Signs and Stability in Higher-Derivative Gravity

Abstract

Perturbatively renormalizable higher-derivative gravity in four space-time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in lorentzian flat space-time for no-tachyons, fixes the signs. Feynman $+iε$ prescription for these sign further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters makes the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space-time dimensions. The theory has a transition point arising from renormalisation group (RG) equations, where the coefficient of $R^2$ diverges without affecting the perturbative quantum field theory. Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations pushes the flow of parameters across the transition point. The flow beyond the transition point is analysed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton's constant $G$ becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a non-perturbative treatment of the theory beyond the transition point.