On unitarity of the Coon amplitude

Abstract

The Coon amplitude is a one-parameter deformation of the Veneziano amplitude. We explore the unitarity of the Coon amplitude through its partial wave expansion using tools from q-calculus. Our analysis establishes manifest positivity on the leading and sub-leading Regge trajectories in arbitrary spacetime dimensions D, while revealing a violation of unitarity in a certain region of (q, D) parameter space starting at the sub-sub-leading Regge order. A combination of numerical studies and analytic arguments allows us to argue for the manifest positivity of the partial wave coefficients in fixed spin and Regge asymptotics.

Positivity of the hypergeometric Coon amplitude

“…This outcome falls within the range where r > −1 and precisely matches the bound r ≥ 0 when m 2 = −1 [23]. Furthermore, we can also consider the limit as r → 0, the partial-wave coefficients that have been proven to be non-negative in the Coon [1,21].…”

“…to represent and solve for the partial-wave coefficients. This method has, also, been very successful in studies of the Veneziano and the Coon [17,21]. Note that [19] use the contour integral representation in studying generalisations of string amplitudes.…”

“…Note that, despite this, we still lack a statement of manifest unitarity derived from the string amplitude perspective below the critical dimension of the (super)string. Subsequent works explored some unitarity properties of the Coon amplitude, including some analytic studies and numerical methods [18][19][20][21][22]. Further studies of positivity of the partial-wave coefficients can be found in [23] for the case of the hypergeometric Veneziano amplitude that was derived in [7].…”