Dynamical equations for a Regge theory with crossing symmetry and unitarity. I. Introduction, and the case of weak coupling

Abstract

A program for construction of a crossing-symmetric unitary Regge theory of meson-meson scattering is proposed. The construction proceeds through solution of a nonlinear functional equation, $\ensuremath{\psi} = G(\ensuremath{\psi})$, for certain partial-wave scattering functions $\ensuremath{\psi}$. The functional equation is analogous to a conventional dynamical equation, in that the scattering amplitude is generated from input functions which describe the primary forces between mesons and possible inelastic effects. A solution of the equation provides a scattering amplitude having Mandelstam analyticity, exact crossing symmetry, exact unitarity below the production threshold, and meromorphy of partial waves in the right-half $l$ plane, with the consequent Regge asymptotics. Inelastic unitarity $[0\ensuremath{\le}\ensuremath{\eta}(l,s)\ensuremath{\le}1]$ is not guaranteed, but may perhaps be achieved through constraints on inputs. In any case, the partial waves are bounded throughout the physical region; such a bound was not ensured in earlier schemes based on the Mandelstam iteration. In this first paper of a series, the equations are formulated for the case of weak coupling, in which no Regge poles enter the right-half $l$ plane. Inelastic effects are described by crossed two-particle processes and assigned input functions. Later papers will treat the case of strong coupling, in which Regge trajectories are generated dynamically, and the extension of the formalism to include many coupled channels.