Counterterms in semiclassical Horava-Lifshitz gravity

TL;DR

This paper investigates the renormalization of semiclassical Hořava-Lifshitz gravity with scalar fields, demonstrating the necessity of six spatial derivatives for UV renormalizability and explicitly calculating counterterms and beta functions.

Contribution

It provides a detailed analysis of counterterms and renormalization procedures in semiclassical Hořava-Lifshitz gravity with scalar fields, emphasizing the role of anisotropic scaling.

Findings

Divergent terms contain up to six spatial derivatives.

Counterterms are explicitly computed up to second adiabatic order.

Beta functions are evaluated in the minimal subtraction scheme.

Abstract

We analyze the semiclassical Ho\v{r}ava-Lifshitz gravity for quantum scalar fields in 3+1 dimensions. The renormalizability of the theory requires that the action of the scalar field contains terms with six spatial derivatives of the field, i.e. in the UV, the classical action of the scalar field should preserve the anisotropic scaling symmetry ($t \to L^{2z}t,$ $\vec{x} \to L^2 \vec{x}$, with $z=3$) of the gravitational action. We discuss the renormalization procedure based on adiabatic subtraction and dimensional regularization in the weak field approximation. We verify that the divergent terms in the adiabatic expansion of the expectation value of the energy-momentum tensor of the scalar field contain up to six spatial derivatives, but do not contain more than two time derivatives. We compute explicitly the counterterms needed for the renormalization of the theory up to second adiabatic order and evaluate the associated $\beta$ functions in the minimal subtraction scheme.