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The leading electromagnetic (e.m.) and strong isospin-breaking corrections tothe $\pi^+ \to \mu^+ \nu[\gamma]$ and $K^+ \to \mu^+ \nu[\gamma]$ leptonicdecay rates are evaluated for the first time on the lattice. The results areobtained using gauge ensembles produced by the European Twisted MassCollaboration with $N_f = 2 + 1 + 1$ dynamical quarks. The relativeleading-order e.m.~and strong isospin-breaking corrections to the decay ratesare 1.53(19)\% for $\pi_{\mu 2}$ decays and 0.24(10)\% for $K_{\mu 2}$ decays.Using the experimental values of the $\pi_{\mu 2}$ and $K_{\mu 2}$ decay ratesand updated lattice QCD results for the pion and kaon decay constants inisosymmetric QCD, we find that the Cabibbo-Kobayashi-Maskawa matrix element $ |V_{us}| = 0.22538(46)$, reducing by a factor of about $1.8$ the correspondinguncertainty in the Particle Data Group review. Our calculation of $|V_{us}|$allows also an accurate determination of the first-row CKM unitarity relation$| V_{ud}|^2 + | V_{us}|^2 + | V_{ub}|^2 = 0.99988(46)$. Theoreticaldevelopments in this paper include a detailed discussion of how QCD can bedefined in the full QCD+QED theory and an improved renormalisation procedure inwhich the bare lattice operators are renormalised non-perturbatively into the(modified) Regularization Independent Momentum subtraction scheme andsubsequently matched perturbatively at $O(\alpha_{em}\alpha_s(M_W))$ into theW-regularisation scheme appropriate for these calculations.

Light-meson leptonic decay rates in lattice QCD+QED