AnI=1ππSum Rule and the Derivatives of theS-WaveππAmplitudes

Abstract

A sum rule for the $I=1\mathrm{}$ $\pi \pi$ amplitude is written at the symmetry point, $s=t=u=\frac{4}{3}{m_{\pi }}^2$, under the assumption that $\frac{\partial A^1(t, V)}{\partial V}$ satisfies an unsubtracted fixed-$t$ dispersion relation, where $t$ and $V$ are the energy and momentum-transfer variables. From the symmetry-point conditions, $\frac{\partial A^0}{\partial t}$ and $\frac{\partial A^2}{\partial t}$ are obtained and compared with the current-algebra estimates.