Exploring Cartan gravity with dynamical symmetry breaking

It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an $SO(1,4)$ or $SO(2,3)$ gauge field $A^{a}_{\phantom{a}bμ}$ and Higgs field $V^{a}$ which acts to break the symmetry down to that of the Lorentz group $SO(1,3)$. This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field $Φ$ of electroweak theory is taken as a genuine dynamical field satisfying a Klein-Gordon equation, the gauge independent norm $V^2\equiv η_{ab}V^{a}V^{b}$ of the Higgs-type field $V^a$ is typically regarded as non-dynamical. Instead, in many treatments $V^a$ does not appear explicitly in the formalism or is required to satisfy $V^2 = \mathrm{const.} \neq 0$ by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection $A^{a}_{\phantom{a}bμ}$ and Higgs field $V^a$ as genuine dynamical fields with no ad-hoc constraints imposed. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. One simple choice leads to the extensively studied Peebles-Ratra rolling quintessence model. Another choice yields a scalar-tensor symmetry broken phase of the theory with positive cosmological constant and an effective mass $M$ of the gravitational Higgs field ensuring the constancy of $V^2$ at low energies and agreement with empirical data if $M$ is sufficiently large. More general cases are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity, each of which yield propagating torsion.