^{1}

^{2}

^{*}

^{1}

^{2}

Edited by: Antonio D. Pereira, Universidade Federal Fluminense, Brazil

Reviewed by: Carlo Pagani, Université Grenoble Alpes, France; Aaron Held, Imperial College London, United Kingdom

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background ^{2}-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.

General relativity taught us to think of gravity in terms of geometric properties of spacetime. The motion of freely falling particles is determined by the spacetime metric _{μν} which, in turn, is determined dynamically from Einstein's equations. It is then an intriguing question what replaces the concept of a spacetime manifold once gravity is promoted to a quantum theory. Typically, the resulting geometric structure is referred to as “quantum geometry” where the precise meaning of the term varies among different quantum gravity programs.

An approach toward a unified picture of the quantum gravity landscape could then build on identifying distinguished properties which characterize the underlying quantum geometry and lend themselves to a comparison between different programs. While this line of research is still in its infancy, a first step in this direction, building on the concept of generalized dimensions, has been very fruitful. In particular, the spectral dimension _{s}, measuring the return probability of a diffusing particle in the quantum geometry, has been computed in a wide range of programs including Causal Dynamical Triangulations [_{s} = 2 rather universally. The interpretation of _{s} as the dimension of a theories momentum space, forwarded in Amelino-Camelia et al. [

Following the suggestion [^{1}^{2}_{0} = 3.986 lent itself to the interpretation that “spacetime could be much more empty than expected.” Recently, Houthoff et al. [

where _{μν}. While it was possible to extract analytic expressions for all γ_{n}, it also became apparent that the single-operator approximation underlying the computation comes with systematic uncertainties. In parallel, the anomalous scaling properties of subvolumes and geodesic distances resulting from the renormalization group fixed points underlying Stelle gravity and Weyl gravity have recently been computed in Becker et al. [

The purpose of present work is two-fold: Firstly, we extend the analysis [_{0} = 3.986 found in Pagani and Reuter [

The rest of this work is organized as follows. Section 2 introduces the composite operator formalism and the propagators entering in our computation. The generating functional determining the matrix of anomalous dimensions is computed in section 3. The link to the stability matrix governing the gravitational renormalization group flow in the vicinity of the Reuter fixed point is made in section 4.1 and the spectral properties of the matrix are analyzed in section 4.2. Section 5 contains our concluding remarks and comments on the possibility of developing a geometric picture of Asymptotic Safety from random geometry. The technical details underlying our computation have been relegated to three appendices:

Functional renormalization group methods provide a powerful tool for investigating the appearance of quantum scale invariance and its phenomenological consequences [

plays a key role in studying the renormalization group (RG) flow of gravity and gravity-matter systems based on explicit computations. It realizes the idea of Wilson's modern viewpoint on renormalization in the sense that it captures the RG flow of a theory generated by integrating out quantum fluctuations shell-by-shell in momentum space. Concretely, Equation (2) encodes the change of the effective average action Γ_{k} when integrating out quantum fluctuations with momentum _{k} is then sourced by the right-hand side where _{k} with respect to the fluctuation fields, the regulator ^{2} ≲ ^{2}, and Tr includes a sum over all fluctuation fields and an integral over loop-momenta. Lowering _{k}. For later convenience, we then also introduce the “RG-time” _{0}) with _{0} an arbitrary reference scale.

In practice, the Wetterich equation allows to extract non-perturbative information about a theories RG flow by restricting Γ_{k} to a subset of all possible interaction monomials and subsequently solving Equation (2) on this subspace. For gravity and gravity-matter systems such computations get technically involved rather quickly. Thus, it is interesting to have an alternative equation for studying the scaling properties of sets of operators _{k}. Within the effective average action framework such an equation is provided by the composite operator equation [_{nm}(

The analogy of _{nm} to a wave-function renormalization then suggests to introduce the matrix of anomalous dimensions

Following the derivation [_{nm} can be computed from the composite operator equation

where _{nn} [c.f.[_{ij} associated with the operators (1).

The computation of γ_{nm} requires two inputs. First, one needs to specify the set of operators _{k} approximated by the Euclidean Einstein-Hilbert (EH) action

supplemented by a suitable choice for the gauge-fixing action (54). In practice, we obtain _{μν} into a background metric _{μν} and fluctuations _{μν}:

In order to simplify the subsequent computation, we then chose the background metric as the metric on the

Moreover, we carry out a transverse-traceless (TT) decomposition of the metric fluctuations [

where the component fields are subject to the differential constraints

The Jacobians associated with the decomposition (9) are taken into account by a subsequent field redefinition

and it is understood that in the sequel all propagators and the matrix elements

We then specify the gauge-fixing introduced in Equation (54) to geometric gauge, setting ρ = 0 and subsequently evoking the Landau limit α → 0. Substituting the general form of the matrix elements listed in _{μ} and the scalar σ drop out from the composite operator equation. As a consequence, the anomalous dimensions are only sourced by the transverse-traceless and conformal fluctuations. The relevant matrix elements are then readily taken from

together with

where

Finally, the matrix entries for the regulator

Here

Substituting the expressions (12)–(15) into the composite operator Equation (5) then yields

where the subscripts _{T} and _{S} is

Equation (16) should then be read as a series expansion in _{nm} are obtained by matching powers of

The structure of the traces appearing in the definition (18) ensures that

where

Before delving into the explicit evaluation of the traces, the following structural remark is in order. Inspecting (17), one observes that the right-hand side associated with the

This structure originates solely from the properties of the operators

The explicit values of the matrix entries (16) are readily computed employing the heat-kernel techniques reviewed in ^{2}, setting the coefficients _{n}, _{nm} this entails that all entries on the diagonal and below (marked in black) are computed exactly while contributions to the terms above the diagonal (marked in blue) will receive additional contributions from higher-orders in the heat-kernel. In particular all entries γ_{nm} with _{T} and _{S} in the transverse-traceless and scalar propagators.

Evaluating (16) based on these approximations then results in an infinite family of generating functionals

Here we introduced the dimensionless couplings

and the anomalous dimension of Newton's coupling

The coefficients

Their counterparts in the scalar sector read

Finally, the

Heat-kernel coefficients

1 | |||

− |

_{d, 2} and δ_{d, 4} are linked to zero modes of the decomposition (9) on the 2- and 4-sphere. The dash −− indicates that the corresponding coefficient is not entering into the present computation

Evaluating (19) for the explicit generating functional (21) then yields the entries of the matrix _{n,n−2}, _{n,n−1},

Equation (21) together with the relation (19) constitutes the main result of this work. They give completely analytic expressions

At this stage, a few remarks are in order.

The entries of the anomalous dimension matrix carry a specific

The entries γ_{n,n−2} are solely generated from the scalar contributions, i.e., the transverse-traceless fluctuations do not enter into these matrix elements. Technically, this feature is associated with the Hessians

Notably, _{nm} with _{S} and thus vanish if

The matrix _{∗} = λ_{∗} = 0 where one recovers the classical scaling of the geometric operators.

Components of the Hessians entering the right-hand side of the composite operator equation (5) and the Wetterich equation evaluated for the Einstein-Hilbert truncation.

^{T}. The off-diagonal terms are symmetric, e.g.,

Starting from the general result (19), we now proceed and discuss its implications for the quantum geometry associated with Asymptotic Safety.

By construction, the matrix

From the definition of the beta function ∂_{t}_{n} = β_{un}(_{i}) and the fact that at a fixed point _{nm}],

Let us denote the eigenvalues of _{n} so that spec(_{n}}. Equation (28) then entails that eigendirections corresponding to eigenvalues with a negative (positive) real part attract (repel) the RG flow when

Formally, one can then derive a relation between

where _{n} = _{nm} obtained at the fixed points (27). Before embarking on this discussion, the following cautious remark is in order though. While the composite operator formalism may allow to obtain information on the stability properties of a fixed point beyond the approximation used for the propagators, it is also conceivable that the formalism becomes unreliable for eigenvalues λ_{n} with _{max}^{3}^{4}

This said, we now investigate the properties of the stability matrix (29). Here we will resort to the following frameworks:

The spectrum of

In the conformally reduced approximation [

The latter choice is motivated by the observation that this framework gives rise to the spec(

We first give the diagonal entries γ_{nn} within framework

These relations exhibit two remarkable features. Firstly, the structure of

We now discuss the properties of the stability matrix evaluated at the Reuter fixed points (27) generated from the functional (21). In practice, we truncate _{n} ∈ spec(

with _{n} denoting the right-eigenvectors of _{n} are real or that the left- and right-eigenvectors of

The structure of

The eigenvector _{1} associated with these eigenvalues is aligned with the volume operator _{n} on the matrix size

The normalized eigenvector associated with

The properties of spec(_{n}) of the stability matrices of size _{n}, _{1} and λ_{3}) are^{5}

Carefully analyzing the _{2} visible in the left diagrams: the oscillations are linked to the appearance of new complex pairs of eigenvalues. Focusing on the four-dimensional case where this feature is most prominent, one finds that singling out the values of λ_{2} _{2}(

so that the fluctuations are reduced by a factor two as compared to the full set 34.

Spec(_{n}) of the eigenvalues found for the stability matrices of sizes _{n} (

Spec(_{n}) of the eigenvalues found for the stability matrices of sizes _{n} (

Spec(_{n}) of the eigenvalues found for the stability matrices of sizes _{n} (

At this stage, it is interesting to compare the averages 34 to the eigenvalue spectrum obtained from the smallest non-trivial stability matrix

Thus we conclude that small values of

We close this section with a general remark on the structure of spec(_{∗}, _{∗}) = (0, 0),

Loosely speaking, the definitions of these domains corresponds to imposing that the quantum corrections are not strong enough to turn more than one classically UV-marginal (

Spectral analysis for the matrices

The boundary of the domains

with the best-fit parameters

Following the ideas [_{eff} which separates perturbative from non-perturbative behavior. Comparing the eigenvalue distributions for the Reuter fixed points shown in

Spectral analysis for the matrix _{n}.

In this work, we applied to composite operator formalism to construct a completely analytic expression for the matrix ^{2}-level can be neglected. On this basis, we derived the generating functional (21) from which the matrix of anomalous dimensions (20) can be generated efficiently.

As illustrated in section 4 the stability matrix

The composite operator approach suggests that in ^{2}-operator into a UV-relevant one. Similarly, the analysis in ^{2}-coupling becomes UV-relevant.

The eigenvectors of

The non-diagonal terms γ_{nm},

The analysis of the spectrum of the stability matrix as a function of the dimensionless Newton coupling _{n} follow an almost Gaussian behavior

where

Putting our results into a broader context, we note that, by now, several classes of consistency tests related to the viability of an RG fixed point for the asymptotic safety program have been put forward. These include, e.g., the stability of the eigenvalue spectrum of _{k} [

Arguably, the most intriguing result of our work is the spectral analysis of the stability matrix showing the distributions of its eigenvalues in the complex plane, c.f. the top-right diagrams of _{n} along their Lee-Yang type orbits in the complex plane could provide a novel tool for testing the convergence of the eigenvalue distribution of

As a by-product our analysis also computed the diagonal entries of the anomalous dimension matrix in geometric gauge [cf. Equation (30)]. It is instructive to compare this result to the value of the diagonal entries obtained in harmonic gauge [

This identifies two features which are robust under a change of gauge-fixing: in both cases, the values of γ_{nn} up to _{nn} are negative definite for all values ^{4} in the curvature expansion. This computation would be “complete” in the sense that it includes

As one of its most intricate features, the composite operator formalism employed in this work could act as a connector between Asymptotic Safety [

All datasets generated for this study are included in the article/supplementary material.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

FS thanks T. Budd, L. Lionni, and M. Reuter for inspiring discussions. Furthermore, we are grateful to W. Houthoff for participating in the earlier parts of the program.

^{2}phase-diagram of QEG and its spectral dimension

The calculation of

These traces can then be evaluated using the Seeley-deWitt expansion of the heat-kernel on the ^{d}:

Here

The expansion (42) is readily generalized to functions of the Laplacian. Introducing the

one has [

In order to write

Here

The arguments of the traces appearing in _{k} ≡ _{k}(

allows to convert the corresponding

Notably, the second set of identities suffices to derive the beta functions of the Einstein-Hilbert truncation while the evaluation of

For maximally symmetric backgrounds the background curvature

Throughout the work, we specify the (scalar) regulator

to the Litim regulator [

with Θ(

Structurally, the composite operator equation provides a map from the couplings contained in the Hessian _{k} in the Einstein-Hilbert truncation supplemented by a general gauge-fixing term. The key result is the position of the Reuter fixed point, Equation (27), which underlies the spectral analysis of section 4. Our analysis essentially follows [

The Einstein-Hilbert truncation approximates the effective average action Γ_{k}[

This ansatz contains two scale-dependent couplings, Newton's coupling _{k} and the cosmological constant Λ_{k}. In the present analysis, we work with a generic gauge-fixing term

where α and ρ are free, dimensionless parameters. The harmonic gauge used in Pagani [

Following the strategy employed in the gravitational sector, c.f. Equation (9), the fields ^{μ} are decomposed into their transverse and longitudinal parts

followed by a rescaling

The part of the ghost action quadratic in the fluctuation fields then becomes

We now proceed by constructing the non-zero entries of the Hessian _{k} to second order in the fluctuation fields, substituting the transverse traceless decomposition (9) and (56), and implementing the field redefinitions (11) and (57). Subsequently taking two functional variations with respect to the fluctuation fields then leads to the matrix elements listed in the middle block of

The final ingredient entering the right-hand side of the Wetterich equation is the regulator

dressing each Laplacian by a scalar regulator _{k}(Δ). The latter then provides a mass for fluctuation modes with momentum ^{2} ≲ ^{2}. In the nomenclature introduced in Codello et al. [

We now have all the ingredients to compute the beta functions resulting from the Wetterich equation projected onto the Einstein-Hilbert action. Adopting the geometric gauge ρ = 0, α → 0 used in the main section, all traces appearing in the equation simplify to the

where the anomalous dimension of Newton's coupling is parameterized by Reuter [

the explicit computation yields

and

Here the threshold functions _{T} and _{S} have been introduced in (23).

It is now straightforward to localize the Reuter fixed point by determining the roots of the beta functions (60) numerically. For the Litim regulator (51) this yields

Analyzing the stability properties of the RG flow in its vicinity, it is found that the fixed point constitutes a UV attractor, with the eigenvalues of the stability matrix given by

These results agree with the ones found in Benedetti et al. [

The expansions of _{k} in the fluctuation fields are readily computed using the xPert extension [_{i} multiplying the curvature terms in

^{1}For related ideas advocated in the context of two-dimensional gravity (see [

^{2}Recently, the formalism has been generalized to the computation of operator product expansions [

^{3}Most conservatively, one may expect that the composite operator formalism allows a qualitatively reliable determination of the stability properties of operators containing two additional spacetime-derivatives on top of the terms included in the propagators. This picture is readily confirmed by comparing the spectrum of

^{4}A second effect which could lead to a stabilization of the spectrum of

^{5}Our errors are purely statistical, giving the standard deviation based on the data set of eigenvalues. An estimate of the systematic errors is highly non-trivial and will not be attempted in this work.