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

TL;DR

This paper develops a mathematical framework using uniformization and Mittag-Leffler expansion to analyze the analytic structure of the three-channel S matrix, representing it as a single-valued function on a torus and including nontrivial pole contributions.

Contribution

It introduces a novel uniformization method transforming the multi-sheeted S matrix onto a torus and expresses the scattering amplitude via Mittag-Leffler expansion with Weierstrass functions, including nontrivial contributions.

Findings

The three-channel S matrix can be represented as a single-valued function on a torus.

The Mittag-Leffler expansion includes both trivial and nontrivial pole contributions.

In a specific model, the scattering amplitude is dominated by contributions from neighboring poles.

Abstract

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 squared complex plane onto a torus, on which the three-channel $S$ matrix is represented single-valued. Secondly, 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, we examine the obtained formula in the context of a simple three-channel model. Taking a simple non-relativistic effective field theory with contact interaction for the $S=-2$, $I=0$, $J^P = 0^+$, $\Lambda\Lambda-N\Xi-\Sigma\Sigma$ coupled-channel scattering, we demonstrate that the scattering amplitude as a function of the uniformization variable is, in fact, given by the Mittag-Leffler expansion with the Weierstrass zeta function and that it is dominated by contributions from neighboring poles.