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.
Highlights
Higher derivative terms play an important role in quantum gravity (QG), as they are required to provide renormalizability
The derivation of one-loop divergences and renormalization group equations has been for long a needful part of the study of models of quantum gravity (QG)
We report on the quantum calculations of the remaining beta functions in a minimal six-derivative model of QG, which includes the versions with real and complex poles corresponding to both normal particles and ghosts
Summary
Higher derivative terms play an important role in quantum gravity (QG), as they are required to provide renormalizability (see, e.g., [1] for introduction). The derivation of one-loop divergences and renormalization group equations has been for long a needful part of the study of models of quantum gravity (QG) The first such calculations were done for the QG based on Einsteinian general relativity with matter in the seminal papers [24,25]. The beta functions obtained in what follows form, together with the previous ones [53], the full set and, enable one to explore how the running affects the positions of the poles of the propagator of metric perturbations on the complex plane From this perspective, the derivation of the full set of equations describing the running is the necessary step in describing the theory at the quantum level. In the special Appendix A we present a detailed analysis of the Wick rotation in local higher derivative quantum field theories
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