Higgs inflation and quantum gravity: An exact renormalisation group approach

We use the Wilsonian functional Renormalisation Group (RG) to study quantum corrections for the Higgs inflationary action including the effect of gravitons, and analyse the leading-order quantum gravitational corrections to the Higgs' quartic coupling, as well as its non-minimal coupling to gravity and Newton's constant, at the inflationary regime and beyond. We explain how within this framework the effect of Higgs and graviton loops can be sufficiently suppressed during inflation, and we also place a bound on the corresponding value of the infrared RG cut-off scale during inflation. Finally, we briefly discuss the potential embedding of the model within the scenario of Asymptotic Safety, while all main equations are explicitly presented.

Related papers:

title: The Standard Model Higgs boson as the inflaton
abstract: We argue that the Higgs boson of the Standard Model can lead to inflation and produce cosmological perturbations in accordance with observations. An essential requirement is the non-minimal coupling of the Higgs scalar field to gravity; no new particle besides already present in the electroweak theory is required.

title: Higgs inflation: consistency and generalisations
abstract: We analyse the self-consistency of inflation in the Standard Model, where the Higgs field has a large non-minimal coupling to gravity. We determine the domain of energies in which this model represents a valid effective field theory as a function of the background Higgs field. This domain is bounded above by the cutoff scale which is found to be higher than the relevant dynamical scales throughout the whole history of the Universe, including the inflationary epoch and reheating. We present a systematic scheme to take into account quantum loop corrections to the inflationary calculations within the framework of effective field theory. We discuss the additional assumptions that must be satisfied by the ultra-violet completion of the theory to allow connection between the parameters of the inflationary effective theory and those describing the low-energy physics relevant for the collider experiments. A class of generalisations of inflationary theories with similar properties is constructed.

title: Planck 2015. XX. Constraints on inflation
abstract: We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to $n_s = 0.9603 \pm 0.0073$, ruling out exact scale invariance at over 5 $\sigma$. Planck establishes an upper bound on the tensor-to-scalar ratio of r 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining $d n_s/d ln k = -0.0134 \pm 0.0090$. Several analyses dropping the slow-roll approximation are carried out, including detailed model comparison and inflationary potential reconstruction. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit $\chi^2$ by ~ 10; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with bounds on $f_\mathrm{NL}$ measured by Planck. The fractional primordial contribution of CDM isocurvature modes in the curvaton and axion scenarios has upper bounds of 0.25% or 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelation can improve $\chi^2$ by approximatively 4 as a result of slightly lowering the theoretical prediction for the $\ell<40$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.

title: Distinguishing between R-2-inflation and Higgs-inflation
abstract: We present three features which can be used to distinguish the R 2 -inflation Higgs-inflation from with ongoing, upcoming and planned experiments, assuming no new physics (apart form sterile neutrinos) up to inflationary scale. (i) Slightly different tilt of the scalar perturbation spectrum n s and ratio r of scalar-to-tensor perturbation amplitudes. (ii) Gravity waves produced within R 2 -model by collapsing, merging and evaporating scalaron clumps formed in the post-inflationary Universe. (iii) Different ranges of the possible Standard Model Higgs boson masses, where the electroweak vacuum remains stable while the Universe evolves after inflation. Specifically, in the R 2 -model Higgs boson can be as light as 116 GeV. These effects mainly rely on the lower reheating temperature in the R 2 -inflation.

title: Brans–Dicke theory in the local potential approximation
abstract: We study the Brans–Dicke theory with arbitrary potential within a functional renormalization group framework. Motivated by the asymptotic safety scenario of quantum gravity and by the well-known relation between f(R) gravity and Brans–Dicke theory at the classical level, we concentrate our analysis on the fixed-point equation for the potential in four dimensions and with Brans–Dicke parameter . For two different choices of gauge, we study the resulting equations by examining both the local and global properties of the solutions, by means of analytical and numerical methods. As a result of our analysis we do not find any nontrivial fixed point in one gauge, but we find a continuum of fixed points in the other one. We interpret such an inconsistency as a result of the restriction to , and thus we suggest that it indicates a failure of the equivalence between f(R) gravity and Brans–Dicke theory at the quantum level.

title: Higgs inflation at the critical point
abstract: Higgs inflation can occur if the Standard Model (SM) is a self-consistent effective field theory up to inflationary scale. This leads to an upper bound on the top Yukawa coupling, y(t)(phys) < y(t)(crit) and thus on the mass of the top quark m(t). If m(t) is more than a few hundred of MeV below the critical value, the Higgs inflation predicts the universal values of inflationary indexes, r similar or equal to 0.003 and n(s) similar or equal to 0.97, independently on the SM parameters. We show that in the vicinity of the critical point y(t)(crit) the inflationary indexes acquire an essential dependence on m(t) and on the mass of the Higgs boson M-h. In particular, the amplitude of the gravitational waves can exceed considerably the universal value.

title: Higgs inflation from Standard Model criticality
abstract: In this chapter, we investigate the possibility that the SM Higgs boson plays the role of an inflation in light of the discovery of the Higgs boson. In 2007, Bezurkov and Shaposhnikov first pointed out this possibility. The successful Higgs inflation is realized if the non-minimal coupling between the Higgs H and scalar curvature \(\mathcal {R}\), \(\xi |H|^2 \mathcal {R}\), is introduced. \(\xi \) is the coupling constant, and very large value, \(\xi \sim 10^5\), is required for successful inflation. However, the observation of the Higgs and determination of its mass changes the situation. As we have seen in Chap. 2, the Higgs self coupling and its beta function becomes zero at very high scale, which means that the Higgs potential is very flat around string/Planck scale. This opens up the new possibilities of the Higgs inflation, which we will present here. In Sect. 3.1, we present the general argument of the Higgs inflation above the SM cutoff \(\Lambda \). In Sect. 3.2, we show that non-minimal coupling \(\xi \) can be as small as \(\mathcal {O}(10)\).

title: The renormalization group for non-renormalizable theories: Einstein gravity with a scalar field
abstract: We develop a renormalization-group formalism for non-renormalizable theories and apply it to Einstein gravity theory coupled to a scalar field with the Lagrangian $L=\sqrt{g} [R U(\phi)-{1/2} G(\phi) g^{\mu\nu} \partial_{\mu}\phi \partial_{\nu}\phi- V(\phi)]$, where $U(\phi), G(\phi)$ and $V(\phi)$ are arbitrary functions of the scalar field. We calculate the one-loop counterterms of this theory and obtain a system of renormalization-group equations in partial derivatives for the functions $U, G$ and $V$ playing the role of generalized charges which substitute for the usual charges in multicharge theories. In the limit of a large but slowly varying scalar field and small spacetime curvature this system gives the asymptotic behaviour of the generalized charges compatible with the conventional choice of these functions in quantum cosmological applications. It also demonstrates in the over-Planckian domain the existence of the Weyl-invariant phase of gravity theory asymptotically free in gravitational and cosmological constants.

title: One-loop renormalization of the four-dimensional theory for quantum dilaton gravity.
abstract: We study one loop renormalization in the most general metric-dilaton theory with second derivative terms only. The classical action includes three arbitrary functions of the dilaton. The general theory can be divided into two classes. Models of one are equivalent to gravity conformally coupled to a scalar field and also to general relativity with a cosmological term. The models of the second class have one extra degree of freedom which corresponds to the dilaton. We calculate the one loop divergences for the models of the second class and find that the theory is not renormalizable off the mass shell. At the same time the arbitrary functions of the dilaton in the starting action can be fine-tuned in such a way that all the higher derivative counterterms disappear on shell. The only structures in both the classical action and counterterms, which survive on shell, are the potential (cosmological) ones. They can be removed by renormalization of the dilaton field that acquires the nontrivial anomalous dimension, which leads to the effective running of the cosmological constant. Another application of our calculations is the following. For some special choice of the arbitrary functions our dilaton model is equivalent to general relativity with an additional ${\mathit{R}}^{2}$ term. Such an equivalence holds at the quantum level if we do not introduce the external source for the dilaton field. Thus our calculations in a general dilaton model in original variables include quantum \ensuremath{\Lambda}+\ensuremath{\alpha}R+\ensuremath{\beta}${\mathit{R}}^{2}$ theory as the particular case.

title: Renormalization Group Flow in Scalar-Tensor Theories. I
abstract: We study the renormalization group flow in a class of scalar-tensor theories involving at most two derivatives of the fields. We show in general that minimal coupling is self consistent, in the sense that when the scalar self couplings are switched off, their beta functions also vanish. Complete, explicit beta functions that could be applied to a variety of cosmological models are given in a five parameter truncation of the theory in d = 4. In any dimension d > 2 we find that the flow has only a “Gaussian Matter” fixed point, where all scalar self interactions vanish but Newton’s constant and the cosmological constant are nontrivial. The properties of these fixed points can be studied algebraically to some extent. In d = 3 we also find a gravitationally dressed version of the WilsonFisher fixed point, but it seems to have unphysical properties. These findings are in accordance with the hypothesis that these theories are asymptotically safe.

title: Renormalization group flow in scalar-tensor theories: II
abstract: We study the UV behaviour of actions including integer powers of scalar curvature and even powers of scalar fields with functional renormalization group techniques. We find UV fixed points where the gravitational couplings have nontrivial values while the matter ones are Gaussian. We prove several properties of the linearized flow at such a fixed point in arbitrary dimensions in the one-loop approximation and find recursive relations among the critical exponents. We illustrate these results in explicit calculations in d = 4 for actions including up to four powers of scalar curvature and two powers of the scalar field. In this setting we note that the same recursive properties among the critical exponents, which were proven at one-loop order, still hold, in such a way that the UV critical surface is found to be five dimensional. We then search for the same type of fixed point in a scalar theory with minimal coupling to gravity in d = 4 including up to eight powers of scalar curvature. Assuming that the recursive properties of the critical exponents still hold, one would conclude that the UV critical surface of these theories is five dimensional.

title: Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity
abstract: We study the fixed point structure of the Higgs-Yukawa model, with its scalar being non-minimally coupled to the asymptotically safe gravity, using the functional renormalization group. We have obtained the renormalization group equations for the cosmological and Newton constants, the scalar mass-squared and quartic coupling constant, and the Yukawa and non-minimal coupling constants, taking into account all the scalar, fermion, and graviton loops. We find that switching on the fermionic quantum fluctuations makes the non-minimal coupling constant irrelevant around the Gaussian-matter fixed point with the asymptotically safe gravity.

title: Gravitational corrections to Yukawa systems
abstract: We compute the gravitational corrections to the running of couplings in a scalar-fermion system, using the Wilsonian approach. Our discussion is relevant for symmetric as well as for broken scalar phases. We find that the Yukawa and quartic scalar couplings become irrelevant at the Gaussian fixed point.

title: EFFECTIVE EQUATIONS OF MOTION AND INITIAL CONDITIONS FOR INFLATION IN QUANTUM COSMOLOGY
abstract: We obtain effective equations of inflationary dynamics for the mean inflation and metric fields — expectation values in the no-boundary and tunneling quantum states of the Universe. The equations are derived in the slow-roll approximation taking the form of the local Schwinger-DeWitt expansion. In this approximation effective equations follow from the Euclidean effective action calculated on the De Sitter gravitational instanton — the basic element of the no-boundary and tunneling cosmological wavefunctions. Effective equations are applied in the model of the inflaton scalar field coupled to the GUT sector of matter fields and also having a strong non-minimal coupling to the curvature. The inverse of its large non-minimal coupling constant, − ξ = ∥ ξ ∥ ⪢ 1, serves as a small parameter of the slow-roll expansion and the semiclassical expansion of quantum gravitational effects. As a source of initial conditions for effective equations we use a sharp probability peak recently obtained in the one-loop approximation for the no-boundary and tunneling quantum states and belonging (in virtue of large ∥ ξ ∥) to the GUT energy scale much below the Planck scale. Cosmological consequences of effective equations in the tunneling quantum state predict a finite duration of the inflationary stage compatible with the observational status of inflation theory, whereas for the no-boundary state they lead to an infinite inflationary epoch with a constant inflaton field.

title: Higgs boson, renormalization group, and naturalness in cosmology
abstract: We consider the renormalization group improvement in the theory of the Standard Model (SM) Higgs boson playing the role of an inflaton with a strong non-minimal coupling to gravity. At the one-loop level with the running of constants taken into account, it leads to a range of the Higgs mass that is entirely determined by the lower WMAP bound on the cosmic microwave background (CMB) spectral index. We find that the SM phenomenology is sensitive to current cosmological data, which suggests to perform more precise CMB measurements as a SM test complementary to the LHC program. By using the concept of a field-dependent cutoff, we show the naturalness of the gradient and curvature expansion in this model within the conventional perturbation theory range of the SM. We also discuss the relation of these results to two-loop calculations and the limitations of the latter caused by parametrization and gauge dependence problems.

title: Asymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field
abstract: We consider the renormalization group improvement in the theory of the Standard Model Higgs boson playing the role of an inflaton with a strong non-minimal coupling to gravity. It suggests the range of the Higgs mass $135.6 {\rm GeV} \lesssim M_H\lesssim 184.5 {\rm GeV}$ compatible with the current CMB data (the lower WMAP bound on $n_s$), which is close to the widely accepted range dictated by the electroweak vacuum stability and perturbation theory bounds. We find the phenomenon of asymptotic freedom induced by this non-minimal curvature coupling, which brings the theory to the weak coupling domain everywhere except at the lower and upper boundary of this range. The renormalization group running of the basic quantity ${\boldmath $A_I$}$ -- the anomalous scaling in the non-minimally coupled Standard Model, which analytically determines all characteristics of the CMB spectrum -- brings ${\boldmath $A_I$}$ to small negative values at the inflation scale. This property is crucial for the above results and may also underlie the formation of initial conditions for the inflationary dynamics in quantum cosmology.

title: Running inflation in the Standard Model
abstract: An interacting scalar field with largish coupling to curvature can support a distinctive inflationary universe scenario. Previously this has been discussed for the Standard Model Higgs field, treated classically or in a leading log approximation. Here we investigate the quantum theory using renormalization group methods. In this model the running of both the effective Planck mass and the couplings is important. The cosmological predictions are consistent with existing WMAP5 data, with 0.967 ≲ n s ≲ 0.98 (for N e = 60 ) and negligible gravity waves. We find a relationship between the spectral index and the Higgs mass that is sharply varying for m h ∼ 120 – 135 GeV (depending on the top mass); in the future, that relationship could be tested against data from PLANCK and LHC. We also comment briefly on how similar dynamics might arise in more general settings, and discuss our assumptions from the effective field theory point of view.

title: Standard Model Higgs boson mass from inflation: Two loop analysis
abstract: We extend the analysis of  of the Standard Model Higgs inflation accounting for two-loop radiative corrections to the effective potential. As was expected, higher loop effects result in some modification of the interval for allowed Higgs masses mmin < mH < mmax, which somewhat exceeds the region in which the Standard Model can be considered as a viable effective field theory all the way up to the Planck scale. The dependence of the index ns of scalar perturbations on the Higgs mass is computed in two different renormalization procedures, associated with the Einstein (I) and Jordan (II) frames. In the procedure I the predictions of the spectral index of scalar fluctuations and of the tensor-to-scalar ratio practically do not depend on the Higgs mass within the admitted region and are equal to ns = 0.97 and r = 0.0034 respectively. In the procedure II the index ns acquires the visible dependence on the Higgs mass and and goes out of the admitted interval at mH below mmin. We compare our findings with the results of .

title: Quantum corrections in Higgs inflation: the real scalar case
abstract: We present a critical discussion of quantum corrections, renormalisation, and the computation of the beta functions and the effective potential in Higgs inflation. In contrast with claims in the literature, we find no evidence for a disagreement between the Jordan and Einstein frames, even at the quantum level. For clarity of discussion we concentrate on the case of a real scalar Higgs. We first review the classical calculation and then discuss the back reaction of gravity. We compute the beta functions for the Higgs quartic coupling and non-minimal coupling constant. Here, the mid-field regime is non-renormalisable, but we are able to give an upper bound on the 1-loop corrections to the effective potential. We show that, in computing the effective potential, the Jordan and Einstein frames are compatible if all mass scales are transformed between the two frames. As such, it is consistent to take a constant cutoff in either the Jordan or Einstein frame, and both prescriptions yield the same result for the effective potential. Our results are extended to the case of a complex scalar Higgs.

title: Quantum corrections in Higgs inflation: the Standard Model case
abstract: We compute the one-loop renormalization group equations for Standard Model Higgs inflation. The calculation is done in the Einstein frame, using a covariant formalism for the multi-field system. All counterterms, and thus the betafunctions, can be extracted from the radiative corrections to the two-point functions; the calculation of higher n-point functions then serves as a consistency check of the approach. We find that the theory is renormalizable in the effective field theory sense in the small, mid and large field regime. In the large field regime our results differ slightly from those found in the literature, due to a different treatment of the Goldstone bosons.

title: Higgs Inflation at NNLO after the Boson Discovery
abstract: We obtain the bound on the Higgs and top masses to have Higgs inflation (where the Higgs field is non-minimally coupled to gravity) at full next-to-next-to-leading order (NNLO). Comparing the result obtained with the experimental values of the relevant parameters we find some tension, which we quantify. Higgs inflation, however, is not excluded at the moment as the measured values of the Higgs and top masses are close enough to the bound once experimental and theoretical uncertainties are taken into account.

title: Classical and quantum initial conditions for Higgs inflation
abstract: We investigate whether Higgs inflation can occur in the Standard Model starting from natural initial conditions or not. The Higgs has a non-minimal coupling to the Ricci scalar. We confine our attention to the regime where quantum Einstein gravity effects are small in order to have results that are independent of the ultraviolet completion of gravity. At the classical level we find no tuning is required to have successful Higgs inflation, provided the initial homogeneity condition is satisfied. On the other hand, at the quantum level we obtain that the renormalization for large non-minimal coupling requires an additional degree of freedom, unless a tuning of the initial values of the running parameters is made. In order to see that this effect may change the predictions we finally include such degree of freedom in the field content and show that Starobinsky's R 2 inflation dominates over Higgs inflation.

title: Covariant one-loop quantum gravity and Higgs inflation
abstract: The quantisation of scalar field theory and Einstein gravity is investigated using a fully covariant background field formalism, including Vilkovisky-DeWitt corrections. The one-loop divergences, which are relevant for the consistency of the low-energy effective theory, differ substantially from non-covariant calculations. The results are applied to the effective action of Higgs inflation, with a non-minimal gravity coupling parameter $\xi$, where the use of a formalism which is independent of field redefinitions is particularly important. Consistency of the one-loop effective action requires the cut-off $\Lambda<M_p/\sqrt{\xi}$ in the small field and $\Lambda<M_p$ in the large field limit.

title: The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure
abstract: `The beginning of the twenty-first century stands a good chance of being identified in history as the time when humankind first came to grips with the Universe ... The emergence of the standard cosmological model had left by the wayside many concepts which in the late 1990s had still had possible relevance for our Universe'. Do these confident assertions by the authors of this new cosmology textbook show that we have reached an `end of history' regarding the big questions of cosmic origins, or is this just more in the line of Landau's quip: `Cosmologists are often in error, but never in doubt'? Despite Landau's comment having some justice at the time, a lot can change in science in over 50 years, and there is indeed a fair case that our overall picture of the Universe has converged. The standard cosmology is dominated by vacuum energy, and contains clumped dark matter whose inhomogeneity was seeded by quantum fluctuations during a period of `inflation', when the vacuum energy was very much higher than it is today. All the theoretical elements of this picture were put in place during a remarkable period of about three years at the start of the 1980s—a revolution surely comparable with the dawn of quantum mechanics in the 1920s. Consistency of this theory with observations hardened progressively during the 1990s, and all serious resistance finally evaporated in 2003 with the first detailed sky maps of the microwave background from NASA's WMAP satellite. The philosophy of this textbook is therefore a good one: rather than dwelling on how we arrived at our current understanding, the aim is to give a clear and clean presentation of the theory, preparing the interpretative framework for the new datasets that are set to arrive in the near future. This represents a deliberate departure from the broader perspective of the book written by the same authors (in the opposite order) in 2000: Lyth and Liddle's view is that the past decade has added so many new observations that the field has changed beyond the point where a simple update is sufficient, and it is hard to disagree. The benefits of the new approach show themselves in the elegance with which the book sets up the formalism for describing metric fluctuations in an expanding Universe, and how such fluctuations are generated during an early phase when the Universe is assumed to be dominated by the energy density of one or more hypothetical scalar fields. The key observables that let us probe this era are the scale-dependence of the density fluctuation spectrum; the tensor fluctuations (primordial gravity waves); and the extent to which the fluctuations deviate from a random-phase Gaussian character. Lyth in particular is an acknowledged master of this material, and has been responsible for creating significant parts of the formalism being described in the book. The fine details of calculating the origin of structure in exotic models are all present, and are set in the context of a nice pedagogical exposition and revision of relevant background in general relativity and field theory. This is a complete guide that can take a graduate student from undergraduate knowledge to calculations at the forefront of research. This book will thus be an indispensable companion for those researchers working in the theory of structure in the early Universe, and those teaching it at the postgraduate level. I have already found it useful for sharpening arguments in lecture notes that I previously regarded as fairly mature. It is not the only book in this area, but it is really the best on the fundamental topics where it concentrates its efforts. On the issue of computing the observational consequences of inflationary fluctuations (i.e. galaxy clustering and anisotropies in the Cosmic Microwave Background), Dodelson's Modern Cosmology (Academic Press) will probably have more appeal to most readers, simply because it treats these matters at greater length and with more use of physical analogy. But readers with some previous exposure to these topics will enjoy the stripped-down clarity of Lyth and Liddle's presentation. Dodelson also has the advantage that he gives outline solutions to his exercises, whereas Lyth and Liddle do not. I do think this is a pity, since students learn nothing from defeat by a problem they cannot solve, but can benefit enormously from struggling and then consulting a hint that helps them see where the key difficulty lay; something for the second edition, perhaps? In all of this, though, we should not lose sight of something a little peculiar. This is a textbook that provides great detail on a mature theory, which has seen great efforts and huge refinements over the past 30 years. And yet we have almost no idea if inflation is the correct theory of the early Universe. Some semi-popular treatments, even quite sophisticated ones, claim that observations of the CMB have proved inflation to be true, and one encounters many students who believe this excessive claim. The enthusiasm of advocates is understandable, since inflation has seen some great successes. Before it was suggested, we knew empirically that the early Universe must have had a mechanism for almost perfectly eliminating spatial curvature in the initial conditions, and generating fluctuations in an acausal manner, so that the Universe is very nearly uniform on large spatial scales, but not quite. The triumph of inflation is that it was designed to solve the first problem, and subsequently realized to offer in addition a possible solution to the second. But we really would like more direct evidence that the energy density was once dominated by a scalar field, and Lyth and Liddle's book is devoted to calculating the three observational signatures: `tilt' of the fluctuation spectrum, relic gravity waves, and non-Gaussianity. The first of these has already been seen, and if the other two signatures can be verified, then we would have a set of interlocking tests that could provide consistent verification of the theory. But as the authors admit in a coda, there are many models where these additional features will never be observable. Inflation would then remain in its current state: provisional acceptance by virtue of lack of an alternative. For the health of cosmology, we have to hope that future observations will be richer than this; as the data come in, Lyth and Liddle will be an essential guide.

title: Quantum Einstein gravity
abstract: We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in quantum Einstein gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the causal dynamical triangulation approach.

title: Aspects of the functional renormalisation group
abstract: We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being N PI quantities. We discuss optimisation and derive a functional optimisation criterion. Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson–Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.

title: Fixed points of quantum gravity and the renormalisation group
abstract: We review the asymptotic safety scenario for quantum gravity and the role and implications of an underlying ultraviolet fixed point. We discuss renormalisation group techniques employed in the fixed point search, analyse the main picture at the example of the Einstein-Hilbert theory, and provide an overview of the key results in four and higher dimensions. We also compare findings with recent lattice simulations and evaluate phenomenological implications for collider experiments.

title: Exact evolution equation for the effective potential
abstract: We derive a new exact evolution equation for the scale dependence of an effective action. The corresponding equation for the effective potential permits a useful truncation. This allows one to deal with the infrared problems of theories with massless modes in less than four dimensions which are relevant for the high temperature phase transition in particle physics or the computation of critical exponents in statistical mechanics.

title: Derivative expansion of the exact renormalization group
abstract: The functional flow equations for the Legendre effective action, with respect to changes in a smooth cutoff, are approximited by a derivative expansion; no other approximation is made. This results in a set of coupled non-linear differential equations. The corresponding differential equations for a fixed point action have at most a countable number of solutions that are well defined for all values of the field. We apply the technique to the fixed points of one-component real scalar field theory in three dimensions. Only two non-singular solutions are found; the gaussian fixed point and an approximation to the Wilson fixed point. The latter is used to compute critical exponents, by carrying the approximation to second order. The results appear to converge rapidly.

title: The local potential approximation in the background field formalism
abstract: A bstractWorking within the familiar local potential approximation, and concentrating on the example of a single scalar field in three dimensions, we show that the commonly used approximation method of identifying the total and background fields, leads to pathologies in the resulting fixed point structure and the associated spaces of eigenoperators. We then show how a consistent treatment of the background field through the corresponding modified shift Ward identity, can cure these pathologies, restoring universality of physical quantities with respect to the choice of dependence on the background field, even within the local potential approximation. Along the way we point out similarities to what has been previously found in the f(R) approximation in asymptotic safety for gravity.

title: En route to Background Independence: Broken split-symmetry, and how to restore it with bi-metric average actions
abstract: The most momentous requirement a quantum theory of gravity must satisfy is Background Independence, necessitating in particular an ab initio derivation of the arena all non-gravitational physics takes place in, namely spacetime. Using the background field technique, this requirement translates into the condition of an unbroken split-symmetry connecting the (quantized) metric fluctuations to the (classical) background metric. If the regularization scheme used violates split-symmetry during the quantization process it is mandatory to restore it in the end at the level of observable physics. In this paper we present a detailed investigation of split-symmetry breaking and restoration within the Effective Average Action (EAA) approach to Quantum Einstein Gravity (QEG) with a special emphasis on the Asymptotic Safety conjecture. In particular we demonstrate for the first time in a non-trivial setting that the two key requirements of Background Independence and Asymptotic Safety can be satisfied simultaneously. Carefully disentangling fluctuation and background fields, we employ a ‘bi-metric’ ansatz for the EAA and project the flow generated by its functional renormalization group equation on a truncated theory space spanned by two separate Einstein–Hilbert actions for the dynamical and the background metric, respectively. A new powerful method is used to derive the corresponding renormalization group (RG) equations for the Newton- and cosmological constant, both in the dynamical and the background sector. We classify and analyze their solutions in detail, determine their fixed point structure, and identify an attractor mechanism which turns out instrumental in the split-symmetry restoration. We show that there exists a subset of RG trajectories which are both asymptotically safe and split-symmetry restoring: In the ultraviolet they emanate from a non-Gaussian fixed point, and in the infrared they loose all symmetry violating contributions inflicted on them by the non-invariant functional RG equation. As an application, we compute the scale dependent spectral dimension which governs the fractal properties of the effective QEG spacetimes at the bi-metric level. Earlier tests of the Asymptotic Safety conjecture almost exclusively employed ‘single-metric truncations’ which are blind towards the difference between quantum and background fields. We explore in detail under which conditions they can be reliable, and we discuss how the single-metric based picture of Asymptotic Safety needs to be revised in the light of the new results. We shall conclude that the next generation of truncations for quantitatively precise predictions (of critical exponents, for instance) is bound to be of the bi-metric type.

title: RG flows of Quantum Einstein Gravity on maximally symmetric spaces
abstract: A bstractWe use the Wetterich-equation to study the renormalization group flow of f (R)-gravity in a three-dimensional, conformally reduced setting. Building on the exact heat kernel for maximally symmetric spaces, we obtain a partial differential equation which captures the scale-dependence of f (R) for positive and, for the first time, negative scalar curvature. The effects of different background topologies are studied in detail and it is shown that they affect the gravitational RG flow in a way that is not visible in finite-dimensional truncations. Thus, while featuring local background independence, the functional renormalization group equation is sensitive to the topological properties of the background. The detailed analytical and numerical analysis of the partial differential equation reveals two globally well-defined fixed functionals with at most a finite number of relevant deformations. Their properties are remarkably similar to two of the fixed points identified within the R2-truncation of full Quantum Einstein Gravity. As a byproduct, we obtain a nice illustration of how the functional renormalization group realizes the “integrating out” of fluctuation modes on the three-sphere.

title: Optimisation of the exact renormalisation group
abstract: A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.

title: Nonperturbative evolution equation for quantum gravity
abstract: A scale-dependent effective action for gravity is introduced and an exact nonperturbative evolution equation is derived which governs its renormalization group flow. It is invariant under general coordinate transformations and satisfies modified Becchi-Rouet-Stora Ward identities. The evolution equation is solved for a simple truncation of the space of actions. In $2+\ensuremath{\varepsilon}$ dimensions, nonperturbative corrections to the \ensuremath{\beta} function of Newton's constant are derived and its dependence on the cosmological constant is investigated. In 4 dimensions, Einstein gravity is found to be ``antiscreening;'' i.e., Newton's constant increases at large distances.

title: Heat kernel expansion: user's manual
abstract: The heat kernel expansion is a very convenient tool for studying one-loop divergences, anomalies and various asymptotics of the effective action. The aim of this report is to collect useful information on the heat kernel coefficients scattered in mathematical and physical literature. We present explicit expressions for these coefficients on manifolds with and without boundaries, subject to local and non-local boundary conditions, in the presence of various types of singularities (e.g., domain walls). In each case the heat kernel coefficients are given in terms of several geometric invariants. These invariants are derived for scalar and spinor theories with various interactions, Yang–Mills fields, gravity, and open bosonic strings. We discuss the relations between the heat kernel coefficients and quantum anomalies, corresponding anomalous actions, and covariant perturbation expansions of the effective action (both “low-” and “high-energy” ones).

title: Functional Renormalization Group Equations, Asymptotic Safety, and Quantum Einstein Gravity
abstract: These lecture notes provide a pedagogical introduction to a specific continuum implementation of the Wilsonian renormalization group, the effective average action. Its general properties and, in particular, its functional renormalization group equation are explained in a simple scalar setting. The approach is then applied to Quantum Einstein Gravity (QEG). The possibility of constructing a fundamental theory of quantum gravity in the framework of Asymptotic Safety is discussed and the supporting evidence is summarized.

title: Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation
abstract: 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 scheme-independent results, and provide several examples of the fact that only dimensionless couplings can have “universal” behavior.

title: Effective Action in Quantum Gravity
abstract: Elements of quantum field theory: The basic models of quantum field theory Effective action in quantum field theory Quantum field theory in curved space-time and effective action: The renormalization and renormalization group equations in curved space-time The renormalization group method in curved space-time with torsion Renormalization group equations and effective action Unique effective action in quantum field theory Effective action of composite fields in curved space-time Selected problems of quantum gravity: Higher derivatives quantum gravity Asymptotically free models of grand unification Effective action in multidimensional quantum gravity Quantum properties of torus compactified membrane References Index

title: Asymptotically Safe Inflation
abstract: Inflation is studied in the context of asymptotically safe theories of gravitation. Conditions are explored under which it is possible to have a long period of nearly exponential expansion that eventually comes to an end.

title: From big bang to asymptotic de Sitter: complete cosmologies in a quantum gravity framework
abstract: Using the Einstein–Hilbert approximation of asymptotically safe quantum gravity we present a consistent renormalization group based framework for the inclusion of quantum gravitational effects into the cosmological field equations. Relating the renormalization group scale to cosmological time via a dynamical cutoff identification this framework applies to all stages of the cosmological evolution. The very early universe is found to contain a period of 'oscillatory inflation' with an infinite sequence of time intervals during which the expansion alternates between acceleration and deceleration. For asymptotically late times we identify a mechanism which prevents the universe from leaving the domain of validity of the Einstein–Hilbert approximation and obtain a classical de Sitter era.

title: Renormalization-group running of the cosmological constant and the fate of the universe
abstract: For a generic quantum field theory we study the role played by the renormalization-group (RG) running of the cosmological constant (CC) in determining the ultimate fate of the universe. We consider the running of the CC of generic origin (the vacuum energy of quantum fields and the potential energy of classical fields), with the RG scale proportional to the (total energy density$\rm{)^{1/4}}$ as the most obvious identification. Starting from the present-era values for cosmological parameters we demonstrate how the running can easily provide a negative cosmological constant, thereby changing the fate of the universe, at the same time rendering compatibility with critical string theory. We also briefly discuss the recent past in our scenario.

title: Running G and Λ at low energies from physics at M X : possible cosmological and astrophysical implications
abstract: The renormalization group (RG) approach to cosmology is an efficient method to study the possible evolution of the cosmological parameters from the point of view of quantum field theory (QFT) in curved space-time. In this work we continue our previous investigations of the RG method based on potential low-energy effects induced from physics at very high energy scales MX . MP . In the present instance we assume that both the Newton constant, G, and the cosmological term, Λ, can be functions of a scale parameter μ. It turns out that G(μ) evolves according to a logarithmic law which may lead to asymptotic freedom of gravity, similar to the gauge coupling in QCD. At the same time Λ(μ) evolves quadratically with μ. We study the consistency and cosmological consequences of these laws when μ ≃ H. Furthermore, we propose to extend this method to the astrophysical domain after identifying the local RG scale at the galactic level. It turns out that Kepler’s third law of celestial mechanics receives quantum corrections that may help to explain the flat rotation curves of the galaxies without introducing the dark matter hypothesis. The origin of these effects (cosmological and astrophysical) could be linked, in our framework, to physics at MX ∼ 1016−17 GeV .

title: Renormalization Group scale-setting in astrophysical systems
abstract: A more general scale-setting procedure for General Relativity with Renormalization Group corrections is proposed. Theoretical aspects of the scale-setting procedure and the interpretation of the Renormalization Group running scale are discussed. The procedure is elaborated for several highly symmetric systems with matter in the form of an ideal fluid and for two models of running of the Newton coupling and the cosmological term. For a static spherically symmetric system with the matter obeying the polytropic equation of state the running scale-setting is performed analytically. The obtained result for the running scale matches the Ansatz introduced in a recent paper by Rodrigues, Letelier and Shapiro which provides an excellent explanation of rotation curves for a number of galaxies. A systematic explanation of the galaxy rotation curves using the scale-setting procedure introduced in this Letter is identified as an important future goal.

title: Cosmology of the Planck era from a renormalization group for quantum gravity
abstract: Homogeneous and isotropic cosmologies of the Planck era before the classical Einstein equations become valid are studied taking quantum gravitational effects into account. The cosmological evolution equations are renormalization group improved by including the scale dependence of Newton's constant and of the cosmological constant as it is given by the flow equation of the effective average action for gravity. It is argued that the Planck regime can be treated reliably in this framework because gravity is found to become asymptotically free at short distances. The epoch immediately after the initial singularity of the Universe is described by an attractor solution of the improved equations which is a direct manifestation of an ultraviolet attractive renormalization group fixed point. It is shown that quantum gravity effects in the very early Universe might provide a resolution to the horizon and flatness problems of standard cosmology, and could generate a scale-free spectrum of primordial density fluctuations.

title: Cosmology with self-adjusting vacuum energy density from a renormalization group fixed point
abstract: Cosmologies with a time dependent Newton constant and cosmological constant are investigated. The scale dependence of G and Λ is governed by a set of renormalization group equations which is coupled to Einstein's equation in a consistent way. The existence of an infrared attractive renormalization group fixed point is postulated, and the cosmological implications of this assumption are explored. It turns out that in the late Universe the vacuum energy density is automatically adjusted so as to equal precisely the matter energy density, and that the deceleration parameter approaches q =−1/4. This scenario might explain the data from recent observations of high redshift type Ia supernovae and the cosmic microwave background radiation without introducing a quintessence field.

title: Inflationary solutions in asymptotically safe f(R) theories
abstract: We discuss the existence of inflationary solutions in a class of renormalization group improved polynomial f(R) theories, which have been studied recently in the context of the asymptotic safety scenario for quantum gravity. These theories seem to possess a nontrivial ultraviolet fixed point, where the dimensionful couplings scale according to their canonical dimensionality. Assuming that the cutoff is proportional to the Hubble parameter, we obtain modified Friedmann equations which admit both power law and exponential solutions. We establish that for sufficiently high order polynomial the solutions are reliable, in the sense that considering still higher order polynomials is very unlikely to change the solution.

title: An effective action for asymptotically safe gravity
abstract: Asymptotically safe theories of gravitation have received great attention in recent times. In this framework an effective action embodying the basic features of the renormalized flow around the non-gaussian fixed point is derived and its implications for the early universe are discussed. In particular, a "landscape" of a countably infinite number of cosmological inflationary solutions characterized by an unstable de Sitter phase lasting for a large enough number of e-folds is found.

title: Renormalization group improvement of scalar field inflation
abstract: We study quantum corrections to Friedmann-Robertson-Walker cosmology with a scalar field under the assumption that the dynamics are subject to renormalisation group improvement. We use the Bianchi identity to relate the renormalisation group scale to the scale factor and obtain the improved cosmological evolution equations. We study the solutions of these equations in the renormalisation group fixed point regime, obtaining the time-dependence of the scalar field strength and the Hubble parameter in specific models with monomial and trinomial quartic scalar field potentials. We find that power-law inflation can be achieved in the renormalisation group fixed point regime with the trinomial potential, but not with the monomial one. We study the transition to the quasi-classical regime, where the quantum corrections to the couplings become small, and find classical dynamics as an attractor solution for late times. We show that the solution found in the renormalisation group fixed point regime is also a cosmological fixed point in the autonomous phase space. We derive the power spectrum of cosmological perturbations and find that the scalar power spectrum is exactly scale-invariant and bounded up to arbitrarily small times, while the tensor perturbations are tilted as appropriate for the background power-law inflation. We specify conditions for the renormalisation group fixed point values of the couplings under which the amplitudes of the cosmological perturbations remain small.

title: Small Cosmological Constant from Running Gravitational Coupling
abstract: In this paper, we explore an idea of having Newton’s constant change its value depending on the curvature scale involved. Such modification leads to a particular scalar-tensor gravity theory, with the Lagrangian derived from renormalization group (RG) flow arguments. Assuming running of the (classical) gravitational coupling is described by the same beta function as the (quantum) one in quantum chromodynamics (QCD), one is led to a simple f(R)-type model with several striking features. The model has unique de Sitter attractor, with effective cosmological constant exponentially suppressed relative to Planck scale. Not only cosmological constant appears dynamically in this model, its value is small precisely because the gravity we observe is almost Einstein-like. With the

title: Asymptotically safe gravity as a scalar-tensor theory and its cosmological implications
abstract: We study asymptotically safe gravity with Einstein-Hilbert truncation taking into account the renormalization group running of both gravitational and cosmological constants. We show the classical behavior of the theory is equivalent to a specific class of Jordan-Brans-Dicke theories with vanishing Brans-Dicke parameter, and potential determined by the renormalization group equation. The theory may be reformulated as an $f(R)$ theory. In the simplest cosmological scenario, we find large--field inflationary solutions near the Planck scale where the effective field theory description breaks down. Finally, we discuss the implications of a running gravitational constant to background dynamics via cosmological perturbation theory. We show that compatibility with General Relativity requires contributions from the running gravitational constant to the stress energy tensor to be taken into account in the perturbation analysis.

title: Asymptotically safe cosmology
abstract: We study quantum modifications to cosmology in a Friedmann-Robertson-Walker universe with and without scalar fields by taking the renormalisation group running of grav- itational and matter couplings into account. We exploit the Bianchi identity to relate the renormalisation group scale with scale factor and derive the improved cosmological evolution equations. We find two types of cosmological fixed points where the renormalisation group scale either freezes in, or continues to evolve with scale factor. We discuss the implications of each of these, and classify the different cosmological fixed points with and without gravity displaying an asymptotically safe renormalisation group fixed point. We state conditions of existence for an inflating ultraviolet cosmological fixed point for Einstein gravity coupled to a scalar field. We also discuss other fixed point solutions such as "scaling" solutions, or fixed points with equipartition between kinetic and potential energies.

title: f(R) Gravity from the renormalisation group
abstract: We explore the cosmological dynamics of an effective f(R) model constructed from a renormalisation group (RG) improvement of the Einstein--Hilbert action, using the non-perturbative beta functions of the exact renormalisation group equation. The resulting f(R) model has some remarkable properties. It naturally exhibits an unstable de Sitter era in the ultraviolet (UV), dynamically connected to a stable de Sitter era in the IR, via a period of radiation and matter domination, thereby describing a non-singular universe. We find that the UV de Sitter point is one of an infinite set, which make the UV RG fixed point inaccessible to classical cosmological evolution. In the vicinity of the fixed point, the model behaves as R^2 gravity, while it correctly recovers General Relativity at solar system scales. In this simplified model, the fluctuations are too large to be the observed ones, and more ingredients in the action are needed.

title: Exact renormalization group with optimal scale and its application to cosmology
abstract: Assuming an effective gravitational action with scale-dependent coupling constants, a consistency condition for the local form of the cut-off scale is derived. The approach is applied to homogeneous cosmology and running couplings with an ultraviolet fixed point. Within the given approach this allows us to derive bounds on the value of the fixed point.

title: Scale Setting for Self-consistent Backgrounds
abstract: Departamento de F´isica,Universidad T´ecnica Federico Santa Mar´ia;Casilla 110-V, Valpara´iso, Chile;(Dated: September 17, 2014)The quest for ﬁndingself-consistent background solutions in quantum ﬁeld theory is closely relatedto the way one decides to set renormalization scale k. This freedom in the choice of the scale settingcan lead to ambiguities and conceptual inconsistencies such as the non-conservation of the stress-energy tensor. In this paper a setting for the “scale-ﬁeld” at the level of eﬀective action proposed,which avoids such inconsistencies by construction. The mechanism and its potential is exempliﬁedfor scalar φ

title: Improved Reissner–Nordström–(A)dS black hole in asymptotic safety
abstract: This paper studies the quantum modifications of the Reissner–Nordstrom–(A)dS black hole within Quantum Einstein Gravity, coupled to an electromagnetic sector. Quantum effects are introduced on the level of the improvements of the classical solution, where the originally constant couplings (G0, Λ0 and α0) are promoted to scale dependent quantities (Gk, Λk and αk). Those running couplings are calculated in the functional renormalization group approach. A crucial point of this so-called “improving solutions” procedure is the scale setting where the arbitrary scale k acquires physical meaning due to a relation to the coordinate scale r. It is proposed to use such scale settings which are stable after iterative improvements. Using this method one finds that for those improved solutions, there is no stable remnant and due to the appearance of a new internal horizon, there is also no necessity to impose a minimal black hole mass for charged black holes, in order to avoid the cosmic censorship hypothesis.

title: Asymptotically safe Starobinsky inflation
abstract: We revisit Starobinsky inflation in a quantum gravitational context, by means of the exact renormalization group (RG). We calculate the nonperturbative beta functions for Newton’s “constant” G and the dimensionless R^2 coupling, and show that there exists an attractive UV fixed point where the latter one vanishes but not the former one, and we provide the corresponding beta functions. The smallness of the R^2 coupling, required for agreement with inflationary observables, is naturally ensured by its vanishing at the UV fixed point, ensuring the smallness of the primordial fluctuations, as well as providing a theoretical motivation for the initial conditions needed for successful inflation in this context. We discuss the corresponding RG dynamics, showing both how inflationary and classical observations define the renormalization conditions for the couplings, and also how the UV regime is connected with lower energies along the RG flow. Finally, we discuss the consistency of our results when higher-order curvature corrections are included, and show that they are robust to the inclusion of R^3 corrections.

title: GAUGE DEPENDENCE OF THE EFFECTIVE AVERAGE ACTION IN EINSTEIN GRAVITY
abstract: We study the gauge dependence of the effective average action Γk and Newtonian gravitational constant using the RG equation for Γk. Then we truncate the space of action functionals to get a solution of this equation. We solve the truncated evolution equation for the Einstein gravity in the De Sitter background for a general gauge parameter α and obtain a system of equatons for the cosmological and Newtonian constants. Analyaing the running of the gravitational constant we find that the Newtonian constant depends strongly on the gauge parameter. This leads to the appearance of antiscreening and screening behavior of the quantum gravity. The resolution of the gauge dependence problem is suggested. For physical gauges like the Landau–DeWitt gauge the Newtonian constant shows an antiscreening.

title: Ultraviolet fixed point and generalized flow equation of quantum gravity
abstract: A new exact renormalization group equation for the effective average action of Euclidean quantum gravity is constructed. It is formulated in terms of the component fields appearing in the transverse-traceless decomposition of the metric. It facilitates both the construction of an appropriate infrared cutoff and the projection of the renormalization group flow onto a large class of truncated parameter spaces. The Einstein-Hilbert truncation is investigated in detail and the fixed point structure of the resulting flow is analyzed. Both a Gaussian and a non-Gaussian fixed point are found. If the non-Gaussian fixed point is present in the exact theory, quantum Einstein gravity is likely to be renormalizable at the nonperturbative level. In order to assess the reliability of the truncation a comprehensive analysis of the scheme dependence of universal quantities is performed. We find strong evidence supporting the hypothesis that 4-dimensional Einstein gravity is asymptotically safe, i.e. nonperturbatively renormalizable. The renormalization group improvement of the graviton propagator suggests a kind of dimensional reduction from 4 to 2 dimensions when spacetime is probed at sub-Planckian length scales.

title: Fixed points of quantum gravity
abstract: Euclidean quantum gravity is studied with renormalization group methods. Analytical results for a nontrivial ultraviolet fixed point are found for arbitrary dimensions and gauge fixing parameters in the Einstein-Hilbert truncation. Implications for quantum gravity in four dimensions are discussed.

title: Renormalization group flow of quantum gravity in the Einstein-Hilbert truncation
abstract: The exact renormalization group equation for pure quantum gravity is used to derive the nonperturbative \ensuremath{\beta}-functions for the dimensionless Newton constant and cosmological constant on the theory space spanned by the Einstein-Hilbert truncation. The resulting coupled differential equations are evaluated for a sharp cutoff function. The features of these flow equations are compared to those found when using a smooth cutoff. The system of equations with a sharp cutoff is then solved numerically, deriving the complete renormalization group flow of the Einstein-Hilbert truncation in $d=4.$ The resulting renormalization group trajectories are classified and their physical relevance is discussed. The nontrivial fixed point which, if present in the exact theory, might render quantum Einstein gravity nonperturbatively renormalizable is investigated for various spacetime dimensionalities.

title: The running gravitational couplings
abstract: We compute the running of the cosmological constant and Newton's constant at sub-Planckian energies, taking into account the effect of quantum fields with any spin between 0 and 2. We find that Newton's constant does not vary appreciably but the cosmological constant can change by many orders of magnitude when one goes from cosmological scales to typical elementary particle scales. In the extreme infrared, zero modes drive a positive cosmological constant to zero.

title: Asymptotic safety and the cosmological constant
abstract: A bstractWe study the non-perturbative renormalisation of quantum gravity in four dimensions. Taking care to disentangle physical degrees of freedom, we observe the topological nature of conformal fluctuations arising from the functional measure. The resulting beta functions possess an asymptotically safe fixed point with a global phase structure leading to classical general relativity for positive, negative or vanishing cosmological constant. If only the conformal fluctuations are quantised we find an asymptotically safe fixed point predicting a vanishing cosmological constant on all scales. At this fixed point we reproduce the critical exponent, ν = 1/3, found in numerical lattice studies by Hamber. Returning to the full theory we find that by setting the cosmological constant to zero the critical exponent agrees with the conformally reduced theory. This suggests the fixed point may be physical while hinting at solution to the cosmological constant problem.

title: Faddeev-Popov ghosts in quantum gravity beyond perturbation theory
abstract: We study the Faddeev-Popov ghost sector of asymptotically safe quantum gravity, which becomes nonperturbative in the ultraviolet. We point out that nonzero matter-ghost couplings and higher-order ghost self-interactions exist at a non-Gaussian fixed point for the gravitational couplings, i.e., in the ultraviolet. Thus the ghost sector in this nonperturbative ultraviolet completion does not keep the structure of a simple Faddeev-Popov determinant. We discuss implications of the new ghost couplings for the renormalization group flow in gravity, the form of the ultraviolet completion, and the relevant couplings, i.e., free parameters, of the theory.

title: Asymptotic safety in Einstein gravity and scalar-fermion matter.
abstract: Within the functional renormalization group approach we study the effective quantum field theory of Einstein gravity and one self-interacting scalar coupled to N(f) Dirac fermions. We include in our analysis the matter anomalous dimensions induced by all the interactions and analyze the highly nonlinear beta functions determining the renormalization flow. We find the existence of a nontrivial fixed point structure both for the gravity and the matter sector, besides the usual Gaussian matter one. This suggests that asymptotic safety could be realized in the gravitational sector and in the standard model. Nontriviality in the Higgs sector might involve gravitational interactions.

title: Effective average action and nonperturbative renormalization group equation in higher derivative quantum gravity
abstract: We study the exact renormalization group (RG) in $R^2$-gravity in the effective average action formalism using the background field method. The truncated evolution equation (where truncation is made to low-derivatives functionals space) for such a theory in a de Sitter background leads to a set of nonperturbative RG equations for cosmological and gravitational coupling constants. The gauge dependence problem is solved by working in the physical Landau-De Witt gauge corresponding to gauge-fixing independent effective action. Approximate solution of nonperturbative RG equations reveals the appearence of antiscreening or screening behaviour of Newtonian coupling, depending on the higher-derivatives coupling constants. The existence of unstable 
UV fixed points is also mentioned.

title: Fixed points of higher-derivative gravity.
abstract: We recalculate the beta functions of higher-derivative gravity in four dimensions using the one-loop approximation to an exact renormalization group equation. We reproduce the beta functions of the dimensionless couplings that were known in the literature, but we find new terms for the beta functions of Newton's constant and of the cosmological constant. As a result, the theory appears to be asymptotically safe at a non-Gaussian fixed point rather than perturbatively renormalizable and asymptotically free.

title: ULTRAVIOLET PROPERTIES OF f(R)-GRAVITY
abstract: 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.

title: On the Renormalization Group Flow of Gravity
abstract: Understanding gravity as a fundamental theory implies understanding its behavior as we move across different length scales. In the present thesis, we have investigated this scale-dependent behavior by studying the renormalization group flow of gravity. A central focus of our investigations was the asymptotic safety scenario, which posits the existence of a non-trivial fixed point of the gravitational renormalization group controlling the behavior of the theory in the ultraviolet and providing it with a predictive and well-defined high-energy limit. Our main tool was a continuous Wilsonian renormalization group technique, the functional renormalization group equation. Computations within this framework rely on truncation approximations, whereby the full renormalization group flow is projected onto a subspace of the couplings of the theory. The reliability of the results thus obtained is assessed by verifying their stability under the gradual extension of the truncation subspace. The simplest, non-trivial gravity truncation one may consider is the Einstein-Hilbert truncation, in which evidence was first found for a non-trivial fixed point of gravity in support of the asymptotic safety scenario. The crucial question is then whether or not this fixed point persists under further enlargement of the truncation subspaces. In this thesis, we have tackled this question by following three complementary strategies. First, in Chapter 3, we investigated the renormalization group flow of gravity restricted to the conformal sector. This simplification allowed us to consider truncations containing terms which would be otherwise intractable for technical reasons, and thus obtain a rough picture of the effect of those terms on the behavior of the complete theory. A second strategy, which we followed in Chapters 4 and 5, returning to the case of full gravity, was to construct a functional renormalization group equation that allowed us to study general truncations spanned by arbitrary functions $f(R)$ of the curvature scalar, and hence investigate large sectors of the full renormalization group flow space. Lastly, in Chapter 6, we included in our truncation terms which are known to be particularly problematic from the point of view of perturbative quantization. First, we moved beyond the $f(R)$-case by explicitly including four-derivative tensorial operators in our truncation, and then added a minimally coupled, massless scalar field to our truncation. Remarkably, in all of these extensions a non-trivial ultraviolet fixed point was found. The coherent picture that arises from these studies provides significant evidence in favor of the asymptotic safety scenario, and suggests that quantized gravity remains predictive at arbitrarily small scales.

title: Taming perturbative divergences in asymptotically safe gravity
abstract: We use functional renormalization group methods to study gravity minimally coupled to a free scalar field. This setup provides the prototype of a gravitational theory which is perturbatively non-renormalizable at one-loop level, but may possess a non-trivial renormalization group fixed point controlling its UV behavior. We show that such a fixed point indeed exists within the truncations considered, lending strong support to the conjectured asymptotic safety of the theory. In particular, we demonstrate that the counterterms responsible for its perturbative non-renormalizability have no qualitative effect on this feature.

title: ASYMPTOTIC SAFETY IN HIGHER-DERIVATIVE GRAVITY
abstract: We study the nonperturbative renormalization group flow of higher-derivative gravity employing functional renormalization group techniques. The nonperturbative contributions to the β-functions shift the known perturbative ultraviolet fixed point into a nontrivial 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.

title: The local potential approximation in quantum gravity
abstract: A bstractWithin the context of the functional renormalization group flow of gravity, we suggest that a generic f (R) ansatz (i.e. not truncated to any specific form, polynomial or not) for the effective action plays a role analogous to the local potential approximation (LPA) in scalar field theory. In the same spirit of the LPA, we derive and study an ordinary differential equation for f (R) to be satisfied by a fixed point of the renormalization group flow. As a first step in trying to assess the existence of global solutions (i.e. true fixed point) for such equation, we investigate here the properties of its solutions by a comparison of various series expansions and numerical integrations. In particular, we study the analyticity conditions required because of the presence of fixed singularities in the equation, and we develop an expansion of the solutions for large R up to order N = 29. Studying the convergence of the fixed points of the truncated solutions with respect to N, we find a characteristic pattern for the location of the fixed points in the complex plane, with one point stemming out for its stability. Finally, we establish that if a non-Gaussian fixed point exists within the full f (R) approximation, it corresponds to an R2 theory.

title: Asymptotic safety goes on shell
abstract: It is well known in quantum field theory that the off-shell effective action depends on the gauge choice and field parametrization used in calculating it. Nevertheless, the typical scheme in which the scenario of asymptotically safe gravity is investigated is an off-shell version of the functional renormalization group equation. Working with the Einstein–Hilbert truncation as a test bed, we develop a new scheme for the analysis of asymptotically safe gravity in which the on-shell part of the effective action is singled out and we show that the beta function for the essential coupling has no explicit gauge dependence. In order to reach our goal, we introduce several technical novelties, including a different decomposition of the metric fluctuations, a new implementation of the ghost sector and a new cut-off scheme. We find a nontrivial fixed point, with a value of the cosmological constant that is independent of the gauge-fixing parameters.

title: A bootstrap towards asymptotic safety
abstract: A search strategy for asymptotic safety is put forward and tested for a simplified version of gravity in four dimensions using the renormalization group. Taking the action to be a high-order polynomial of the Ricci scalar, a self-consistent ultraviolet fixed point is found where curvature invariants become increasingly irrelevant with increasing mass dimension. Intriguingly, universal scaling exponents take near-Gaussian values despite the presence of residual interactions. Asymptotic safety of metric gravity would seem in reach if this pattern carries over to the full theory.

title: Asymptotically safe Lorentzian gravity.
abstract: The gravitational asymptotic safety program strives for a consistent and predictive quantum theory of gravity based on a nontrivial ultraviolet fixed point of the renormalization group (RG) flow. We investigate this scenario by employing a novel functional renormalization group equation which takes the causal structure of space-time into account and connects the RG flows for Euclidean and Lorentzian signature by a Wick rotation. Within the Einstein-Hilbert approximation, the β functions of both signatures exhibit ultraviolet fixed points in agreement with asymptotic safety. Surprisingly, the two fixed points have strikingly similar characteristics, suggesting that Euclidean and Lorentzian quantum gravity belong to the same universality class at high energies.

title: Fixed Functionals in Asymptotically Safe Gravity
abstract: We summarize the status of constructing fixed functionals within the f(R)-truncation of Quantum Einstein Gravity in three spacetime dimensions. Focusing on curvatures much larger than the IR-cutoff scale, it is shown that the fixed point equation admits three different scaling regimes: for classical and quantum dominance the equation becomes linear and has power-law solutions, while the balanced case gives rise to a generalized homogeneous equation whose order is reduced by one and whose solutions are non-analytical.

title: Asymptotic safety in the f(R) approximation
abstract: A bstractIn the asymptotic safety programme for quantum gravity, it is important to go beyond polynomial truncations. Three such approximations have been derived where the restriction is only to a general function f(R) of the curvature R > 0. We confront these with the requirement that a fixed point solution be smooth and exist for all R ≥ 0. Singularities induced by cutoff choices force the earlier versions to have no such solutions. However, we show that the most recent version has a number of lines of fixed points, each supporting a continuous spectrum of eigen-perturbations. We uncover and analyse the first five such lines. Sensible fixed point behaviour may be achieved if one consistently incorporates geometry/topology change. As an exploratory example, we analyse the equations analytically continued to R < 0, however we now find only partial solutions. We show how these results are always consistent with, and to some extent can be predicted from, a straightforward analysis of the constraints inherent in the equations.

title: Redundant operators in the exact renormalisation group and in the f (R) approximation to asymptotic safety
abstract: A bstractIn this paper we review the definition and properties of redundant operators in the exact renormalisation group. We explain why it is important to require them to be eigenoperators and why generically they appear only as a consequence of symmetries of the particular choice of renormalisation group equations. This clarifies when Newton’s constant and or the cosmological constant can be considered inessential. We then apply these ideas to the Local Potential Approximation and approximations of a similar spirit such as the f (R) approximation in the asymptotic safety programme in quantum gravity. We show that these approximations can break down if the fixed point does not support a ‘vacuum’ solution in the appropriate domain: all eigenoperators become redundant and the physical space of perturbations collapses to a point. We show that this is the case for the recently discovered lines of fixed points in the f (R) flow equations.

title: Further evidence for asymptotic safety of quantum gravity
abstract: The asymptotic safety conjecture is examined for quantum gravity in four dimensions. Using the renormalisation group, we find evidence for an interacting UV fixed point for polynomial actions up to the 34th power in the Ricci scalar. The extrapolation to infinite polynomial order is given, and the self-consistency of the fixed point is established using a bootstrap test. All details of our analysis are provided. We also clarify further aspects such as stability, convergence, the role of boundary conditions, and a partial degeneracy of eigenvalues. Within this setting we find strong support for the conjecture.

title: On the number of relevant operators in asymptotically safe gravity
abstract: The asymptotic safety scenario of gravity conjectures that i) the quantum field theory of gravity exists thanks to the presence of a non-trivial ultraviolet fixed point of the renormalization group, and that ii) the fixed point has only a finite number of relevant perturbations, i.e., a finite number of UV-stable directions (or, in other words, a finite number of free parameters to be fixed experimentally). Within the f(R) approximation of the functional renormalization group equation of gravity, we show that assuming the first half of the conjecture to be true, the remaining half follows from general arguments, that is, we show that assuming the existence of a non-trivial fixed point, the fact that the number of relevant directions is finite is a general consequence of the structure of the equations.

title: A proper fixed functional for four-dimensional Quantum Einstein Gravity
abstract: A bstractRealizing a quantum theory for gravity based on Asymptotic Safety hinges on the existence of a non-Gaussian fixed point of the theory’s renormalization group flow. In this work, we use the functional renormalization group equation for the effective average action to study the fixed point underlying Quantum Einstein Gravity at the functional level including an infinite number of scale-dependent coupling constants. We formulate a list of guiding principles underlying the construction of a partial differential equation encoding the scale-dependence of f(R)-gravity. We show that this equation admits a unique, globally well-defined fixed functional describing the non-Gaussian fixed point at the level of functions of the scalar curvature. This solution is constructed explicitly via a numerical double-shooting method. In the UV, this solution is in good agreement with results from polynomial expansions including a finite number of coupling constants, while it scales proportional to R2, dressed up with non-analytic terms, in the IR. We demonstrate that its structure is mainly governed by the conformal sector of the flow equation. The relation of our work to previous, partial constructions of similar scaling solutions is discussed.

title: Flow equation for $f(R)$ gravity and some of its exact solutions
abstract: We write a Renormalization Group (RG) equation for the function f in a theory of gravity in the f(R) truncation. Our equation differs from previous ones due to the exponential parametrization of the quantum fluctuations and to the choice of gauge. The cutoff procedure depends on three free parameters, and we find that there exist discrete special choices of parameters for which the flow equation has fixed points where f=f_0+f_1 R+f_2 R^2. For other values of the parameters the solution seems to be continuously deformed.

title: Renormalization Group Flow in Scalar-Tensor Theories. I
abstract: We study the renormalization group flow in a class of scalar-tensor theories involving at most two derivatives of the fields. We show in general that minimal coupling is self consistent, in the sense that when the scalar self couplings are switched off, their beta functions also vanish. Complete, explicit beta functions that could be applied to a variety of cosmological models are given in a five parameter truncation of the theory in d = 4. In any dimension d > 2 we find that the flow has only a “Gaussian Matter” fixed point, where all scalar self interactions vanish but Newton’s constant and the cosmological constant are nontrivial. The properties of these fixed points can be studied algebraically to some extent. In d = 3 we also find a gravitationally dressed version of the WilsonFisher fixed point, but it seems to have unphysical properties. These findings are in accordance with the hypothesis that these theories are asymptotically safe.

title: Search of scaling solutions in scalar–tensor gravity
abstract: We write new functional renormalization group equations for a scalar nonminimally coupled to gravity. Thanks to the choice of the parametrization and of the gauge fixing they are simpler than older equations and avoid some of the difficulties that were previously present. In three dimensions these equations admit, at least for sufficiently small fields, a solution that may be interpreted as a gravitationally dressed Wilson–Fisher fixed point. We also find for any dimension $$d>2$$d>2 additional analytic scaling solutions which we study for $$d=3$$d=3 and $$d=4$$d=4. One of them corresponds to the fixed point of the Einstein–Hilbert truncation, the others involve a nonvanishing minimal coupling.

title: On unimodular quantum gravity
abstract: Unimodular gravity is classically equivalent to standard Einstein gravity, but differs when it comes to the quantum theory: the conformal factor is non-dynamical, and the gauge symmetry consists of transverse diffeomorphisms only. Furthermore, the cosmological constant is not renormalized. Thus the quantum theory is distinct from a quantization of standard Einstein gravity. Here we show that within a truncation of the full renormalization group flow of unimodular quantum gravity, there is a non-trivial ultraviolet (UV)-attractive fixed point, yielding a UV completion for unimodular gravity. We discuss important differences to the standard asymptotic-safety scenario for gravity, and provide further evidence for this scenario by investigating a new form of the gauge-fixing and ghost sector. Communicated by P R L V Moniz

title: The Renormalization Group flow of unimodular f(R) gravity
abstract: A bstractUnimodular gravity is classically equivalent to General Relativity. This equivalence extends to actions which are functions of the curvature scalar. At the quantum level, the dynamics could differ. Most importantly, the cosmological constant is not a coupling in the unimodular action, providing a new vantage point from which to address the cosmological constant fine-tuning problem. Here, a quantum theory based on the asymptotic safety scenario is studied, and evidence for an interacting fixed point in unimodular f (R) gravity is found. We study the fixed point and its properties, and also discuss the compatibility of unimodular asymptotic safety with dynamical matter, finding evidence for its compatibility with the matter degrees of freedom of the Standard Model.

title: Essential nature of Newton’s constant in unimodular gravity
abstract: We point out that in unimodular gravity Newton’s constant is an essential coupling, i.e. it is independent of field redefinitions. We illustrate the consequences of this fact by a calculation in a standard simple approximation, showing that in this case the renormalization group flow of Newton’s constant is gauge and parametrization independent.

title: Asymptotic Safety, Fractals, and Cosmology
abstract: These lecture notes introduce the basic ideas of the Asymptotic Safety approach to Quantum Einstein Gravity (QEG). In particular they provide the background for recent work on the possibly multifractal structure of the QEG space-times. Implications of Asymptotic Safety for the cosmology of the early Universe are also discussed.

title: Asymptotically safe cosmology
abstract: We study quantum modifications to cosmology in a Friedmann-Robertson-Walker universe with and without scalar fields by taking the renormalisation group running of grav- itational and matter couplings into account. We exploit the Bianchi identity to relate the renormalisation group scale with scale factor and derive the improved cosmological evolution equations. We find two types of cosmological fixed points where the renormalisation group scale either freezes in, or continues to evolve with scale factor. We discuss the implications of each of these, and classify the different cosmological fixed points with and without gravity displaying an asymptotically safe renormalisation group fixed point. We state conditions of existence for an inflating ultraviolet cosmological fixed point for Einstein gravity coupled to a scalar field. We also discuss other fixed point solutions such as "scaling" solutions, or fixed points with equipartition between kinetic and potential energies.

title: Asymptotically Safe Higgs Inflation
abstract: We construct a new inflation model in which the standard model Higgs boson couples minimally to gravity and acts as the inflaton. Our construction of Higgs inflation incorporates the standard model with Einstein gravity which exhibits asymptotic safety in the ultraviolet region. The slow roll condition is satisfied at large field value due to the asymptotically safe behavior of Higgs self-coupling at high energies. We find that this minimal construction is highly predictive, and is consistent with both cosmological observations and collider experiments.

title: Asymptotic safety of gravity and the Higgs boson mass
abstract: There are indications that gravity is asymptotically safe. The Standard Model (SM) plus gravity could be valid up to arbitrarily high energies. Supposing that this is indeed the case and assuming that there are no intermediate energy scales between the Fermi and Planck scales we address the question of whether the mass of the Higgs boson mH can be predicted. For a positive gravity induced anomalous dimension Aλ > 0 the running of the quartic scalar self interaction λ at scales beyond the Planck mass is determined by a fixed point at zero. This results in mH = mmin = 126 GeV, with only a few GeV uncertainty. This prediction is independent of the details of the short distance running and holds for a wide class of extensions of the SM as well. For Aλ 0 is favored by explicit computations existing in the literature.

title: Inflationary Potentials from the Exact Renormalisation Group
abstract: We show that an inflationary slow-roll potential can be derived as an IR limit of the non-perturbative exact renormalisation group equation for a scalar field within the mean-field approximation. The result follows without having to specify a Lagrangian in the UV, which we take to be somewhere below the Planck scale to avoid discussing quantum gravity effects. We assume that the theory contains a scalar mode with suppressed coupling to other fields, and that higher derivative couplings are suppressed. In this framework the exact RG equation becomes a one-dimensional Schrodinger equation, which we solve. The effective IR potential is then dominated by the eigen-states of the RG Hamiltonian with the highest eigenvalues. We find that these potentials can generically give rise to slow-roll inflation, which is fully consistent with recent observations. As an example of how the proposed renormalisation group procedure works, we perform an explicit calculation in the ϕ4 theory in an appendix.

title: Optimisation of the exact renormalisation group
abstract: A simple criterion to optimise coarse-grainings for exact renormalisation group equations is given. It is aimed at improving the convergence of approximate solutions of flow equations. The optimisation criterion is generic, as it refers only to the coarse-grained propagator at vanishing field. In physical terms, it is understood as an optimisation condition for amplitude expansions. Alternatively, it can be interpreted as the requirement to move poles of threshold functions away from the physical region. The link to expansions in field amplitudes is discussed as well. Optimal parameters are given explicitly for a variety of different coarse-grainings. As a by-product it is found that the sharp cut-off regulator does not belong to the class of such optimal coarse-grainings, which explains the poor convergence of amplitude expansions based on it.

title: On the scheme dependence of gravitational beta functions
abstract: We discuss the arbitrariness in the choice of cutoff scheme in calculations of beta functions. We define a class of "pure" cutoff schemes, in which the cutoff is completely independent of the parameters that appear in the action. In a sense they are at the opposite extreme of the "spectrally adjusted" cutoffs, which depend on all the parameters that appear in the action. We compare the results for the beta functions of Newton's constant and of the cosmological constant obtained with a typical cutoff and with a pure cutoff, keeping all else fixed. We find that the dependence of the fixed point on an arbitrary parameter in the pure cutoff is rather mild. We then show in general that if a spectrally adjusted cutoff produces a fixed point, there is a corresponding pure cutoff that will give a fixed point in the same position.

title: Physics and the choice of regulators in functional renormalisation group flows
abstract: The Renormalisation Group is a versatile tool for the study of many systems where scale-dependent behaviour is important. Its functional formulation can be cast into the form of an exact flow equation for the scale-dependent effective action in the presence of an infrared regularisation. The functional RG flow for the scale-dependent effective action depends explicitly on the choice of regulator, while the physics does not. In this work, we systematically investigate three key aspects of how the regulator choice affects RG flows: (i) We study flow trajectories along closed loops in the space of action functionals varying both, the regulator scale and shape function. Such a flow does not vanish in the presence of truncations. Based on a definition of the length of an RG trajectory, we suggest a practical procedure for devising optimised regularisation schemes within a truncation. (ii) In systems with various field variables, a choice of relative cutoff scales is required. At the example of relativistic bosonic two-field models, we study the impact of this choice as well as its truncation dependence. We show that a crossover between different universality classes can be induced and conclude that the relative cutoff scale has to be chosen carefully for a reliable description of a physical system. (iii) Non-relativistic continuum models of coupled fermionic and bosonic fields exhibit also dependencies on relative cutoff scales and regulator shapes. At the example of the Fermi polaron problem in three spatial dimensions, we illustrate such dependencies and show how they can be interpreted in physical terms.

title: Investigating the ultraviolet properties of gravity with a Wilsonian renormalization group equation
abstract: 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 scheme-independent results, and provide several examples of the fact that only dimensionless couplings can have “universal” behavior.

title: Standard Model Higgs boson mass from inflation
abstract: We analyse one-loop radiative corrections to the inflationa ry potential in the theory, where inflation is driven by the St andard Model Higgs field. We show that inflation is possible provided the Higgs mass mH lies in the interval mmin < mH < mmax, where mmin = [136.7 + (mt− 171.2)× 1.95] GeV, mmax = [184.5 + (mt− 171.2)× 0.5] GeV and mt is the mass of the top quark. In the renormalization scheme associated with the Einstein frame the predictions of the spectral index of scalar fluctuati ons and of the tensor-to-scalar ratio practically do not depend on the Higgs mass within the admitted region and are equal to ns = 0.97 and r = 0.0034 correspondingly.

title: Living beyond the edge: Higgs inflation and vacuum metastability
abstract: The measurements of the Higgs mass and top Yukawa coupling indicate that we live in a very special universe, at the edge of the absolute stability of the electroweak vacuum. If fully stable, the Standard Model (SM) can be extended all the way up to the inflationary scale and the Higgs field, nonminimally coupled to gravity with strength xi, can be responsible for inflation. We show that the successful Higgs inflation scenario can also take place if the SM vacuum is not absolutely stable. This conclusion is based on two effects that were overlooked previously. The first one is associated with the effective renormalization of the SM couplings at the energy scale M-P/xi, where M-P is the Planck scale. The second one is a symmetry restoration after inflation due to high temperature effects that leads to the (temporary) disappearance of the vacuum at Planck values of the Higgs field.

title: Higgs boson mass and new physics
abstract: A bstractWe discuss the lower Higgs boson mass bounds which come from the absolute stability of the Standard Model (SM) vacuum and from the Higgs inflation, as well as the prediction of the Higgs boson mass coming from the asymptotic safety of the SM. We account for the three-loop renormalization group evolution of the couplings of the SM and for a part of the two-loop corrections that involve the QCD coupling αs to the initial conditions for their running. This is one step beyond the current state-of-the-art procedure (“one-loop matching-two-loop running”). This results in a reduction of the theoretical uncertainties in the Higgs boson mass bounds and predictions, associated with the SM physics, to 1–2 GeV. We find that with the account of existing experimental uncertainties in the mass of the top quark and αs (taken at the 2σ level) the bound reads MH ≥ Mmin (equality corresponds to the asymptotic-safety prediction), where $ {{M}_{{\min }}}=\left( {129\pm 6} \right) $ GeV. We argue that the discovery of the SM Higgs boson in this range would be in agreement with the hypothesis of the absence of new energy scales between the Fermi and Planck scales, whereas the coincidence of MH with Mmin would suggest that the electroweak scale is determined by Planck physics. In order to clarify the relation between the Fermi and Planck scales a construction of an electron-positron or muon collider with a center-of-mass energy ~ (200 + 200 GeV) (Higgs and t-quark factory) would be needed.

title: Why should we care about the top quark Yukawa coupling?
abstract: In the cosmological context, for the Standard Model to be valid up to the scale of inflation, the top quark Yukawa coupling yt should not exceed the critical value ytcrit, coinciding with good precision (about 0.2‰) with the requirement of the stability of the electroweak vacuum. So, the exact measurements of yt may give an insight on the possible existence and the energy scale of new physics above 100 GeV, which is extremely sensitive to yt. We overview the most recent theoretical computations of and the experimental measurements of ytcrit and the experimental measurements of yt. Within the theoretical and experimental uncertainties in yt, the required scale of new physics varies from 107 GeV to the Planck scale, urging for precise determination of the top quark Yukawa coupling.

title: Higgs mass and vacuum stability in the Standard Model at NNLO
abstract: A bstractWe present the first complete next-to-next-to-leading order analysis of the Standard Model Higgs potential. We computed the two-loop QCD and Yukawa corrections to the relation between the Higgs quartic coupling (λ) and the Higgs mass (Mh), reducing the theoretical uncertainty in the determination of the critical value of Mh for vacuum stability to 1 GeV. While λ at the Planck scale is remarkably close to zero, absolute stability of the Higgs potential is excluded at 98 % C.L. for Mh 
 < 126 GeV. Possible consequences of the near vanishing of λ at the Planck scale, including speculations about the role of the Higgs field during inflation, are discussed.

title: Higgs mass implications on the stability of the electroweak vacuum
abstract: We update instability and metastability bounds of the Standard Model electroweak vacuum in view of the recent ATLAS and CMS Higgs results. For a Higgs mass in the range 124–126 GeV, and for the current central values of the top mass and strong coupling constant, the Higgs potential develops an instability around 10 11 GeV, with a lifetime much longer than the age of the Universe. However, taking into account theoretical and experimental errors, stability up to the Planck scale cannot be excluded. Stability at finite temperature implies an upper bound on the reheat temperature after inflation, which depends critically on the precise values of the Higgs and top masses. A Higgs mass in the range 124–126 GeV is compatible with very high values of the reheating temperature, without conflict with mechanisms of baryogenesis such as leptogenesis. We derive an upper bound on the mass of heavy right-handed neutrinos by requiring that their Yukawa couplings do not destabilize the Higgs potential.

title: Higgs Mass Bounds from Renormalization Flow for a simple Yukawa model
abstract: We study the functional renormalization group flow of a Higgs-Yukawa toy model mimicking the top-Higgs sector of the standard model. This approach allows for treating arbitrary bare couplings. For the class of standard bare potentials of \phi^4-type at a given ultraviolet cut-off, we show that a finite infrared Higgs mass range emerges naturally from the renormalization group flow itself. Higgs masses outside the resulting bounds cannot be connected to any conceivable set of bare parameters in this standard-model \phi^4 class. By contrast, more general bare potentials allow to diminish the lower bound considerably. We identify a simple renormalization group mechanism for this depletion of the lower bound. If active also in the full standard model, Higgs masses smaller than the conventional infrared window do not necessarily require new physics at low scales or give rise to instability problems.

title: The Higgs mass and the scale of new physics
abstract: A bstractIn view of the measured Higgs mass of 125 GeV, the perturbative renormalization group evolution of the Standard Model suggests that our Higgs vacuum might not be stable. We connect the usual perturbative approach and the functional renormalization group which allows for a straightforward inclusion of higher-dimensional operators in the presence of an ultraviolet cutoff. In the latter framework we study vacuum stability in the presence of higher-dimensional operators. We find that their presence can have a sizable influence on the maximum ultraviolet scale of the Standard Model and the existence of instabilities. Finally, we discuss how such operators can be generated in specific models and study the relation between the instability scale of the potential and the scale of new physics required to avoid instabilities.

title: Vacuum stability of asymptotically safe gauge-Yukawa theories
abstract: A bstractWe study the phase diagram and the stability of the ground state for certain four-dimensional gauge-Yukawa theories whose high-energy behaviour is controlled by an interacting fixed point. We also provide analytical and numerical results for running couplings, their crossover scales, the separatrix, and the Coleman-Weinberg effective potential. Classical and quantum stability of the vacuum is established.