Study ofKe4Decays

Abstract

It is assumed that the $K\to \pi +\pi +e+\nu$ 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 $K_{e4\mathrm{}}$ amplitude is related to the $K^{*}$ width and $K\to \mu +\nu$ 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-Omnès type are obtained, which are then solved using various assumed forms for the pion-pion $T=0\mathrm{}$ interaction. The effect of the $\rho$ resonance is included in the $T=1\mathrm{}$ $P$-wave amplitude. We obtain agreement with the experimental rate for ${K_{e4\mathrm{}}}^{+}$ decay when the $S$-wave pion-pion interaction is described by a scattering-length approximation with a scattering length of ${\alpha }_0=(1\mathrm{}\pm 0.3)$ pion Compton wavelengths. With this value of ${\alpha }_0$, 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 $\sigma$ meson ($T=0\mathrm{}$, $S$-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.