π−πScattering with Unitarity and Crossing

Abstract

A model of pion-pion scattering is presented in which each partial-wave amplitude in the $s$ channel satisfies unitarity including inelastic states. This is accomplished by the use of the $\frac{N}{D}$ technique. The full amplitude, i.e., the explicit, continued, partial-wave sum does not satisfy unitarity in the $t$ and $u$ channels, but it does have the correct branch points in these channels. These characteristics are guaranteed by construction and do not depend on arbitrary parameters occurring in the $N$ functions. The parameters are fixed by imposing the crossing relations and by demanding the existence of a Pomeranchuk trajectory in the $T=0\mathrm{}$ channel. The existence of the $\rho$ is not assumed. Since crossing is not satisfied exactly, the parameters are determined as those that yield the best fit to the crossing relations. Once the parameters have been fixed, the behavior of each partial wave in each isospin channel is determined. Phase shifts are presented for the $S$, $P$, and $D$ waves. The $P$ wave is repulsive at low energies but becomes quite attractive at higher energies due to inelastic effects. A resonance occurs at $s=33.7\mathrm{}{\mu }^2$ with a width about three times that experimentally observed; this width is extremely sensitive to the parameters, the position is not. An $S$-wave ghost occurs at $s=-57.1\mathrm{}$ ${\mathrm{\mu }}^2$ (whose residue is zero) as well as the Pomeranchuk trajectory at $s=0\mathrm{}$. The $S$ waves are strongly repulsive at low energies. In fact, they are so repulsive that rather broad peaks are produced in their cross sections near 400 MeV when the phase shifts pass through $-\frac{\pi }{2}$. The peak in the $T=0\mathrm{}$ channel may very well correspond to the Abashian, Booth, and Crowe (ABC) anomaly. There are no resonances in the $D$ waves. In particular, the existence of an $f_0$ is incompatible with any choice of the parameters unless it is accompanied by another strong $D$-wave resonance at low energy, and even this possibility violates crossing badly. This may be due to a poor choice for our trial function.