On the power-counting renormalizability of a Lifshitz-type QFT in configuration space

Recently, Hořava (Phys. Rev. D. 79, 084008, 2009) proposed a theory of gravity in 3+1 dimensions with anisotropic scaling using the traditional framework of quantum field theory (QFT). Such an anisotropic theory of gravity, characterized by a dynamical critical exponent z, has proven to be power-counting renormalizable at a z=3 Lifshitz Point. In the present article, we develop a mathematically precise version of power-counting theorem in Lorentz violating theories and apply this to the Hořava-Lifshitz (scalar field) models in configuration space. The analysis is performed under the light of the systematic use of the concept of extension of homogeneous distributions, a concept tailor-made to address the problem of the ultraviolet renormalization in QFT. This becomes particularly transparent in a Lifshitz-type QFT. In the specific case of the ϕ44-theory, we show that is sufficient to take z=3 in order to reach the ultraviolet finiteness of the S-matrix in all orders