The CW mechanism in a semi-definite system

We study the $\phi^6 - \hat{\phi}^4$ model with $O(N)$-symmetry near three dimensions. This model has a sextic bulk-interaction and a quartic boundary-interaction. The bulk two-point correlator is found upto two-loops by solving the equation of motion and applying the boundary conditions. Finally we apply the Coleman-Weinberg mechanism to this model, which allows us to flow along the renormalization group to a first-ordered phase transition. At one-loop order only the boundary receives a non-trivial effective potential, giving the scalar on the boundary a vacuum expecation value. However, due to the boundary operator product expansion, the bulk one-point function is non-zero as well. This leads to a spontaneous symmetry breaking of the original $O(N)$-symmetry.