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Gravitational-gauge vector interaction in the Hořava-Lifshitz framework

An anisotropic model describing gravity--vector gauge coupling at all energy scales is presented. The starting point is the 4+1 dimensional non--projectable Hořava--Lifshitz gravity theory subject to a geometrical restriction. Renormalizability arguments require all possible interactions in the potential up to terms with $z=4$ spatial derivatives on the geometrical tensor fields: the Riemann and Weyl tensors. The latter being necessary on a 4+1 dimensional formulation. The dimensional reduction to 3+1 dimensions give rise to a model invariant under {foliation--preserving diffeomorphisms} (FDiff) and $U(1)$ symmetry groups. The reduced theory on the {kinetic conformal} (KC) point ($λ=1/3$), propagates the same spectrum of the Einstein--Maxwell theory. Moreover, at low energies, on the IR point $α=0$, $β=1$, its field equations are exactly the Einstein--Maxwell ones in a particular gauge condition. The Minkowski ground state is stable provided several restrictions on the coupling parameters are satisfied, they are explicitly obtained. The quantum propagators of the physical degrees of freedom are obtained and after an analysis of the first and second class constraints the renormalizability by power counting is proved, provided that the aforementioned restrictions on the coupling parameters are satisfied.