O(N) BCFT: new data from conformal partial wave expansions

The $\mathsf{O}(N)$ $\mathsf{B}\mathsf{C}\mathsf{F}\mathsf{T}$ is analyzed in the large $N$ expansion in a generic bulk dimension $2<d+1<4$. Focus is on boundary conditions corresponding to the ordinary transition, however techniques used can be generalized to special or extraordinary transitions. We study the system of bulk $2$-pt functions $⟨\phi \phi ⟩$, $⟨{\phi }^2{\phi }^2⟩$ and $⟨\sigma \sigma ⟩$, where $\sigma$ is the Hubbard–Stratonovich field. They are expanded in bulk/boundary conformal partial waves. The coefficient functions of this expansion – the spectral functions – encode $\mathsf{B}\mathsf{C}\mathsf{F}\mathsf{T}$ data and determine bulk/boundary conformal block expansions. We prove conjectures about boundary spectral functions in Dujava et al. [arXiv:2503.16345] and reproduce the bulk expansion of $⟨\phi \phi ⟩$ in Giombi et al. [arXiv:2007.04955]. The bulk expansion of $⟨\sigma \sigma ⟩$ is new and allows to extract the leading large $N$ expression for an unknown OPE coefficient $C_{\sigma \sigma {\sigma }^3}$ (the computation outputs as a byproduct also a mixing coefficient, which matches an existing result by Derkachov et al. [arXiv:hep-th/9705020], providing further support for the main result). Besides these derived data infinitely many constraints are produced, though due to operator mixing growing in complexity as degeneracy increases, they cannot be disentangled without further input.