Radiative corrections to the $K_{e3}^{\pm}$ decay revised

Abstract.We consider the lowest order radiative corrections for the decay $K^{\pm}\to \pi^0 e^{\pm} \nu$, usually referred to as $K_{e3}^{\pm}$ decay. This decay is the best way to extract the value of the Vus element of the CKM matrix. The radiative corrections become crucial if one wants a precise value of Vus. The existing calculations were performed in the late 60’s and are in disagreement. The calculation by Ginsberg turns out to be ultraviolet cutoff sensitive. The necessity of precise knowledge of Vus and the contradiction between the existing results constitute the motivation of our paper.We remove the ultraviolet cutoff dependence by using Sirlin’s prescription; we set it equal to the W mass. We establish the whole character of the small lepton mass dependence based on the renormalization group approach. In this way we can provide a simple explanation of Kinoshita-Lee-Nauenberg cancellation of singularities in the lepton mass terms in the total width and pion spectrum. We give an explicit evaluation of the structure-dependent photon emission based on ChPT in the lowest order. We estimate the accuracy of our results to be at the level of $1\%$. The corrected total width is $\Gamma = \Gamma_0(1 + \delta)$ with $\delta = 0.02\pm0.0002$. Using the form factor value $f_ + (0) = 0.9842\pm 0.0084$ calculated by Cirigliano et al. leads to $\vert V_{us}\vert = 0.2172 \pm 0.0055$.