The $O(N)$ model with $\phi^6$ potential in ${\mathbb R}^2 \times {\mathbb R}^+$

TL;DR

This paper analyzes the large N limit of an O(N) scalar field theory with a marginal  potential in three dimensions, exploring boundary conditions, phase stability, and boundary anomaly coefficients in a conformal setting.

Contribution

It introduces a large N analysis of the  theory with boundary conditions, computes effective potentials, and determines boundary anomaly coefficients and correlation functions.

Findings

Identifies different phases based on boundary conditions.

Calculates boundary anomaly coefficients depending on the  coupling.

Provides boundary and bulk conformal block decompositions of stress tensor correlators.

Abstract

We study the large $N$ limit of $O(N)$ scalar field theory with classically marginal $\phi^6$ interaction in three dimensions in the presence of a planar boundary. This theory has an approximate conformal invariance at large $N$. We find different phases of the theory corresponding to different boundary conditions for the scalar field. Computing a one loop effective potential, we examine the stability of these different phases. The potential also allows us to determine a boundary anomaly coefficient in the trace of the stress tensor. We further compute the current and stress-tensor two point functions for the Dirichlet case and decompose them into boundary and bulk conformal blocks. The boundary limit of the stress tensor two point function allows us to compute the other boundary anomaly coefficient. Both anomaly coefficients depend on the approximately marginal $\phi^6$ coupling.