  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 $\Phi$, in $4D$ , we can define $\Phi
=\varepsilon^{\mu\nu\alpha\beta}\partial_{\mu}A_{\nu\alpha\beta}$ that gives a
new interesting mechanism for breaking scale symmetry in 4-D theories of
gravity plus matter fields, through the $A_{\nu\alpha\beta}$ equations of
motion which lead to an integration constant the breaks the scale symmetry,
while introducing terms of the form $eG ln K$ , $e$ being the determinant of
the vierbein, $G$ being the Gauss Bonnet scalar and $K$ being scalar functions
of the fields transforming like $K \rightarrow 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 $\varepsilon^{\mu\nu\alpha\beta}F^a_{\mu\nu}F^a_{\alpha\beta}$
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.
