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

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,thecorrectionsontheorderof$\omega$ 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 $K_{l3\gamma }^0$ decays are calculated for the first time. The values of the \xi-odd asymmetry constitute $-1\times {10}^{-4}$ and $-4.5\times {10}^{-4}$ in the $K^0\to {\pi }^{-}{\mu }^{+}{\nu }_{\mu }\gamma$ and $K^0\to {\pi }^{-}{e}^{+}{\nu }_e\gamma$ decays, respectively.