On the unitarity and low energy expansion of the Coon amplitude

TL;DR

This paper investigates the unitarity and low energy behavior of the Coon amplitude, revealing positivity regimes, extending Veneziano amplitudes, and connecting these findings to the EFT-hedron and low spin dominance.

Contribution

It derives positivity conditions and low energy expansions for the Coon amplitude, revealing critical dimensions and extending Veneziano amplitudes to massive scalars.

Findings

Amplitude switches from positive in all dimensions to critical dimensions

Veneziano amplitudes extended to massive scalars with m^2 up to 1/3

First couplings computed using q-deformed Riemann zeta values

Abstract

The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switches from a regime where it is positive in all dimensions to a regime with critical dimensions, that connects to the known $d = 26, 10$ when the deformation is removed. En passant, we find that the Veneziano amplitudes can be extended to massive scalars of masses up to $m^2 = 1/3$, where it has critical dimension 6.3. On the low-energy side, we compute the first few coupligs of the theory in terms of $q$-deformed analogues of the standard Riemann zeta values of the string expansion. We locate their location in the EFT-hedron, and find agreement with a recent conjecture that theories with accumulation points populate this space. We also discuss their relation to low spin dominance.