A set of model amplitudes, characterized by two functions $h\left(x\right)\mathrm{}$ and $\omega \mathrm{}\left(x\right)\mathrm{}$, is given which satisfies the conditions of duality, crossing symmetry, and Regge behavior. These model amplitudes can be written as $G\left(\alpha \mathrm{}\right(s), \alpha \mathrm{}(t\left)\right)=\int 0^1\mathrm{dx}h\left(x\right)\mathrm{}{x}^{-\alpha \mathrm{}\left(t\right)\mathrm{}}{\left[\omega \mathrm{}\right(x\left)\right]}^{-\alpha \mathrm{}\left(s\right)\mathrm{}}$ and are generalizations of the beta function. The model has resonances of zero width, like Veneziano's model. We prove that, under certain conditions, the model amplitude has Regge behavior as $s\to \infty$ with $\arg \mathrm{}(-s)<\pi$ and $t$ fixed, and that it dies exponentially as $s\to \infty$ with $0<\left|\arg \mathrm{}\right(s\left)\right|<\mathrm{}\pi$ and $u$ fixed, as does the amplitude in Veneziano's model. We give specific examples of $\omega \mathrm{}\left(x\right)\mathrm{}$ and $h\left(x\right)\mathrm{}$ which satisfy all the required conditions, and show that the models may be generalized to models which bear the same relation to the model $\frac{\Gamma (1-\alpha \mathrm{}(s\left)\right)\Gamma (1-\alpha \mathrm{}(t\left)\right)}{\Gamma (1-\alpha \mathrm{}(s)-\alpha \mathrm{}(t\left)\right)}$ as the above models bear to the beta function.

• Received 28 August 1969

DOI:https://doi.org/10.1103/PhysRevD.1.2888