The Baker-Coon-Romans $N$-point amplitude and an exact field theory limit of the Coon amplitude

We study the $N$-point Coon amplitude discovered first by Baker and Coon in the 1970s and then again independently by Romans in the 1980s. This Baker-Coon-Romans (BCR) amplitude retains several properties of tree-level string amplitudes, namely duality and factorization, with a $q$-deformed version of the string spectrum. Although the formula for the $N$-point BCR amplitude is only valid for ${q > 1}$, the four-point case admits a straightforward extension to all ${q \geq 0}$ which reproduces the usual expression for the four-point Coon amplitude. At five points, there are inconsistencies with factorization when pushing ${q < 1}$. Despite these issues, we find a new relation between the five-point BCR amplitude and Cheung and Remmen's four-point basic hypergeometric amplitude, placing the latter within the broader family of Coon amplitudes. Finally, we compute the $q \to \infty$ limit of the $N$-point BCR amplitudes and discover an exact correspondence between these amplitudes and the field theory amplitudes of a scalar transforming in the adjoint representation of a global symmetry group with an infinite set of non-derivative single-trace interaction terms. This correspondence at $q = \infty$ is the first definitive realization of the Coon amplitude (in any limit) from a field theory described by an explicit Lagrangian.