A sum rule for the $I=2\mathrm{}$ amplitude at the symmetry point, $s=t=u=\frac{4}{3}m_{\pi }^{}{}_{}{}^2$, is written down with the assumption that the amplitude satisfies an unsubtracted fixed-energy dispersion relation. This sum rule can be saturated by the $S$ waves and the $\rho$ and $f^0$ mesons. The known crossing relations at the symmetry point connecting the value and the derivative of the $I=0\mathrm{}$ amplitude to those of the $I=2\mathrm{}$ amplitude are then used to show that the $I=0\mathrm{}$ $S$-wave scattering length must be small and negative.

• Received 25 March 1968

DOI:https://doi.org/10.1103/PhysRevLett.21.54