Quantizing non-projectable Ho\v{r}ava gravity with Lagrangian path integral

TL;DR

This paper develops a Lagrangian path integral formulation for non-projectable Hořava gravity, performs a one-loop analysis in (2+1) dimensions, and derives beta functions for key couplings, advancing understanding of its quantum properties.

Contribution

It introduces a local measure formulation with auxiliary fields, performs explicit one-loop calculations without truncations, and verifies divergence cancellations in non-projectable Hořava gravity.

Findings

Successfully isolated and canceled linear-in-frequency divergences.

Derived beta functions for Newton constant and coupling λ.

Established a framework for analyzing perturbative renormalizability.

Abstract

We formulate the quantum version of non-projectable Ho\v{r}ava gravity as a Lagrangian theory with a path integral in the configuration space with an ultra-local in time, but non-local in space, field-dependent measure. Using auxiliary fields, we cast the measure into a local form satisfying several bosonic and fermionic symmetries. We perform an explicit one-loop computation in the theory in $(2+1)$ dimensions, using for the case study the divergent part of the action on a background with non-trivial shift vector; the background spatial metric is taken to be flat and the background lapse function is set to 1. No truncations are assumed at the level of perturbations, for which we develop a diagrammatic technique and a version of the heat-kernel method. We isolate dangerous linear-in-frequency divergences in the two-point function of the shift, which can lead to spatial non-localities, and explicitly verify their cancellation. This leaves a fully local expression for the divergent part of the quadratic effective action, from which we extract the beta functions for the Newton constant and the essential coupling $\lambda$ in the kinetic term of the metric. We formulate the questions that need to be addressed to prove perturbative renormalizability of the non-projectable Ho\v{r}ava gravity.