Approximate Method for Determining the Elastic-Scattering Amplitude for Strong Interactions

A method for calculating the elastic-scattering amplitude in the $S$-matrix theory of strong interactions is proposed which does not require a partial-wave expansion of the amplitude. Crossing symmetry is satisfied by the amplitude, but unitarity is imposed only approximately. Equations are derived for the case of scattering of two spinless particles of unequal mass. The special case where the masses are equal is considered in detail for the input assumption that the scattering is predominantly $S$ wave. Crossing symmetry introduces higher partial-wave contributions to the amplitude. The amplitude calculated in this way is in good agreement with the input assumption. The amount of violation of unitarity is least near threshold, but is only on the order of a percent at $s=8{m}^{2}$. In spite of this, there are serious problems with low-energy resonances and bound states. It is concluded that both unitarity and crossing symmetry are important in the production of resonances and bound states and that the modification of either may lead to difficulties. The total cross section derived from the approximate amplitude is compared with that obtained using the partial-wave expansion and keeping only the $S$ wave. The results are in good agreement with each other.