Model for PionS-Wave Phase Shifts from Current Algebra and Partial Conservation of Axial-Vector Current

From the algebra of axial-vector charges and divergences, using extensively the principle of pion-pole dominance, we derive a set of sum rules for ($\pi \pi$) $S$-wave interactions in the isospin-zero channel. In a single-particle approximation, they provide information on the conjectured $\sigma$ meson. In the approximation of elastic unitarity, the sum rules take the form of an integral equation for a vertex function $〈\pi \left|\sigma \right|\pi 〉$, which leads to the inequality $0<\mathrm{}{\delta }_0<\pi$ for the ($\pi \pi$) $S$-wave phase shift ${\delta }_0$ in the isospin-zero channel, whenever it is approximately valid. Exact solutions of the approximate integral equation, however, cannot be constructed, and we have to introduce an effective cutoff function as a correction to our approximation of the partial conservation of axial-vector current. The solutions for ${\delta }_0$ give a scattering length slightly larger than Weinberg's result and a broad maximum of the phase shift around 700 MeV; its height is sensitive to the cutoff parameter.