Renormalization group in super-renormalizable quantum gravity

Renormalization group in super-renormalizable quantum gravity The calculation of quantum corrections always had a very special role in quantum theories of gravity. The first relevant calculation was done by t’Hooft and Veltman [1], who derived the one-loop divergences in the quantum version of general relativity, including coupling to the minimal scalar field. Soon after similar calculations were performed for gravity-vector and gravity-fermion systems [2]. These first calculations have a great merit, regardless of the fact that the output was shown to be gauge-fixing dependent [3]. Later on one could learn a lot from the two-loop calculations in general relativity [4, 5]. Technically more complicated are calculations in four-derivative gravity, which were first performed in [6] and with some corrections in [7], [8] and finally in [9], where some extra control of the calculations was introduced and the hypothesis of the non-zero effect of the topological Gauss-Bonnet term [10] explored. Let us also mention similar calculations in the conformal version of the four-derivative theory [7, 11, 12]. The importance of four-derivative quantum gravity is due to its renormalizability [13], which is related to the presence of massive unphysical ghosts, typical in the higher derivative field theories. Naturally, there were numerous and interesting works trying to solve the unitarity problem in this theory [14–16]. The mainstream approach is based on the expectation that the loop corrections may transform the real massive unphysical pole in the tensor sector of the theory into a pair of complex conjugate poles, which do not spoil unitarity within the Lee-Wick quantization scheme [17]. However, it was shown that the definite knowledge of whether this scheme works or not requires an exact non-perturbative beta-function for the coefficient of the Weyl-squared term and for the Newton constant [18]. The existing methods to obtain such a non-perturbative result give some hope [19], but unfortunately they are not completely reliable1 and, therefore, the situation with unitarity in the four-derivative quantum gravity is not certain, at least. Another interesting aspect of quantum corrections in models of gravity is related to the running of the cosmological constant Λcc and especially Newton constant G. These quantum effects may be relevant in cosmology and astrophysics (see, e.g. [21, 22]) and can be explored in different theoretical frameworks, such as semiclassical gravity [23], higher derivative quantum gravity [7, 15], low-energy effective quantum gravity [24], induced gravity [25] and functional renormalization group [26]. Indeed, the status of the corresponding types of quantum corrections is different, but there are also some common points. In particular, in many cases one can formulate general restrictions on the running of the Newton constant G, which are based on covariance and dimensional arguments [27]. The beta-function for the ∗Electronic address: lmodesto@sustc.edu.cn, lmodesto1905@icloud.com †Electronic address: grzerach@gmail.com ‡Electronic address: shapiro@fisica.ufjf.br 1 One of the reasons is a strong gauge-fixing dependence of the on-shell average effective action, which was discussed in Yang-Mills theory [20] and is expected to take place also in quantum gravity. 2 inverse Newton constant which follows from these condition has the form µ d dµ 1 G = X ij Aij mimj , (1) where mi are masses of the fields or more general parameters in the action with the dimensions of masses and Aij are given by series in coupling constants of the theory. In the perturbative quantum gravity case there may be one more complication. In the model based on EinsteinHilbert’s gravity there is no beta-function for G, and in the four-derivative model this beta-function is dependent on the choice of gauge fixing condition [7, 28, 29]. Only a dimensionless combination of G and Λcc has well-defined running, but this is not sufficient for the mentioned applications to cosmology and astrophysics. Recently there was a significant progress in development of perturbative quantum gravity models which have very different properties. If the action of the theory includes local covariant terms that have six or more derivatives, this theory may be i) super-renormalizable [30]; ii) unitary, in case of only complex conjugate massive poles in the treelevel propagator [31, 32], and iii) have gauge-fixing and field reparametrization independent beta-functions for both G and Λcc. The theory is unitary at any order in the perturbative loop expansion when the CLOP [33] prescription is implemented or non-equivalently if the theory is defined through a non-analytic Wick rotation from Euclidean to Minkowskian signature [34–36]. This means that such a theory satisfies the minimal set of consistency conditions and deserves a detailed investigation at both classical and quantum levels. The classical aspects of the theory started to be explored recently in [37], where it was shown that the version with real simple poles has no singularity in the modified Newtonian potential. Quite recently this result was generalized for more general cases including multiple and complex poles [38, 39]. Furthermore, in [38, 40] the detailed analysis of light bending in six-derivative models was given. Another generalization of simple higher derivative model is nonlocal gravity, where we allow for nonlocal functions of differential operators [41]. Moreover, there exists a class of nonlocal theories in which UV behaviour is exactly the same like in polynomial higher derivative theories. Therefore, they also satisfy the above three points, namely they are quantum super-renormalizable models and the analysis of divergences and RG running presented here apply to these theories as well. Until now, the unique example of quantum calculations in the super-renormalizable quantum gravity was the derivation of the beta-function for the cosmological constant in [30]. Here we start to explore the models further and derive the most relevant phenomenologically one-loop beta-function for the Newton constant G. The work is organized as follows. In Sect. II one can find a brief general review of the super-renormalizable models [30], including power counting and gauge-fixing independence of the beta-functions. In Sect. III, we describe the oneloop calculations. Some of the relevant bulky formulas are separated into Appendix A, to provide a smooth reading of the main text. In Sect. IV, two important classes of the nonlocal models of quantum gravity are considered. It turns out that the derivation of one-loop divergences by taking a limit in the results for a polynomial models meets serious difficulties, which can be solved only for a special class of nonlocal theories, which are asymptotically polynomial in the ultraviolet regime. But still these theories are super-renormalizable or even finite. In Sect. V, the renormalization group for the Newton and cosmological constants are discussed, within the minimal subtraction scheme of renormalization. Finally, in Sect. VI we draw our conclusions and outline general possibilities for further work.