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

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.