Pion-pion scattering in the strip approximation - I

Abstract

In this paper, results are presented of calculations on pionpion scattering in the strip approximation. A cut-off is made to avoid divergences at high energy. The starting point is an assumed form for the low partial waves based on experiment, supplemented by partial-wave solutions of the pion-pion problem. The spectral functions are calculated and two sets of predictions are made. Firstly, higher elastic partial waves, in particular theD-waves, are evaluated, as are also the low-energy inelastic partial waves. These predictions are relatively insensitive to the cut-off. Secondly, Regge trajectories are deduced from the asymptotic behaviour of the spectral functions. These do depend critically on the cut-off and the object here is so to adjust the action of the cut-off as to reproduce the expected properties of the trajectories. This is partially successful. Solutions are found in which the I=0 trajectory passes approximately throughl = l att = 0 and which at the same time gives a reasonable reproduction of the ρ resonance in the I=1 state. The bootstrap mechanism for the ρ resonance is only possible when a substantial I= 0D-wave at fairly high energy is included in the input. This is identified with the f0 resonance recently observed. The f0 itself is not reproduced although the low-energyI=0D-wave is probably