Light-meson leptonic decay rates in lattice QCD+QED

Abstract

The leading electromagnetic (e.m.) and strong isospin-breaking corrections to the ${\pi }^{+}\to {\mu }^{+}\nu [\gamma ]$ and $K^{+}\to {\mu }^{+}\nu [\gamma ]$ 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 $N_f=2+1+1$ dynamical quarks. The relative leading-order e.m.~and strong isospin-breaking corrections to the decay rates are 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 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 $|V_{us}|=0.22538(46)$, reducing by a factor of about $1.8$ the corresponding uncertainty in the Particle Data Group review. Our calculation of $|V_{us}|$ allows also an accurate determination of the first-row CKM unitarity relation $|V_{ud}{|}^2+|V_{us}{|}^2+|V_{ub}{|}^2=0.99988(46)$. Theoretical developments in this paper include a detailed discussion of how QCD can be defined in the full QCD+QED theory and an improved renormalisation procedure in which the bare lattice operators are renormalised non-perturbatively into the (modified) Regularization Independent Momentum subtraction scheme and subsequently matched perturbatively at $O({\alpha }_{em}{\alpha }_s(M_W))$ into the W-regularisation scheme appropriate for these calculations.

63 pages, 10 figures and 2 tables. Version matches the published paper

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