Theory of Low-Energyπ−πScattering

We make a self-contained examination of the $\pi -\pi$ scattering problem on the basis of the $S$-matrix conjecture, using analyticity, elastic unitarity, and crossing symmetry. The crossing-symmetry relations are derived without using the double-dispersion relations. The nearest singularity hypothesis is adopted. Inelastic effects and distant left-hand singularities are crudely incorporated by means of polynomials. Nearby left-hand singularities are chosen in such a way the functional forms are correct near the branch point with unknown parameters. The amplitude so constructed satisfies elastic unitarity exactly in one channel; the parameters are adjusted through self-consistency conditions derived from crossing symmetry. By use of the inverse amplitude, we are able to construct simple amplitudes and find solutions by hand calculation. A $p$-wave resonance is generated for negative values of $\lambda$ with $\left|\lambda \right|<\mathrm{}0.39\mathrm{}$. Solutions exhibit the development of $I=0\mathrm{}$ $s$-wave bound states for negative values of $\lambda$ with $\left|\lambda \right|\ge 0.44$ and of $I=1\mathrm{}$ $p$-wave bound states for negative values of $\lambda$ with $\left|\lambda \right|\ge 0.39$. We find the range of $\lambda$, $-0.3\mathrm{}\lesssim \lambda <0.1\mathrm{}$, to be self-consistent.