Boundary anomalous dimensions from BCFT: $\phi^{3}$ theories with a boundary and higher-derivative generalizations

We consider the bulk $\phi^3$ deformation of the free boundary conformal field theory in the $\epsilon$ expansion. We determine the leading corrections to the scaling dimensions of boundary fundamental operators and some boundary operator expansion coefficients. Our procedure combines the conformal multiplet recombination with the boundary crossing symmetry. The results cover both the single field case and the multi-field case with $S_{N+1}$ global symmetry, which are associated with the Yang-Lee model and the $(N+1)$-state Potts model respectively. These semi-infinite models describe branched polymers, percolation, and spanning forest at a surface. We generalize these results to some higher derivative theories. In addition, we study the $\phi^{2n+1}$ theories with $n>1$, but only obtain some boundary operator expansion coefficients.