Conditions that must be satisfied by crossing-symmetric finite-energy sum-rules

It is shown that there can be no exact solution of a crossing-symmetric finite-energy sum-rule in terms of a finite number of towers of zero-width resonances. If the Regge behaviour in a channel is to come from the resonances in the same channel, and not from some background term, there must either be appreciable overlap of broad resonances, or an infinite number of Regge trajectories. If complete crossing-symmetry is to be achieved, Regge cuts, as well as poles, must be incorporated. The above statements are supported by mathematical models; and the conclusion is that finite-energy sum-role bootstrap calculations with a finite number of towers of resonances should be treated with extreme caution, for they can, at best, be only very approximate.