Group Representations for Complex Angular Momentum

Abstract

The notion of complex angular momentum has come to the fore in S-matrix theory since the work of Regge, who showed that for a complete description of potential scattering, an analytical continuation of the scattering amplitude into the complex angular momentum had to be made; but the idea is well-recognized in mathematics, where the Schlafli integral representation, with suitable cuts, is a solution of the Legendre equation for arbitrary values of l, the angular momentum quantum number. Attempts at extending the description in terms of Regge poles to the relativistic problem of elementary particle interactions have not been uniformly successful, though there are interactions (such as proton-proton or K + mesonproton scattering) in which the characteristic shrinking of the diffraction peak at high energies predicted by the Regge pole hypothesis is observed, and though there are situations such as that in the multiperipheral model of Amati et al., where the summation over the ladder diagrams gives rise to a Regge pole behavior for the scattering amplitude. One would like to have such a behavior, because the Regge pole moving on its trajectory can describe a bound state or a resonance and also the high-energy behavior of cross sections—all that one could wish for from a scattering theory.