Regge Surfaces and Singularities in a Relativistic Theory

Abstract

The analyticity properties of a partial wave amplitude $a(l,s)$ are discussed in a relativistic theory. A domain of holomorphy in $l$ exists when the full amplitude $A(s,t)$ is bounded by a polynomial in $t$ and satisfies the Mandelstam representation. The amplitude $a(l,s)$ is unique provided it is required to satisfy the asymptotic conditions on $l$ needed for Carlson's theorem.The types of singularity of $a(l,s)$ are studied by tracing them along the Regge surfaces in ($l,s$) space on which they lie. We divide singularities into different classes that depend on the nature of the Regge surfaces. For one class inelastic (production) processes are unimportant, and we deduce that these singularities must be poles. Another class is related to the possibility of a Regge surface going into an inelastic unphysical sheet via the elastic unphysical sheet. It is shown that knowledge of this remote sheet is very important to the problem of establishing meromorphy domains of $a(l,s)$.By means of unitarity and dominant Born terms we study the general form of the real sections of Regge surfaces with reference in particular to the occurrence of resonances.