretation of the measurements of the future muon g-2 experiments at Fermilab and J-PARC.