Probing multi-particle unitarity with the Landau equations

TL;DR

This work examines the nonperturbative analytic structure of 2→2 scattering for identical massive scalars by mapping Landau curves arising from multi-particle unitarity in the region $16 m^2 le s,t < 36 m^2$. Using a graph-theoretic generation and a numerically solved set of Landau equations for alpha-positive leading singularities, the authors uncover an infinite family of Landau curves that accumulate at a finite locus on the physical sheet, indicating a generic feature of multi-particle unitarity. They discuss implications for the lightest-particle maximal analyticity hypothesis, extended elastic unitarity, and the S-matrix bootstrap program, and outline avenues for extending the analysis to higher multi-particle channels. The results illuminate how multi-particle unitarity shadows constrain the elastic amplitude and reveal a rich, hierarchical landscape of singularities beyond the elastic regime.

Abstract

We consider the $2\to 2$ scattering amplitude of identical massive particles. We identify the Landau curves in the multi-particle region $16m^2 \leq s, t < 36m^2$. We systematically generate and select the relevant graphs and numerically solve the associated Landau equations for the leading singularity. We find an infinite sequence of Landau curves that accumulates at finite $s$ and $t$ on the physical sheet. We expect that such accumulations are generic for $s,t > 16m^2$. Our analysis sheds new light on the complicated analytic structure of nonperturbative relativistic scattering amplitudes.

Figures (26)

• Figure 1: The Landau curves in the elastic region $4m^2 \leq s,t < 16 m^2$ are known thanks to elastic unitarity. The first of these are plotted in this figure. In the gray region, the double discontinuity is equal to zero. The main purpose of the present paper is to explore the structure of the Landau curves in the multi-particle region $s,t \geq 16 m^2$.
• Figure 2: A few simplest examples of graphs that represent various singularities of the $2 \to 2$ scattering amplitude. a) The bubble diagram represents multi-particle normal thresholds. b) The two-particle box diagram. It represents a Landau curve along which the scattering amplitude develops double discontinuity. c) The four-particle box diagram. This diagram corresponds to four-particle scattering both in the $s$- and in the $t$-channel. In this paper we systematically study the graphs of this type and the corresponding Landau curves.
• Figure 3: The planar cross and the non-planar cross (open envelope) graphs. Each of the diagrams is the first one in an infinite chain of diagrams, see figure \ref{['triangles']}, that generates the Landau curves on the physical sheet, in the region $16 m^2 \leq s,t < 36 m^2$.
• Figure 4: The planar, a), and non-planar, b), triangle chain graphs. Remarkably, each of the graphs involves four-particle scattering both in the $s$- and in the $t$- channel. As the number of triangles grows, the corresponding Landau curves quickly accumulate around the locus \ref{['eq:accumulation']} on the physical sheet. Notice that adding a single triangle to each chain increases the number of vertices $V$ by $2$. Closely related diagrams appeared before in Bjorken:1963zzeden1960analyticBros:1983vf.
• Figure 5: The Landau curves in the $2 \to 4$ multi-particle region. To each diagram corresponds a pair of crossing symmetry-related curves. For crossing symmetric diagrams, there is only one curve. The red curve, given by equation \ref{['eq:accumulation']}, is an accumulation point of infinitely many Landau curves. The uppermost curve (black #1) is given by the non-planar cross diagram, figure \ref{['fig:phi4box']} (b), while the planar cross, figure \ref{['fig:phi4box']} (a), gives the lowermost curve (black #2). The planar triangle chain graphs (#2, #4, #6,...), figure \ref{['triangles']} (a), approach the red curve from below while non-planar triangle chain graphs (#1, #3, #5,...), figure \ref{['triangles']} (b), approach it from above. As shown in the inset panel, the approach is fast (see table \ref{['table:2']}). We expect that the Landau curves presented here are all the curves that cross the square $16m^2 <s,t < 36m^2$ on the physical sheet. We collect explicit equations for some of the Landau curves in appendix \ref{['app:multiLC']}.