Crossing-Symmetric Amplitudes with Arbitrary Regge Behavior Based on Elementary Symmetric Functions

A sequence of amplitudes for $\pi \pi \to \pi \omega$ is constructed exhibiting exact crossing symmetry, Regge asymptotic behavior, direct-channel poles, and arbitrary nonlinear trajectories. The sequence, constructed out of the elementary symmetric functions of $s$, $t$, and $u$, possesses some of the desired features in a limiting sense only. In the complex angular momentum plane, any member of the sequence contains a leading finite array of simple poles spaced by two units, and additional nonleading arrays of higher-order singularities. A similar amplitude for the elastic scattering of neutral scalar bosons is proposed to test the unitarizability of the representation by finding a suitable nonlinear trajectory.