Determination ofS-Wave Pion-Pion Scattering Lengths

The $S$-wave pion-pion scattering lengths $a_0$ and $a_2$ 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 $\rho$ and $f$ resonances in respective channels, that these are the only $\pi \pi$ 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 2${\mathrm{\mu }}^{-1\mathrm{}}$ (where ${\mathrm{\mu }}^{-1\mathrm{}}$ is the pion Compton wavelength and the pion-pion force range is expected to be 0.5${\mathrm{\mu }}^{-1\mathrm{}}$ because of 2-pion exchange). The scattering lengths are determined as $\mu a_0=0.25\mathrm{}\pm 0.08$ and $\mu a_2=0.0\mathrm{}\pm 0.03$. The uncertainties are based upon the variations in $a_0$ and $a_2$ due to changes in the parametrization of the $\pi \pi$ scattering used in the present determination. It is found that the unknown details of high-energy scattering are relatively unimportant in this determination of $a_0$ and $a_2$. It is shown that the above values of $a_0$ and $a_2$ 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 ($\sigma$ and $\epsilon$) 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.