S-wave interaction in a three-pion system and the decay of K_π3

Abstract

A model of s-wave (T = 0,2) π-π interaction is proposed from certain formal analogies with field theory, on lines similar to a recent paper (16) by one of us (ANM). The model leads in an exact fashion to a relativistic effective range formula of the two phase shifts δ₀⁰ and δ₀², from which a simple relation is obtained between the s-wave scattering lengths and the strength parameters of the corresponding interactions. The model is then used to solve a three-body Schrödinger equation in momentum space for the isovector 3π wave function in a purely outgoing state so as to make the results applicable to K_π3-decay. The possible symmetries of the 3π system (with two pions identical) are fully taken into account. This wave function is then employed for evaluation of the energy distribution of the unlike pion in τ-decay, on the usual assumption of zero range for the weak interaction. The results are qualitatively similar to those known from earlier dispersion theoretic treatments. However attractive π-π interactions seem to have little effect on the spectrum, suggesting that the structure of the weak interaction itself plays the dominant role.