Lorentz symmetry violating Lifshitz-type field theories

TL;DR

This paper explores 3+1 dimensional Lifshitz-type field theories with anisotropic scaling, examining their renormalization, symmetry breaking, and implications for the hierarchy problem, highlighting the emergence of gauge and Lorentz symmetries at low energies.

Contribution

It introduces a framework for Lifshitz-type theories with z=3, analyzing their renormalization properties and the conditions for emergent gauge and Lorentz symmetries.

Findings

Scalar self-interactions and Yukawa couplings are asymptotically free.

Gauge symmetry cannot be maintained with the momentum-dependent vertices.

Lorentz and gauge symmetries emerge below a certain energy scale M.

Abstract

We discuss the ultraviolet sector of 3+1 dimensional Lifshitz-type anisotropic higher derivative scalar, fermion and gauge field theories, with anisotropy exponent z=3 and with explicit breaking of Lorentz symmetry. By discarding from the action all momentum dependent vertex operators, which is essential to avoid phenomenologically unacceptable deformations of the light cone, we find that renormalizable scalar self-interaction and Yukawa-like couplings are, in general, asymptotically free. However, the requirement of cancelling momentum dependent vertex operators is incompatible with gauge symmetry and, therefore, for this kind of theories, gauge symmetry as well as Lorentz symmetry are recovered only as emergent properties below some energy scale M, that must be constrained from experiments. The quantum corrections to the scalar mass and their impact on the hierarchy problem are also analyzed.