Evaluation of theI=0Pion-Pion Scattering Length Using a Forward-Scattering Dispersion Relation

Abstract

The $I=0\mathrm{}$ $\pi \pi$ scattering length is evaluated with a forward-direction elastic-scattering dispersion relation. The high-energy contribution to the dispersion integral is obtained on the assumption that the high-energy behavior of the forward scattering amplitude is dominated by a few leading crossed-channel Regge poles, while any available experimental information on total $\pi \pi$ cross sections is used to compute the low-energy contribution. The scattering length is found to be negative, with a calculated value of ${-1.7\mathrm{}}_{-0.5\mathrm{}}^{+1.3\mathrm{}}$ (in pion Compton wavelengths). Evaluation of the $I=1\mathrm{}$ amplitude at threshold yields the value -0.4, which is found to be consistent with zero; this indicates that the method used in the evaluation of the scattering length is not unreasonable.