Asymptotic safety in higher-derivative gravity

We study the non-perturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques. The non-perturbative contributions to the β-functions shift the known perturbative ultraviolet fixed point into a non-trivial fixed point with three UV-attractive and one UV-repulsive eigendirections, consistent with the asymptotic safety conjecture of gravity. The implication of this transition on the unitarity problem, typically haunting higher-derivative gravity theories, is discussed. Among the many approaches to quantum gravity, a special place is occupied by higherderivative gravity, which, besides the Einstein-Hilbert term, also includes fourth-order operators in the action. Indeed, the higher-derivative propagators soften the divergences encountered in the perturbative quantization, rendering the theory perturbatively renormalizable [1] and asymptotically free at the one-loop level [2, 3, 4, 5, 6]. Unfortunately, the extra terms responsible for the improved UV behavior also induce massive negative norm states [7], so-called “poltergeists”, which led to the belief that the theory is not unitary. Several arguments suggest that this shortcoming can be cured by quantum effects [2, 8], but the lack of non-perturbative methods has made it hard to substantiate such claims. Recently, the question of renormalizability has received renewed attention due to mounting evidence in favor of the non-perturbative renormalizability, or asymptotic safety (AS), of gravity [9, 10, 11, 12]. In this scenario, the ultraviolet (UV) behavior of the theory is controlled by a non-Gaussian fixed point (NGFP) of the renormalization group flow, with a finite number of UV-attractive (relevant) directions. This fixed point is supposed to provide a continuum limit which is predictive and safe from divergences. While initially found only for gravity in 2 + ǫ space-time dimensions [13, 14] or in the 1/N expansion [15, 16], a candidate for such a NGFP has been recently observed in four dimensions thanks to the application of functional renormalization group equations (FRGE) to gravitational actions of the f(R)-type [17, 18, 19, 20, 21, 22, 23, 24]. Further support for AS comes from lattice simulations [25, 26]. Since higher-derivative terms play a central role in the renormalizability and unitarity of gravity theories in the perturbative setting, it is of great interest to study their impact on the AS scenario. To this effect, in this letter we investigate the RG flow of gravity including all fourderivative interactions and present, for the first time, non-perturbative results in this setting. In particular, our work augments the previous f(R) findings by including the second power-counting marginal coupling related to the Weyl-squared interaction. This constitutes an important step in the program of studying the non-perturbative RG flow of gravity in a derivative expansion, since, besides higher-derivative vertices studied previously, the approximation also includes a four-derivative kinetic term for the graviton propagator. Our main result is the existence of a NGFP with three UV-attractive and one UV-repulsive eigendirections. Together with the condition that this fixed point provides the continuum limit for gravity, the latter implies a relation between the coupling constants of the theory at high energy. Tracing the RG flow to the IR, this translates into a relation between the four-derivative couplings in the effective field theory which could, in principle, be tested experimentally. Furthermore, the transition from asymptotic freedom to asymptotic safety may realize an old proposal [2, 8] for the removal of the poltergeists from the spectrum of the theory, providing the exciting perspective of asymptotically safe quantum gravity being unitary. The key tool in our investigation is the effective average action Γk and its FRGE [30] (see, e.g., [28, 10, 11] for reviews). This setup provides a continuum analogue of Wilsons lattice renormalization group, realizing the idea of integrating out all fluctuation modes with momentum larger than a certain cutoff k (high momentum modes), and taking them into account through a modified dynamics for the remaining fluctuations with momentum smaller than k (low momentum modes). This is implemented by means of a cutoff Rk which, for a given k, suppresses the contributions of the low momentum modes, so that only the high momentum modes are integrated out in the path integral. The functional Γk then defines an effective field theory valid near the scale k and, evaluated at tree level, describes all quantum effects originating from the high-momentum modes. In particular, Γk essentially interpolates between the bare action Sbare ≃ limk→∞ Γk and the standard effective action Γ = limk→0 Γk. (For more details on the reconstruction of the bare action from Γk, see [31].) The k-dependence of Γk is governed by a FRGE which takes the form [29] ∂tΓk[Φ] = 1 2STr [ ( δΓk δΦδΦ + Rk ) −1 ∂tRk ]