Forward Scattering Amplitude and Univalent Functions

Abstract

Starting with the relativistic crossing-symmetric forward scattering amplitude, we constructed previously a function $g\left(E\right)\mathrm{}$ of energy $E$ which is both analytic and univalent (schlicht) in the upper-half energy plane. In this paper we exploit the univalence of $g\left(E\right)\mathrm{}$ to obtain information on the analytic properties of the forward-scattering amplitude. Various inequalities satisfied by $g\left(E\right)\mathrm{}$ are derived, making use of powerful theorems on univalent functions. In particular, we have established several theorems which relate the asymptotic behavior of the phase of $g\left(E\right)\mathrm{}$ to that of $\mathrm{}\left|g\right(E\left)\right|$ itself. We have also obtained several inequalities for $g\left(E\right)\mathrm{}$ which may be useful in an experimental test of the consequences of local field theory. We start with only those properties of the forward scattering amplitude that have already been proved in axiomatic local field theory. The only extra assumption used that has not yet been proved in field theory is the physical assumption that the forward scattering amplitude does not become relatively real in the high-energy limit.