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Propagator in the Horava-Lifshitz gravity

In this paper it is studied the propagator for the modified theory of gravity proposed by Ho\vrava. We first calculate the propagator in the $λ=1$ case and show that the main poles that arise correspond to the spin two particle and scalar particle, already known in the literature. The presence of a bad ultraviolet behaving term spoils renormalizability of the theory but it is eliminated by imposing the detailed balance condition, although just a soft version of this condition is actually needed. The problems of a negative mass term and a residue with undefined sign, which is due to the presence of the cosmological constant, is verified at the tree level demanding a complete elimination of the tadpole in order to be fully analyzed. However, in the absence of such constant, the extra scalar degree of freedom has no dynamics, at least at the tree-level, and the theory posses only two dynamical degrees of freedom. Also, to understand the implications of $λ$, we analyse a simplified model, the $λR$ theory, and verify that it becomes non unitary, being a strong argument to set $λ=1$.