Composite operators near the boundary

TL;DR

This paper uses renormalization group techniques to analyze boundary composite operators in conformal field theories, relating boundary data to bulk divergences and computing conformal data at the Wilson-Fisher fixed point.

Contribution

It develops a formalism connecting boundary operator data to bulk divergences and computes boundary conformal data in a scalar theory at the Wilson-Fisher fixed point.

Findings

Boundary stress-energy tensor and displacement operator have zero anomalous dimensions.

The boundary stress tensor decouples at the fixed point, satisfying Cardy's condition.

The formalism relates boundary operator anomalous dimensions to beta functions.

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 $d=3-\epsilon$ 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 $\epsilon-$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 $\epsilon$.