### abstract

- 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 \(\Phi \) in four dimensions defined in terms of scalars \(\varphi _a\) by \(\Phi =\varepsilon ^{\mu \nu \rho \sigma } \varepsilon _{abcd} (\partial _{\mu }\varphi _a)(\partial _{\nu }\varphi _b) (\partial _{\rho }\varphi _c) (\partial _{\sigma }\varphi _d)\). With the help of a dilaton field \(\phi \) we construct theories that are globally scale invariant. In particular, by introducing couplings of the dilaton \(\phi \) to the Gauss–Bonnet (GB) topological density \(\, {\sqrt{-g}} \, \phi \left( R_{\mu \nu \rho \sigma }^2 - 4 R_{\mu \nu }^2 + R^2 \right) \,\) we obtain a theory that is scale invariant up to a total divergence. Integration of the \(\varphi _a\) 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.