Discontinuity calculus and applications to two-body coupled-channel scattering

TL;DR

The paper introduces a discontinuity calculus to systematically compute analytic continuations and map the Riemann-surface topology of complex functions, focusing on $S$-matrix elements in two-body coupled-channel scattering. It defines the discontinuity operator $D$, continuation kernels $K_i$, and continuation generators $T_i$, and applies them to the scalar two-point Green's function $G(s)$ and the two-body $T$-matrix, deriving explicit discontinuities and multi-sheet structures. Key results include ${D} G(s) = -2 i \rho(s) \theta(s-s_+)$, a countably infinite Riemann-surface structure for $G(s)$ and $\mathbf{G}(s)$, and a genus formula $g_{n_c}=(n_c-3)2^{n_c-2}+1$ that connects to uniformization mappings (rational for $g=0$, elliptic for $g=1$, and automorphic for $g\ge 2$). The work links analytic structure to the uniformization theorem and argues for extensions to multi-body and multivariable scattering problems, offering a systematic toolkit for exploring resonances and pole effects on invariant-mass distributions.

Abstract

We present a novel method, termed discontinuity calculus, for computing discontinuities of complex functions. This framework enables a systematic investigation of both analytic continuation and the topological structure of Riemann surfaces. We apply this calculus to analyze the analytic continuation of partial-wave amplitudes in two-body coupled-channel scattering problems and discuss their uniformization of the corresponding Riemann surfaces. This methodology offers new perspectives and tools for analyzing coupled-channel scattering problems in quantum scattering theory.

Figures (5)

• Figure 1: Schematic diagram illustrating the cuts of the Green's function $G(s)$ in the complex $s$-plane. The red line on the real axis represents Cut-1, while the blue line represents Cut-2. The two Riemann sheets corresponding to the Green's functions are interconnected along their respective cuts, as indicated by the pink arrows.
• Figure 2: Left: Riemann surface $\overline{\operatorname{RS}}\{G(s)\}$, which consists of countably infinite Riemann sheets, with the corresponding Green's function labeled in the plot. Middle: Riemann surface $\overline{\operatorname{RS}}\{G(s)\}$ when the analytic continuation along Cut-2 is not considered; the originally simply connected Riemann surface is divided into multiple independent branches, each consisting of two Riemann sheets. Right: When the analytic continuation along Cut-2 is disregarded, the branch in the middle plot containing the physical sheet consists of two Riemann sheets; notably, the entire branch is topologically equivalent to a sphere. Here, the red lines represent Cut-1 on the physical sheet, the blue lines represent Cut-2 on the unphysical sheets, the orange lines represent Cut-1 on the unphysical sheets, and the black dots represent the branch points of the Green's functions.
• Figure 3: The diagram illustrates the connection patterns among different Riemann sheets following analytic continuation of the partial-wave scattering matrix ${\bm{T}}(s)$ for $n_c=2$ (left) and $n_c=3$ (right). The red lines indicate Cut-1 on the physical sheet, while the orange lines indicate Cut-1 on unphysical sheets. The black dots indicate the branch points of the Green's functions.
• Figure 4: An intuitive schematic diagram illustrating the construction of the Riemann surface $\mathrm{RS}_{1}\to\mathrm{RS}_{2}\to\mathrm{RS}_{3}$ through three steps: cutting, duplicating, and gluing. The red lines indicate Cut-1 on the physical sheet, while the orange lines indicate Cut-1 on the unphysical sheet. The black dots indicate the branch points of the partial-wave scattering matrix ${\bm{T}}(s)$.
• Figure 5: Illustrative explanation of the last equality in Eq. \ref{['eq:disc-I']}. The red lines represent integration contours, with arrowheads indicating the integration direction. The black and blue dots denote the two poles in the integrand. (a) The contour avoids being pinched when both singularities coalesce and lie in the same half-plane. As a result, no singularities emerge in the integral. (b) If the poles are positioned on opposite sides of the contour, their coalescence pinches the contour, leading to singularities in the integral. (c) By deforming the contour in panel (b), the two poles can be relocated to the same side (e.g., by moving the blue pole from the upper to the lower half-plane). To preserve the continuity of the integral, an additional integration over a closed contour with winding number $+1$ encircling the blue pole is introduced. Even in this configuration, pole coalescence continues to pinch the contour, thereby generating singularities. (d) When the poles approach each other while being enclosed by sub-contours with identical winding numbers (e.g., both with winding number $+1$), no pinching occurs, and the integral remains free of singularities.