Phase Contours of Scattering Amplitudes. I. Phase Contours, Zeros, and High-Energy Behavior

We define phase contours as curves along which the phase of an invariant scattering amplitude is a constant. These curves are sections of a complex surface, which we take either in the real ($s,t$) plane or in the complex plane of one of the variables. The relation between phase contours, zeros of the amplitude, and high-energy behavior is discussed. Characteristic features of phase contours are investigated in a variety of special models, and it is seen that basic assumptions about high-energy behavior can readily be expressed in terms of the topology of the phase contours. Specific illustrations of phase contours are given for pionnucleon scattering based on phase-shift solutions at low and medium energies and on an extrapolation from Regge solutions at high energies.