Application of Hard-Pion Four-Point Functions to Pion-Pion Scattering

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Application of hard-pion four-point functions is made to $\ensuremath{\pi}\ensuremath{\pi}$ scattering on the basis of the $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ current algebra, a conserved vector current, a partially conserved axial-vector current, and the hypothesis of single-meson dominance of intermediate states in $T$ products. The calculation uses techniques previously developed for exploiting the content of the current algebra and the subsidiary conditions for an $N$-point process. The $\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitude is shown to include, besides the well-known pole diagrams, a set of seagull terms. The Weinberg scattering lengths and effective ranges are found to be accurate to within a few percent, since the hard-pion corrections at threshold are only of $O(\frac{m_{\ensuremath{\pi}}^{}{}_{}{}^{2}}{m_{\ensuremath{\rho}}^{}{}_{}{}^{2}})$. In the low-energy region, the scattering-phase shifts are seen to be generally small and essentially model-independent, while at the $K$-meson mass we find $\ensuremath{\delta}_{0}^{}{}_{}{}^{0}\ensuremath{-}\ensuremath{\delta}_{0}^{}{}_{}{}^{2}\ensuremath{\simeq}{35}^{\ensuremath{\circ}}$. All existing data (up to 1 GeV) can be fitted by adjusting one model-dependent parameter.