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Register a new userRadiative corrections to semileptonic decay rates
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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 $K_{ell3}$ 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$_mathrm{,L}$ 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 $f^pm(q^2)$ but also on their derivatives $df^pm/dq^2$. (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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RBC/UKQCD is preparing a calculation of leptonic decay rates including isospin breaking corrections using a perturbative approach to include NLO contributions from QED effects. We present preliminary numerical results for a contribution to the leptonic pion decay rate and report on exploratory studies of computational techniques based on all-to-all propagators.
The leading-order electromagnetic and strong isospin-breaking corrections to the ratio of $K_{mu 2}$ and $pi_{mu 2}$ decay rates are evaluated for the first time on the lattice, following a method recently proposed. The lattice results are obtained using the gauge ensembles produced by the European Twisted Mass Collaboration with $N_f = 2 + 1 + 1$ dynamical quarks. Systematics effects are evaluated and the impact of the quenched QED approximation is estimated. Our result for the correction to the tree-level $K_{mu 2} / pi_{mu 2}$ decay ratio is $-1.22,(16) %$ to be compared to the estimate $-1.12,(21) %$ based on Chiral Perturbation Theory and adopted by the Particle Data Group.
We review some recent progress in the theory of electroweak radiative corrections in semileptonic decay processes. The resurrection of the so-called Sirlins representation based on current algebra relations permits a clear separation between the perturbatively-calculable and incalculable pieces in the $mathcal{O}(G_Falpha)$ radiative corrections. The latter are expressed as compact hadronic matrix elements that allow systematic non-perturbative analysis such as dispersion relation and lattice QCD. This brings substantial improvements to the precision of the electroweak radiative corrections in semileptonic decays of pion, kaon, free neutron and $J^P=0^+$ nuclei that are important theory inputs in precision tests of the Standard Model. Unresolved issues and future prospects are discussed.
An expression for the Dalitz plot of the semileptonic decay of a charged kaon, including radiative corrections to order O[alpha pi)(q/M_1)], where q is the four-momentum transfer and M_1 is the mass of the decaying kaon, is obtained. Contributions of both the three- and four-body regions are accounted for. Besides, the emitted muon is considered to be polarized so the analysis is also focused on evaluating the radiative corrections to the longitudinal, transverse, and normal polarization components of the muon. The final formulas, with the triple integration of the bremsstrahlung photon variables ready to be performed numerically, are general enough to be used in model-independent experimental analyses whether or not the real photon is discriminated. With the numerical values of the weak form factors and slope parameters of the process, the radiative corrections to the components of the muon polarization are found to be very small compared to their respective uncorrected values.
In Carrasco et al. we have recently proposed a method to calculate $O(e^2)$ electromagnetic corrections to leptonic decay widths of pseudoscalar mesons. The method is based on the observation that the infrared divergent contributions (that appear at intermediate stages of the calculation and that cancel in physical quantities thanks to the Bloch-Nordsieck mechanism) are universal, i.e. depend on the charge and the mass of the meson but not on its internal structure. In this talk we perform a detailed analysis of the finite-volume effects associated with our method. In particular we show that also the leading $1/L$ finite-volume effects are universal and perform an analytical calculation of the finite-volume leptonic decay rate for a point-like meson.
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