Small dark energy and stable vacuum from Dilaton–Gauss–Bonnet coupling in TMT

In two measures theories (TMT), in addition to the Riemannian measure of integration, being the square root of the determinant of the metric, we introduce a metric-independent density Φ in four dimensions defined in terms of scalars φ a by Φ = ε μ ν ρ σ ε a b c d ( ∂ μ φ a ) ( ∂ ν φ b ) ( ∂ ρ φ c ) ( ∂ σ φ d ) . With the help of a dilaton field ϕ we construct theories that are globally scale invariant. In particular, by introducing couplings of the dilaton ϕ to the Gauss–Bonnet (GB) topological density - g ϕ R μ ν ρ σ 2 - 4 R μ ν 2 + R 2 we obtain a theory that is scale invariant up to a total divergence. Integration of the φ a field equation leads to an integration constant that breaks the global scale symmetry. We discuss the stabilizing effects of the coupling of the dilaton to the GB-topological density on the vacua with a very small cosmological constant and the resolution of the ‘TMT Vacuum-Manifold Problem’ which exists in the zero cosmological-constant vacuum limit. This problem generically arises from an effective potential that is a perfect square, and it gives rise to a vacuum manifold instead of a unique vacuum solution in the presence of many different scalars, like the dilaton, the Higgs, etc. In the non-zero cosmological-constant case this problem disappears. Furthermore, the GB coupling to the dilaton eliminates flat directions in the effective potential, and it totally lifts the vacuum-manifold degeneracy.