Radiative corrections to semileptonic decay rates

TL;DR

This paper develops a theoretical framework for calculating radiative corrections to semileptonic decay rates, especially for $K_{ ext{l}3}$ decays, addressing new issues like unphysical exponential terms and finite-volume effects in lattice QCD.

Contribution

It extends previous methods for leptonic decays to semileptonic decays, including the treatment of unphysical terms and finite-volume corrections in lattice simulations.

Findings

Identified and proposed subtraction of unphysical exponential terms.

Showed that finite-volume corrections are universal with QED_L treatment.

Outlined perturbative calculations for finite-volume effect subtraction.

Abstract

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.