We review and extend in several directions recent results on the "asymptotic safety" approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation.We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton's constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences. We then apply the Renormalization Group to higher derivative gravity. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four-derivative terms, while Newton's constant and of the cosmological constant have a nontrivial fixed point. In the case of actions that are polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV-attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three-dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme-dependent and schemeindependent results, and provide several examples of the fact that only dimensionless couplings can have "universal" behavior. * Electronic address: a.codello@gmail.com †

We discuss the existence and properties of a nontrivial fixed point in f (R)-gravity, where f is a polynomial of order up to six. Within this seven-parameter class of theories, the fixed point has three ultraviolet-attractive and four ultraviolet-repulsive directions; this brings further support to the hypothesis that gravity is nonperturbatively renormalizable.

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We study O(N ) models with power-law interactions by renormalization group (RG) methods: when the wave function renormalization is not present or not field dependent, their critical exponents can be computed from the ones of the corresponding short-range O(N ) models at an effective fractional dimension. Explicit results in 2 and 3 dimensions are given for the exponent ν. We propose an improved RG to describe the full theory space of the models where both short-range and long-range interactions are present and competing, and no a priori choice among the two in the RG flow is done: the eigenvalue spectrum of the full theory for all possible fixed points is drawn and the effective dimension shown to be only approximate. A full description of the fixed points structure is given, including multicritical long-range universality classes. PACS numbers: 11.10.Hi, 05.70.Fh, 11.10.Kk Preprint: CP 3 -Origins-2014-22 DNRF90 and DIAS-2014-22 O(N ) models are celebrated and tireless workhorses of statistical mechanics and play a key role in the field of critical phenomena: from one side the interest for their properties motivated the developments of numerous -analytical and numerical -techniques, from the other side they are concretely used as a test ground to benchmark the validity of new techniques for critical phenomena and lattice models.Among the interactions studied in the context of O(N ) models an important and paradigmatic role is played by long-range (LR) interactions, having the form of power-law decaying couplings.A first reason is that the results can be contrasted with the findings obtained for short-range (SR) interactions, to explore how universal and non-universal quantities change increasing the range where S i denote a unit vector with N components in the site i of a lattice in dimension d, J is a coupling energy and d + σ is the exponent of the power-law decay (we refer in the following to cubic lattices). When σ ≤ 0 a diverging energy density is obtained and to well define the thermodynamic limit it is necessary to rescale the coupling constant J [5]. When σ > 0 the model may have a phase transition of the second order, in particular as a function of the parameter σ three different regimes occur [6,7]: (i) for σ ≤ d/2 the mean-field approximation is valid even at the critical point; (ii) for σ greater than a critical value, σ * , the model has the same critical behaviour of the SR model (formally, the SR model is obtained in the limit σ → ∞); (iii) for d/2 < σ ≤ σ * the system exhibits peculiar LR critical exponents. For the Ising model in d = 1[2-4] the value σ * = 1 is found, and for σ = σ * a phase transition of the Berezinskii-KosterlitzThouless universality class occur [8][9][10] (see more references in [11]). Many efforts have been devoted to the determination of σ * and to the characterization of the universality classes in the region d/2 < σ ≤ σ * for general N in dimension d ≥ 2, which is the case we are going to consider in this paper. In the classical paper [6] the expression η = 2 − σ...

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We construct a consistent closure for the beta functions of the cosmological and Newton's constants by evaluating the influence that the anomalous dimensions of the fluctuating metric and ghost fields have on their renormalization group flow. In this generalized framework we confirm the presence of a UV attractive non-Gaussian fixed point, which we find characterized by real critical exponents. Our closure method is general and can be applied systematically to more general truncations of the gravitational effective average action.

We propose a novel derivation of the non-local heat kernel expansion, first studied by Barvinsky, Vilkovisky and Avramidi, based on simple diagrammatic equations satisfied by the heat kernel. For Laplace-type differential operators we obtain the explicit form of the non-local heat kernel form factors to second order in the curvature. Our method can be generalized easily to the derivation of the non-local heat kernel expansion of a wide class of differential operators.Comment: 23 pages, 1 figure, 31 diagrams; references added; to appear in JM

We show how the use of standard perturbative RG in dimensional regularization allows for a renormalization group-based computation of both the spectrum and a family of coefficients of the operator product expansion (OPE) for a given universality class. The task is greatly simplified by a straightforward generalization of perturbation theory to a functional perturbative RG approach. We illustrate our procedure in the -expansion by obtaining the nextto-leading corrections for the spectrum and the leading corrections for the OPE coefficients of Ising and Lee-Yang universality classes and then give several results for the whole family of renormalizable multi-critical models φ 2n . Whenever comparison is possible our RG results explicitly match the ones recently derived in CFT frameworks.