Composite operators near the boundary

Abstract

We use renormalization group methods to study composite operators existing at a boundary of an interacting conformal field theory. In particular we relate the data on boundary operators to short-distance (near-boundary) divergences of bulk two-point functions. We further argue that in the presence of running couplings at the boundary the anomalous dimensions of certain composite operators can be computed from the relevant beta functions and remark on the implications for the boundary (pseudo) stress-energy tensor. We apply the formalism to a scalar field theory in d = 3 − dimensions with a quartic coupling at the boundary whose beta function we determine to the first non-trivial order. We study the operators in this theory and compute their conformal data using −expansion at the Wilson-Fisher fixed point of the boundary renormalization group flow. We find that the model possesses a non-zero boundary stress-energy tensor and displacement operator both with vanishing anomalous dimensions. The boundary stress tensor decouples at the fixed point in accordance with Cardy's condition for conformal invariance. We end the main part of the paper by discussing the possible physical significance of this fixed point for various values of . A. General form of a bulk-boundary correlator 32 B. Four-point correlator 33 C. Master integral 34 D. Stress-energy tensor correlator at orderĝ 35

O(N) models with boundary interactions and their long range generalizations

“…Note added. After completion of this paper, we became aware of [61] which has some overlap with parts of our work.…”

Why boundary conditions do not generally determine the universality class for boundary critical behavior

“…[34] and their extension to systems with boundaries (see, e.g., [35][36][37][38] and their references). These advances and the ongoing great interest in the AdS/CFT correspondence have spurred considerable recent activity by the high-energy theory community in the field of boundary conformal field theories [39][40][41][42][43][44]. Using these techniques, the series expansions to second order in = 4 − d about the upper critical dimension d * = 4 of the surface correlation exponents η ord and η sp at the ordinary and special transitions, which have been known since the beginning of the 1980s [8][9][10][11][12][13], were recovered [37,38].…”

Bootstrapping boundary-localized interactions

“…The computation of the beta functions for bulk-boundary couplings in terms of the data of the CFT d−1 was performed in[32] for some examples of perturbations around Dirichlet and Neumann. Some perturbative constructions of interacting boundary conditions for free theories can also be found in[25,33,34].12 This is the correct scaling with if the three-point function of the operator Ω vanishes for the free boundary conditions, as in the examples we will consider below. One can also consider cases in which the…”

“…O(N ) invariant conformal boundary conditions for multiple free scalars were studied recently in[25,34].…”