Abstract
We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars. Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof. Combining functional renormalisation with high order polynomial approximations and full numerical integration we derive the renormalisation group flow for all couplings and analyse their fixed points, scaling exponents, and the fixed point effective action as a function of the background Ricci curvature. The theory is characterised by three relevant couplings. Higher-dimensional couplings show near-Gaussian scaling with increasing canonical mass dimension. We find that Ricci tensor invariants stabilise the UV fixed point and lead to a rapid convergence of polynomial approximations. We apply our results to models for cosmology and establish that the gravitational fixed point admits inflationary solutions. We also compare findings with those from $f(R)$-type theories in the same approximation and pin-point the key new effects due to Ricci tensor interactions. Implications for the asymptotic safety conjecture of gravity are indicated.
Highlights
It is widely acknowledged that for quantum field theories to be fundamental and predictive up to highest energies, their short-distance behavior should be controlled by an ultraviolet (UV) fixed point under the renormalization group [1,2]
We investigate the asymptotic safety conjecture for quantum gravity including curvature invariants beyond Ricci scalars
Our strategy is put to work for families of gravitational actions which depend on functions of the Ricci scalar, the Ricci tensor, and products thereof
Summary
It is widely acknowledged that for quantum field theories to be fundamental and predictive up to highest energies, their short-distance behavior should be controlled by an ultraviolet (UV) fixed point under the renormalization group [1,2]. Much unlike in non-Abelian gauge theories, proofs or theorems for asymptotic safety are presently not at hand This is largely due to the dimensional nature of Newton’s coupling whose canonical mass dimension must be compensated by large anomalous dimensions. In other words, evaluating the renormalization group equations on spaces with constant curvature is sufficient to find the running for all polynomial couplings of the theory, as well as the nonperturbative renormalization group flow of the functions F and Z. By and large, these ideas are similar to what had been done previously for fðRÞ type actions.
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