One-loop divergences in higher-derivative gravity

TL;DR

This paper reviews the calculation of one-loop divergences in higher-derivative gravity, providing explicit formulas, discussing gauge fixing, and analyzing renormalization group behavior, including asymptotic freedom of couplings.

Contribution

It introduces general formulas for one-loop divergences in higher-derivative gravity and extends the analysis to theories with arbitrary functions of curvature invariants.

Findings

Higher-derivative gauge fixing introduces additional ghosts.

Dimensionless couplings in these theories are asymptotically free.

Results are independent of metric parametrization and gauge on shell.

Abstract

We give a review of the one-loop divergences in higher derivative gravity theories. We first make the bilinear expansion in the quantum fluctuation on arbitrary backgrounds, introduce a higher-derivative gauge fixing and show that higher-derivative gauge fixing must have ghosts in addition to those naively expected. We give general formulae for the one-loop divergences in such theories, and give explicit results for theories with quadratic curvature terms. In this calculation, we need the heat kernel coefficients for the four-derivative minimal operators and two-derivative nonminimal vector operators, which are summarized. We also discuss the beta functions in the renormalization group, and show that the dimensionless couplings are asymptotically free. The calculation is also extended to the theories with arbitrary functions of $R$ and $R_{\mu\nu}^2$. We show that the result is independent of metric parametrization and gauge on shell.