On the dynamical determination of the Regge pole parameters

Abstract

The purpose of this article is to outline a method, based on the principles of analyticity and unitarity in the t channel, which may help to determine some dynamical properties of the Regge parameters α( t) and β( t). In the introduction we list various applications of this method, and discuss the role of crossing symmetry and unitarity in all three channels in relation to the uniqueness of the solutions. We derive the unitarity condition satisfied by the partial wave amplitude A ( l, t), for l complex, for a relativistic, two-body scattering process. Upon neglecting intermediate states of more than two particles, the unitarity condition can be expressed in terms of Regge parameters. An approximate form for the unitarity condition, accurate at low t, is next derived. This form will be used for numerical work. In Section III we show, in the relativistic case, that the functions α( t) and β( t), describing a boson Regge pole, are analytic with only right-hand cuts, in addition to those arising from the crossing of Regge trajectories. Our proof is based on two assumptions: (1) validity of the Mandelstam representation; (2) analyticity of A ( l, t) in the whole l plane, with at most poles and essential singularities. The consequence of the existence of essential singularities at l = −1, −2, −3, … in relation to α( t) and β( t) is especially discussed. Finally, we note in this section how the preceding results are modified if the Regge pole being considered is a Fermion. In Section IV we write dispersion relations for α( t) and β( t). These, together with the unitarity condition of Section II, constitute a tentative method for the dynamical determination of the Regge parameters. We outline an extension of our method which is appropriate for discussions of Fermion Regge poles. The behavior of α( t) and β( t) at the elastic or inelastic thresholds is derived, and applied to perform subtractions in the dispersion relation for β( t). Finally, in Section V, we turn specifically to π-π scattering and discuss an approximation which might possibly lead to a reasonably accurate description of this process. Estimates of the range of validity of the approximation are made.