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

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 $q\ge 0$ and discover a new transcendental structure in its low-energy expansion. We show that there is no analogous $q$-deformation of the Virasoro amplitude.

arXiv commentv1, 23 pages; v2, fixed minor typos

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• Cutting the Coon Amplitude
• On (Scalar QED) Gravitational Positivity Bounds
• On the positivity of the hypergeometric Veneziano amplitude
• Uniqueness Criteria for the Virasoro-Shapiro Amplitude
• Positivity of the Hypergeometric Coon Amplitude
• Asymptotic safety, quantum gravity, and the swampland: a conceptual assessment
• Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers
• Lectures in Quantum Gravity
• Strings from Almost Nothing
• Worldsheet for Generalized Veneziano Amplitudes
• A glimpse into the Ultrametric spectrum