In the case of discretely degenerate vacua, for example in the double well potential, there are instantonic corrections to the energies. The degeneracy is lifted, and the true vacuum becomes a spatially even linear combination of the left- and right-vacua. A great introduction/reference is Coleman's Aspects of Symmetry, in Chapter 7, ``The uses of instantons''.
Are there similar corrections to the energies when the degeneracy is continuous, for example in the case of a Mexican-Hat potential? If so, please provide a reference.