Simple Model for Low-Energy Pion-Pion Scattering

A simple model for low-energy pion-pion scattering is given, which expresses the $S$-wave phase shifts in terms of the $\rho$-meson parameters. A vital assumption is that the isospin amplitudes are as free as possible from singularities—that is, that the threshold branch points are weak, and can be treated by successive approximation. The present version of the model obeys elastic unitarity above threshold. If a parameter is chosen in a range such that the amplitudes are crossing-symmetric below threshold to a high degree of accuracy, and such that the ${\pi }^0{\pi }^0$ $S$-wave amplitude obeys a set of constraints proven from axiomatic field theory, then the numerical predictions agree reasonably with experiment. A method of improving the model using partial-wave dispersion relations is indicated. Comparison with other, more complicated, models suggests that these often contain many superfluous features which obscure the essential origins of their predictions. It is concluded, for example, that there is no real difference between duality and interference models, as far as prediction of $S$-wave $\pi -\pi$ scattering lengths are concerned.