Boundary anomalous dimensions from BCFT: O($N$)-symmetric $\phi^{2n}$ theories with a boundary and higher-derivative generalizations

We investigate the $\phi^{2n}$ deformations of the O($N$)-symmetric (generalized) free theories with a flat boundary, where $n\geqslant 2$ is an integer. The generalized free theories refer to the $\Box^k$ free scalar theories with a higher-derivative kinetic term, which is related to the multicritical generalizations of the Lifshitz type. We assume that the (generalized) free theories and the deformed theories have boundary conformal symmetry and O($N$) global symmetry. The leading anomalous dimensions of some boundary operators are derived from the bulk multiplet recombination and analyticity constraints. We find that the $\epsilon^{1/2}$ expansion in the $\phi^6$-tricritical version of the special transition extends to other multicritical cases with larger odd integer $n$, and most of the higher derivative cases involve a noninteger power expansion in $\epsilon$. Using the analytic bootstrap, we further verify that the multiplet-recombination results are consistent with boundary crossing symmetry.