Spontaneous breaking of conformal invariance in theories of conformally coupled matter and Weyl gravity

We study the theory of Weyl conformal gravity with matter degrees of freedom in a conformally invariant interaction. Specifically, we consider a triplet of scalar fields and SO(3) non-Abelian gauge fields, i.e. the Georgi-Glashow model conformally coupled to Weyl gravity. We show that the equations of motion admit solutions spontaneously breaking the conformal symmetry and the gauge symmetry, providing a mechanism for supplying a scale in the theory. The vacuum solution corresponds to anti-de Sitter spacetime, while localized soliton solutions correspond to magnetic monopoles in asymptotically anti-de Sitter spacetime. This mechanism strengthens the reasons for considering conformally invariant matter-gravity theory, which has shown promising indications concerning the problem of missing matter in galactic rotation curves.