Renormalization Group in Six-derivative Quantum Gravity

TL;DR

This paper derives the one-loop beta functions for a six-derivative quantum gravity model, demonstrating its asymptotic freedom and analyzing the behavior of ghost states through renormalization group equations.

Contribution

It provides the first complete set of beta functions for a super-renormalizable six-derivative quantum gravity theory using the Barvinsky-Vilkovisky technique.

Findings

The theory is asymptotically free in the UV regime.

Ghostlike states appear as complex conjugate pairs at all energy scales.

Running may cause ghosts to become tachyons, suggesting possible extensions.

Abstract

The exact one-loop beta functions for the four-derivative terms (Weyl tensor squared, Ricci scalar squared and the Gauss-Bonnet) are derived for the minimal six-derivative quantum gravity (QG) theory in four spacetime dimensions. The calculation is performed by means of the Barvinsky and Vilkovisky generalized Schwinger-DeWitt technique. With this result we gain, for the first time, the full set of the relevant beta functions in a super-renormalizable model of QG. The complete set of renormalization group (RG) equations, including also those for the Newton and the cosmological constant, is solved explicitly in the general case and for the six-derivative Lee-Wick (LW) quantum gravity proposed in a previous paper by two of the authors. In the ultraviolet regime, the minimal theory is shown to be asymptotically free and describes free gravitons in Minkowski or (anti-) de Sitter ((A)dS) backgrounds, depending on the initial conditions for the RG equations. The ghostlike states appear in complex conjugate pairs at any energy scale consistently with the LW prescription. However, owing to the running, these ghosts may become tachyons. We argue that an extension of the theory that involves operators cubic in Riemann tensor may change the beta functions and hence be capable of overcoming this problem.