Two-Variable Expansion of the Scattering Amplitude for any Mass and Spin and Crossing Symmetry for Partial Waves

Abstract

We derive an infinite number of sum rules for the process $a+b\to c+d$, where $a$, $b$, $c$, and $d$ are particles of arbitrary mass and spin. Each of the sum rules involves a finite number of partial waves. They are implied by the crossing symmetry of the system and are complete. A classification of these relations into an independent set suggests a basis for the two-variable expansion of the scattering amplitude. The basis has the further virtue that it explicitly displays the kinematic singularities of the amplitude and the threshold and pseudothreshold zeros of the partial waves. The formalism is also valid for a decay process. The partial waves in the sum rules then refer to amplitudes where two of the three final particles are in a state of definite angular momentum, while the region of expansion becomes the Dalitz plot.