Phase Contours of Scattering Amplitudes. II. Crossing Symmetry, Resonance Poles, and High-Energy Behavior

Abstract

Phase contours are used to study consistency conditions for a symmetric scattering amplitude having a given high-energy behavior. The latter is taken to correspond to dominance by Regge poles that have continuously rising trajectories. By means of crossing symmetry we deduce the existence of a sequence of real zeros of the amplitude that lie along lines of symmetry in our model, $s=u$, in $t<0$, for example. The leading zero in this sequence is related to the scattering length. Under quite general conditions these zeros may be related, via complex surfaces of zeros, to the zeros of Regge residues below threshold. By considering complex sections of phase-contour surfaces, we show that these zeros may also be related to a sequence of zeros at complex points that are due to interference between resonance poles on unphysical sheets above threshold.