Effective action of the Hořava theory: Cancellation of divergences

We compute the one-loop effective action of the Hořava theory, in its nonprojectable formulation. We take the quantization of the ($2+1$)-dimensional theory in the Batalin-Fradkin-Vilkovisky formalism, and comment on the extension to the ($3+1$) case. The second-class constraints and the appropriate gauge-fixing condition are included in the quantization. The ghost fields associated with the second-class constraints can be used to get the integrated form of the effective action, which has the form of a Berezinian. We show that all irregular loops cancel between them in the effective action. The key for the cancellation is the role of the ghosts associated with the second-class constraints. These ghosts form irregular loops that enter in the denominator of the Berezinian, eliminating the irregular loops of the bosonic nonghost sector. Irregular loops produce dangerous divergences; hence their cancellation is an essential step for the consistency of the theory. The cancellation of this kind of divergences is in agreement with the previous analysis done on the ($2+1$) quantum canonical Lagrangian and its Feynman diagrams.