Operator expansions, layer susceptibility and two-point functions in BCFT

We show that in boundary CFTs, there exists a one-to-one correspondence between the boundary operator expansion of the two-point correlation function and a power series expansion of the layer susceptibility. This general property allows the direct identification of the boundary spectrum and expansion coefficients from the layer susceptibility and opens a new way for efficient calculations of two-point correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function $\langle\phi_i \phi^i\rangle$ of the O(N) model at the extraordinary transition in 4-$ϵ$ dimensional semi-infinite space to order $O(ϵ)$. The bulk operator product expansion of the two-point function gives access to the spectrum of the bulk CFT. In our example, we obtain the averaged anomalous dimensions of scalar composite operators of the O(N) model to order $O({ϵ}^2)$. These agree with the known results both in $ϵ$ and large-N expansions.