Time Dependence ofμPolarization in the Leptonic Decay of theK0Meson

Abstract

The transverse polarization of the muon in the ($\pi \mu \nu$) decay of the $K^0$ is calculated as a function of time under the assumption of complex form factors, i.e., breakdown of $T$ invariance. It is shown that the nature of the time dependence depends on relative phases and magnitudes of the four form factors $f_{+}$ and $f_{-}$ for $\Delta S=\Delta Q$ and $g_{+}$ and $g_{-}$ for $\Delta S=-\Delta Q$. If $g_{+}=g_{-}=0\mathrm{}$, the relative transverse polarization is constant in time and its value depends on the relative phase of $f_{+}$ and $f_{-}$. In order for the polarization to vary with time, it is necessary that either $g_{+}$ or $g_{-}$ be different from zero; hence a measurement of the time dependence provides a direct test of the $\Delta S=\Delta Q$ rule. Even if the form factors are real (i.e., $T$ invariance is valid), the transverse polarization will be nonvanishing for a time comparable to several $K_1^{}{}_{}{}^0$ lifetimes as long as $f_{+}{g}_{-}-f_{-}{g}_{+}$ is different from zero.