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 $S=-2$, $I=0$, $J^P=0^{+}$, $\mathrm{\Lambda }\mathrm{\Lambda }-N\mathrm{\Xi }-\mathrm{\Sigma }\mathrm{\Sigma }$ 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.

Figure 1

Two-sheeted complex $z_{12}$ plane for the three-channel S matrix. Eight Riemann sheets of the complex $\sqrt{s}$ plane are specified by a set of complex channel momenta, e.g., $[ttb{]}_{+}$ means $\mathrm{Im}{q}_1>0$, $\mathrm{Im}{q}_2>0$, $\mathrm{Im}{q}_3<0$, and $\mathrm{Im}\sqrt{s}>0$. $T_1^{++}$, $T_2^{++}$, and $T_3^{++}$ are physical thresholds of channels 1, 2, and 3, respectively. $T_3^{++}$ together with its unphysical counterparts, $T_3^{+-}$, $T_3^{-+}$, and $T_3^{--}$ form the set of branch points.

Figure 2

Torus representation of the three-channel S matrix.

Figure 3

Real and imaginary parts of the $\mathrm{\Lambda }\mathrm{\Lambda }\to \mathrm{\Lambda }\mathrm{\Lambda }$ elastic scattering amplitude, ${\mathcal{A}}_{11}$, for cases (a)–(d). Lines represent the amplitude of direct model calculation (blue, solid), reconstructed uniformized Mittag-Leffler expansion with all four poles $1+2+3+4$ (red, dashed), contributions from pole 1 (pink, dot-dot-dashed), pole 2 (green, dotted), and their sum $1+2$ (orange, dot-dashed), respectively.

Figure 4

Contour plot of $|{\mathcal{A}}_{11}{|}^2$ on the torus for cases (a)–(d). The red line corresponds to the physical domain. Labels $\mathrm{\Lambda }\mathrm{\Lambda }$, $N\mathrm{\Xi }$, and $\mathrm{\Sigma }\mathrm{\Sigma }$ represent the corresponding thresholds, and $\infty$ corresponds to infinity point on the physical sheet.