Unitarity Corrections to Current-Algebra Calculations of thes-Wave Pion-Pion Scattering Lengths

A method of imposing unitarity on the amplitude obtained from current algebra is examined as a possible extrapolation procedure for the calculation of the $s$-wave pion-pion scattering lengths. This method consists of two steps: The current-algebra result is incorporated to fix the amplitude at the symmetry point in the unphysical region, and the dispersion relation based on rigorous grounds is used to extrapolate up to the physical threshold. This procedure allows us to derive a set of exact sum rules for the $s$-wave scattering lengths which enables us to estimate the unitarity corrections to the current-algebra calculations.