Unitarity, Crossing Symmetry and Duality in the scattering of ${\cal N}=1$ Susy Matter Chern-Simons theories

TL;DR

<3-5 sentences>We address the scattering problem in the most general renormalizable ${\cal N}=1$ $U(N)$ Chern-Simons theory with a single fundamental multiplet, and compute the $2\times2$ S-matrix at leading order in the planar limit, valid for all orders in the ’t Hooft coupling. Using a supersymmetric light-cone gauge, we derive an exact matter propagator and solve a linear integral equation for the offshell four-point function, enabling the onshell S-matrix in three channels and a conjectured form for the singlet channel that preserves unitarity through a modified crossing rule. The results are fully consistent with the proposed strong-weak duality, map bosonic and fermionic S-matrices into each other in the T/U channels, and, in the singlet channel, exhibit a rich analytic structure including a massless bound state at a critical coupling $w_c(\lambda)$ and a threshold pole as $w$ approaches certain limits. These findings support the modified crossing framework of Jain et al. and point to intriguing IR dynamics for the nearly massless states, with potential extensions to higher ${\cal N}$ theories and connections to known large-$N boson/fermion dualities.

Abstract

We study the most general renormalizable ${\cal N}=1$ $U(N)$ Chern-Simons gauge theory coupled to a single (generically massive) fundamental matter multiplet. At leading order in the 't Hooft large $N$ limit we present computations and conjectures for the $2 \times 2$ $S$ matrix in these theories; our results apply at all orders in the 't Hooft coupling and the matter self interaction. Our $S$ matrices are in perfect agreement with the recently conjectured strong weak coupling self duality of this class of theories. The consistency of our results with unitarity requires a modification of the usual rules of crossing symmetry in precisely the manner anticipated in arXiv:1404.6373, lending substantial support to the conjectures of that paper. In a certain range of coupling constants our $S$ matrices have a pole whose mass vanishes on a self dual codimension one surface in the space of couplings.

Figures (6)

• Figure 1: Scalar superfield propagator
• Figure 2: Gauge superfield propagator, the arrow indicates direction of momentum flow
• Figure 3: Integral equation for self energy
• Figure 4: The diagrams in the first line pictorially represents the Schwinger-Dyson equation for offshell four point function (see \ref{['int4pt']}). The second line represents the tree level contributions from the gauge superfield interaction and the quartic interactions.
• Figure 5: Four point function in superspace