Radiative Corrections to the $K_{e3}^{\pm}$ Decay Revised

We consider the lowest order radiative corrections for the decay $K^{\pm}\to \pi^0 e^{\pm} \nu$, usually referred as $K_{e3}^{\pm}$ decay. This decay is the best way to extract the value of the $V_{us}$ element of the CKM matrix. The radiative corrections become crucial if one wants a precise value of $V_{us}$. The existing calculations were performed in the late 60's \cite{B,G} and are in disagreement. The calculation in \cite{G} turns out to be ultraviolet cutoff sensitive. The necessity of precise knowledge of $V_{us}$ and the contradiction between the existing results constitute the motivation of our paper. We remove the ultraviolet cutoff dependence by using A.Sirlin's prescription; we set it equal to the $W$ mass. We establish the whole character of 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 formfactor value $f_+(0)=0.9842\pm 0.0084$ calculated in \cite{CKNRT} leads to $|V_{us}|=0.2172 \pm0.0055$.