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

Application of hard-pion four-point functions is made to $\pi \pi$ scattering on the basis of the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ 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 $\pi \pi$ 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 $O\left(\frac{m_{\pi }^{}{}_{}{}^2}{m_{\rho }^{}{}_{}{}^2}\right)$. 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 $\delta _0^{}{}_{}{}^0-\delta _0^{}{}_{}{}^2\simeq {35}^{\circ }$. All existing data (up to 1 GeV) can be fitted by adjusting one model-dependent parameter.