Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon

TL;DR

This paper explores the mathematical properties of key string scattering amplitudes, revealing new structures and deformations, especially focusing on the Coon amplitude as a $q$-deformation of the Veneziano amplitude.

Contribution

It introduces a $q$-analysis approach to the Coon amplitude, establishing it as a $q$-deformation of the Veneziano amplitude and uncovering new transcendental structures.

Findings

Coon amplitude is a $q$-deformation of the Veneziano amplitude for all $q \\geq 0$.

Identifies a new transcendental structure in the low-energy expansion of the Coon amplitude.

No $q$-deformation exists for the Virasoro amplitude.

Abstract

We detail the properties of the Veneziano, Virasoro, and Coon amplitudes. These tree-level four-point scattering amplitudes may be written as infinite products with an infinite sequence of simple poles. Our approach for the Coon amplitude uses the mathematical theory of $q$-analysis. We interpret the Coon amplitude as a $q$-deformation of the Veneziano amplitude for all $q \geq 0$ and discover a new transcendental structure in its low-energy expansion. We show that there is no analogous $q$-deformation of the Virasoro amplitude.