Determination ofS-Wave Pion-Pion Scattering Lengths

The S-wave pion-pion scattering lengths a₀ and a₂ in the channels of total isospin 0 and 2, respectively, are determined by requiring that the high-energy limit of the pion-pion total cross section be the same in all isospin channels. The determination consists of using the once-subtracted dispersion relation and the phase representation which are satisfied by the crossing-symmetric forward pion-pion amplitudes and also the unsubtracted dispersion relation valid for the crossing-antisymmetric amplitude. The specific approximations to be made are that the scattering becomes asymptotic fairly rapidly above the ρ and f resonances in respective channels, that these are the only ππ resonances in the energy region up to the f resonance, that the S wave dominates below the resonances, and that the conventional effective-range expansion is valid for the S wave with the effective range between zero and 2μ^-1 (where μ^-1 is the pion Compton wavelength and the pion-pion force range is expected to be 0.5μ^-1 because of 2-pion exchange). The scattering lengths are determined as μa₀=0.25±0.08 and μa₂=0.0±0.03. The uncertainties are based upon the variations in a₀ and a₂ due to changes in the parametrization of the ππ scattering used in the present determination. It is found that the unknown details of high-energy scattering are relatively unimportant in this determination of a₀ and a₂. It is shown that the above values of a₀ and a₂ are consistent with the partially conserved axial-vector current sum rule due to Adler. This is contrary to the conclusion of previous authors; we attribute the difference to a different use of the sum rule. When one of the conjectured resonances (σ and ε) is added as a true resonance, no solution is found to make the high-energy limit of the total cross section the same with the parametrization of the phase and the cross section considered in the present work.