Crossing as a Group and Elimination of Exotic Channels

Crossing transformations constitute a group of permutations under which the scattering amplitude is invariant. Using Mandelstam's analyticity, we decompose the amplitude into irreducible representations of this group. The usual quantum numbers, such as isospin or $\mathrm{SU}\left(3\right)\mathrm{}$, are "crossing-invariant." Thus no higher symmetry is generated by crossing itself. However, elimination of certain quantum numbers in intermediate states is not crossing-invariant, and higher symmetries have to be introduced to make it possible. The current literature on exchange degeneracy is a manifestation of this statement. To exemplify application of our analysis, we show how, starting with $\mathrm{SU}\left(3\right)\mathrm{}$ invariance, one can use crossing and the absence of exotic channels to derive the quark-model picture of the tensor nonet. No detailed dynamical input is used.