Precision radiative corrections to the muon polarisation in the semileptonic decay of a charged kaon

Abstract

An expression for the Dalitz plot of the semileptonic decay of a charged kaon, including radiative corrections to order $${\mathscr {O}}[(\alpha /\pi )(q/M_1)]$$ , 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. Additionally, the emitted muon is considered to be polarised, so the analysis is also focused on evaluating the radiative corrections to the longitudinal, transverse, and normal polarisation 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 known numerical values of the weak form factors and slope parameters of the process, the radiative corrections to the components of the muon polarisation are found to be very small compared to their respective uncorrected values over most of the allowed kinematical region. Similarly, the fully integrated radiative corrections to the components of the muon polarisation vector are also rather small, around $$0.2\%$$ . This might suggest that radiative corrections to the muon polarisation can be safely ignored in an actual analysis of $$K_{\mu 3}^+$$ decays.