Higher derivative quantum gravity near four dimensions

Abstract

We investigate the role of the Gauss-Bonnet term for the n = 4 and n = 4 − ε renormalization group, for both conformal and general versions of the theory. The cancellation of the quantum effects of the Gauss-Bonnet term in the n = 4 limit represents an efficient test for the correctness of previous calculations and also resolves two long-standing problems concerning quantum corrections in quantum gravity. In the case of n = 4 − ε renormalization group there is a number of new nontrivial fixed points, that may indicate to a rich nonperturbative structure of the theory. At the same time, if we do not treat ε as a small parameter, the renormalization group is spoiled by an extensive gauge fixing ambiguity.It is well known that the conventional way of quantization of General Relativity leads to a non-renormalizable theory [1][2][3]. An alternative way of quantizing gravity is to introduce some higher derivative terms into a classical action, treating them at the same footing as the lower-derivative (Einstein-Hilbert and cosmological) terms, in such a way that the new quantum theory is renormalizable [4] (see also [5]). The renormalizability of higher derivative quantum gravity (HDQG) enables one to establish the asymptotic freedom in the UV limit [6][7][8][9] and explore the possible role of quantum gravity in the asymptotic behavior for GUT-like models [10] (see [11] for the general introduction to the subject).The price for the renormalizability of HDQG is the unphysical pole in the spin-two sector of the tree-level propagator of the renormalizable quantum gravity. The pole corresponds to a state with negative kinetic energy or with a negative norm in the state space. Such unphysical states (massive ghosts) [4] have masses with a magnitude of the Planck order, still they spoil unitarity of the S-matrix, making the whole approach of HDQG rather doubtful. A number of attempts to overcome the problem of massive ghosts has been undertaken. In particular, one can construct ghost-free models of R 2 -type gravity with torsion [12], but the renormalizability is lost in these models [13]. A very interesting possibility is to investigate the nonperturbative structure of the theory, e.g., through the resummation of quantum corrections to the graviton propagator or through the 1/N approximation [14,15]. Unfortunately none of these approaches can give definite answer related to the presence of ghost in the asymptotic states of the exact propagator [16]. However, due to the importance of the problem, it looks reasonable to consider other possible non-perturbative approaches to HDQG, in particular the n = 4 − ε renormalization group which proved to be a powerful tool in statistical mechanics [17], quantum field theory [18] and also in quantum gravity, where the n = 2 − ε approach was quite fruitful [19][20][21][22].The classical action of HDQG in n = 4 − ε dimensions has * Also at Tomsk State Pedagogical University, Russia the formwhere µ is a dimensional parameter,is the square of Weyl tensor, E = R 2 µναβ − 4R 2 αβ + R 2 is th...

Universally finite gravitational and gauge theories

“…Moreover all these couplings are renormalizable (in the framework of two-derivative standard model and quadratic gravity). These results of semiclassical computation strongly suggest that a consistent theory of quantum gravity in D = 4 must be formulated as a theory with higher derivative actions (at least with four derivatives) [45]. An higher derivative extension of the standard model of particle physics makes it possible to restore super-renormalizability for all fundamental interactions because all the kinetic and interaction operators in the action for matter, gravity and gauge bosons have the same scaling in the ultraviolet regime.…”

Effective action from the functional renormalization group

“…Leaving this problem aside, we may still analyze beta functions of the couplings. Due to the renormalizability of this model of gravity coupled to matter [72][73][74], the number of UV-divergent terms is smaller. We note that in gravitational models, in which there are no perturbative UVdivergences (and they are UV-finite [75,76]) by the above argumentation we will not find any logarithmic term in the effective action 0 .…”

Effective Action from the Functional Renormalization Group

“…Leaving this problem aside, we may still analyze beta functions of the couplings. Due to the renormalizability of this model of gravity coupled to matter [57][58][59], the number of UV-divergent terms is smaller. We note that in gravitational models, in which there are no perturbative UV-divergences (and they are UV-finite [60,61]) by the above argumentation we will not find any logarithmic term in the effective action Γ 0 .…”