8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory

Observing that the Hamiltonian of the renormalisable scalar field theory on 4-dimensional Moyal space AA is the square of a Dirac operator DD of spectral dimension 8, we complete (A,D)(A,D) to a compact 8-dimensional spectral triple. We add another Connes–Lott copy and compute the spectral action of the corresponding U(1)U(1)-Yang–Mills–Higgs model. We find that in the Higgs potential the square ϕ2ϕ2 of the Higgs field is shifted to ϕ⋆ϕ+const⋅Xμ⋆Xμ, where XμXμ is the covariant coordinate. The classical field equations of our model imply that the vacuum is no longer given by a constant Higgs field, but both the Higgs and gauge fields receive non-constant vacuum expectation values.