Cutting the Coon Amplitude

Abstract

The Coon amplitude is a $q$-deformed generalization of the Veneziano amplitude exhibiting a semi-infinite sequence of poles that converge on an accumulation point, from which a branch cut emerges. A number of recent papers have provided compelling evidence that the residues of this amplitude satisfy the positivity requirements imposed by unitarity. This paper investigates whether positivity is also satisfied along the branch cut. It is found that positivity violations occur in a region of the branch cut exponentially close to the accumulation point according to a scale set by $q$. The closing section of the paper discusses possible interpretations of this fact and strategies for excising negativity from the partial wave coefficients. An appendix presents derivations of instrumental identities relating the $q$-gamma and $q$-polygamma functions to the Weierstrass elliptic and quasiperiodic functions.

v2: fixed typo in equation (55), fixed Figure 1, added two references, made the summary of section 2 in the introduction more precise, edited discussion in second bullet point in section 3,

References in corpus (11)

• Gluon scattering amplitudes at strong coupling
• Dynamics of Flux Tubes in Large N Gauge Theories
• On Unitarity of Tree-Level String Amplitudes
• Veneziano Variations: How Unique are String Amplitudes?
• Generalized Veneziano and Virasoro amplitudes
• On the unitarity and low energy expansion of the Coon amplitude
• Accumulation-Point Amplitudes in String Theory
• UV-Complete Gravity Amplitudes and the Triple Product
• Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon
• Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d\ge 2$
• On the positivity of Coon amplitude in D=4

Cited by in corpus (12)

• Bespoke Dual Resonance
• The Stringy S-matrix Bootstrap: Maximal Spin and Superpolynomial Softness
• Multiparticle Factorization and the Rigidity of String Theory
• Bootstrap Principle for the Spectrum and Scattering of Strings
• On the positivity of the hypergeometric Veneziano amplitude
• Uniqueness Criteria for the Virasoro-Shapiro Amplitude
• Positivity of the Hypergeometric Coon Amplitude
• Coon unitarity via partial waves or: how I learned to stop worrying and love the harmonic numbers
• Strings from Almost Nothing
• A glimpse into the Ultrametric spectrum
• Higher-Spin and Higher-Point Constraints on Stringy Amplitudes
• Worldsheet for Generalized Veneziano Amplitudes