Comments on epsilon expansion of the O$(N)$ model with boundary

TL;DR

This paper investigates the boundary critical behavior of the O(N) model near four dimensions using epsilon expansion, extending conformal field theory methods to boundary cases and calculating boundary operator dimensions.

Contribution

It extends the Rychkov and Tan framework to boundary conformal field theory and computes boundary operator dimensions at leading order in epsilon expansion.

Findings

Consistent boundary operator dimensions obtained from diagrammatic and axiomatic approaches.

Extension of conformal bootstrap methods to boundary theories.

Identification of boundary fixed points in the O(N) model.

Abstract

The O$(N)$ vector model in the presence of a boundary has a non-trivial fixed point in $(4-\epsilon)$ dimensions and exhibits critical behaviors described by boundary conformal field theory. The spectrum of boundary operators is investigated at the leading order in the $\epsilon$-expansion by diagrammatic and axiomatic approaches. In the latter, we extend the framework of Rychkov and Tan for the bulk theory to the case with a boundary and calculate the conformal dimensions of boundary composite operators with attention to the analyticity of correlation functions. In both approaches, we obtain consistent results.