New relations between analyticity, Regge trajectories, Veneziano amplitude, and Moebius transformations

TL;DR

This paper explores the deep connections between analyticity in scattering amplitudes, Regge trajectories, the Veneziano model, and Moebius transformations using conformal mapping and geometric interpretations.

Contribution

It introduces a novel geometric framework linking Regge trajectories and automorphic functions via Moebius transformations, enriching the theoretical understanding of scattering amplitudes.

Findings

Established a connection between Regge trajectories and triangle angles in conformal maps.

Linked automorphic functions and Moebius group generators to scattering amplitude properties.

Provided new relations that integrate analyticity, geometry, and group theory in particle physics.

Abstract

In this paper we use the analyticity properties of the scattering amplitude in the context of the conformal mapping techniques. The Schwarz-Christoffel and Riemann-Schwarz functions are used to map the upper half -plane onto a triangle. We use the known asymptotic and threshold behaviors of the scattering amplitude to establish a connection between the values of the Regge trajectory functions and the angles of the triangle. This geometrical interpretation allows a link between values of the Regge trajectory functions and the generators of the invariance group of Moebius transformations associated with the underlying automorphic function. The formalism provides useful new relations between analyticity, geometry, Regge trajectory functions, Veneziano model, groups of Moebius transformations and automorphic functions. It is hoped that they will provide avenues for further work.