Faddeev Equations for the Three-Pion Final-State Interaction inτ,τ′, andηDecays

Abstract

The anisotropy in the pion energy spectra for the various three-pion decay modes are assumed capable of providing information concerning the $\pi -\pi$ interaction parameters. Such determinations have so far yielded values that differ radically from those obtained from.the analysis of low-energy pion-nucleon elastic scattering as well as production phenomena. The problem is reconsidered in the light of the complete three-body scattering amplitude given according to the prescription of Faddeev. The assumptions made regarding the kernel of the integral equations include (i) a simple off-shell extension of the two-body scattering amplitude, separable in the initial and final momenta, (ii) a nonrelativistic framework except for the pion kinematics, and (iii) an attractive $s$-wave $\pi -\pi$ interaction in the absence of bound states or resonances. Besides the pion scattering-length parameters, we have a momentum cutoff which, over a wide range of values, has an insignificant effect on the shape of the pion energy spectrum. The integral equations are solved numerically and the parameters that best fit the pion spectra for $\tau$, ${\tau }^{\prime }$, and $\eta$ decay are obtained. In each case the $s$-wave scattering length $a_2$ is found to be much larger than $a_0$. This confirms the conjecture that the inclusion of the three-particle effects does not significantly alter the general features of earlier work.