Spontaneous breaking of scale invariance in a D = 3 U(N ) model with Chern-Simons gauge fields

William A Bardeen,Moshe Moshe

doi:10.1007/jhep06(2014)113

Abstract

We study spontaneous breaking of scale invariance in the large N limit of three dimensional $U(N)_\kappa$ Chern-Simons theories coupled to a scalar field in the fundamental representation. When a $\lambda_6(\phi^\dagger\cdot\phi)^3$ self interaction term is added to the action we find a massive phase at a certain critical value for a combination of the $\lambda_6$ and 't Hooft's $\lambda=N/\kappa$ couplings. This model attracted recent attention since at finite $\kappa$ it contains a singlet sector which is conjectured to be dual to Vasiliev's higher spin gravity on $AdS_4$. Our paper concentrates on the massive phase of the 3d boundary theory. We discuss the advantage of introducing masses in the boundary theory through spontaneous breaking of scale invariance.

Highlights

We study spontaneous breaking of scale invariance in the large N limit of three dimensional U(N )κ Chern-Simons theories coupled to a scalar field in the fundamental representation
We find it appealing to further analyse the Chern-Simons matter theory with masses introduced through spontaneous breaking of scale invariance which assures the introduction of masses into the theory but leaves the boundary d=3 theory conformal to O(N −1)
We presented in this paper several aspects of spontaneous breaking of scale invariance in the large N limit of a three dimensional Chern-Simons theory coupled to a scalar field in the fundamental representation of U(N )

Summary

The vertex in the massive phase at order λ

Where l2 and p2 denote the variables in the two dimensional space, and after integrating on l3 , V (p2, k3) is given by:. One notes that at momentum transfer k+ = 0 V (p2, k3) depends only on the two dimensional vector p and on the momentum transfer k3. The contribution to the vertex of diagrams a1-2, b1-2 and c1-2 in figure 3 is:. The self interaction of the scalar fields contributes to the vertex the term (d1−2)(p, k) λ6. When all vertex contributions are added at k+ = 0, diagrams a1-2, b1-2, c1-2, d1-2 result in:. The combined contribution of order λ2 and λ6 results in a local vertex. The mutual cancelations which are presented in eqs.

Vertex and “bubble” graph at leading N

The exact full planar vertex and “bubble graph”

J0J0 correlator and the dilaton

Explicit breaking of scale invariance and the pseudo-dilaton

The J0J0J0 correlator and the three dilatons interaction

Summary and conclusions