Low-Energyp-Waveπ−πInteraction

A sum rule, based on forward dispersion relations, is discussed which makes possible the calculation of the $p$-wave $\pi -\pi$ scattering amplitude near threshold using as input principally the parameters of the $\rho$-meson resonance. This sum rule contains kinematic factors which suppress both the $s$-wave amplitude at threshold and the high-energy behavior. The $p$-wave scattering length is found to be $a_1=0.040\mathrm{}\pm 0.005$ (units of ${m_{\pi }}^{-3\mathrm{}}$) which is compared with predictions based on partially conserved axial-vector current and with the $\rho$-meson "tail."