# Composite operators near the boundary

2019-12-13

We use renormalization group methods to study composite operators existing at a boundary of an interacting conformal field theory. By reinterpreting the boundary operator expansion as operator renormalization we relate the boundary conformal data to short-distance (near-boundary) divergences of bulk two-point functions. We clarify the difference between different regularization schemes and show that in the presence of boundary cutoff one can encounter power divergences implying additive renormalization of massive parameters akin to the divergences that appear in effective field theories with scalars. 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. The methods used in this paper can therefore serve as complement to the current boundary conformal bootstrap techniques in the regime where conformal symmetry is broken by quantum effects. We apply the formalism to a scalar field theory in $d=3-\epsilon$ 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-energy tensor decouples at the fixed point in accordance with the Cardy's condition for conformal invariance. We end the main part of the paper by discussing the possible physical significance of this model for various values of $\epsilon$.