  String theory offers an elegant and concrete realization of how to
consistently couple states of arbitrarily high spin. But how unique is this
construction? In this paper we derive a novel, multi-parameter family of
four-point scattering amplitudes exhibiting i) polynomially bounded high-energy
behavior and ii) exchange of an infinite tower of high-spin modes, albeit with
a finite number of states at each resonance. These amplitudes take an
infinite-product form and, depending on parameters, exhibit mass spectra that
are either unbounded or bounded, thus corresponding to generalizations of the
Veneziano and Coon amplitudes, respectively. For the bounded case, masses
converge to an accumulation point, a peculiar feature seen in the Coon
amplitude but more recently understood to arise naturally in string theory.
Importantly, our amplitudes contain free parameters allowing for the
customization of the slope and offset of the spin-dependence in the Regge
trajectory. We compute all partial waves for this multi-parameter class of
amplitudes and identify unitary regions of parameter space. For the unbounded
case, we apply similar methods to derive new deformations of the Veneziano and
Virasoro-Shapiro amplitudes.
