Radiative Corrections toKe30Decays and theΔI=12Rule

The radiative corrections to the Dalitz plot in ${K_{e3\mathrm{}}}^0$ decays are calculated assuming a phenomenological weak $K-\pi$ vertex and using perturbation theory. The answer depends logarithmically on a cutoff, as is the case for nuclear $\beta$ decay. An interesting feature of these decays is that they offer a means of measuring the $q^2$ dependence of the form factors $f_{\pm }\left(q^2\right)$. The radiative corrections contribute an additional energy dependence which cannot be separated experimentally. It is found that the radiative corrections are considerable, i.e., greater than 3% in absolute magnitude, over a large portion of the Dalitz plot, and are not particularly sensitive to a reasonable choice of cutoff. The corrections to the lepton and pion spectra, and the decay rate are also given. A comparison with previous results for ${K_{e3\mathrm{}}}^0$ reveals that the ${K_{e3\mathrm{}}}^{+}$ correction is of the same order of magnitude but everywhere more positive. In particular, the ratio of the decay rates $\frac{\Gamma \left({K_{e3\mathrm{}}}^0\right)}{\Gamma \left({K_{e3\mathrm{}}}^{+}\right)}$, which is equal to 2 according to the $\Delta I=\frac{1}{2}$ rule, must be modified by a factor $(1+\delta )$ due to the radiative corrections. It is found that $\delta \simeq 1\frac{1}{4}\%$ and is independent of the cutoff.