Determination of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi></mml:math>-Wave Pion-Pion Scattering Lengths

Abstract

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 $\ensuremath{\rho}$ and $f$ resonances in respective channels, that these are the only $\ensuremath{\pi}\ensuremath{\pi}$ 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${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$ (where ${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$ is the pion Compton wavelength and the pion-pion force range is expected to be 0.5${\mathrm{\ensuremath{\mu}}}^{\ensuremath{-}1}$ because of 2-pion exchange). The scattering lengths are determined as $\ensuremath{\mu}{a}_{0}=0.25\ifmmode\pm\else\textpm\fi{}0.08$ and $\ensuremath{\mu}{a}_{2}=0.0\ifmmode\pm\else\textpm\fi{}0.03$. The uncertainties are based upon the variations in ${a}_{0}$ and ${a}_{2}$ due to changes in the parametrization of the $\ensuremath{\pi}\ensuremath{\pi}$ 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 ($\ensuremath{\sigma}$ and $\ensuremath{\epsilon}$) 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.