Liouville-Lifshitz theory in $3+1$ dimensions

We consider a four-dimensional theory in the $z=3$ Lifshitz context, with an exponential (Liouville) potential. We determine the exact renormalized potential of the theory and derive the nonperturbative relation between the renormalized and bare couplings. In addition, we show that Lorentz symmetry is naturally generated by quantum fluctuations in the infrared regime, and conclude that the model can be relevant to high energy physics.

Figure 1

Diagrams contributing to infinities in the Liouville theory from the $n$-point vertex: (a) one-loop infinity renormalized by an $(n-2)$-point counterterm. (b) $k$-loop infinity renormalized by the $(n-2k)$-point countertem, (c)