Asymptotic Safety in quantum gravity

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Curator: Frank Saueressig

Asymptotic Safety (sometimes also referred to as nonperturbative renormalizability) is a concept in quantum field theory which aims at finding a consistent and predictive quantum theory of the gravitational field. Its key ingredient is a nontrivial fixed point of the theory’s renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences. Although originally proposed by Steven Weinberg to find a theory of quantum gravity the idea of a nontrivial fixed point providing a possible UV completion can be applied also to other field theories, in particular to perturbatively nonrenormalizable ones.

The essence of Asymptotic Safety is the observation that nontrivial renormalization group fixed points can be used to generalize the procedure of perturbative renormalization. In an asymptotically safe theory the couplings do not need to be small or tend to zero in the high energy limit but rather tend to finite values: they approach a nontrivial UV fixed point. The running of the coupling constants, i.e. their scale dependence described by the renormalization group (RG), is thus special in its UV limit in the sense that all their dimensionless combinations remain finite. This suffices to avoid unphysical divergences, e.g. in scattering amplitudes. The requirement of a UV fixed point restricts the form of the bare action and the values of the bare coupling constants, which become predictions of the Asymptotic Safety program rather than inputs.

As for gravity, the standard procedure of perturbative renormalization fails since Newton's constant, the relevant expansion parameter, has negative mass dimension rendering general relativity perturbatively nonrenormalizable. This has driven the search for nonperturbative frameworks describing quantum gravity, including Asymptotic Safety which — in contrast to other approaches — is characterized by its use of quantum field theory methods, without depending on perturbative techniques, however.



Gravity, at the classical level, is described by Einstein's field equations of general relativity, \(\textstyle R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = 8 \pi G \, T_{\mu \nu}\). These equations combine the spacetime geometry encoded in the metric \(g_{\mu\nu}\) with the matter content comprised in the energy momentum tensor \(T_{\mu\nu}\). The quantum nature of matter has been tested experimentally, for instance quantum electrodynamics is by now one of the most accurately confirmed theories in physics. For this reason quantization of gravity seems plausible, too. Unfortunately the quantization cannot be performed in the standard way: Already a simple power-counting consideration signals the perturbative nonrenormalizability since the mass dimension of Newton's constant is \(-2\). The problem occurs as follows. According to the traditional point of view renormalization is implemented via the introduction of counterterms that should cancel divergent expressions appearing in loop integrals. Applying this method to gravity, however, the counterterms required to eliminate all divergences proliferate to an infinite number. As this inevitably leads to an infinite number of free parameters to be measured in experiments, the program is unlikely to have predictive power beyond its use as a low energy effective theory.

It turns out that the first divergences in the quantization of general relativity which cannot be absorbed in counterterms consistently (i.e. without the necessity of introducing new parameters) appear already at one-loop level in the presence of matter fields ('t Hooft and Veltman, 1974). At two-loop level the problematic divergences arise even in pure gravity (Goroff and Sagnotti, 1986). In order to overcome this conceptual difficulty the development of nonperturbative techniques was required, providing various candidate theories of quantum gravity. For a long time the prevailing view has been that the very concept of quantum field theory — even though remarkably successful in the case of the other fundamental interactions — is doomed to failure for gravity. By way of contrast, the idea of Asymptotic Safety retains quantum fields as the theoretical arena and instead abandons only the traditional program of perturbative renormalization.

History of Asymptotic Safety

After having realized the perturbative nonrenormalizability of gravity, physicists tried to employ alternative techniques to cure the divergence problem, for instance resummation or extended theories with suitable matter fields and symmetries, all of which come with their own drawbacks. In 1976, Weinberg proposed a generalized version of the condition of renormalizability, based on a nontrivial fixed point of the underlying renormalization group (RG) flow for gravity (Weinberg, 1976). This was called Asymptotic Safety (Weinberg, 1979). The idea of a UV completion by means of a nontrivial fixed point was proposed even earlier in scalar field theory (Wilson and Kogut, 1974). The applicability to perturbatively nonrenormalizable theories was first demonstrated explicitly for a variant of the Gross-Neveu model (Gawędzki and Kupiainen, 1985).

As for gravity, the first studies concerning this new concept were performed in \(d=2+\epsilon\) spacetime dimensions in the late seventies. In exactly two dimensions there is a theory of pure gravity that is renormalizable according to the old point of view. (In order to render the Einstein-Hilbert action \(\textstyle {1 \over 16\pi G} \int \mathrm{d}^2 x \sqrt{g} \, R\) dimensionless, Newton's constant \(G\) must have mass dimension zero.) For small but finite \(\epsilon\) perturbation theory is still applicable, and one can expand the beta-function (\(\beta\)-function) describing the renormalization group running of Newton's constant as a power series in \(\epsilon\). Indeed, in this spirit it was possible to prove that it displays a nontrivial fixed point (see e.g. Weinberg, 1979).

However it was not clear how to do a continuation from \(d=2+\epsilon\) to \(d=4\) dimensions as the calculations relied on the smallness of the expansion parameter \(\epsilon\). The computational methods for a nonperturbative treatment were not at hand by this time. For this reason the idea of Asymptotic Safety in quantum gravity was put aside for some years. Only in the early 90s, aspects of \(2+\epsilon\) dimensional gravity have been revised in various works, but still not continuing the dimension to four.

As for calculations beyond perturbation theory, the situation improved with the advent of new functional renormalization group methods, in particular the so-called effective average action (a scale dependent version of the effective action). Introduced in 1993 by Wetterich for scalar theories (Wetterich, 1993), and by Reuter and Wetterich for general gauge theories on flat Euclidean space (Reuter and Wetterich, 1994), it is similar to a Wilsonian action (coarse grained free energy) (Wilson and Kogut, 1974) but differs at a deeper level (see e.g. Berges et. al., 2002). The cutoff scale dependence of this functional is governed by a functional flow equation which, in contrast to earlier attempts, can easily be applied in the presence of local gauge symmetries also.

In 1996, Reuter constructed a similar effective average action and the associated flow equation for the gravitational field (Reuter, 1998). It complies with the requirement of background independence, one of the fundamental tenets of quantum gravity. This work provides for the first time the possibility of nonperturbative computations in quantum gravity for arbitrary spacetime dimensions. It was shown that at least for the Einstein-Hilbert truncation, the simplest ansatz for the effective average action, a nontrivial fixed point is indeed present.

These results mark the starting point for many Asymptotic Safety related studies on quantum gravity that followed. Since it was not clear in the pioneer work by Reuter to what extend the findings depended on the truncation ansatz considered, the next obvious step consisted in enlarging the truncation. This process was initiated by Percacci and collaborators, starting with the inclusion of matter fields (Dou and Percacci, 1998). Up to the present many different works by a continuously growing community — including, e.g., \(f(R)\)- and Weyl tensor squared truncations — have confirmed independently that the Asymptotic Safety scenario is actually possible: The existence of a nontrivial fixed point was shown within each truncation studied so far. (For reviews on Asymptotic Safety with comprehensive lists of references see Further reading.) Although still lacking a final proof, there is mounting evidence that the Asymptotic Safety program can ultimately lead to a consistent and predictive quantum theory of gravity within the general framework of quantum field theory.

Asymptotic Safety: The main idea

Theory space

Figure 1: Trajectories of the renormalization group flow in theory space, parametrized by infinitely many coupling constants. By convention, the arrows of the vector field (and the one on the green trajectory) point from UV to IR scales. The set of actions which lie inside theory space and are pulled into the fixed point under the inverse RG flow (i.e., going in the direction opposite to the arrows) is referred to as UV critical surface. The Asymptotic Safety hypothesis is that a trajectory can only be realized in Nature if it is contained in the UV critical surface since only then it has a well-behaved high energy limit (orange, blue, and magenta trajectories, by way of example). Trajectories outside this surface escape theory space for \(k\rightarrow\infty\) since they develop unacceptable divergences in the UV, while going to lower scales they approach the UV critical surface. This situation is represented by the green trajectory which lies above the surface and runs away from it for increasing RG scale (opposite to the green arrow).

The Asymptotic Safety program adopts a modern Wilsonian viewpoint on quantum field theory. Here the basic input data to be fixed at the beginning are, firstly, the kinds of quantum fields carrying the theory's degrees of freedom and, secondly, the underlying symmetries. For any theory considered, these data determine the stage the renormalization group dynamics takes place on, the so-called theory space. It consists of all possible action functionals depending on the fields selected and respecting the prescribed symmetry principles. Each point in this theory space thus represents one possible action. Often one may think of the space as spanned by all suitable field monomials. In this sense any action in theory space is a linear combination of field monomials, where the corresponding coefficients are the coupling constants, \(\{g_\alpha\}\). (Here all couplings are assumed to be dimensionless. Couplings can always be made dimensionless by multiplication with a suitable power of the RG scale.)

Renormalization group flow

The renormalization group (RG) describes the change of a physical system due to smoothing or averaging out microscopic details when going to a lower resolution. This brings into play a notion of scale dependence for the action functionals of interest. Infinitesimal RG transformations map actions to nearby ones, thus giving rise to a vector field on theory space. The scale dependence of an action is encoded in a "running" of the coupling constants parametrizing this action, \(\{g_\alpha\} \equiv \{g_\alpha(k)\}\), with the RG scale \(k\). This gives rise to a trajectory in theory space (RG trajectory), describing the evolution of an action functional with respect to the scale. Which of all possible trajectories is realized in Nature has to be determined by measurements.

Taking the UV limit

The construction of a quantum field theory amounts to finding an RG trajectory which is infinitely extended in the sense that the action functional described by \(\{g_\alpha(k)\}\) is well-behaved for all values of the momentum scale parameter \(k\), including the infrared limit \(k \rightarrow 0\) and the ultraviolet (UV) limit \(k \rightarrow \infty\). Asymptotic Safety is a way of dealing with the latter limit. Its fundamental requirement is the existence of a fixed point of the RG flow. By definition this is a point \(\{g_\alpha^*\}\) in theory space where the running of all couplings stops, or, in other words, a zero of all \(\beta\)-functions: \(\beta_\gamma(\{g_\alpha^*\})=0\) for all \(\gamma\). In addition that fixed point must have at least one UV-attractive direction. This ensures that there are one or more RG trajectories which run into the fixed point for increasing scale. The set of all points in the theory space that are "pulled" into the UV fixed point by going to larger scales is referred to as UV critical surface. Thus the UV critical surface consists of all those trajectories which are safe from UV divergences in the sense that all couplings approach finite fixed point values as \(k \rightarrow \infty\), see Figure 1. The key hypothesis underlying Asymptotic Safety is that only trajectories running entirely within the UV critical surface of an appropriate fixed point can be infinitely extended and thus define a fundamental quantum field theory. It is obvious that such trajectories are well-behaved in the UV limit as the existence of a fixed point allows them to "stay at a point" for an infinitely long RG time.

With regard to the fixed point, UV-attractive directions are called relevant, UV-repulsive ones irrelevant, since the corresponding scaling fields increase and decrease, respectively, when the scale is lowered. Therefore, the dimensionality of the UV critical surface equals the number of relevant couplings. An asymptotically safe theory is thus the more predictive the smaller is the dimensionality of the corresponding UV critical surface.

For instance, if the UV critical surface has the finite dimension \(n\) it is sufficient to perform only \(n\) measurements in order to uniquely identify Nature's RG trajectory. Once the \(n\) relevant couplings are measured, the requirement of Asymptotic Safety fixes all other couplings since the latter have to be adjusted in such a way that the RG trajectory lies within the UV critical surface. In this spirit the theory is highly predictive as infinitely many parameters are fixed by a finite number of measurements.

In contrast to other approaches, a bare action which should be promoted to a quantum theory is not needed as an input here. It is the theory space and the RG flow equations that determine possible UV fixed points. Since such a fixed point, in turn, corresponds to a bare action, one can consider the bare action a prediction in the Asymptotic Safety program. This may be thought of as a systematic search strategy among theories that are already "quantum" which identifies the "islands" of physically acceptable theories in the "sea" of unacceptable ones plagued by short distance singularities.

Gaussian and non-Gaussian fixed points

A fixed point is called Gaussian if it corresponds to a free theory. Its critical exponents agree with the canonical mass dimensions of the corresponding operators which usually amounts to the trivial fixed point values \(g_\alpha^* = 0\) for all essential couplings \(g_\alpha\). Thus standard perturbation theory is applicable only in the vicinity of a Gaussian fixed point. In this regard Asymptotic Safety at the Gaussian fixed point is equivalent to perturbative renormalizability plus asymptotic freedom. Due to the arguments presented in the introductory sections, however, this possibility is ruled out for gravity.

In contrast, a nontrivial fixed point, that is, a fixed point whose critical exponents differ from the canonical ones, is referred to as non-Gaussian. Usually this requires \(g_\alpha^* \neq 0\) for at least one essential \(g_\alpha\). It is such a non-Gaussian fixed point that provides a possible scenario for quantum gravity. As yet, studies on this subject thus mainly focused on establishing its existence.

Quantum Einstein Gravity (QEG)

Quantum Einstein Gravity (QEG) is the generic name for any quantum field theory of gravity that (regardless of its bare action) takes the spacetime metric as the dynamical field variable and whose symmetry is given by diffeomorphism invariance. This fixes the theory space and the RG flow of the effective average action defined over it, but it does not single out a priori any specific action functional. However, the flow equation determines a vector field on that theory space which can be investigated. If it displays a non-Gaussian fixed point by means of which the UV limit can be taken in the "asymptotically safe" way, this point acquires the status of the bare action.

Implementation via the effective average action

The primary tool for investigating the gravitational RG flow with respect to the energy scale \(k\) at the nonperturbative level is the effective average action \(\Gamma_k\) for gravity (Reuter, 1998). It is a scale dependent version of the effective action where in the underlying functional integral field modes with covariant momenta below \(k\) are suppressed while only the remaining high momentum modes are integrated out.

Dynamical vs. background fields

For later use it is assumed here that the fundamental degrees of freedom are carried by a dynamical metric \(\gamma_{\mu\nu}\), although the arguments presented in the following hold more generally for arbitrary theory spaces with other choices of the dynamical field. The requirement of background independence can be established by means of the background field method, where the dynamical field is split into a fixed but arbitrary background \(\bar{g}_{\mu\nu}\) and a fluctuation field \(h_{\mu\nu}\): \[ \gamma_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu} . \] The classical action is assumed to be invariant under general coordinate transformations \(\delta\gamma_{\mu\nu}=\mathcal{L}_\xi \gamma_{\mu\nu}\), where vector fields \(\xi\) generate diffeomorphisms on the manifold considered, the Lie derivative \(\mathcal{L}_\xi\) appearing in their infinitesimal representation. Due to the fact that the description depends on two fields now, there is some freedom in splitting the gauge transformation: both \(\delta\bar{g}_{\mu\nu}\) and \(\delta h_{\mu\nu}\) can be transformed independently as long as their sum equals \(\delta\gamma_{\mu\nu}\). Two possible choices are the true gauge transformations \(\big\{\delta\bar{g}_{\mu\nu}=0\), \(\delta h_{\mu\nu}=\mathcal{L}_\xi(\bar{g}_{\mu\nu} + h_{\mu\nu})\big\}\) and the background gauge transformations \(\big\{\delta \bar{g}_{\mu\nu}=\mathcal{L}_\xi \bar{g}_{\mu\nu}\), \(\delta h_{\mu\nu} = \mathcal{L}_\xi h_{\mu\nu}\big\}\). The former are gauge fixed in the functional integral defining the effective average action which, however, is left invariant by the latter and thus satisfies background Ward identities. The true gauge transformations are accounted for by generalized BRST Ward identities. They reduce to the usual ones at vanishing RG scale, \(k=0\), but get modified for higher scales due to the mode suppression term introduced in the next section (Reuter, 1998).

The gravitational effective average action

Figure 2: Modes with covariant momenta above the cutoff scale are integrated out, those below are suppressed. Lowering the scale means integrating out the additional modes in the corresponding interval.

The idea is to start with a (formal) functional integral, however, furnished with an implicit mode suppression cutoff that suppresses infrared modes with covariant momenta below a cutoff scale \(k\) while the others are integrated out. Lowering the scale from \(k\) to \(k'<k\) means integrating out modes in the interval \([k',k]\), see Figure 2.

Technically the cutoff is implemented via an additional term in the action, the cutoff action \(\Delta S_k\) so that the integrand of the functional integral contains an extra factor \(e^{-\Delta S_k}\) suppressing those modes for which \(\Delta S_k\) is large. It is given by \[ \Delta S_k \propto \int \text{d}^d x \sqrt{\bar{g}} \, h_{\mu\nu}(x) \mathcal{R}_k[\bar{g}]^{\mu\nu\rho\sigma} h_{\rho\sigma}(x) \] plus a similar term for the Faddeev-Popov ghost fields \(C^\mu\) and \(\bar{C}_\mu\) resulting from the gauge fixed true gauge transformations. The cutoff operator \(\mathcal{R}_k\) thus acts like a generalized mass term and has the structure \[ \mathcal{R}_k[\bar{g}]^{\mu\nu\rho\sigma} = Z_k^{\mu\nu\rho\sigma} k^2 R^{(0)}\left(-\frac{\bar{D}^2}{k^2}\right) , \] where \(Z_k^{\mu\nu\rho\sigma}\) is a tensor constructed from the background metric \(\bar{g}_{\mu\nu}\), \(\bar{D}^2 = \bar{g}^{\mu\nu} \bar{D}_\mu \bar{D}_\nu\) is the covariant Laplacian with respect to \(\bar{g}_{\mu\nu}\), and \(R^{(0)}(z)\) is a dimensionless cutoff shape function. It is this shape function that dictates the details of the mode suppression, interpolating between \(R^{(0)}(z=0) = 1\) (in order to suppress modes with low momenta) and \(R^{(0)}(z \rightarrow \infty) = 0\) (such that high momentum modes are untouched by the cutoff and thus integrated out completely). Typical shape functions are shown in Figure 3, indicating the separation between high (\(p^2 > k^2\)) and low (\(p^2 < k^2\)) momentum modes.

Figure 3: Typical cutoff shape functions that separate modes with momenta below and above \(k\). The hatched area signals the range of a substantial mode suppression.

Based on these ingredients one constructs a \(k\)-dependent generating functional by means of the functional integral \[ \tag{1} \begin{split} Z_{\color{blue}{k}}[t^{\mu\nu},\sigma^\mu,\bar{\sigma}_\mu;\bar{g}_{\mu\nu}] = \int & \mathcal{D}h_{\mu\nu} \mathcal{D}C^\mu \mathcal{D}\bar{C}_\mu \exp\Big\{ -S[\bar{g}+h] -S_{\text{gf}}[h;\bar{g}] \\ & -S_{\text{gh}}[h,C,\bar{C};\bar{g}]-{\color{blue}{\Delta S_k[h,C,\bar{C};\bar{g}]}} + S_{\text{source}} \Big\} . \end{split} \] Here \(S[\bar{g}+h] = S[\gamma]\) is the bare action, \(S_\text{gf}\) denotes the gauge fixing action, and \(S_\text{gh}\) is the corresponding ghost action. Furthermore, the fields \(h_{\mu\nu}\), \(C^\mu\) and \(\bar{C}_\mu\) are coupled to the sources \(t^{\mu\nu}\), \(\bar{\sigma}_\mu\) and \(\sigma^\mu\), respectively, by virtue of the source action \[ S_\text{source} = \int \text{d}^d x \sqrt{\bar{g}} \, \Big\{ t^{\mu\nu}h_{\mu\nu} + \bar{\sigma}_\mu C^\mu + \sigma^\mu \bar{C}_\mu \Big\} . \] Taking the logarithm of \(Z_k\) leads to a scale dependent generating functional for the connected Green's functions, \[ W_k[t^{\mu\nu},\sigma^\mu,\bar{\sigma}_\mu;\bar{g}_{\mu\nu}] = \ln Z_k[t^{\mu\nu},\sigma^\mu,\bar{\sigma}_\mu;\bar{g}_{\mu\nu}] . \] Functional derivatives with respect to the sources give rise to the \(k\)-dependent expectation values \[ \bar{h}_{\mu\nu} \equiv \langle h_{\mu\nu} \rangle = \frac{1}{\sqrt{\bar{g}}} \frac{\delta W_k}{\delta t^{\mu\nu}}, \qquad\; \xi^\mu \equiv \langle C^\mu \rangle = \frac{1}{\sqrt{\bar{g}}} \frac{\delta W_k}{\delta \bar{\sigma}_\mu}, \qquad\; \bar{\xi}_\mu \equiv \langle \bar{C}_\mu \rangle = \frac{1}{\sqrt{\bar{g}}} \frac{\delta W_k}{\delta \sigma^\mu} \, . \] These relations are formally solved for the sources in order to calculate the Legendre transform of \(W_k\). Subtracting the cutoff action from this Legendre transform yields the effective average action \[ \Gamma_k[\bar{h},\xi,\bar{\xi};\bar{g}] \equiv \int \text{d}^d x \sqrt{\bar{g}} \Big \{ t^{\mu\nu}\bar{h}_{\mu\nu} + \bar{\sigma}_\mu\xi^\mu + \sigma^\mu\bar{\xi}_\mu \Big\} - W_k[t,\sigma,\bar{\sigma};\bar{g}] - \Delta S_k[\bar{h},\xi,\bar{\xi};\bar{g}] , \] or, equivalently, by introducing the metric \(g_{\mu\nu} \equiv \langle\gamma_{\mu\nu}\rangle \equiv \bar{g}_{\mu\nu} + \bar{h}_{\mu\nu}\), \[ \tag{2} \Gamma_k[g,\bar{g},\xi,\bar{\xi}] \equiv \Gamma_k[\bar{h} = g-\bar{g},\xi,\bar{\xi};\bar{g}] . \] To summarize the preceding construction, one may consider \(\Gamma_k\) a modification of the standard effective action \(\Gamma\): \[ \Gamma_k = \Gamma\Big|_{S\mapsto S+\Delta S_k} - \Delta S_k . \]

Thus the effective average action \(\Gamma_k\) is a scale dependent version of the effective action \(\Gamma\): All quantum effects induced by high momentum modes above the scale \(k\) are incorporated in \(\Gamma_k\) since these modes have already been integrated out. Only the low momentum modes below \(k\) are not yet fully taken into account. At vanishing cutoff scale, \(k = 0\), all modes are integrated out completely, and thus the effective average action equals the effective action: \[ \Gamma_{k=0} = \Gamma . \] On the other hand, \(\Gamma_k\) in the limit \(k \rightarrow \infty\) is basically given by the bare (microscopic) action. The precise dependence of \(\Gamma_k\) on the scale \(k\) is governed by the functional flow equation presented in the following section.

Exact functional renormalization group equation

For a given theory space, let \(\Phi\) and \(\bar{\Phi}\) denote the set of dynamical and background fields, respectively. Differentiating the above definition of \(\Gamma_k\) with respect to \(k\) one can show that the following functional RG equation (FRGE) is satisfied: \[ \tag{3} k \partial_k \Gamma_k\big[\Phi, \bar{\Phi}\big] = \frac{1}{2}\, \mbox{STr}\Big[\big(\Gamma_k^{(2)}[\Phi, \bar{\Phi}\big] + \mathcal{R}_k[\bar{\Phi}]\big)^{-1} k \partial_k \mathcal{R}_k[\bar{\Phi}] \Big], \] (Wetterich, 1993; Reuter and Wetterich, 1994, Reuter, 1998). Here \(\Gamma_k^{(2)}\) is the second functional derivative of \(\Gamma_k\) with respect to the quantum fields \(\Phi\) at fixed \(\bar{\Phi}\). By the above definition the mode suppression operator \(\mathcal{R}_k[\bar{\Phi}]\) provides a \(k\)-dependent mass-term for fluctuations with covariant momenta \(p^2 \ll k^2\) and vanishes for \(p^2 \gg k^2\). Its appearance in the numerator and denominator renders the supertrace \( (\mbox{STr})\) both infrared and UV finite, peaking at momenta \(p^2 \approx k^2\). The FRGE is an exact equation without any perturbative approximations. Given an initial condition it determines \(\Gamma_k\) for all scales uniquely. Due to the properties of the effective average action, the solutions of the FRGE interpolate between the bare action at \(k \rightarrow \infty\) and the effective action \(\Gamma[\Phi] = \Gamma_{k=0}\big[\Phi, \bar{\Phi}=\Phi\big]\) at \(k \rightarrow 0\). They can be visualized as trajectories in the underlying theory space. Any exact solution of (3) automatically satisfies the generalized BRST Ward identities. Note that the FRGE (3) has become independent of the bare action \(S\). One can search for solutions of the FRGE rather than evaluate the path integral (1) directly. In a sense the effective average action can be found independently of its path integral construction and, in particular, independently of its bare action. For an asymptotically safe theory, the pertinent bare action is then determined by the fixed point functional \(\Gamma_* = \Gamma_{k\rightarrow\infty}\).

Truncations of the theory space

Let us assume there is a set of basis functionals \(\{P_\alpha[\,\cdot\,]\}\) spanning the theory space under consideration so that any action functional, i.e. any point of this theory space, can be written as a linear combination of the \(P_\alpha\)'s. Then solutions \(\Gamma_k\) of the FRGE have expansions of the form

\[ \Gamma_k[\Phi,\bar{\Phi}] = \sum\limits_{\alpha=1}^{\infty} g_\alpha(k) P_\alpha[\Phi,\bar{\Phi}] . \]

Inserting this expansion into the FRGE (3) and expanding the trace on its right-hand side in order to extract the \(\beta\)-functions, one obtains the exact RG equation in component form: \(k \partial_k g_\alpha(k) = \beta_\alpha(g_1,g_2,\cdots)\). Together with the corresponding initial conditions these equations fix the evolution of the running couplings \(g_\alpha(k)\), and thus determine \(\Gamma_k\) completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since there are infinitely many couplings, and the \(\beta\)-functions can depend on all of them. This makes it very hard to solve the system in general.

A possible way out is to restrict the analysis on a finite-dimensional subspace as an approximation of the full theory space. In other words, such a truncation of the theory space sets all but a finite number of couplings to zero, considering only the reduced basis \(\{P_\alpha[\,\cdot\,]\}\) with \(\alpha=1,\cdots,N\). This amounts to the ansatz

\[ \Gamma_k[\Phi,\bar{\Phi}] = \sum\limits_{\alpha=1}^N g_\alpha(k) P_\alpha[\Phi,\bar{\Phi}] , \]

leading to a system of finitely many coupled differential equations, \(k\partial_k g_\alpha(k) = \beta_\alpha(g_1,\cdots,g_N)\), which can now be solved employing analytical or numerical techniques.

Clearly a truncation should be chosen such that it incorporates as many features of the exact flow as possible. Violations of the generalized BRST Ward identities, for instance, should be negligible for an appropriate truncation. Although it is an approximation, the truncated flow still exhibits the nonperturbative character of the FRGE, and the \(\beta\)-functions can contain contributions from all powers of the couplings.

In the gravitational case for a truncation with \(k\)-independent ghost sector the supertrace in (3) decomposes into two parts, involving the reduced effective average action \(\breve{\Gamma}_k \equiv \Gamma_k - S_{\text{gh}}\) on the one hand and a ghost part on the other hand, such that the FRGE amounts to \[ \tag{4} k \partial_k \Gamma_k\big[g, \bar{g}\big] = \frac{1}{2}\,\mbox{Tr}\Big[\big(\breve{\Gamma}_k^{(2)}[g, \bar{g}\big] + \mathcal{R}_k^\text{grav}[\bar{g}]\big)^{-1} k \partial_k \mathcal{R}_k^\text{grav}[\bar{g}] \Big] - \mbox{Tr}\Big[\big(-\mathcal{M}[g, \bar{g}\big] + \mathcal{R}_k^\text{gh}[\bar{g}]\big)^{-1} k \partial_k \mathcal{R}_k^\text{gh}[\bar{g}] \Big] , \] where the first term on the right-hand side is the graviton contribution, and the second term, including the Faddeev-Popov operator \(\mathcal{M}\), stems from the ghosts.

Evidence for Asymptotic Safety from truncated flow equations

The Einstein-Hilbert truncation

As described in the previous section, the FRGE lends itself to a systematic construction of nonperturbative approximations to the gravitational \(\beta\)-functions by projecting the exact RG flow onto subspaces spanned by a suitable ansatz for \(\Gamma_k \). In its simplest form, such an ansatz is given by the Einstein-Hilbert action where Newton's constant \(G_k\) and the cosmological constant \(\Lambda_k\) depend on the RG scale \(k\). Let \(g_{\mu\nu}\) and \(\bar{g}_{\mu\nu}\) denote the dynamical and the background metric, respectively. Then \(\Gamma_k \) reads, for arbitrary spacetime dimension \(d\),

\[ \tag{5} \Gamma_k[g,\bar{g},\xi,\bar{\xi}] = \frac{1}{16\pi G_k} \int\text{d}^d x\, \sqrt{g}\, \big( -R(g) + 2\Lambda_k \big) + \Gamma_k^\text{gf}[g,\bar{g}] + \Gamma_k^\text{gh}[g,\bar{g},\xi,\bar{\xi}] . \]

Here \(R(g)\) is the scalar curvature constructed from the metric \(g_{\mu\nu}\). Furthermore, \(\Gamma_k^\text{gf}\) denotes the gauge fixing action, and \(\Gamma_k^\text{gh}\) the ghost action with the ghost fields \(\xi\) and \(\bar{\xi}\). Note that \(\bar{g}_{\mu\nu}\) separately appears only in the gauge fixing and ghost sector.

In order to determine the \(k\)-dependence of \(\Gamma_k\), the truncation ansatz (5) is to be inserted into the FRGE (4). The corresponding \(\beta\)-functions, describing the evolution of the dimensionless Newton constant \(g_k=k^{d-2} G_k\) and the dimensionless cosmological constant \(\lambda_k=k^{-2}\Lambda_k\), have been first derived by Reuter (1998) for any value of the spacetime dimensionality \(d\). They read

\[ \tag{6} \begin{split} k\partial_k g_k &= \beta_g(g_k,\lambda_k) , \\ k\partial_k \lambda_k &= \beta_\lambda(g_k,\lambda_k) , \end{split} \] with \[ \tag{7} \beta_g(g,\lambda) = (d-2+\eta_N)g , \] and \[ \tag{8} \beta_\lambda(g,\lambda) = -(2-\eta_N)\lambda + \frac{1}{2} g (4\pi)^{1-d/2} \Big[ 2d(d+1)\Phi_{d/2}^1(-2\lambda) - 8d\Phi_{d/2}^1(0) - d(d+1)\eta_N \widetilde{\Phi}_{d/2}^1(-2\lambda) \Big] . \] Here \(\eta_N\) denotes the anomalous dimension of Newton's constant, \[ \eta_N(g,\lambda) = \frac{g B_1(\lambda)}{1-g B_2(\lambda)} , \] where \(B_1\) and \(B_2\) are functions of the cosmological constant, \[ \begin{split} B_1(\lambda) &= \frac{1}{3}(4\pi)^{1-d/2} \Big[ d(d+1) \Phi^1_{d/2-1}(-2\lambda) - 6d(d-1)\Phi^2_{d/2}(-2\lambda) - 4d\Phi^1_{d/2-1}(0)-24\Phi^2_{d/2}(0) \Big], \\ B_2(\lambda) &= -\frac{1}{6}(4\pi)^{1-d/2} \Big[ d(d+1) \widetilde\Phi^1_{d/2-1}(-2\lambda) - 6d(d-1)\widetilde\Phi^2_{d/2}(-2\lambda) \Big] , \end{split} \] and the so-called threshold functions \(\Phi\) and \(\widetilde{\Phi}\), incorporating a dependence on the cutoff shape function \(R^{(0)}\), are given by \[ \begin{split} \Phi^p_n(x) &= \frac{1}{\Gamma(n)}\int_0^\infty dz\,z^{n-1}\frac{R^{(0)}(z) - z R^{(0)\,\prime}(z)}{[z+R^{(0)}(z)+x]^p} \, , \\ \widetilde\Phi^p_n(x) &= \frac{1}{\Gamma(n)}\int_0^\infty dz\,z^{n-1}\frac{R^{(0)}(z)}{[z+R^{(0)}(z)+x]^p} \, . \end{split} \]

Figure 4: QEG flow diagram for the Einstein-Hilbert truncation. Arrows point from UV to IR scales. Dark background color indicates a region of fast flow, in regions of light background the flow is slow or even zero. The latter case includes a vicinity of the Gaussian fixed point in the origin, and the NGFP in the center of the spiralling arrows, respectively. The cross-over trajectory tangent to the green arrows connects the non-Gaussian to the Gaussian fixed point and plays the role of a separatrix. (Phase portrait first obtained by Reuter and Saueressig, 2002.)

The RG flow the \(\beta\)-functions (7) and (8) give rise to can be analyzed for arbitrary dimensionality \(d\) and for any appropriate choice of the cutoff shape. In particular, in \(d=4\) dimensions and for a sharp cutoff (i.e., a constant function with a step at \(p^2=k^2\) such that it vanishes for all \(p^2>k^2\)) the \(\beta\)-functions (7) and (8) assume the simple form (Reuter and Saueressig, 2002) \[ \tag{9} \begin{split} \beta_g (g,\lambda) &= ( 2+\eta_N) g , \\ \beta_\lambda(g,\lambda) &= -(2-\eta_N)\lambda -\frac{g}{\pi}\bigg [ 5 \ln(1-2\lambda) - 2\zeta(3) + \frac{5}{4} \eta_N \bigg ] , \end{split} \] with the anomalous dimension \[ \tag{10} \eta_N(g,\lambda) = - \frac{2 g}{6\pi +5 g} \bigg [ \frac{18}{1-2\lambda} + 5 \ln(1-2\lambda) - \zeta(2) + 6 \bigg ] . \] Here \(\zeta\) denotes the Riemann zeta function. The system (9), (10) results in the flow diagram shown in Figure 4.

The most important result is the existence of a non-Gaussian fixed point suitable for Asymptotic Safety. It is UV-attractive both in \(g\)- and in \(\lambda\)-direction. This fixed point is related to the one found in the seventies in \(d=2 + \epsilon\) dimensions by perturbative methods in the sense that it is recovered in the nonperturbative approach presented here by inserting \(d=2 + \epsilon\) into the \(\beta\)-functions (7), (8) and expanding in powers of \(\epsilon\) (Reuter, 1998). Since the \(\beta\)-functions were shown to exist and were explicitly computed for any real, i.e., not necessarily integer value of \(d\), no analytic continuation is involved here. The fixed point in \(d=4\) dimensions, too, is a direct result of (7) and (8), and, in contrast to the earlier attempts, no extrapolation in \(\epsilon\) is required.

Extended truncations

Subsequently, the existence of the fixed point found within the Einstein-Hilbert truncation has been confirmed in subspaces of successively increasing complexity. The next step in this development was the inclusion of an \(R^2\)-term in the truncation ansatz (Lauscher and Reuter, 2002). This has been extended further by taking into account polynomials of the scalar curvature \(R\) (so-called \(f(R)\)-truncations) (Codello et. al., 2009), and the square of the Weyl curvature tensor (Benedetti et. al., 2009). The contact to perturbation theory has been established by Niedermaier (2009). Moreover, the impact of various kinds of matter fields has been investigated (Dou and Percacci, 1998). While in all those truncations only the gauge fixing and ghost terms contained \(\bar{g}_{\mu\nu}\) separately, the considerably more general class of "bimetric truncations" allows for an arbitrary \(\bar{g}_{\mu\nu}\)-dependence (Manrique and Reuter, 2010). Also computations based on a field reparametrization invariant effective average action seem to recover the crucial fixed point (Donkin and Pawlowski, 2012). In combination these results constitute strong evidence that gravity in four dimensions is a nonperturbatively renormalizable quantum field theory, indeed with a UV critical surface of reduced dimensionality, coordinatized by only a few relevant couplings. (For reviews see Further reading.)

The microscopic structure of spacetime

Asymptotic Safety related investigations indicate that the effective spacetimes of QEG have fractal-like properties on microscopic scales. It is possible to determine, for instance, their spectral dimension and argue that they undergo a dimensional reduction from 4 dimensions at macroscopic distances to 2 dimensions microscopically. The same dimensional reduction is seen in the effective dimensionality \(d_\text{eff} = d+\eta_N\) which controls the distance dependence of the graviton propagator on flat space: \(\mathcal{G}(x;y) \propto |x-y|^{d_\text{eff}\,-2}\). Directly at the fixed point, \(\eta_N = 2-d\), whence, for all \(d\), \(d_\text{eff} = 2\), and as a result the short distance singularity becomes softened to a logarithm, \(\mathcal{G}(x;y) \propto \ln (x-y)^2\), which is reminiscent of a classical boson propagator on a 2-dimensional spacetime. In momentum space the logarithm amounts to a large momentum behavior \(\propto 1/p^d\) (Lauscher and Reuter, 2001; Lauscher and Reuter, 2005). This benign short distance behavior can be transferred from the gravitational into the matter sector where then the propagators of all free matter fields are softened to logarithmic ones (Niedermaier and Reuter, 2006). Computing the scale dependence of the spectral dimension it might be possible to draw the connection to other approaches to quantum gravity, e.g. to causal dynamical triangulations, and compare the results (Reuter and Saueressig, 2012).


Summarizing one can say that while the Asymptotic Safety program makes essential use of nonperturbative concepts and tools, it does not rely on any unproven physical assumptions, such as special symmetries, extra fields, higher dimensions, or extended objects like strings and branes. Also, it requires no unification of gravity with the other fundamental forces. The analyses of the RG flow on the theory space of QEG which are available by now are all consistent with the theory's conjectured nonperturbative renormalizability, or, Asymptotic Safety.

See also


Further reading

External links

  • The Asymptotic Safety FAQs — A collection of questions and answers about Asymptotic Safety and a comprehensive list of references.

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