Tree-Unitarity and renormalizability in Lifshitz-scaling theory -- as a toy model of Ho\v{r}ava's gravity theory

TL;DR

This paper investigates the conditions for tree-unitarity and renormalizability in Lifshitz-scaling theories, highlighting how anisotropic scaling modifies these conditions compared to relativistic theories, and emphasizing the role of symmetries.

Contribution

It demonstrates that the conditions for unitarity and renormalizability in Lifshitz-scaling theories are equivalent to those in relativistic theories, despite the lack of Lorentz symmetry.

Findings

Unitarity conditions are stronger due to frame dependence.

Renormalizability requires extended power counting.

Symmetries are crucial for renormalizability.

Abstract

We study tree-unitarity and renormalizability in Lifshitz-scaling theory, which is characterized by an anisotropic scaling between the spacial and time directions. Due to the lack of the Lorentz symmetry, the conditions for both unitarity and renormalizability are modified from those in relativistic theories. For renormalizability, the conventional discussion of the power counting conditions has to be extended. Because of the dependence of $S$-matrix elements on the reference frame, unitarity requires stronger conditions than those in relativistic cases. We show that the conditions for unitarity and renormalizabilty are identical as in relativistic theories. We discuss the importance of symmetries for a theory to be renormalizable.