Kl3γ0decays: Branching ratios andT-odd momenta correlations

Abstract

The branching ratios of the ${K}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{l}^{+}{\ensuremath{\nu}}_{l}\ensuremath{\gamma}$ ($l=e$, $\ensuremath{\mu}$) decays, and the $T$-odd triple momenta correlations $\ensuremath{\xi}=\stackrel{\ensuremath{\rightarrow}}{q}\ifmmode\cdot\else\textperiodcentered\fi{}[{\stackrel{\ensuremath{\rightarrow}}{p}}_{l}\ifmmode\times\else\texttimes\fi{}{\stackrel{\ensuremath{\rightarrow}}{p}}_{\ensuremath{\pi}}]/{M}_{K}^{3}$, due to the electromagnetic final state interaction, in these processes are calculated. The contributions on the order of ${\ensuremath{\omega}}^{\ensuremath{-}1}$ and ${\ensuremath{\omega}}^{0}$ to the corresponding amplitudes are treated exactly. For the branching ratios and $T$-odd correlation in ${K}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}\ensuremath{\gamma}$ decay, the corrections on the order of $\ensuremath{\omega}$ 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 ${K}_{l3\ensuremath{\gamma}}^{0}$ decays are calculated for the first time. The values of the $\ensuremath{\xi}$-odd asymmetry are on the order of ${10}^{\ensuremath{-}3}$ and ${10}^{\ensuremath{-}2}$ in the ${K}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}{\ensuremath{\nu}}_{\ensuremath{\mu}}\ensuremath{\gamma}$ and ${K}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{e}^{+}{\ensuremath{\nu}}_{e}\ensuremath{\gamma}$ decays, respectively.