Analytic Map of Three-Channel S Matrix: Generalized Uniformization and Mittag-Leffler Expansion

We explore the analytic structure of the three-channel S matrix by generalizing uniformization and making a single-valued map for the three-channel S matrix. First, by means of the inverse Jacobi's elliptic function we construct a transformation from eight Riemann sheets of the center-of-mass energy complex plane onto a torus, on which the three-channel S matrix is represented single-valued. Second, we show that the Mittag-Leffler expansion, a pole expansion, of the three-channel scattering amplitude includes not only topologically trivial but also nontrivial contributions and is given by the Weierstrass zeta function. Finally, taking a simple nonrelativistic effective field theory with contact interaction for the S=-2, I=0, J^{P}=0^{+}, ΛΛ-NΞ-ΣΣ coupled-channel scattering, we demonstrate that as a function of the uniformization variable the scattering amplitude is, in fact, given by the Mittag-Leffler expansion and is dominated by contributions from neighboring poles.