IsoscalarS-WaveππPhase Shift from a Unitary Current-Algebra Model

Abstract

Using the equal-time commutation relations of chiral $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$, we derive a Ward identity relating the vertex function $〈\pi \left|\sigma \right|\pi 〉$ to the $\sigma$ propagator. Adopting a model for the primitive three-point function of two axial-vector currents and an isoscalar $\sigma$ operator, we derive an integral equation for the vertex function by inserting elastic unitarity into the identity. In the approximation of neglecting inelastic effects, we solve the integral equation exactly. The solution contains three arbitrary constants which we fix from current-algebra and partially conserved axial-vector current considerations. We then obtain an isoscalar, $S$-wave $\pi -\pi$ phase shift which is positive and rises to 180° at infinity. Moreover, it corresponds to a scattering length $a_0$ just 5% larger than Weinberg's and, together with the $\rho$ and $f$ contributions, saturates the $\pi -\pi$ Adler-Weisberger relation to within 10%. Finally, we predict $m_{\sigma }=725\mathrm{}\pm 55$ MeV and ${\Gamma }_{\sigma \to \pi \pi }=455\mathrm{}\pm 55$ MeV.