Determine boundary-channel structure constants for continuous diagonal boundaries

Determine the boundary-channel structure constants for disc two-point functions of bulk fields in critical loop models when the diagonal boundary condition yields a continuous boundary spectrum, i.e., when the boundary-channel decomposition is an integral over boundary momenta.

Background

For diagonal boundaries, the authors develop bulk-channel expansions and boundary-channel decompositions. In the continuous boundary case, the boundary spectrum is not discrete, and the decomposition involves an integral over boundary momenta.

They state explicitly that the boundary structure constants are unknown in the continuous case, preventing numerical tests and full crossing checks. Determining these constants would allow testing and potentially proving crossing symmetry with continuous boundaries.