The asymptotic safety scenario for gravity and matter

The outlook of the Thesis is the following. In Chapter 2, we will give a detailed account of asymptotic safety, describing how the RG can be used to analyze UV divergences in QFT's. We will explicitly show that, when the Gaussian FP is taken to perform the continuum limit, asymptotic safety is equivalent to perturbative renormalisation (plus the condition of asymptotic freedom). When one considers a non-Gaussian FP, asymptotic safety can then be viewed as a generalization of standard renormalisability. After recalling the basic principles of Wilson's RG in the context of perturbation theory for a scalar theory, in Section 2.4 we will discuss the subtle issue of the asymptotic behaviour of gravitational couplings at a FP. It should be clear from first principles that applications of the RG to gravity differ from applications to any other QFT. This is because the metric is needed to measure lengths (and therefore also momenta). Thus: \vherever the cutoff scale k appears in the action it must necessarily appear in a combination involving the metric. In other QFT's: the cutoff k is defined using some external background metric, so the dependence of rk on the cutoff does not involve the dynamical variables at all. VVe give a proper analysis of this point, leading to the determination of the scaling behaviour of gravitational couplings at a FP. This has to be regarded as a definition of a gravitational FP, irrespective of any techniques and approximations that may be used in later calculations. It agrees with the behaviour that had been looked for by earlier authors, but it had not been properly justified until now. We then discuss two rather striking consequences of this fact: the existence of a maximum momentum in Planck units, and the fact that the anomalous dimension of the graviton field must be exactly equal to 2. The second result had been derived before, but we point out that both results must be true at any gravitational NGFP, independently of approximations. In Chapter 3, we will treat the Wilson RG in a non-perturbative way, reviewing the formalism of Exact RG Equations. VVe will discuss both types of ERGEs that exist in the literature, namely the Polchinski equation for the VVilsonian effective action and the Wetterich equation for a modified Legendre effective action, named average effective action. vVe will show that the two approaches are equivalent, the equations being related to each other by a Legendre transform. We will focus mostly on the Legendre effective action and the Wetterich equation, which is the one that has been used in explorations of the asymptotic safety scenario. As a an example, we will discuss the case of a scalar theory in Section 3.3. In Chapter 4, we will introduce the ERG Es for pure gravity in the approach of the average effective action, following mainly the work of Reuter and collaborators [18- 21]. We will discuss the gauge-fixing procedure, adopting the background technique which allows one to have a diffeomorphism invariant average effective action. This is a crucial point to know, at least formally, the exact form of the effective action, and take consistent truncations thereof. We will discuss in detail how the ERGE is derived through a suitable decomposition of the metric field, which is well adapted to the use of heat-kernel techniques to evaluate the functional traces. Then we will report on the results for· the Einstein-Hilbert truncation (plus a cosmological constant term), and show the presence of a non-Gaussian FP which has good properties for the application to asymptotic safety. We will finally discuss the arguments that point to a physical reliability of the result. In Chapter 5: ·we 'Nill discuss the inclusion of matter in the picture. After introducing matter fields in the ERGK ·we ·will start off by discussing our results for a truncation involving massless, minimally coupled matter fields. \Ne 1Nill describe in detail the procedure (numerical and analytical) that 1:ve have followed to look for NGFP's as the number of matter fields is varied. As a consequence, we will see that the NGFP is not always present, so that bounds can be put on the matter content of the theory. When there exists a NG FP, there is actually only one that possesses good physical properties. Next, we will look at the important example of a self-interacting scalar field. vVe will discuss a class of actions depending on two infinite series of coupling constants. Even if pure gravity was proven to be asymptotically safe, it is not a priori guaranteed that a FP will exist in this extended theory, nor that it will have a finite number of attractive directions. This would mean that the predictivity of the theory is spoilt. A numerical search for NGFP convinces us that there is only one NGFP, at which the purely gravitational coupling constants are nonzero, whereas the other ones vanish. Since it generalizes the GFP of pure matter theory, we call it Gaussian-Matter FP {GMFP). It has a finite number of attractive directions, so that the asymptotic safety conditions are satisfied. A remarkable byproduct of this analysis is that the coupling to g;ravity solves the triviality problem of the scalar field theory. Then we add to the action other minimally coupled, massless fields and study their effect on the GMFP. The result is that it survives for certain combinations of matter fields, and the dimension of the UV critical surface is always finite. Finally, we discuss the cutoff and gauge dependence of physical results. The outcome of the analysis of the coupled system is that matter can be coupled to gravity to produce an asymptotically safe theory, provided the matter content satisfies some constraints. The Standard Model and many popular (supersymmetric) GUT theories fit well in this scenario. Finally, in Chapter 6 we discuss the results of this Thesis and draw the conclusions. The original contribution to this Thesis is contained in [22-25). The content of [22) is reported in Section 2.4. The contents of [23-25] cover all the discussion in Chapter 5.