Poles in the $S$-Matrix of Relativistic Chern-Simons Matter theories from Quantum Mechanics

TL;DR

This paper demonstrates that in a specific scaling limit, the relativistic $S$-matrix of large N Chern-Simons matter theories with a scalar matches exactly with a non-relativistic quantum mechanics description, confirming the conjectured $S$-matrix structure.

Contribution

The paper establishes a precise correspondence between the relativistic $S$-matrix and a non-relativistic quantum mechanics model in a particular scaling limit, validating the conjectured all-orders $S$-matrix formula.

Findings

The pole in the $S$-matrix is near threshold in the scaling limit.

The non-relativistic Schrödinger equation reproduces the $S$-matrix exactly.

The pole structures of the relativistic and non-relativistic $S$-matrices match perfectly.

Abstract

An all orders formula for the $S$-matrix for 2 $\rightarrow$ 2 scattering in large N Chern-Simons theory coupled to a fundamental scalar has recently been conjectured. We find a scaling limit of the theory in which the pole in this $S$-matrix is near threshold. We argue that the theory must be well described by non-relativistic quantum mechanics in this limit, and determine the relevant Schroedinger equation. We demonstrate that the $S$-matrix obtained from this Schroedinger equation agrees perfectly with this scaling limit of the relativistic $S$-matrix; in particular the pole structures match exactly. We view this matching as a nontrivial consistency check of the conjectured field theory $S$-matrix.