Exponential asymptotic behaviour and low-energy pion-pion scattering

Abstract

A new approach to low-energy π-π scattering is described in which it is assumed that, because of the very short range of the important forces in this problem, the distant singularity structure is relatively important. It is remarked that the exponential decrease of the amplitude at fixed angle observed experimentally, which also corresponds to the maximum rate of decrease allowed theoretically in a reasonableS-matrix theory, gives information about this structure. The high-energy and low-energy phenomena are connected by using the fixed-angle method to calculate the partial-wave amplitudes at low energy. A representation is conjectured which contains the correct asymptotic behaviour as well as the nearby part of the left-hand cut which is known from crossing symmetry. A formal solution, analogous to the full determinantalN/D method, is presented. In a simple, but fairly realistic model an additional attractive potential is produced by the asymptotic left-hand cut; it is suggested that this may be responsible for the missing attraction in the ρ bootstrap mechanism.