Study ofKe4Decays

Abstract

It is assumed that the $K\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}+e+\ensuremath{\nu}$ process has a Born term dominated by ${K}^{*}$ exchange. By assuming a "nearly conserved" axial current we obtain a Goldberger-Treiman-type relation by which the ${K}_{e4}$ amplitude is related to the ${K}^{*}$ width and $K\ensuremath{\rightarrow}\ensuremath{\mu}+\ensuremath{\nu}$ amplitude. This is used to determine the left-hand cut for a set of partial-wave dispersion relations. With the assumption of elastic unitarity for pion-pion scattering, integral equations of the Muskhelishvili-Omn\`es type are obtained, which are then solved using various assumed forms for the pion-pion $T=0$ interaction. The effect of the $\ensuremath{\rho}$ resonance is included in the $T=1$ $P$-wave amplitude. We obtain agreement with the experimental rate for ${{K}_{e4}}^{+}$ decay when the $S$-wave pion-pion interaction is described by a scattering-length approximation with a scattering length of ${\ensuremath{\alpha}}_{0}=(1\ifmmode\pm\else\textpm\fi{}0.3)$ pion Compton wavelengths. With this value of ${\ensuremath{\alpha}}_{0}$, the two-pion invariant mass distribution is in good agreement with experiment, and the total $P$-wave contribution to the total rate is predicted to be 18%. If a $\ensuremath{\sigma}$ meson ($T=0$, $S$-wave resonance at 400 MeV) is assumed to dominate the $S$-wave pion-pion interaction, the calculated rate becomes larger than the experimental one by two orders of magnitude. The possibility of $T$ violation is discussed.