Abstract
We study the large N limit of O(N ) scalar field theory with classically marginal ϕ6 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 ϕ6 coupling.
Highlights
We are interested in the O(N ) scalar field theory with a classically marginal (φ2)3 interaction, described by the Lagrangian density
We study the large N limit of O(N ) scalar field theory with classically marginal φ6 interaction in three dimensions in the presence of a planar boundary
We find different phases of the theory corresponding to different boundary conditions for the scalar field
Summary
We begin with an analysis of the Lagrangian density (2.4) which describes the behavior of a free scalar field with a position dependent mass. Symmetry associated with a Euclidean boundary conformal field theory in three dimensions. As it is not more difficult, let us work in general dimension. We expect to recover the usual two-point function for a massless free field, κ. We can re-express this three point function in terms of the corresponding propagators and the three point vertex σφ in the effective Lagrangian. We will use the Gφ propagator to compute a one-loop effective potential while in sections 3 and 4, we will use these Feynman rules to study the Jμαβ(x)Jνγδ(x ) and T μν(x)T σρ(x ) correlation functions at leading order in N
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