Scale symmetry breaking from total derivative densities and the cosmological constant problem

The use in the action integral of totally divergent densities in generally coordinate invariant theories can lead to interesting mechanisms of spontaneous symmetry breaking of scale invariance. With dependence in the action on a metric independent density Φ, in 4D, we can define Φ=εμναβ∂μAναβ that gives a new interesting mechanism for breaking scale symmetry in 4D theories of gravity plus matter fields, through the Aναβ equations of motion which lead to an integration constant the breaks the scale symmetry, while introducing terms of the form eGlnK, e being the determinant of the vierbein, G being the Gauss–Bonnet scalar and K being scalar functions of the fields transforming like K→cK (where c is a constant) under a scale transformation. Such a term is invariant only up to a total divergence and therefore leads to breaking of scale invariance due to gravitational instantons. The topological density constructed out of gauge field strengths εμναβFμνaFαβa can be coupled to the dilaton field linearly to produce a scale invariant term up to a total divergence. The scale symmetry can be broken by Yang–Mills instantons which lead to a very small vacuum energy for our Universe.