Dispersion Relations and High Energy Limits in Quantum Field Theory. III

Abstract

It is shown that pion-nucleon forward scattering amplitude must have zeros on the com­ plex energy plane. This is derived by imposing the necessary condition for the existence of solutions on the dispersion relation for piort-nucleon forward scattering. By extending this result to the case of pion propagator, discussions against perturbation theory are made. Finally by applying this necessary condition to the dispersion relation given by Goldberger, it is concluded that either the pion-nucleon total cross sections increase at very high energies or Goldberger's dispersion relation has no solution. 2 > if Low's equation or dispersion relation has a unique solution. Some authors insisted that the solution of dispersion rela­ tion can be determined uniquely by requiring the possibility of power series expansion of the amplitude or propagator with respect to coupling constants. This requirement is equivalent to the exclusion of zeros from the amplitude or propagator in the complex energy planes. In § 2 of this paper, we shall show that the pion-nucleon forward scat­ tering amplitude must have zeros on the complex energy plane. It should be stressed that this result is independent of the assumption on the high energy limit of the amplitude. It is a mere consequence of a necessary condition for the existence of the solution of the dispersion relation for pion-nucleon forwarcl scattering. Such a condition has been derived for pion propagator by, Lehmann, Symanzik and Zimmermann 3 > and a famous theorem on the damping of vertex function at high energy limit has been obtained, but problems relating to the existence of zeros remain unsolved. The reason why we can draw the answer to the problems of the existence of zeros only for the forward scattering ampli­ tude but not for the propagator is that we have some amount of knowledge estab­ lished experimentally for pion-nucleon scattering, for example, the S-wave scat­ tEring lengths, total cross sections, whereas we have not any experiiLental information on the propagator.