Tree-level(π,K)amplitude and analyticity

I consider the tree-level amplitude describing all three channels of the binary $(\pi ,K)$ reaction, as a meromorphic, polynomially bounded function of three dependent complex variables. Using the Mittag-Leffler theorem, I construct three convergent partial fraction expansions, each one being applied in the corresponding domain. Noting that the mutual intersections of those domains are nonempty, I employ analytical continuation. It is shown that the necessary conditions to make such a continuation feasible are the following: (1) The only parameters completely determining the amplitude are the on-shell couplings and masses; (2) these parameters are restricted by a certain (infinite) system of bootstrap equations; (3) the full cross-symmetric amplitude takes the dual form even when the Pomeron contribution is taken into account; (4) this latter contribution corresponds to a nonresonant background which, in turn, is expressed in terms of cross-channel resonance parameters. Also, it is demonstrated that chiral symmetry provides a unique scale for the mentioned parameters, the resonance saturation effect appearing as a direct consequence of the above results.