Analysis of s-t symmetric classical S-matrices

TL;DR

This paper investigates the analytic structure of classical four-particle S-matrices with s-t crossing symmetry, revealing a regular pole spectrum and extending results to more complex cases with accumulating poles, including connections to the Coon S-matrix.

Contribution

It demonstrates that under certain conditions, classical S-matrices have equally spaced poles and extends these findings to cases with accumulating poles, linking to the Coon S-matrix.

Findings

S-matrices exhibit equally spaced poles under analytic conditions

Extension to S-matrices with accumulating poles matches Coon S-matrix spectrum

Encountered functions with non-isolated singularities resembling natural boundaries

Abstract

We analyze the complex analytic properties of Classical (tree-level) S-matrices for four scalar particles with s-t crossing symmetry, involving an infinite number of exchanges. Under suitable analytic conditions, we demonstrate that such S-matrices exhibit a spectrum of poles that is equally spaced. We extend this result to S-matrices with accumulating poles, proving that under analogous conditions, their pole spectrum coincides with that of the Coon S-matrix. The boundedness of the S-matrix in the Regge limit is not essential for our results. While studying S-matrices that do not meet the conditions of our theorems, we encounter functions that have novel non-isolated singularities akin to what is called the natural boundary.