Ultraviolet properties of Lifshitz-type scalar field theories

TL;DR

This paper studies Lifshitz-type scalar theories with anisotropic scaling, showing that Lorentz symmetry violations are suppressed at low energies, and explores their quantum corrections, divergence behavior, and potential for asymptotic freedom.

Contribution

It demonstrates how Lorentz symmetry violations in Lifshitz scalar theories are suppressed in the infrared and provides exact summation of quantum corrections in a specific case, revealing asymptotic freedom.

Findings

Lorentz violations are suppressed at low energies

Quantum corrections are mostly logarithmic and small

The theory can be asymptotically free in a special case

Abstract

We consider Lifshitz-type scalar field theories that exhibit anisotropic scaling laws near the ultraviolet fixed point, with explicit breaking of Lorentz symmetry. It is shown that, when all momentum dependent vertex operators are discarded, actions with anisotropy parameter z=3 in 3+1 dimensions generate Lorentz symmetry violating quantum corrections that are suppressed by inverse powers of the momentum, so that the symmetry is sensibly restored in the infrared region. In the ultraviolet region, the singular behavior of the corrections is strongly smoothened: only logarithmic divergences show up, producing very small changes of the couplings over a range of momentum of many orders of magnitude. In the particular case where all couplings are equal, the theory shows a Liouville-like potential and quantum corrections are exactly summable, giving an asymptotically free theory. However, the observed weakening of the divergences is not sufficient to avoid a residual fine tuning of the mass parameter at a very high energy scale, in order to recover a physically acceptable mass in the infrared region.