Dyson's Equations for Quantum Gravity in the Hartree-Fock Approximation

Unlike scalar and gauge field theories in four dimensions, gravity is not perturbatively renormalizable and as a result perturbation theory is badly divergent. Often the method of choice for investigating nonperturbative effects has been the lattice formulation, and in the case of gravity the Regge-Wheeler lattice path integral lends itself well for that purpose. Nevertheless, lattice methods ultimately rely on extensive numerical calculations, leaving a desire for alternate calculations that can be done analytically. In this work we outline the Hartree-Fock approximation to quantum gravity, along lines which are analogous to what is done for scalar fields and gauge theories. The starting point is Dyson's equations, a closed set of integral equations which relate various physical amplitudes involving graviton propagators, vertex functions and proper self-energies. Such equations are in general difficult to solve, and as a result not very useful in practice, but nevertheless provide a basis for subsequent approximations. This is where the Hartree-Fock approximation comes in, whereby lowest order diagrams get partially dressed by the use of fully interacting Green's function and self-energies, which then lead to a set of self-consistent integral equations. Specifically, for quantum gravity one finds a nontrivial ultraviolet fixed point in Newton's constant G for spacetime dimensions greater than two, and nontrivial scaling dimensions between d=2 and d=4, above which one obtains Gaussian exponents. In addition, the Hartree-Fock approximation gives an explicit analytic expression for the renormalization group running of Newton's constant, suggesting gravitational antiscreening with Newton's G slowly increasing on cosmological scales.


Introduction
The traditional approach to quantum field theory is generally based on Feynman diagrams, which involve a perturbative expansion in some suitable small coupling constant. In cases like QED this procedure works rather well, since the vacuum about which one is expanding is rather close, in its physical features, to the physical vacuum (weakly coupled electrons and photons). In other cases, such as QCD, subtleties arise because of effects which are non-analytic in the coupling constant and cannot therefore be seen to any order in perturbation theory. Indeed, in QCD it is well known that the perturbative ground-state describes free quarks and gluons, and as result the fundamental property of asymptotic freedom is easily derived in perturbation theory. Nevertheless, important features such as gluon condensation, quark confinement and chiral symmetry breaking remain invisible to any order in perturbation theory. But the fact remains that the formulation of quantum field theory based on the Feynman path integral is generally not linked in any way to a perturbative expansion in terms of diagrams. As a result, the path integral generally makes sense even in a genuinely nonperturbative context, provided that it is properly defined and regulated, say via a spacetime lattice and a suitable Wick rotation.
Already at the level of nonrelativistic quantum mechanics, examples abound of physical system for which perturbation theory entirely fails to capture the essential properties of the model. Which leads to the reason why well-established nonperturbative methods such as the variational trial wave function method, numerical methods or the Hartree-Fock (HF) self-consistent method were developed early on in the history of quantum mechanics, motivated by the need to go beyond the -often uncertain -predictions of naive perturbation theory. In addition, many of these genuinely nonperturbative methods are relatively easy to implement, and in many cases lead to significantly improved physical predictions in atomic, molecular and nuclear physics. Going one step further, examples abound also in relativistic quantum field theory and many-body theory, where naive perturbation theory clearly fails to give the correct answer. Interesting, and physically very relevant, cases of the latter include the theory of superconductivity, superfluidity, screening in a Coulomb gas, turbulent fluid flow, and generally many important aspects of critical (or cooperative) phenomena in statistical physics. A common thread within many of these widely different systems is the emergence of coherent quantum mechanical behavior, which is not easily revealed by the use of perturbation theory about some non-interacting ground state. However, some of the approximate methods mentioned above seem at first to be intimately tied to the existence of a Hamiltonian formulation (as in the case of the variational method), and are thus not readily available if one focuses on the Feynman path integral approach to quantum field theory.
Nevertheless, it is possible, in some cases, to cleverly modify perturbation theory in such a way as to make the diagrammatic perturbative approach still viable and useful. One such method was pioneered by Wilson, who suggested that in some theories a double expansion in both the coupling constant and the dimensionality (the so called ǫ-expansion) could be performed [1][2][3].
These expansion methods turn out to be quite powerful, but they also generally lead to series that are known to be asymptotic, and in some cases (such as the nonlinear sigma model) can have rather poor convergence properties, mainly due to the existence of renormalon singularities along the positive real Borel transform axis. Nevertheless, these investigations lead eventually to the deep insight that perturbatively nonrenormalizable theories might be renormalizable after all, if one is willing to go beyond perturbation theory [4][5][6]. In addition, lately direct numerical methods have acquired a more prominent position, largely because of the increasing availability of very fast computers, allowing many of the foregoing ideas to be rigorously tested. A small subset of the vast literature on the subject of field theory methods combined with modern renormalization group ideas, as applied mostly to Euclidean quantum field theory and statistical physics, can be found in [7][8][9][10][11].
Turning to the gravity context, it is not difficult to see that the case perhaps closest to quantum gravity is non-Abelian gauge theories, and specifically QCD. In the latter case one finds that, besides the established Feynman diagram perturbative approach in the gauge coupling g, there are very few additional approximate methods available, mostly due to the crucial need to preserve exact local gauge invariance. It is not surprising therefore that in recent years, for these theories, the method of choice has become the spacetime lattice formulation, which attempts to evaluate the Feynman path integral directly and exactly by numerical methods. The above procedure then leads to answers which, at least in principle, can be improved arbitrarily given a large enough lattice subdivision and sufficient amounts of computer time. More recently lattice methods have also been applied to the case of quantum gravity, in the framework of the simplicial Regge-Wheeler discretization [12,13,14], where they have opened the door to accurate determinations of critical points and nontrivial scaling dimensions.
It would seem nevertheless desirable to be able to derive certain basic results in quantum gravity using perhaps approximate, but largely analytical methods. Unfortunately, in the case of gravity perturbation theory in Newton's G is even less useful than for non-Abelian gauge theories and QCD, since the theory is unequivocally not perturbatively renormalizable in four spacetime dimensions [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. To some extent it is possible to partially surmount this rather serious issue, by developing Wilson's 2+ ǫ diagrammatic expansion [3] for gravity, with ǫ = d− 2. Such an expansion is constructed around the two-dimensional theory, which, apart from being largely topological in nature, also leads in the end to the need to set ǫ = 2 (clearly not a small value) in order to finally reach the physical theory in d = 4 [30,31,32]. Consequently, one finds significant quantitative uncertainties regarding the interpretation of the results in four spacetime dimensions, especially given the yet unknown convergence properties of the asymptotic series in ǫ. Lately, approximate truncated renormalization group (RG) methods have been applied directly in four dimensions, which thus bypass the need for Wilson's dimensional expansion. These nevertheless can lead to some significant uncertainties in trying to pin down the resulting truncation errors, in part due to what are often rather drastic renormalization group truncation procedures, which in turn generally severely limit the space of operators used in constructing the renormalization group flows.
In this work we will focus instead on the development of the Hartree-Fock approximation for quantum gravity. The Hartree-Fock method is of course well known in non-relativistic quantum mechanics and quantum chemistry, where it has been used for decades as a very successful tool for describing correctly (qualitatively, and often even quantitatively to a high degree) properties of atoms and molecules, including, among other things, the origin of the periodic table [33]. In the Hartree-Fock approach the direct interaction of one particle (say an electron in an atom) with all the other particles is represented by some sort of average field, where the average field itself is determined self-consistently via the solution of a nonlinear, but single-particle Schroedinger-like equation. The latter in turn then determines, via the single-particle wave functions, the average field exerted on the original single particle introduced at the very start of the procedure. As such, it is often referred to as a self-consistent method, where an explicit solution to the effective nonlinear coupled wave equations is obtained by successive (and generally rapidly convergent) iterations.
It has been known for a while that the Hartree-Fock method has a clear extension to a quantum field theory (QFT) context. In quantum field theory (and more generally in many-body theory) it is possible to derive the leading order Hartree-Fock approximation [33][34][35][36][37][38][39][40][41][42][43][44] as a truncation of the Dyson-Schwinger (DS) equations [45,46,47]. For the latter a detailed, modern exposition can be found in the classic quantum field theory book [48], as well as (for non-relativistic applications) in [38]. The Hartree-Fock method is in fact quite popular to this day in many-body theory, where it is used, for example, as a way of deriving the gap equation for a superconductor in the Bardeen-Cooper-Schrieffer (BCS) theory [34,35,36]. There the Hartree-Fock method was originally used to show that the opening of a gap ∆ at the Fermi surface is directly related to the existence of a Cooper pair condensate. Many more applications of the Hartree-Fock method in statistical physics can be found in [37,38]. Later on, the successful use of the Hartree-Fock approach in many body theory spawned many useful applications to relativistic quantum field theory [39][40][41][42][43][44], including one of the earliest investigations of dynamical chiral symmetry breaking. There is also a general understanding that the Hartree-Fock method, since it is usually based on the notion of a (self-consistently determined) average field, cannot properly account for the regime where large fluctuations and correlations are important. The precursor to these kind of reasonings is the general validity of mean field theory for spin systems and ferromagnet above four dimensions (as described by the Landau theory), but not in low dimensions where fluctuations become increasingly important.
In statistical field theory the Hartree-Fock approximation is seen to be closely related to the large-N expansion, as described for example in detail in [44]. There one introduces N copies for the fields in question (the simplest case is for a scalar field, but it can be applied to Fermions as well), and then expands the resulting Green's functions in powers of 1/N . The expansion thus generally starts out from some sort of extended symmetry group theory, which is taken to be O(N ) or SU (N ) invariant. Then it is easy to see that the leading order in the 1/N expansion reproduces what one obtains to lowest order in the Hartree-Fock approximation for the same theory. Furthermore, the next orders in the 1/N expansion generally correspond to higher order corrections within the Hartree-Fock approximation. The large N expansion can be developed for SU (N ) gauge theories as well, and to leading order it gives rise to the 't Hooft planar diagram expansion. In view of the previous discussion it would seem therefore rather natural to consider the analog of the Hartree-Fock approximation for QCD as well [49,50]. Nevertheless, this avenue of inquiry has not been very popular lately, due mainly to its rather crude nature, and because of the increasing power (and much greater reliability) of ab initio numerical lattice approaches to QCD.
Unfortunately, in the case of quantum gravity there is no obvious large N expansion framework. This can be seen from the fact that one starts out with clearly only one spacetime metric, and introducing multiple N copies of the same metric seems rather unnatural. Such multiple metric formulation would cause some rather serious confusion on which of the N metrics should be used to determine the spacetime interval between events, and thus provide the needed spacetime arguments for gravitational n-point functions. In addition, and unlike QCD, there is no natural global (or local) symmetry group that would relate these different copies of the metric to each other. As will be shown below, it is nevertheless possible (and quite natural) to develop the Hartree-Fock approximation for quantum gravity, in a way that is rather similar to the procedure followed in the case of the nonlinear sigma model, gauge theories and general many-body theories. In the gravitational case one starts out, in close analogy to what is done for these other theories, from the Dyson-Schwinger equations, and then truncates them to the lowest order terms, replacing the bare propagators by dressed ones, the bare vertices by dressed ones etc. Thus, the Hartree-Fock approximation for quantum gravity can be written out without any reference to a 1/N expansion, which as discussed above is often the underlying justification for other non-gravity theories. In addition, the leading Hartree-Fock result represents just the lowest order term in a modified diagrammatic expansion, which in principle can be carried out systematically to higher order.
As a reference, for the scalar λφ 4 theory the Hartree-Fock approximation was already outlined some time ago in [44], where its relationship to the 1/N expansion was laid out in detail. One important aspect of both the Hartree-Fock approximation and the 1/N expansion discussed in the references given above is the fact that it can usually be applied in any dimensions, and is therefore not restricted in any way to the spacetime (or space) dimension in which the theory is found to be perturbatively renormalizable. It represents therefore a genuinely nonperturbative method, of potentially widespread application. In the scalar field case one can furthermore clearly establish, based on the comparison with other methods, to what extent the Hartree-Fock approximation is able to capture the key physical properties of the underlying theory, such as correlation functions and universal scaling exponents. These investigations in turn provide a partial insight into what physical aspects the Hartree-Fock approximation can correctly reproduce, and where it fails. Generally, the conclusion appears to be that, while quantitatively not exceedingly accurate, the Hartree-Fock approximation and the 1/N expansion tend to reproduce correctly at least the qualitative features of the theory in various dimensions, including, for example, the existence of an upper and a lower critical dimension, the appearance of nontrivial fixed points of the renormalization group, as well as the dependence of nontrivial scaling exponents on the dimension of spacetime.
In this work we first review the properties of the Hartree-Fock approximation, as it applies initially to some of the simpler field theories. In the case of the scalar field theory, with the scalar field having N components, one can see that the Hartree-Fock approximation is easily obtained from the large N saddle point [44]. The resulting gap equation is generally valid in any dimension, and once solved provides an explicit expression for the mass gap (the inverse correlation length) as a function of the bare coupling constant. Solutions to the gap equation show some significant sensitivity to the number of dimensions. For the scalar field a clear transition is seen between the lower critical dimension d = 2 (for N > 2) and the upper critical dimension d = 4, above which the critical exponents attain the Landau theory values. Indeed, the Hartree-Fock approximation reproduces correctly the fact that for d ≤ 2 the O(N ) symmetric nonlinear sigma model for N > 2 exhibits a non-vanishing gap for bare coupling g > 0, and that for d > 4 it reduces to a noninteracting (Gaussian) theory in the long-distance limit (also known as the triviality of λ φ 4 in d > 4). Then, as a warm-up to the gravity case, it will pay here to discuss briefly the SU (N ) gauge theory case. Here again one can see that the Hartree-Fock result correctly reproduces the asymptotic freedom running of the gauge coupling g, and furthermore shows that d = 4 is the lower critical dimension for SU (N ) gauge theories. This last statement is meant to refer to the fact that above four dimensions non-Abelian gauge theories are known to have a nontrivial ultraviolet fixed point at some g c , separating a weak coupling Coulomb-like phase from a strong coupling confining phase.
Finally, in the gravitational case one starts out by following a procedure which closely parallels the gauge theory case. Nevertheless, at the same time one is faced with the fact that the gravitational case is considerably more difficult, for a number of (mostly well known) reasons which we proceed to enumerate here. Unlike the scalar field and gauge theory case in four dimensions, gravity is not perturbatively renormalizable in four dimensions, and as a result perturbation theory is badly divergent, and thus not very useful. Furthermore, as mentioned previously, there is no sensible way of formulating a large N expansion for gravity. That leaves as the method of choice, at least for investigating nonperturbative properties of the theory, the lattice formulation, and specifically the elegant simplicial lattice formulation due to Regge and Wheeler [12,13] and it's Euclidean Feynman path integral extension. One disadvantage is that the lattice methods nevertheless ultimately rely on extensive numerical calculations, leaving a desire for alternate methods that can be pursued analytically. As mentioned previously, it is also possible to develop for quantum gravity the 2 + ǫ dimensional expansion [30,31,32], in close analogy to Wilson's 4 − ǫ and 2 + ǫ dimensional expansion for scalar field theories [3]; the ability to perform such an expansion for gravity hinges on the fact that Einstein's theory becomes formally perturbatively renormalizable in two dimensions.
But there are significant problems associated with this expansion, notably the fact that ǫ = d − 2 has to be set equal to two at the end of the calculation. One more possible approach is via the Wheeler-DeWitt equations, which provide a useful Hamiltonian formulation for gravity, both in the continuum and on the lattice, see [51] and references therein. Nevertheless, while it has been possible to obtain some exact results on the lattice in 2 + 1 dimensions, the 3 + 1 case still appears rather difficult. As a result, generally the number of reasonably well-tested nonperturbative methods available to investigate the vacuum structure of quantum gravity seem rather limited. It is therefore quite remarkable that it is possible to develop a Hartree-Fock approximation to quantum gravity, in a way that in the end is completely analogous to what is done in scalar field theories and gauge theories. In the following we briefly outline the features of the Hartree-Fock method as applied to gravity.
For any field theory it is possible to write down a closed set of integral equations, which relate various physical amplitudes involving particle propagators, vertex functions, proper self-energies etc. These equations follow in a rather direct way from their respective definitions in terms of the Feynman path integral, and the closely connected generating function Z[J]. One particularly elegant and economic way to obtain these expressions is to apply suitable functional derivatives to the generating function in the presence of external classical sources. The resulting exact relationship between n-point functions, known as the set of Dyson-Schwinger equations, are, by virtue of their derivation, not reliant in any way on perturbation theory, and thus genuinely nonperturbative.
Nevertheless, the set of coupled Dyson-Schwinger equations for the propagators, self-energies and vertex functions are in general very difficult to solve, and as a result not very useful in practice. This is where the Hartree-Fock approximation comes in, whereby lowest order diagrams get partially dressed by the use of fully interacting Green's function and self-energies. Since the resulting coupled equations involve fully dressed propagators and full proper self-energies to some extent, they are referred to as a set of self-consistent equations. In principle the Hartree-Fock approximation to the Dyson-Schwinger equations can be carried out to arbitrarily high order, by examining the contribution of increasingly complicated diagrams involving dressed propagators and dressed self-energies etc. Of particular interest here will be therefore the lowest order Hartree-Fock approximation for quantum gravity. As in the scalar and gauge theory case, it will be necessary, in order to derive the Hartree-Fock equations, to write out the lowest order contributions to the graviton propagator, to the gravitational self-energy, and to the gravitational vertex function, and for which the lowest order perturbation theory expressions are known as well. Indeed, one remarkable aspect of most Hartree-Fock approximations lies in the fact that ultimately the calculations can be shown to reduce to the evaluation of a single one-loop integral, involving a self-energy contribution, which is then determined self-consistently.
Nevertheless, quite generally the resulting diagrammatic expressions are still ultraviolet divergent, and the procedure thus requires the introduction of an ultraviolet regularization, such as the one provided by dimensional regularization, an explicit momentum cutoff, or a lattice as in the Regge-Wheeler lattice formulation of gravity [12,13]. Once this is done, one then finds a number of significant similarities of the Hartree-Fock result with the (equally perturbatively nonrenormalizable in d > 2) nonlinear sigma model. Specifically, in the quantum gravity case one finds that the ultraviolet fixed point in Newton's constant G is at G = 0 in two spacetime dimensions, and moves to a non-zero value above d = 2. Nontrivial scaling dimensions are found between d = 2 and d = 4, above which one obtains Gaussian (free field) exponents, thus suggesting that, at least in the Hartree-Fock approximation, the upper critical dimension for gravity is d = 4. One useful and interesting aspect of the above calculations is that the Hartree-Fock results are not too far off from what one obtains by either numerical investigations within the Regge-Wheeler lattice formulation, or via the 2 + ǫ dimensional expansion, or by truncated renormalization group methods. Furthermore, as will be shown below, the Hartree-Fock approximation provides an explicit expression for the renormalization group (RG) running of Newton's constant, just like the Hartree-Fock leads to similar results for the nonlinear sigma model. As in the lattice case, the Hartree-Fock results suggest a two-phase structure, with gravitational screening for G < Gc, and gravitational antiscreening for G > Gc in four dimensions. Due to the known nonperturbative instability of the weak coupling phase of G < G c , only the G > G c phase will need to be considered further, which then leads to a gravitational coupling G(k) that slowly increases with distance, with a momentum dependence that can be obtained explicitly within the Hartree-Fock approximation.
The paper is organized as follows. In Section 2 we recall the main features of Dyson's equations and how they lead to the Hartree-Fock approximation. We point out the important and general result that Dyson's equations can be easily derived using the elegant machinery of the Feynman path integral combined with functional methods. The Hartree-Fock approximation then follows from considering a specific set of sub-diagrams with suitably dressed propagators, all to be determined later via the solution of a set of self-consistent equations. In Section 3 we discuss in detail, as a first application and as preamble to the quantum gravity case, the case of the O(N )-symmetric nonlinear sigma model, initially here from the perturbative perspective of Wilson's 2 + ǫ expansion.
A summary of the basic results will turn out to be useful when comparing later to the Hartree-Fock approximation. In Section 4 we briefly recall the main features of the nonlinear sigma model in the large-N expansion, which can be carried out in any dimension. Again, the discussion here is with an eye towards the later, more complex discussion for gravity. In Section 5 we discuss the main features of the Hartree-Fock result for the O(N ) nonlinear sigma model in general dimensions 2 ≤ d ≤ 4, and point out its close relationship to the large-N results described earlier. The section ends with a comparison of the three approximations, and how they favorably relate to each other and to other results, such as the lattice formulation and actual experiments. In Section 6 we proceed to a more difficult case, namely the discussion of some basic results for the Hartree-Fock approximation for non-Abelian gauge theories in general spacetime dimensions 4 ≤ d ≤ 6. The basic starting point is again a general gap equation for a nonperturbative, dynamically generated mass scale. Here again we will note that the Hartree-Fock results capture many of the basic ingredients of nonperturbative physics, while still missing out on some (such as confinement). In Section 7 we move on to the gravitational case, and discuss in detail the results of the Hartree-Fock approximation in general spacetime dimensions 2 ≤ d ≤ 4. For the gravity case, the basic starting point will be again a gap equation for the nonperturbatively generated mass scale, whose solution will lead to a number of explicit results, including an explicit expression for the renormalization group running of Newton's constant G. In Section 8 we then present a sample application of the Hartree-Fock results to cosmology, where the running of G is compared to current cosmological data, specifically here the Planck-18 Cosmic Background Radiation (CMB) data. Finally, Section 9 summarizes the key points of our study and concludes the paper.

Dyson's Equations and the Hartree-Fock Approximation
One of the earliest applications of the Hartree-Fock approximation to solving Dyson's equations for propagators and vertex functions was in the context of the BCS theory for superconductors [34,35,36]. A few years later it was applied to the (perturbatively nonrenormalizable) relativistic theory of a self-coupled Fermion, where it provided the first convincing evidence for a dynamical breaking of chiral symmetry and the emergence of Nambu-Goldstone bosons [39,40]. A necessary preliminary step involved in deriving the Hartee-Fock approximation to a given theory is writing down Dyson's equations (often referred to as the Schwinger-Dyson equations) for the field, or fields, in question. Dyson's equations have a nice, almost self-explanatory, form when written in terms of Feynman diagrams for various propagators, self-energies and vertex functions associated with the particles appearing in the theory, such as for example electrons and photons in QED. These generally describe a set of coupled integral equations for the various dressed, fully interacting, n-point functions, but whose solution is generally rather difficult, if not impossible. As a result, they are usually only discussed as a method for deriving exact Ward identities that follow from local gauge invariance, with their practical application then usually restricted in most cases to perturbation theory only. Nevertheless, as will be shown below, they play a key role in deriving the Hartree-Fock approximations for the fields in question, thus providing a method that is essentially nonperturbative, involving a partial re-summation of an infinite class of diagrams.
In non-relativistic many body theory Dyson's equation take on a simple form when describing the static electromagnetic interaction between electrons [38]. 3 In terms of the electron propagator 3 It is often customary in many body theory to describe the interaction of electrons close to the Fermi surface in G(p) one has at first the decomposition where G 0 (p) is the bare (tree level) propagator, G(p) the dressed one and Σ(p) the electron selfenergy. The above should really be regarded as a matrix equation, with indices appropriate for the particle being exchanged intended, and not written out in this case. If one regards the self-energy Σ as being constructed out of repeated iterations of a proper self-energy Σ * (p), then Dyson's equation takes on the form with solution Since generally G(p) is regarded as a matrix with spin indices, the above should be interpreted as the matrix inverse. This then leads to the identification of Σ * (0) as an effective mass correction for the electron, induced, perturbatively or nonperturbatively, by radiative corrections.
Similarly, the boson (photon or phonon, depending of the interaction being considered) exchange between electrons leads to the expansion for the interaction term where U 0 (p) is the bare (tree level) contribution, U (p) the dressed one and Π(p) the self-energy contribution. Again, the above should be regarded as a matrix equation, with indices appropriate for the particle being exchanged. If one considers the above self-energy Π(p) as being constructed out of repeated iterations of a proper self-energy Π * (p), then Dyson's equation in this case takes on the form with solution again intended as a proper matrix inverse. If the interaction is spin-independent, as in the Coulomb case, then the above matrices can all be regarded as diagonal. This last result then leads to the identification of Π * (0) as the effective mass (or inverse range) associated with the interaction described by U (p). The ratio κ(p) = U 0 (p)/U (p) is referred to, in the electromagnetic case, as an effective generalized dielectric function [38].
Alternatively, it is possible to derive Dyson's equations directly from the Feynman path integral using the rather elegant machinery of functional differentiation of the generating function Z[J] with respect to a suitable source terms J(x). It is this latter approach that we follow here, as it generalizes rather straightforwardly to the case of quantum gravity. For a scalar field one has the statement that the integral of a derivative is zero for suitable boundary conditions, or where L is the Lagrangian density for the scalar field in question. As a consequence If the scalar field potential is generally denoted by V (φ), then the previous expression can be re-cast, defining as The latter then plays the role of the quantum Heisenberg equations of motion. Successive applications of functional derivatives with respect to J(x) then leads to a sequence of simple identities at zero source and so on. More generally, for n-point functions one obtains Within the current discussion of Dyson's equations, a step up in difficulty is clearly represented by the gauge theory case, with QED as the simplest example. Since the resulting equations share some similarity with quantum gravity, it will be useful here to recall briefly the main aspects.
Compared to scalar field theory, the first new ingredient is the presence of multiple fields, for QED in the form of photon and matter fields. The generating function is with where I(A, ψ,ψ) is the QED action. Again, for suitable boundary conditions the integral of a derivative is zero, which leads to the identity and consequently For the case of the QED action one has where λ is the gauge parameter for a gauge fixing term 1 2 λ(∂ · A) 2 , and g µν here the flat Lorentz metric. The resulting equation can then be regarded as the quantum version of Maxwell's equations [48]. In QED and gauge theories in general it is often useful to introduce the generating function Γ of one-particle-irreducible (1PI) Green's functions, defined as the Legendre transform of G, Then the introduction of the classical fields A µ , ψ andψ via the definitions and conversely, as a consequence of the definition of Γ, which allows one to rewrite Eq. (19) simply as Equivalently, one has in terms of the generating function Γ exclusively A very similar procedure can be carried out for non-Abelian gauge theories and gravity, nevertheless the complexity increases greatly due to the proliferation of Lorentz indices, color indices and a multitude of interaction vertices. We will skip here almost entirely a detailed discussion of gauge theories, and only refer to the relevant diagrams and corresponding expressions, where appropriate. Also, for ease of exposition, the quantum gravity case will be postponed until the Hartree-Fock approximation for gravity is developed.

The Case of the Nonlinear Sigma Model
The question then arises, are there any field theories where the standard perturbative treatment fails, yet for which one can find alternative methods and from them develop consistent predictions?
The answer seems unequivocally yes [4,5,6]. Indeed, outside the quantum gravity framework, there are two notable examples of field theories: the nonlinear sigma model and the self-coupled fermion model. Both are not perturbatively renormalizable for d > 2, and yet lead to consistent, and in some instances precision-testable, predictions above d = 2.
The key ingredient to all of these results is, as recognized originally by Wilson, the existence of a nontrivial ultraviolet fixed point (also known as a phase transition in the statistical field theory context) with nontrivial universal scaling dimensions [1,2,3,52]. Furthermore, three different calculational approaches are by now available for comparing predictions: the 2+ǫ expansion, the large-N limit, and the lattice approach.
From within the lattice approach, several additional techniques are available: the strong coupling expansion, the weak coupling expansion and the numerically exact evaluation of the lattice path integral.
The O(N )-symmetric nonlinear sigma model provides an instructive (and rich) example of a theory which, above two dimensions, is not perturbatively renormalizable in the traditional sense, and yet can be studied in a controlled way via Wilson's 2 + ǫ expansion [53][54][55][56][57][58][59][60]. Such a framework provides a consistent way to calculate nontrivial scaling properties of the theory in those dimensions where it is not perturbatively renormalizable (e.g. d = 3 and d = 4), which can then be compared to nonperturbative results based on the lattice theory. In addition, the model can be solved exactly in the large N limit for any d, without any reliance on the 2 + ǫ expansion. In all three approaches the model exhibits a nontrivial ultraviolet fixed point at some coupling g c (a phase transition in statistical mechanics language), separating a weak coupling massless ordered phase from a massive strong coupling phase. Finally, the results can then be compared to experiments, since in d = 3 the model describes either a ferromagnet or superfluid helium in the vicinity of its critical point.
The nonlinear sigma model is described by an N -component scalar field φ a satisfying a unit constraint φ 2 (x) = 1, with functional integral given by The action is taken to be O(N )-invariant Here Λ is an ultraviolet cutoff (as provided for example by a lattice), and g the bare dimensionless coupling at the cutoff scale Λ; in a statistical field theory context the coupling g 2 plays the role of a temperature.
In perturbation theory one can eliminate one φ field by introducing a convenient parametrization where π a is an N − 1-component field, and then solving In the framework of perturbation theory in g the constraint |π(x)| < 1 is not important as one is restricting the fluctuations to be small. Accordingly the π integrations are extended from −∞ to +∞, which reduces the development of the perturbative expansion to a sequence of Gaussian integrals. Values of π(x) ∼ 1 give exponentially small contributions of order exp(−const./g) which are considered negligible to any finite order in perturbation theory. In term of the π field the original action S becomes The change of variables from φ(x) to π(x) gives rise to a Jacobian which is necessary for the cancellation of spurious tadpole divergences. In the presence of an explicit ultraviolet cutoff δ(0) ≃ (Λ/π) d . The combined functional integral for the unconstrained π field is then given by with In perturbation theory the above action is then expanded out in powers of π, and the propagator for the π field can be read off from the quadratic part of the action, In the weak coupling limit the π fields correspond to the Goldstone modes of the spontaneously broken O(N ) symmetry, the latter broken spontaneously in the ordered phase by a non-vanishing vacuum expectation value π = 0. Since the π field has mass dimension 1 2 (d − 2), and thus the interaction ∂ 2 π 2n consequently has dimension n(d−2)+2, one finds that the theory is perturbatively renormalizable in d = 2, and perturbatively non-renormalizable above d = 2. Potential infrared problems due to massless propagators are handled by introducing an external h-field term for the original composite σ field, which then can be seen to act as a regulating mass term for the π field.
One can write down the same field theory on a lattice, where it corresponds to the O(N )symmetric classical Heisenberg model at a finite temperature T ∼ g 2 . The simplest procedure is to introduce a hypercubic lattice of spacing a, with sites labeled by integers n = (n 1 . . . n d ), which introduces an ultraviolet cutoff Λ ∼ π/a. On the lattice, field derivatives are replaced by finite and the discretized path integral then reads The above expression is recognized as the partition function for a ferromagnetic O(N )-symmetric lattice spin system at finite temperature. Besides ferromagnets, it can be used to describe systems which are related to it by universality, such as superconductors and superfluid helium transitions.
In two dimensions one can compute the renormalization of the coupling g from the action of Eq. (30) and one finds after a short calculation [55,56] for small g  [61]. For N > 2 as g(µ) flows toward increasingly strong coupling it eventually leaves the regime where perturbation theory can be considered reliable.
Above two dimensions, d − 2 = ǫ > 0, one can redo the same type of perturbative calculation to determine the coupling constant renormalization. There one finds [55,56] for the Callan-Symanzik The latter determines the scale dependence of g for an arbitrary momentum scale µ, and from the differential equation µ ∂g 2 ∂µ = β(g(µ)) one determines how g(µ) flows as a function of momentum scale µ. The scale dependence of g(µ) is such that if the initial g is less than the ultraviolet fixed point value g c , with then the coupling will flow towards the Gaussian fixed point at g = 0. The new phase that appears when ǫ > 0 and corresponds to a low temperature, spontaneously broken phase with non-vanishing order parameter. On the other hand, if g > g c then the coupling g(µ) flows towards increasingly strong coupling, and eventually out of reach of perturbation theory.
The one-loop running of g as a function of a sliding momentum scale µ = k and ǫ > 0 are obtained by integrating Eq. (36), and one finds which is identified with the correlation length appearing in n-point functions. The multiplicative constant in front of the expression on the right hand side arises as an integration constant, and cannot be determined from perturbation theory in g. The quantity m = 1/ξ is usually referred to as the mass gap of the theory, that is the energy difference between the ground state (vacuum) and the first excited state.
In the vicinity of the fixed point at g c one can do the integral in Eq. (39), using the linearized expression for the β-function in the vicinity of the ultraviolet fixed point, and one has for the inverse correlation length with correlation length

Nonlinear Sigma Model in the Large-N Limit
A rather fortunate circumstance is provided by the fact that in the large N limit the nonlinear sigma model can be solved exactly [62][63][64][65][66][67]. This allows an independent verification of the correctness of the general ideas developed in the previous section, as well as a direct comparison of explicit results for universal quantities. The starting point is the functional integral of Eq. (25), and T ≡ g 2 /Λ d−2 , with g dimensionless and Λ the ultraviolet cutoff. The constraint on the φ field is implemented via an auxiliary Lagrange multiplier field α(x). One has with Since the action is now quadratic in φ(x) one can integrate over N − 1 φ-fields (denoted previously by π). The resulting determinant is then re-exponentiated, and one is left with a functional integral over the remaining first field φ 1 (x) ≡ σ(x), as well as the Lagrange multiplier field α(x), with now In the large N limit one neglects, to leading order, fluctuations in the α and σ fields. For a constant α field, α(x) = m 2 , the last (trace) term can be written in momentum space as and the function Ω d (m) given by the integral The above integral can be evaluated explicitly in terms of hypergeometric functions, One only needs the large cutoff limit, m ≪ Λ, in which case one finds the more useful expression with c 1 and c 2 some d-dependent coefficients, given below. From Eq. (49) one notices that at weak coupling and for d > 2 a non-vanishing σ-field expectation value implies that m, the mass of the π field, is zero. If one sets (N − 1) Ω d (0) = 1/T c , one can then write the first expression in Eq. (49) as which shows that T c is the critical coupling at which the order parameter σ vanishes. Above T c the order parameter σ vanishes, and m(T ) is obtained, from Eq. (49) by solving the nonlinear gap Using the definition of the critical coupling T c , one can now write, in the interval 2 < d < 4, for the common mass of the σ and π fields .
with the numerical coefficient given by  The latter is valid again in the vicinity of the fixed point at g c , due to the assumption, used in Eq. (55), that m ≪ Λ. Then Eq. (57) gives the momentum dependence of the coupling at fixed cutoff, and upon integration one finds   Figure 2 shows the typical perturbative expansion for the proper self-energy of a self-interacting scalar field.
To derive the Hartree-Fock approximation, one can first write down Dyson's equations for the dressed propagator and the dressed vertices, as shown in Figure 3. The Hartree-Fock approximation to the self-energy is then obtained by replacing the bare propagator with a dressed one in the lowest order loop diagram, as shown in Figure 4. There the dressed scalar propagator is to be determined self-consistently from the solution of the nonlinear Hartree-Fock equations. It should be noted that in the lowest order approximation the loop integrals involve the dressed scalar propagator ∝ (k 2 + m 2 ) −1 , while the scalar vertex is still the bare one. This is a characteristic feature of the Hartree-Fock approximation both in many-body theories [38] and in QCD [49,50].
One important aspect that needs to be brought up at this stage is the dependence of the results on the specific choice of ultraviolet cutoff. In the foregoing discussion the momentum integrals were cut off, in magnitude, at some large virtual momentum Λ, which resulted in an explicit expression  [1][2][3][4][5][6], the comprehensive monographs [7][8][9][10][11], and the many more references therein.
In view of the later discussion on quantum gravity, it will be useful here to next look explicitly at a few individual cases, mainly as far as the dimension d is concerned. These follow from the general expression for the loop integral given above with S d = 1/2 d−1 π d/2 Γ(d/2). One finds for the integral itself in general dimensions Here one recognizes that the only remnant of the large N expansion is the overall coefficient β 0 = N − 1; perhaps a better, physically motivated, choice would have been a factor of N − 2, since It now pays to look at some specific dimensions individually, just as will be done later in the case of quantum gravity. Then specifically, for dimension d = 2, on has leading to the d = 2 gap equation The solution is given by The self-consistent Hartree-Fock gap equation for m in the nonlinear sigma model shares some analogies with the gap equation for a superconductor. There one finds, in the Hartree-Fock approximation, the following nonlinear gap equation [37,38], where g is the electron-phonon coupling constant, ωD the upper phonon Debye (cutoff) frequency, ∆ is the electron energy gap at the Fermi surface, T the temperature and ξ a shifted energy variable defined as In the above expression µ is the chemical potential, and the quantity N (0) = m kF /2π 2 3 indicates the density of states for one spin projection near the Fermi surface, with kF the Fermi wavevector. The above equation then leads, in close analogy to what is done in the nonlinear sigma model, to an estimate for the critical temperature at which the electron gap ∆ vanishes Note that the role of the ultraviolet cutoff here is played by ωD, and that the critical temperature is non-analytic in the electron-phonon coupling g. Furthermore the role of the mass gap m in the nonlinear sigma model is played here by the energy gap at the Fermi surface ∆. The condensed spin zero electron Cooper pairs then act as massless Goldstone bosons, and later induce a dynamical Higgs mechanism in the presence of an external electromagnetic field, commonly referred to as the Meissner effect.
so that here the constant of proportionality between the mass gap m and the scaling parameter (N−1)T is exactly one for weak coupling T = g 2 ≪ 1. Then the above result corresponds to a correlation length exponent ν = ∞ in d = 2. On the other hand, in the opposite strong coupling so that, as expected, the correlation length ξ = 1/m approaches zero in this limit.
Moving up one dimension, in three dimensions, d = 3, one has for the basic integral leading to the d = 3 gap equation with critical point at for which the mass gap m = 0. In the vicinity of the critical point one can solve explicitly for the so that in d = 3 the universal correlation length exponent is ν = 1. Equivalently, in terms of the dimensionless coupling g, for which T = g 2 /Λ, one has g c = √ 2 π/ √ N − 1 and therefore Again moving up in dimensions, one finds that four dimensions (d = 4) represents a marginal case and one obtains leading to the d = 4 gap equation with critical point and therefore g c = 4π/ √ N − 1 where m = 0. The four-dimensional case is slightly more complex, and here one has to determine the solution to the gap equation recursively. In the vicinity of the critical point one finds for the mass gap m so that in d = 4 the universal correlation length exponent is given by the Landau theory value ν = 1/2, up to logarithmic corrections. This last result is in agreement with triviality arguments for scalar field theories in and above four dimensions [1,2,3]. Also, the logarithmic correction has the right form with the correct power −1/2, in agreement with the exact universal result for the Ising model (given here by the N =1 case) in four dimensions [71].
Going up one more dimension, in five dimensions (d = 5) one has for the integral leading to the d = 5 gap equation with critical point at and therefore g c = 6 π 3/2 / √ N − 1, at which point again m = 0. In the vicinity of the critical point one finds for the mass gap m m = Λ 2 6 1/6 π so that in d = 5 (and above) the universal correlation length exponent stays at ν = 1/2, again in agreement with the Landau theory prediction for all scalar field theories above d = 4 [1,2,3].
More generally, the last set of results agree with the general formula for the critical point T c , valid for both d < 4 and d ≥ 4, and similarly for the dimensionless critical coupling and is given for d < 4 explicitly by the expression For d > 4 the amplitude can be computed explicitly as well, and is given instead by which shows again that d = 4 is indeed a special case, and needs to be treated with some care.
One also notes that both the d < 4 and the d > 4 amplitudes vanish as one approaches d = 4 due to the infrared divergence in this case, which leads to the log correction described above.
The previous, rather detailed, discussion shows that the Hartee-Fock approximation (and, in this case, the equivalent large-N limit), based on a single one loop tadpole diagram, leads to a number of basically correct and nontrivial analytical results in various dimensions, which we proceed to enumerate here. The first one is the fact that it correctly predicts the absence of a phase transition, and thus asymptotic freedom in the coupling g, for any N > 1 in two dimensions.
This result is indeed known to be correct up to N = 2, the latter representing the special case Furthermore, for dimension d between two (lower critical) and four (upper critical), one has where η is the field (or propagator) anomalous dimension, G(p) ∼ 1/p 2−η in the vicinity of the nontrivial fixed point, and γ the zero-field magnetic susceptibility exponent, χ ∼ ξ γ/ν , again in the vicinity of the nontrivial fixed point at g c . In four dimensions the Hartree-Fock approximation correctly reproduces the expectation of mean field exponents (ν = 1/2 here) up to a logarithmic correction. This is not surprising, since some sort of mean field theory is incorporated into the parameter. The latter is then determined self-consistently by a suitable nonlinear gap equation.
There are a number of aspects which are quite similar to the nonlinear sigma model case, which will be helpful here in vastly streamlining the discussion. In particular, in gauge theories the gluon propagator is modified by the gauge and matter vacuum polarization contribution δ ab Π µν (p) with with p µ Π µν (p) = 0. In the Fermi-Feynman gauge the gluon propagator then becomes In perturbation theory, and to all orders, the gluon stays massless as a consequence of gauge invariance, with p 2 Π(p 2 ) ∼ p→0 0 (here we will assume no spontaneous symmetry breaking via the Higgs mechanism). In more detail, the relevant one loop integral has the form where the dot here indicates a dot product over the relevant Lorentz indices associated with the three-gluon, matter or ghost vertices. Additional factors involve a symmetry factor 1 2 for the diagram, the gauge coupling g 2 weight, and an overall group theory color factor. Figure 5 shows the lowest order Feynman diagrams contribution to the vacuum polarization tensor in gauge theories, with again S d = 1/2 d−1 π d/2 Γ(d/2) and σ = 0 in the gauge theory case. Here Λ is an ultraviolet cutoff, such as the one implemented in Wilson's lattice gauge theory formulation, for which Λ ∼ π/a with a the lattice spacing. Note the insertion of a mass parameter m, to be determined later selfconsistently by the gap equation given below.
One finds for the integral itself in general dimensions the explicit result with 2 F 1 (a, b, c, z) the generalized hypergeometric function. The general gap equation for the dynamically generated mass scale m then reads with g the gauge coupling and β 0 an overall N -dependent numerical coefficient resulting from combined group theory weights, Lorentz traces and diagram symmetry factors.
As in the case of the nonlinear sigma model discussed previously, it will be instructive to look in detail at individual dimensions. In four dimensions (d = 4) the gap equation reduces to The solution to the above equation is Here at the end the correct expression β 0 = 11N/3 for SU (N ) gauge theories was inserted, as obtained from one loop perturbation theory. The above explicit results then implies, within the current Hartree-Fock approximation, that the mass gap m is by a factor e − 1 2 = 0.6065 smaller than the scaling violation parameter Λ exp(−1/2 β 0 g 2 ), with the latter appearing on the r.h.s. of Eq. (94). Note that one could have obtained the overall coefficient of the one loop contribution, as used here in the Hartree-Fock approximation, from the Nielsen-Hughes formula [72,73]. There the one-loop β-function contribution coefficient arising from a particle of spin s running around the The renormalization group running of the gauge coupling g(µ) again follows from the requirement that the mass gap m be scale independent, which then gives the expected asymptotic freedom running of g(µ) in the vicinity of the ultraviolet fixed point at g = 0, One observes that the quantity m here plays a role analogous to the ΛM S ≈ 330MeV parameter For small mass gap parameter m the gap equation in five dimensions then reads The solution for g > g c is then given by There is a nontrivial fixed point at g 2 c = 12π 3 /Λ, and the correlation length exponent is ν = 1, just like for the d = 3 nonlinear sigma model discussed previously. For the gauge theory case one expects that the critical point at g c separates a weak coupling Coulomb phase from a confining strong coupling case. Then, as d approaches four from above, the weak coupling massless gluon Coulomb phase disappears.
The renormalization group running of the gauge coupling g(µ) close to the nontrivial ultraviolet fixed point then follows again from the requirement that the mass gap m be scale independent, µ d m d µ = 0. In the vicinity of the five-dimensional ultraviolet fixed point, and within the strong coupling phase (g > g c ) one obtains Note that in this case the power associated with the running in momentum space is determined again by the exponent 1/ν = 1, and that the coefficient of the running term here is in fact independent The solution for m is then, for g > g c and d > 6, Again, the renormalization group running of g(µ) follows from the requirement that the mass gap m be scale independent, and in the vicinity of the ultraviolet fixed point, and in the strong coupling phase g > g c , one finds here It follows that the power associated with the running in momentum space is always given by the it is known that d H = 2, as for regular Brownian motion [75,77,76].
A few words should be spent here on the physical interpretation of m in gauge theories. When one refers to the mass gap, it means a gap in energy between the ground state and the first excited stated, m = E 1 − E 0 , as derived from the quantum Hamiltonian (or transfer matrix) describing, in this case, the nonlinear sigma model. It's relationship with the Euclidean correlation length ξ is provided by the Lehman representation (i.e. completeness) for the two point function (as an example), which then gives m = 1/ξ. In addition, ξ and thus m are related by scaling to a multitude of other observables, such as the order parameter or magnetization σ in the low temperature phase for spin system, and the gluon condensate F 2 µν for gauge theories. In the gauge theory case one has a fundamental relationship between the nonperturbative scale ξ = m −1 and a nonvanishing vacuum expectation value for the gluon field [83,84].
In QCD this last result is obtained from purely dimensional arguments, once the existence of a fundamental correlation length ξ (inversely related to the mass gap) is established. Actual physical values for the QCD condensates are well known; current lattice and phenomenological estimates cluster around α S π F 2 µν ≃ (440 MeV) 4 [85,86]. In gauge theories an additional physically related quantity is provided by the quark field condensate whose physical value is estimated at ψ ψ ≃ (280 MeV) 3 [87]. Again, the power of ξ here is fixed by the canonical dimension of the corresponding fermion field.
A possible physical explicit value for the mass gap parameter m in N =3 QCD is provided by the J P C = 0 ++ spin zero 500 MeV σ or f 0 (500) glueball state, a very broad resonance seen in ππ and γγ scattering [88]. If this particle is roughly considered as a spin-zero bound state of two Conversely, if one uses the current world average of Λ M S ≈ (332 ± 17) MeV [89] for three flavor QCD, then one obtains for the mass of the σ / f 0 meson m σ ≈ 403 MeV. These results seem to suggest that quantitatively the lowest order Hartree-Fock result is not expected to be better than a factor or two or so. Many of these considerations will be useful later when the gravity case is discussed, due to the many deep analogies between gauge theories and quantum gravity.

The Quantum Gravity Case
Upon gaining confidence in this approach and technique, one can perform again a similar type of analysis in the case of quantum gravity, as outlined in detail in the preceding sections. In the end, one major result which makes the calculation feasible is the reliance on well-known results for simple one loop diagrams. This important insight makes it possible to avoid lengthy and complex one-loop gravity calculations, and rely instead, as in all the previous cases, on suitably modified known diagrams via the insertion of an effective, dynamically generated mass parameter m. The latter is then determined self-consistently via the solution of the resulting nonlinear gap equation, as discussed in detail for the nonlinear sigma model (in Section 5), and for gauge theories (in Section 6).
A number of new ingredients arise in the quantum gravity calculation, which we proceed to enumerate here. The first component is the (gauge choice dependent) graviton propagator with the above expression denoting in general the dressed propagator, and D 0 αβ,µν (p) the bare (tree-level) one. An explicit form for the tree-level graviton propagator in covariant gauges, as well as the three-graviton and four-graviton vertex, can be found in the Feynman rules section of [14], and references therein, and for brevity will not be reproduced here. Occasionally, in the following it will be convenient to suppress Lorentz indices altogether in order to avoid unnecessary extensive cluttering. Via loop corrections, the tree-level graviton propagator is then modified by gauge and matter vacuum polarization contributions Π µν,ρσ (p), with the latter written as in terms of the scalar quantity Π(p 2 ). Here η denotes the flat spacetime metric. The vacuum polarization contribution in quantum gravity is shown pictorially in Figure 10. By virtue of energymomentum conservation, one then has the transversality condition p α Π αβ,µν (p) = p µ Π αβ,µν (p) = 0, and the graviton propagator (in the harmonic gauge [21,14]) can be written as with I αβ,µν = η αµ η βν + η αν η βµ − η µν η αβ . As a consequence of gauge invariance (here more properly described as general coordinate, or diffeomorphism, invariance) in perturbation theory, and to all orders, the graviton is expected to stay massless, p 2 Π(p 2 ) ∼ p→0 0. Here we will find it quite = + Figure 10: Dressed graviton propagator D αβ,µν (p), with a graviton proper vacuum polarization insertion Π αβ,µν (p). The thin wavy line denotes the tree-level graviton propagator D 0 αβ,µν (p).
useful that an explicit form for the vacuum polarization contribution Π αβ,µν (p) in the context of perturbative quantum gravity was given some time ago in [22], and such an explicit form will be used below.
In addition, one has the dressed three-graviton vertex (shown here in Figure 11) αβ,µν,ρσ (p, q, r) and the dressed four-graviton vertex (shown here in Figure 12) αβ,γδ,µν,ρσ (p, q, r, s) , and similarly for the O(h n ) higher order graviton vertices, as they arise in the weak field expansion of the Einstein-Hilbert action about flat space, g µν = η µν +h µν . Recall that for the Einstein-Hilbert action √ g R written in terms of the weak field metric field h µν one has with trace h = h µ µ , and consequently (up to total derivatives) one obtains schematically The latter shows the origin of the trilinear (h 3 ) vertex, with weight proportional to momenta squared, ∼ k 2 in Fourier space, as well as the appearance of the (infinitely many) higher order momentum-dependent vertices O(h n ).
· D(q 1 ) · D(p) The next step is the derivation of the Hartree-Fock approximation for quantum gravity. As should be clear from the various cases discussed earlier, namely the nonlinear sigma model (discussed in Section 5) and gauge theories (discussed in Section 6), one needs to focus on just the lowest order loop diagrams. Nevertheless, these will have bare graviton propagators replaced by dressed ones, later to be determined self-consistently. Figure 16 illustrates the Hartree-Fock approximation for gravity, carried out to the lowest order by just considering the lowest order graviton loop diagram, the subject of the current investigation. On the other hand, figure 17 illustrates the next order Hartree-Fock approximation for gravity, which would include both the lowest order graviton loop diagram, as well as the next order graviton two-loop diagrams. These will not be considered further here, but could in the future provide an improved answer, as well as useful quantitative estimates for the overall uncertainty.
In practice, for quantum gravity the relevant one loop integral for the vacuum polarization has where the dot here indicates a generalized dot product over the relevant multitude of Lorentz indices associated with the three-graviton, matter and ghost vertices. Additional ingredients involve a symmetry factor 1 2 for the diagram, and of course the gravitational coupling G. In spite of the complexity of the one-loop expression for the graviton vacuum polarization contribution, the actual integral that needs to be evaluated here has a rather simple form. In the Hartree-Fock approximation, and in close analogy to the treatment of the nonlinear sigma model and gauge theories described previously, one is generally left with an evaluation of the following integral, with a combined spacetime volume factor S d = 1/2 d−1 π d/2 Γ(d/2), but now with σ = 2, specific to the gravitational case. The latter steep momentum dependence is of course at the root of the perturbative nonrenormalizability of quantum gravity in four dimension [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] : the significant weight of the high momentum region in loop integrals renders perturbation theory in Newton's G badly divergent.
Going back to the required integral of Eq. (115), one finds for the momentum integral itself, in a, b, c, z) the generalized hypergeometric function. Here again Λ is the ultraviolet cutoff, such as one implemented in the Regge-Wheeler lattice formulation of gravity [12,13], for which Λ ∼ π/a, with a defined as an average lattice spacing a 2 ≡ l 2 . Indeed, as in all the previous examples, an explicit ultraviolet cutoff is required in order to be able to write down a meaningful form of the Hartree-Fock equation. Generally, for any quantum mechanical system a lattice cutoff is required to make the Feynman path integral well defined, as discussed already in great detail some time ago by the inventor himself [75], and in some more recent monographs [77,76]. The root cause of this situation is that the dominant quantum paths contributing to the path integral are in general nowhere differentiable, a key realization that motivated Regge and Wheeler to develop a lattice formulation for gravity [12,13]. For more details on the Regge-Wheeler lattice theory of gravity see [14], and further references therein. The use of an explicit momentum cutoff leads to some subtle differences in the treatment of ultraviolet divergences, some of which are ordinarily sanctioned to be zero by the formal rules of dimensional regularization, such as These nevertheless can be non-zero (and possibly highly divergent) when evaluated explicitly with both an infrared (µ) and an ultraviolet (Λ) cutoff [18,14].
In quantum gravity another subtlety arises from the fact that one can easily reabsorb some troublesome ultraviolet divergences by a simple rescaling of the metric [14]. Consider the metric field redefinition with ω a constant. For pure gravity with a cosmological constant term proportional to λ, one writes for the Lagrangean Under a rescaling of the metric as in Eq. (118), one obtains which just amounts to a rescaling of Newton's constant and the cosmological constant leaving the dimensionless combination G d λ d−2 unchanged. As a consequence, it seems physically meaningless to discuss separately the renormalization properties of G and λ, as they are both individually gauge-dependent in the sense just illustrated. To some extent these arguments should contribute to clarifying why in the following it will be sufficient to focus on the renormalization properties of Newton's constant G. In addition, one issue we have not touched here at all is the role played by the gravitational functional measure (normally taken to be the De Witt one, see [14] and references therein) in cancelling spurious divergences. The issue is a rather subtle one, whose importance was already emphasized in the earlier discussion of the non-linear sigma model, where the role of the measure in canceling spurious tadpole divergences was pointed out in Eq. (29).
From the integral in Eq. (116) one then obtains the general nonlinear gap equation for the parameter m in d dimensions, which reads Here G is Newton's constant, and β 0 an overall numerical coefficient resulting from Lorentz index traces, diagram multiplicities and individual diagram symmetry factors. In the above equation G is dimensionful, with mass dimension d − 2, G ∼ Λ 2−d . The next step is to discuss the solution to this nonlinear equation in various dimensions, both to draw parallels to the nonlinear sigma model and gauge theory cases outlined earlier, and to explore the sensitivity of the results to the dimension d.
It will be useful here to digress very briefly about the physics of quantum gravity in d < 4.
In general it is possible at least in principle to define quantum gravity in any d > 2. There while the static gravitational potential involves just the spatial Fourier transform In two spacetime dimensions there are no genuine gravitational (transverse-traceless) degrees of freedom, the Einstein-Hilbert action gives a topological invariant proportional to the Euler characteristic of the manifold. The only surviving degree of freedom is the conformal mode, which in two dimensions enters the gravitational action as a total derivative. In the absence of matter, the only residual interaction is then generated by the non-local Fadeev-Popov determinant associated with the choice of conformal gauge. Nevertheless it is possible to consider, as was done some time ago in [30,31,32] and later in [90][91][92][93][94][95], two dimensions as the ǫ ≡ d − 2 → 0 limit for gravity formulated in 2 + ǫ dimensions, with ǫ considered a small quantity. This followed a similar procedure, the dimensional expansion, pioneered earlier by Wilson for scalar fields [1,2,3]. It was also realized early on that in the case of gravity this limit is rather subtle, due to confluent kinematic singularities as d → 2.
In two spacetime dimensions (d = 2) the gravitational Hartree-Fock gap equation of Eq. (122) reduces to A spurious quadratic divergence in the integral given above needs to be subtracted, in order to achieve consistency with the known, more recent perturbative result for gravity in 2 + ǫ dimensions given in [91][92][93][94][95], discussed in greater detail further below. The critical point (or ultraviolet fixed point) in this case is located at the origin The solution to the above d = 2 gap equation then has the form where at the end the correct expression β 0 = 25/3 for pure gravity in d = 2 derived in [95] has been inserted. Here G ≡ G(Λ) is the bare coupling at the cutoff scale Λ. The above results then implies, within the current approximation, that the dynamically induced gravitational mass gap m is by a factor e − 1 4 = 0.7788 smaller than the gravitational scaling parameter Λ exp(−π/β 0 G).
The above result then leads to some rather immediate consequences, again very much in line with the previous discussion for scalar fields in Section 5, and for gauge theories in Section 6.
Thus the renormalization group running of the gravitational coupling G(µ) follows again from the requirement that the mass gap parameter m be scale independent, This then gives the expected asymptotic freedom-like running of the gravitational coupling G(µ) in the vicinity of the d = 2 ultraviolet fixed point located at G = 0, namely One notes here that the quantity m plays a role quite analogous to the ΛM S of QCD and, more generally, SU (N ) gauge theories, as discussed here earlier in Section 6. Indeed, the quantum gravity result in d = 2 looks rather similar to the result for the two-dimensional nonlinear sigma model (for N > 2) and to the gauge theory result in d = 4, in the sense that all three models have a mass gap that vanishes exponentially at zero coupling. This last fact is of course already well known from the simplest application of perturbation theory. Nevertheless, the gravity result in two dimensions can be regarded as of very limited physical interest, given that the Einstein-Hilbert action is just a topological invariant in d = 2. It is also worthy of note here that, overall, the main results follow from the structure of the one loop integral -and from the sign of β 0 -but are otherwise universal, in the sense that a specific value for the magnitude of β 0 does not play much of a role in the scaling laws, in the sense that quite generally the mass gap has an essential singularity at G = 0 in d = 2.
Before proceeding to discuss the Hartree-Fock results in higher dimensions, it will be useful here to recall the main results of the 2 + ǫ expansion for gravity [91][92][93][94][95]. To first order in G one where c is the central charge for the massless matter fields. Later the original one-loop calculation was laboriously extended to two loops [95], with the result again for c massless real scalar fields minimally coupled to gravity. At the nontrivial ultraviolet fixed point, for which β(G c ) = 0, the theory becomes scale invariant by definition. In statistical field theory language, the fixed point corresponds to a phase transition where the correlation length ξ = 1/m diverges in units of the cutoff. If one defines the correlation length exponent ν in the usual way by then one finds the expansion in ǫ This then gives, for pure gravity without matter (c = 0) in four dimensions (d=4), to lowest order ν −1 = 2, and ν −1 ≈ 4.4 at the next order. One can integrate the β-function equation in Eq. (131) to obtain the renormalization group invariant quantity which is identified with the mass gap m, or equivalently the inverse correlation length ξ. The multiplicative constant in front of the last expression on the right hand side arises as an integration constant of the renormalization group equation, and thus cannot be determined from perturbation theory in G alone. From Eq. (131), in the 2 + ǫ expansion the running of Newton's G in the strong coupling (antiscreening) phase with ǫ > 0 and G > G c is given by where the dots indicate higher order radiative corrections. Again, it is noteworthy here that the quantum amplitude c 0 arises as an integration constant of the renormalization group equations, and thus (unlike the Hartree-Fock result) remains undetermined in perturbation theory.
Next consider the case of quantum gravity in a spacetime dimension d = 3. Even in three dimensions it is known that there are no genuine gravitational (transverse-traceless) degrees of freedom, with the only dynamical surviving remnant being the conformal mode [96]. In d = 3 the gravitational gap equation of Eq. (122) for small m reduces to Again, a spurious cubic divergence in the integral needs to be subtracted, as discussed earlier, in order to achieve consistency with the known perturbative results for gravity in 2 + ǫ dimensions.
The solution to the above equation then has the form What is new here is the appearance of a critical point in G, located at Again, a spurious quartic divergence in the integral needs to be subtracted in order to achieve consistency with the known perturbative results for gravity in 2+ǫ dimensions; as mentioned previously, in the 2 + ǫ expansion the latter is removed by a suitable rescaling of the metric. Furthermore, one would need to consider here the role played by the gravitational functional measure (normally taken to be the De Witt one, see [14] and references therein) in cancelling spurious δ(0) ≃ (Λ/π) d tadpole divergences. The measure's effect was already emphasized earlier in the discussion of the non-linear sigma model, where its role in canceling spurious tadpole divergences was pointed out in the discussion surrounding Eq. (29).
The solution to the above d = 4 gap equation then has the form In four dimensions one finds that the critical point in Newton's G is located at and for the mass gap one now has an exponent ν = 1/2. This last result is again universal, in the sense that it does not depend on the magnitude of β 0 , nor on the value of G c .
The above d = 4 gravity results share some significant similarity with the corresponding results for the d = 4 nonlinear sigma model, and for gauge theories in d = 6. Recall that the main result here follows generally from the structure of the one loop integral -and from the sign of β 0 -but is otherwise universal, in the sense that a specific value for the magnitude of β 0 does not play any role in the scaling laws: the mass gap in d = 4 goes to zero with a square root singularity (up to log corrections) at G c , at least within the Hartree-Fock approximation.
For gravity one can use again the Nielsen-Hughes formula [72,73] to provide an estimate for the numerical coefficient β 0 in d = 4. The one-loop β-function contribution coefficient arising from a particle of spin s running around the loop is For spin s = 2 the above formula then gives for pure gravity (no matter or radiation fields) in four dimensions β 0 = 47/3, quite a bit larger than the gauge theory case (and with an overall negative sign in the beta function, correctly taken into account previously). This then gives from Eq. (144) G c = 5.035/Λ 2 , compared to the lattice gravity value in d = 4 G c = 1.069/Λ 2 given in [101].
again in the vicinity of the ultraviolet fixed point, and thus up to higher order corrections in m/µ.
Next consider the universal scaling exponent ν, which is ν = 1 2 for the Hartree-Fock approximation in four dimensions, as can be seen from Eq. (142). As a direct consequence, one finds a power of 1/ν = 2 in the momentum dependence of G(µ) in Eq. (145). The value of ν can then be compared to the lattice gravity estimate, where one finds in four dimensions ν ≃ 1/3, which then implies a larger power of 3 for the running of G(µ) (as given below). Indeed it is useful at this stage to compare the Hartree-Fock result for G(µ) to the corresponding result in the Regge-Wheeler lattice theory of gravity [101], where one finds for the running of Newton's G One notes that the structure of the running is similar to the Hartree-Fock result, nevertheless the powers are different (two versus three) and there is some difference in the numerical value of the amplitude as well. The amplitude in the lattice case, c 0 ≃ 16.0 [101], is thus somewhat larger (by a factor of five) than the Hartree-Fock value c 0 = 3/2 given above. A recent comparison of the running of G with current cosmological observations, to be discussed later here, leads to useful observational constraints on the amplitude c 0 , suggesting a value c 0 ≈ 2.29 and thus smaller than the current lattice estimate, and more in line with the above Hartree-Fock result.
It pays to look briefly at a few cases of dimension greater than four. In five spacetime dimensions (d = 5) the gravitational gap equation of Eq. (122) for small m reduces to again with the spurious leading Λ 5 divergence subtracted in order to achieve consistency with the known perturbative result for gravity in 2 + ǫ dimensions. The solution to the above equation then has the form The critical point in Newton's G is now located at For the mass gap exponent one has again the universal value ν = 1/2, independent of β 0 or G c .
As before, the above relation between m and bare G can be inverted to give the renormalization group running of G(µ), from µ d m d µ = 0. This then gives for G(µ) in the vicinity of the d = 5 gravity ultraviolet fixed point with, in this case, no log correction. As expected the power in the running is 1/ν = 2. In turn the result of Eq. (151) implies that here in real space the quantum correction to Newton's constant initially grows quadratically with distance in the strong coupling phase, δG(r) ∼ (r/ξ) 2 , with ξ = 1/m. Here the quantum amplitude coefficient in d = 5 is c 0 = 9/2 = 4.5 in the Hartree-Fock approximation, which is about one fourth the value found on the lattice in d = 4, c 0 ≃ 16.0, see [102,101] and references therein. We will return to this point later on.
Overall, the gravity results here share some similarity with the corresponding results for the imply d H = 2 if the arguments in [80,81,82] are followed here as well. In comparison, for a Gaussian scalar field (a free particle with no spin) it has been known that d H = 2, as for regular Brownian motion [75,77,76]. Indeed, a large-d dimensional expansion was pursued for lattice gravity in [103,104]. Nevertheless, one finds that in this limit the lowest order results are not particularly illuminating regarding to what might happen in four dimensions. We should mention that a similar large-d expansions were pursued for scalar field theories in [68,69,70] and for gauge theories in [79], and in both cases the theory was formulated on the lattice. Some additional papers that also looked at quantum gravity in the continuum from the large d point of view can be found in [105,106].
It is possible to derive a general analytical expression for the mass gap m in general dimensions.
In d dimensions, with d > 4, one finds from Eq. (122) for the critical point and for the mass gap This then leads in d > 4 to the renormalization group running of Newton's constant The dimensionless quantum amplitude is given here in general by c 0 = 3(d − 2)/2(d − 4) for d > 4, and the pole in d = 4 is then seen as the cause of the log correction seen earlier in Eq. (142).
Conversely, for d < 4 the general d result from Eq. (122) is that ν = 1/(d − 2). The running of Newton's constant for this case is then given by . The On the other hand, for non-Abelian gauge theories in four dimensions one has the corresponding result, describing a non-vanishing vacuum expectation value for the chromo-electric and chromomagnetic fields [83,84,85,86] F a µν F a µν ≃ In QCD this last result can be obtained from purely dimensional arguments, once the existence of a fundamental correlation length ξ (which for QCD is given by the inverse mass of the lowest spin zero glueball) is established. Note that the latter result is completely gauge invariant, in spite of the appearance of the quantity ξ, which, by virtue of the Hartree-Fock gap equation and its dependence on the gauge-fixed gluon propagator, could have inherited some gauge dependence.
The power of ξ in the two cases above is thus determined largely by the canonical dimension of the primary fields φ or A a µ . The close analogy between non-Abelian gauge theories and gravity then suggests that a similar identification should be true in gravity for the Ricci scalar, as argued in [107,108,102], and reference therein. Via the classical (or quantum Heisenberg) vacuum field equations R = 4λ one can then relate the above quantities to the observed (scaled) cosmological constant λ obs (up to a numerical constant of proportionality, expected to be of order unity). One obtains In this picture the latter quantity is regarded as the quantum gravitational condensate, a measure of the vacuum energy, and thus of the intrinsic curvature of the vacuum [107,108].
Irrespective of the specific value of ξ, this would indicate that generally the recovery of classical GR results takes place at distances much smaller than the correlation length ξ so that coherent quantum gravity effects become negligible on distance scales r ≪ ξ. 5 In particular, the static Newtonian potential is expected to acquire a tiny quantum correction from the running of G [see Eqs. (145) and (147)], V (r) ∼ − G(r)/r. Figure 19 provides a direct comparison between the continuum analytical self-consistent Hartee-Fock quantum gravity result for G(q) in Eq. (145), and the Regge-Wheeler lattice gravity running of G(q) of Eq. (147) as given in [101], (162) Figure 20 later provides a direct comparison between the two results, but in real space. In generating this last graph, we have simply substituted µ → 1/r. This was done in order to avoid at this early stage the complexities of having to deal, in a general coordinate space, with a renormalization group running Newton's G( ) via the covariant substitution with ≡ g µν ∇ µ ∇ ν , as discussed in detail elsewhere, see for example [110,111].
To conclude this section, one can raise the legitimate concern of how these results are changed by quantum fluctuations of various matter fields coupled to gravity (scalars, fermions, vector bosons, spin-3/2 fields etc.). These would enter in the vacuum polarization loop diagrams containing these fields. Their contribution will appear through many additional loops, and will therefore affect the value of the coefficient β 0 appearing in Eq. (122). Nevertheless one would expect that significant changes to the result of Eqs. (145) will arise from matter fields which are light enough to compete with gravity, and whose Compton wavelength is therefore comparable to the scale of the gravitational vacuum condensate, or equivalently the observed cosmological constant, so that for these fields m −1 ∼ 1/ λ/3. At present the number of candidate fields that could fall into this category is rather limited, with the photon and a near-massless gravitino belonging to this category. Note that in the 2 + ǫ perturbative expansion for quantum gravity one encounters factors of 25 − c in the renormalization groups β function, where c is the central charge associated with the (massless) matter fields [94,95]. In four dimensions similar factors involve 48 − c [72,73], which would again lend support to the argument that such effects should be rather small in four nowadays is usually analyzed in the framework of the standard ΛCDM model. Included in the usual assumptions is the fact that Newton's G does not run with scale. If such an assumption were to be relaxed, it could affect a number of cosmological parameters, including λ obs , whose value could then perhaps change significantly. In the following the estimate of Eq. (160) is used as a sensible starting point. On cosmological scales, a running of G leads to a multitude of physically observable effects, ranging from modifications to early cosmological evolution, the primordial growth of matter perturbations, and the cosmic background radiation (CMB). We chose here to focus on the latter for two main reasons. The first one is the sensitivity of the CMB data to extremely large scales, clearly comparable in magnitude to the value for ξ given in Eq. (160). The second rather obvious reason is the availability of very accurate recent satellite data presented in the Planck15 and Planck18 surveys [109].
Here we will focus almost exclusively on one well-studied physical observable, the matter power spectrum P m (k), for which detailed graphs have been produced by the aforementioned Planck collaboration. The matter power spectrum relates to the cosmologically observed matter density where r = |x − y|, and is the matter density contrast, which measures the fractional overdensity of matter denstiy ρ above an average background densityρ. In the literature, this correlation is more often studied in wavenumber-space, G ρ (k; t, t ′ ) ≡ δ(k, t) δ(−k, t ′ ) , via a Fourier transform. It is also common to bring these measurements to the same time, say t 0 , so that one can compare density fluctuations of different scales as they are measured and appear today. The resultant object P m (k) is referred to as the matter power spectrum, where δ(k, t) ≡ F (t) ∆(k, t 0 ). The factor F (t) then simply follows the standard GR evolution formulas as governed by the Friedmann-Robertson-Walker (FRW) metric. As a result, P m (k) can be related to, and extracted from, the real-space measurements via the inverse transform It is common to parameterize these correlators by a so-called scale-invariant spectrum, which includes an amplitude and a scaling index, conventionally written as or equivalently with the powers related by s = (d − 1) − γ = 3 − γ via the Fourier transform. Nevertheless, such simple parametrizations generally only apply to specific regions in wave-vector space (specifically the galaxy regime at larger k, or the primordial CMB regime at smaller k), in spite of the fact that different regimes are known to be related to each other by standard classical cosmological evolution, via the so-called transfer function.
In the matter-dominated era, such as the one where galaxies and clusters are formed, the energy momentum tensor follows a perfect pressureless fluid to first approximation. Hence, the trace equation reads (For any perfect fluid the trace gives T = 3p − ρ, and thus T ≃ −ρ for a non-relativistic fluid.) Since λ is a constant, the curvature and matter variations, and hence their correlations, are directly related δR(x) δR(y) = (8πG) 2 δρ(x) δρ(y) .
A number of subtleties arise here, such as the fact that to extend beyond the linear matter dominated regime, the trace equation alone becomes insufficient (since the trace of the energy momentum tensor for radiation vanishes), and the full tensor field equations have to be used. Furthermore, in a real universe with multiple fluid components, interactions and transient behaviors have to be taken into account, which are governed by coupled nonlinear Boltzmann equations. However, these classical procedures have been fully worked out in standard cosmology texts [112,113], as well as in currently popular sophisticated computer codes, such as CAMB and MGCAMB [114,115,116], CLASS and MGCLASS [117,118], ISiTGR [119,120,121] and COSMOMC [121] (for a detailed comparison of features in various codes see [122]).
Quite generally, the matter power spectrum can be decomposed as two parts -an initial condition known as the primordial spectrum R o k , and an interpolating function between the domains known as a transfer function T (k) [112]. As a result, the full matter power spectrum P m (k) beyond the galaxy (larger k) domain will take the form where C 0 ≡ 4(2π) 2 C 2 (Ω Λ /Ω M )/25 Ω 2 M H 4 0 is a combined constant of cosmological parameters, and the k 4 factor is purely for convenience. The transfer function is usually written in terms of κ ≡ √ 2k/k eq , a scaled dimensionless wavenumber, with k eq being the wavenumber at matterradiation-equality. With this decomposition, the transfer function is a fully classical solution of the set of Friedmann and Boltzmann equations, capturing the nonlinear dynamics. This leaves the initial primordial function, which is usually parameterized as a scale-invariant spectrum involving an overall amplitude N 2 and a spectral index n s . k R is commonly referred to as the "pivot scale", a reference scale conventionally taken to be k R = 0.05 Mpc −1 in cosmology. While the transfer function T (κ) -the solution to the coupled and nonlinear set of Friedmann, Boltzmann and continuity differential equations -is difficult to obtain as an explicit function, it is nevertheless in principle fully determined from classical dynamics. Moreover, assuming standard ΛCDM cosmology dynamics and evolution, a semi-analytical interpolating formula for T (κ) [112] is known. As a result, if the initial spectrum R o k , or more specifically the parameters N and n s , are fixed, then P m (k) is fully determined.
Here, in this section, the main concern will be the inclusion of the effects of a running of Newton's G, as given in Eqs. (161), (162) and (159) in the matter power spectrum, as measured for example in current CMB data. Modifications to the matter power spectrum P m (k) at small k originating in a running G(k) can be done either analytically using the transfer function [123,124] or by relying on more comprehensive numerical programs [125]. Analytically, the effect of a running Newton's constant [Eq. (161)] can be included via dimensional analysis for the correct factors of G to include, P m (k) → (G 0 /G(k)) 2 P m (k) [124]. Here we will focus instead on the more comprehensive, and more accurate, results that can be obtained by consistently incorporating the running of G in all relevant cosmological equations, and in many more cosmological observables, such as temperature and polarization correlations [125].
The key resultant prediction for the matter power spectrum P m (k) is found here in Figure   21, showing (not surprisingly) still an almost perfect fit to all observational data for k ≫ m ≃ 2.8 × 10 −4 hMpc −1 . Nevertheless, for scales of k comparable to m, additional quantum effects become significant due to the G(k), enough to cause significant deviations from the classical ΛCDM result for P m (k). In Figure 21, the middle solid orange curve shows the Hartree-Fock expression for the running of Newton's constant G, while the bottom dashed green curve and the top blue dotted curve show the lattice result of a running Newton's constant G (with the lattice coefficient c 0 = 16.0), as well as the result for no running respectively for reference. It seems that the Hartree-Fock running of G is in reasonably good consistency with the lattice expression, except for the eventual unwieldly upturn below k < 2 × 10 −4 hMpc −1 . However, this upturn is most likely an artifact from the Hartree-Fock expression being just a first-order analytical approximation after all (as discussed earlier, the lowest order Hartree-Fock approximation can be extended to higher order, by including increasingly complex higher loop diagrams, with dressed propagators and vertices still determined self-consistently by a truncated version of the Schwinger-Dyson equations).
Nevertheless, the Hartree-Fock approximation shows good consistency with both the latest available observational data sets, as well as with the lattice result. The fact that it exhibits a gentler dip at small k perhaps also provides support for a potentially smaller lattice running coefficient of Eq. (162) of approximate value c 0 ≈ 2.29. More details for the most recent constraints on this lattice parameter c 0 are discussed in [124,125].
Following similar analysis to determining P m (k), other spectra such as the angular temperature spectrum C T T l , should be fully derivable from the primordial function R o k , or specifically n s , which is set by the scaling of gravitational curvature fluctuations ν.

Conclusion
In this work we have shown how the Hartree-Fock approximation to quantum gravity can be carried out, by considering what ultimately reduces to a set of relatively easy to evaluate one-loop integrals.
The main feature of such an approach is that the complete analytic expression for the one-loop integrals allows one to write down a gap equation for the dynamically generated mass scale in the strongly coupled phase and, more importantly, for any spacetime dimension. This result can be viewed as analogous to the situation in the nonlinear sigma model, where a nonlinear gap equation is obtained explicitly, and can then in principle be solved for any dimension. Due to the inherent approximation of a mean-field type approach the calculation is expected to be rather crude, but nevertheless provides interesting insights, and can furthermore provide a useful starting point for improved, higher loop self-consistent calculations.
There are a number of interesting features that arise from the results presented earlier, which will be summarized in the following paragraphs. The first notable aspect is the appearance of a nontrivial renormalization group fixed point (referred to as a critical point in statistical field theory language) G c in Newton's constant, for any spacetime dimension d greater than two. As a consequence, above two dimensions the theory exhibits two phases, a Coulomb-like phase with gravitational screening for G < G c , and a strongly coupled phase with gravitational antiscreening for G > G c . Since the Hartree-Fock method retains some vestiges of a perturbative diagrammatic calculation, the above results allow do not allow one to determine whether both phases are indeed physical. On the other hand current lattice calculations, which are genuinely nonperturbative, suggest that gravitational screening is impossible (no stable ground state exists G < G c ) and that, as in Yang Mills theories, only the gravitational antiscreening phase for G > G c is physically realized in this theory.
For the latter phase the Hartree-Fock calculation gives, once suitable renormalization group arguments are applied, an explicit expression for the running of the gravitational Newton's constant as a function of scale. The latter can be expressed as a G(k) in wavevector space, or, if one wishes, more generally as a covariantly formulated G( ), with covariant d'Alembertian ≡ g µν ∇ µ ∇ ν .
Unlike perturbatively renormalizable theories such as QCD, the RG running of G in the gravitational case is not logarithmic, but instead follows a power law in the relevant scale. In addition, the Hartree-Fock results suggest that for quantum gravity the lower critical dimension is two (as expected based on naive dimensional arguments, and also on the basis of the lattice results, and initially from Wilson's 2 + ǫ gravity perturbative expansion as well), and furthermore that the upper critical dimension is six. This would correlate with the fact that, within the Hartree-Fock approximation, above six spacetime dimensions scaling dimensions and critical exponents flow into their Gaussian, free field value. How that would explicitly reflect on the nature of various (local and non-local) gravitational correlations is at this point still not entirely clear at this point.
More importantly, in the physically relevant case of four spacetime dimensions one finds that the gravitational coupling grows like a distance squared, up to logarithmic corrections (between d = 2 and d = 4 the relevant power varies as 1/(d − 2), and stays constant above that). This result is similar to, but nevertheless still quantitatively different from, the Regge-Wheeler lattice result, which gives a renormalization group growth of the gravitational coupling G with the cube of the distance, see [101,102] and references therein. Nevertheless, in either case the reference scale for the growth of Newton's constant as a function of distance is determined by a new, genuinely nonperturbative quantity ξ = 1/m. The latter is non-analytic in the bare couplings, and arises naturally as an integration constant of the renormalization group equations. It was argued elsewhere [107,102] that this new nonperturbative scale, specific to quantum gravity, should be identified with the gravitational vacuum condensate R = 2λ via λ = 3/ξ 2 , where λ is the observed (tiny) scaled cosmological constant, in this picture more properly referred to as the gravitational vacuum energy.
The latter would then provide the needed reference scale for the RG running of the gravitational coupling G. Since ξ is very large (from current astrophysical observations of λ [109]) one has ξ ∼ 5300Mpc), which would then lead to a very slow rise of G, observable only on very large, cosmological scales.
It is of some interest here to point out that the Hartree-Fock approximate calculation does not just give values for exponents and scaling dimensions, it also provides an explicit analytic expression for quantum amplitudes. Such as, for example, the overall amplitude of the quantum correction to G(k) when referred to the new nonperturbative scale ξ. This is quite different from ordinary perturbation theory say in the 2 + ǫ expansion, where no information can ever be gained on the quantum amplitudes themselves (only on some ratios of critical amplitudes, as is already the case in the nonlinear sigma model).
What is the physical interpretation of the self-consistent Hartree-Fock type calculation? For suggestions, one can look at the analogous results in condensed matter physics, or from the nonlinear sigma model, to try to gain some insight into what type of physical process is taken into account in the Hartree-Fock self-consistent, mean field-type approximation. Perturbation theory initially only takes into account single graviton exchange (at the tree level), then to higher order two graviton exchanges in loop diagrams, then three graviton exchanges etc.. On the other hand, in the Hartree-Fock approximation all these multi-graviton effects are included into one single, self-consistently determined, diagrammatic contribution. The latter sums up infinitely many perturbative diagrams, and thus can lead to entirely novel features such as a dynamically generated nonperturbative scale. The lowest order Hartree-Fock self-consistent approximation should therefore be viewed as a (physically motivated) re-summation of a certain subset of infinitely many diagrams. The next order Hartree-Fock correction then self-consistently re-sums a second set of topologically distinct more complex diagrams, leading presumably to a further improvement over the lowest order Hartree- Heisenberg equations of motion) clearly appears as a mass-like term in the weak field expansion of the gravitational action. Nevertheless, in the gravity case one has the well-known subtlety that such a mass-like term remains fully consistent with general covariance (and thus entirely independent of gauge choice), since the λ term in the gravitational action retains that property.