$K^0_{l3\gamma}$ decays: branching ratios and T-odd momenta correlations

Abstract

The branching ratios of the $K^0 \to \pi^- l^+ \nu_l \gamma (l = e, \mu)$ decays, and the T-odd triple momenta correlations $\xi=\vec{q}\cdot[\vec{p}_l \times \vec{p}_\pi]/M^3_K$, due to the electromagnetic final state interaction, in these processes are calculated. The contributions on the order of $\omega^{-1}$ and $\omega^0$ to the corresponding amplitudes are treated exactly. For the branching ratios and T-odd correlation in $K^0 \to \pi^- e^+ \nu_e \gamma$ decay, the corrections on the order of $\omega$ are estimated and demonstrated to be small. The results for the branching ratios are in good agreement with the previous ones. The T-odd triple momenta correlations in $K^0_{l3\gamma}$ decays are calculated for the first time. The values of the \xi-odd asymmetry constitute $-1\times10^{-4}$ and $-4.5\times10^{-4}$ in the $K^0 \to \pi^- \mu^+ \nu_\mu \gamma$ and $K^0 \to \pi^- e^+ \nu_e\gamma$ decays, respectively.