In [181] the coupling flow (4.45) implied by the Einstein–Hilbert truncation has been analyzed in detail, using both analytical and numerical methods. In particular all trajectories of this system of equations have been classified, and examples have been computed numerically. The most important classes of trajectories in the phase portrait on the – plane are shown in Figure 2. The trajectories were obtained by numerically solving the system (4.50) for a sharp cutoff; using a smooth one all qualitative features remain unchanged.

The RG flow is dominated by two fixed points : a Gaussian fixed point (GFP) at , and a nonGaussian fixed point (NGFP) with and . There are three classes of trajectories emanating from the NGFP: Trajectories of Type Ia and IIIa run towards negative and positive cosmological constants, respectively, and the single trajectory of Type IIa (“separatrix”) hits the GFP for . The short–distance properties of Quantum Einstein Gravity are governed by the NGFP; for , in Figure 2 all RG trajectories on the half–plane run into this fixed point – its unstable manifold is twodimensional. Note that near the NGFP the dimensionful Newton constant vanishes for according to . The conjectured nonperturbative renormalizability of Quantum Einstein Gravity is due to this NGFP: If it was present in the untruncated RG flow it could be used to construct a microscopic quantum theory of gravity by taking the limit of infinite UV cutoff along one of the trajectories running into the NGFP, implying that the theory does not develop uncontrolled singularities at high energies [227].
The trajectories of Type IIIa cannot be continued all the way down to the infrared () but rather terminate at a finite scale . (This feature is not resolved in Figure 2.) At this scale the functions diverge. As a result, the flow equations cannot be integrated beyond this point. The value of depends on the trajectory considered. The trajectory terminates when the dimensionless cosmological constant reaches the value . This is due to the fact that the functions and – for any admissible choice of – have a singularity at , and because in all terms of the –functions. In Equations (4.50) the divergence at is seen explicitly. The phenomenon of terminating RG trajectories is familiar from simpler theories, such as Yang–Mills theories. It usually indicates that the truncation becomes insufficient at small .
Here we collect the evidence for asymptotic safety obtained from the Einstein–Hilbert and truncations, Equation (4.34) and Equation (4.35), respectively, of the flow equations in Section 4.2 [133, 131].
The details of the flow pattern depend on a number of adhoc choices. It is crucial that the properties of the flow which point towards the asymptotic safety scenario are robust upon alterations of these choices. This robustness of the qualitative features will be discussed in more detail below. Here let us only recapitulate the three main ingredients of the (truncated) flow equations that can be varied: The shape functions in Equation (4.31) can be varied, the gauge parameter in Equation (4.33) can be varied, and the vector and transversal parts in the traceless tensor modes can be treated differently (type and cutoffs).
Picking a specific value for the gauge parameter has a somewhat different status than the other two choices. The truncations are actually oneparameter families of truncations labelled by ; in a more refined treatment would be a running parameter itself determined by the FRGE.
In practice the shape function was varied within the class (4.51) of exponential cutoffs and a similar oneparameter class of cutoffs with compact support [133, 131]. Changing the cutoff function at fixed may be thought of as analogous to a change of scheme in perturbation theory.
The main qualitative properties of the coupling flow can be summarized as follows:
We proceeded to discuss various aspects of the evidence for asymptotic safety in more detail, namely the structure of unstable manifold and the robustness of the qualitative features of the flow. Finally we offer some comments on the full FRGE dynamics.
This can be studied in the vicinity of the fixed point by a standard linearized stability analysis. We summarize the results for the nonGaussian fixed point, first in the Einstein–Hilbert truncation and then in the more general truncation. To set the notation recall that for a flow equation of the form the linearized flow near the fixed point is governed by the stability matrix with components ,
The general solution to this equation reads where the ’s are the righteigenvectors of with eigenvalues , i.e. . The ’s are constants of integration, is a reference scale, and decreasing is the direction of coarse graining. Since is not symmetric in general the ’s are not guaranteed to be real. In principle could also be degenerate in which case the linearized analysis would only put some constraints on the structure of the unstable manifold in the vicinity of the fixed point. As a matter of fact is nondegenerate for both the Einstein–Hilbert truncation and for the truncation. In such a situation, the eigendirections with are irrelevant; they die out upon coarse graining and span the tangent space of the fixed point’s stable manifold. The remaining eigendirections with are relevant perturbations which span the tangent space of the fixed point’s unstable manifold. The eigenvalues play a role similar to the “critical exponents” in the theory of critical phenomena. Guided by this analogy one expects them to be rather insensitive to changes in the cutoff action .As explained in Section 2.1 it is often convenient to set (which is to be read as in the presence of an ultraviolet cutoff ) and ask “where a coarse graining trajectory comes from” by formally sending to (while the coarse graining flow is in the direction of increasing ). The tangent space to the unstable manifold has its maximal dimension if all the essential couplings taken into account hit the fixed point as is sent to : The fixed point is ultraviolet stable in the direction opposite to the coarse graining. This is the case for both the Einstein–Hilbert truncation and the truncation, as we shall describe now in more detail.
Linearizing the flow equation (4.45) according to Equation (4.54) we obtain a pair of complex conjugate eigenvalues with negative real part and imaginary parts . In terms of the general solution to the linearized flow equations reads
with being an arbitrary complex number and the righteigenvector of with eigenvalue . Equation (4.54) implies that, due to the positivity of , all trajectories hit the fixed point as is sent to . The nonvanishing imaginary part has no impact on the stability. However, it influences the shape of the trajectories which spiral into the fixed point for . In summary, for any modecutoff employed the nonGaussian fixed point is found to be ultraviolet attractive in both directions of the plane.Solving the full, nonlinear flow equations numerically [181] shows that the asymptotic scaling region where the linearization (4.56) is valid extends from down to about with the Planck mass defined as . Here marks the lower boundary of the asymptotic scaling region. We set so that the asymptotic scaling regime extends from about to .
The nonGaussian fixed point of the truncation likewise proves to be ultraviolet attractive in any of the three directions of the tangent space for all cutoffs used. The linearized flow in its vicinity is always governed by a pair of complex conjugate eigenvalues with , and a real negative one . The linearized solution may be expressed as
with arbitrary complex , arbitrary real . Here and are the righteigenvectors of the stability matrix with eigenvalues and , respectively. Clearly the conditions for ultraviolet stability are and . They are indeed satisfied for all cutoffs. For the exponential shape function with , for instance, we find , , , and , , . (The vectors are normalized such that .) The trajectories (4.57) comprise three independent normal modes with amplitudes proportional to , , and , respectively. The first two are of the spiral type again, the third one is a straight line.For any cutoff, the numerical results have several quite remarkable properties. They all indicate that, close to the nonGaussian fixed point, the flow is rather well approximated by the Einstein–Hilbert truncation:
Due to the large value of , the new scaling field is ‘very relevant’. However, when we start at the fixed point and raise it is only at the low energy(!) scale that reaches unity, and only then, i.e. far away from the fixed point, the new scaling field starts growing rapidly.
Summarizing the three points above we can say that very close to the fixed point the flow seems to be essentially twodimensional, and that this twodimensional flow is well approximated by the coupling flow of the Einstein–Hilbert truncation. In Figure 3 we show a typical trajectory which has all three normal modes excited with equal strength , . All its way down from to about it is confined to a very thin box surrounding the plane.

As explained before the details of the coupling flow produced by the various truncations of Equation (4.19) depend on the choice of the cutoff action (, type A vs. B) and the gauge parameter . Remarkably the qualitative properties of the flow, in particular those features pointing towards the asymptotic safety scenario are unchanged upon alterations of the computational scheme. Here we discuss these robustness properties in more detail. The degree of insensitivity of quantities expected to be “universal” can serve as a measure for the reliability of a truncation.
We begin with the very existence of a nonGaussian fixed point. Importantly, both for type A and type B cutoffs the nonGaussian fixed point is found to exists for all shape functions . This generalizes earlier results in [205]. Indeed, it seems impossible to find an admissible modecutoff which destroys the fixed point in . This is nontrivial since in higher dimensions the fixed point exists for some but does not exist for other modecutoffs [181] (see however [79]).
Within the Einstein–Hilbert truncation also a RG formalism different from (and in fact much simpler than) that of the average action was used [39]. The fixed point was found to exist already in a simple RG improved 1loop calculation with a proper time cutoff.
We take this as an indication that the fixed point seen in the Einstein–Hilbert [204, 133, 136, 39] and the truncations [131] is the projection of a genuine fixed point and not just an artifact of an insufficient truncation.
Support for this interpretation comes from considering the product of the fixed point coordinates. Recall from Section 2.3.2 that the product is a dimensionless essential coupling invariant under constant rescalings of the metric [116]. One would expect that this combination is also more robust with respect to scheme changes.
In Figure 4 we show the fixed point coordinates for the family of shape functions (4.51) and the type B cutoff. For every shape parameter , the values of and are almost the same as those obtained with the Einstein–Hilbert truncation. Despite the rather strong scheme dependence of and separately, their product has almost no visible dependence for not too small values of ! For , for instance, one obtains from the Einstein–Hilbert truncation and from the generalized truncation. One can also see that the coupling at the fixed point is uniformly small throughout the family of exponential shape functions (4.51).
A similar situation is found upon variation of the gauge parameter . Within the Einstein–Hilbert truncation the analysis has been performed in ref. [133] for an arbitary constant gauge parameter , including the ‘physical’ value . For example one finds
The differences between the ‘physical’ (fixed point) value of the gauge parameter, , and the technically more convenient are at the level of about 10 to 20 percent. In view of this the much more involved analysis in the truncation has been performed in the simpler gauge only [131]. The product with is then found to differ slightly from the corresponding value in the Einstein–Hilbert truncation, however the deviation is of the same size as the difference between the and the results of the Einstein–Hilbert truncation. Taken together the analysis suggests the universal value .Next we consider the (in)dependence of the “critical exponents” , in Equation (4.56, 4.57). Within the Einstein–Hilbert truncation the eigenvalues are found to be reasonably constant within about a factor of 2. For and , for instance, they assume values in the ranges , and , , respectively. The corresponding results for the truncation are shown in Figure 5. It presents the dependence of the critical exponents, using the family of shape functions (4.51). For the cutoffs employed and assume values in the ranges and , respectively. While the scheme dependence of is weaker than in the case of the Einstein–Hilbert truncation one finds that it is slightly larger for . The exponent suffers from relatively strong variations as the cutoff is changed, , but it is always significantly larger in modulus than .

In summary, the qualitative properties listed above (, , etc.) hold for all cutoffs. The ’s have a stronger scheme dependence than , however. This is most probably due to having neglected further relevant operators in the truncation so that the matrix we are diagonalizing is still too small.
Finally one can study the dimension dependence of these results. The beta functions produced by the truncated FRGE are continuous functions of the spacetime dimension and it is instructive to analyze them for . This was done for the Einstein–Hilbert truncation in [181, 79], with the result that the coupling flow is quantitatively similar to the 4dimensional one for not too large . The robustness features have been explored with varios cutoffs with the result that the sensitity on the cutoff parameters increases with increasing . In [181] a strong cutoff dependence was found for larger than approximately , for two versions of the sharp cutoff (with ) and for the exponential cutoff with . In [79] a number of different cutoffs were employed and no sharp increase in sensitivity to the cutoff parameters was reported for .
Close to the results of the expansion are recovered. Indeed, the fixed point of Section 1 originally found in the expansion is recovered in the present framework [179],
The coefficient for coincides with the one found in the expansion using the volume operator as a reference. In the expression for , is a scheme dependent positive constant. Of course here only parameterizes the dimension and does not serve double duty also as an ultraviolet regulator.This concludes our analysis of the robustness properties of the truncated RG flow. For further details the reader is referred to Lauscher et al. [133, 131, 132]. On the basis of these robustness properties we believe that the nonGaussian fixed point seen in the Einstein–Hilbert and truncations is very unlikely to be an artifact of the truncations. On the contrary there are good reasons to view it as the projection of a fixed point of the full FRGE dynamics. It is especially gratifying to see that within the scheme dependence the additional term has a quantitatively small impact on the location of fixed point and its unstable manifold.
In summary, we interpret the above results and their mutual consistency as quite nontrivial indications supporting the conjecture that 4dimensional Quantum Einstein Gravity possesses a RG fixed point with precisely the properties needed for its asymptotic safety.
The generalization of the previous results to more complex truncations would be highly desirable, but for time being it is out of computational reach. We therefore add some comments on what one can reasonably expect to happen.
The key issue obviously is the dimension and the structure of the unstable manifold. For simplicity let us restrict the discussion to the ansatz (4.29, 4.33) in which the bimetric character of the functionals and the evolution of the ghost sector are neglected. Morally speaking the following remarks should however apply equally to generic functionals . Within the restricted functional space (4.29, 4.33) only the ansatz for can be successively generalized. A generic finitedimensional truncation ansatz for has the form
where is the averaged metric and the are ‘wellchosen’ local or nonlocal reparameterization invariant functionals of it.Let us first briefly recall the scaling pattern based on the perturbative Gaussian fixed point. As described in Section 3.3 in a perturbative construction of the effective action the divergent part of the loop contribution is always local and thus can be added as a counter term to a local bare action , where the sum is over local curvature invariants . The scaling pattern of the monomials with respect to the perturbative Gaussian fixed point will thus reflect those of the in the effective action and vice versa. As explained in Section 2.3 the shortdistance behavior of the perturbatively defined theory will be dominated by the ’s with the largest number of derivatives acting upon . In a local invariant containing the Riemann tensor to the th power and covariant derivatives acting on it, the number of derivatives acting on is . If one starts with just a few ’s and performs loop calculations one discovers that higher ’s are needed as counter terms. As a consequence the high energy behavior is dominated by the bottomless chain of invariants with more and more derivatives.
As already argued in Section 2.3 in an asymptotically safe Quantum Gravidynamics the situation is different. The absence of a blowup in the couplings is part of the defining property. The dominance of the high energy behavior by the bottomless chain of high derivative local invariants is replaced with the expectation that all invariants should be about equally important in the extreme ultraviolet.
This can be seen from the FRGE for the effective average action via the following heuristic argument. Assume that , where the sum runs over a (dynamically determined) subset of all local and nonlocal invariants. The existence of a nontrivial fixed point means that the dimensionless couplings approach constant values for . As a consequence, the dimensionful couplings have the following dependence in the fixed point regime:
Obviously for any . The traces on the righthandside of the exact flow equation (4.31) are a compact representation of the beta functions for all ’s. They contain the Hessian . Let us perform the traces in the eigenbasis of , denoting its eigenvalues by . At least when all Hessians can be expressed in terms of they will be diagonal in this basis, with eigenvalues of the form , where the constants ’s are of order unity. Now, as explained previously, due to the factor the functional traces receive significant contributions only from a small band of eigenvalues near . Hence is effectively the same as under the trace, and the corresponding Hessian is If the coefficients were constant, the high energy limit of Equation (4.62) would indeed be dominated by the higher derivative invariants, their importance growing as . However the couplings are dependent by themselves and near a NGFP they run according to Equation (4.61). As a result the growing factor is compensated by the which stems from the fixed point running. Therefore is essentially a sum of the form in which the higher order invariants are merely equally significant as the lower ones, and the same is true for the beta functions. The bottomless chain of higher derivative invariants is replaced on both sides of the FRGE (4.31) by quantities which could have a welldefined limit as .Clearly the above argument can be generalized to action functionals depending on all the fields , , . Also the choice of field variables is inessential and the argument should carry over to other types of flow equations. It suggests that there could indeed be a fixed point action which is welldefined when expressed in terms of dimensionless quantities and which describes the extreme ultraviolet dynamics of Quantum Einstein Gravity. By construction the unstable manifold of this fixed point action would be nontrivial.
Without further insight unfortunately little can be said about its dimension. Among the local invariants in Equation (4.57) arguably only a finite number should be relevant. This is because the power counting dimensions of the infinite set of irrelevant local invariants may receive large positive corrections which makes them relevant with respect to the NGFP. An example for this phenomenon is provided by the invariant. It is power counting marginal () but with respect to the NGFP the scaling dimension of the associated dimensionless coupling is shifted to a large positive value of . Nevertheless it seems implausible that this will happen to couplings with arbitrarily large negative power counting dimension as correspondingly large corrections would be required.
On the other hand this reasoning does not apply to nonlocal invariants. For example arbitrary functions of the above are likewise power counting marginal, and on account of a similar positive shift they too would become relevant with respect to the NGFP. While such terms would not occur in the perturbative evaluation of the effective action, in the present framework they are admissible and their importance has to be estimated by computation. As a similar an ansatz of the form
was considered in [180, 182]. Here is the volume of the Riemannian manifold, is an arbitrary scale dependent function, and is the ordinary Newton constant whose evolution is neglected here. The results obtained point towards a ‘quenching’ of the cosmological constant similar to but more pronounced as in the mechanism of [211].Also scalar modes like the conformal factor have vanishing power counting dimension. The way how such dimensionless scalars enter the effective action is then not constrained by the above ‘implausibility’ argument. An unconstrained functional occurance however opens the door to a potentially infinitedimensional unstable manifold.
Another core issue are of course the positivity properties of the Quantum Einstein Gravity defined through the FRGE. As already explained in Section 1.5 the notorious problems with positivity and causality which arise within standard perturbation theory around flat space in higherderivative theories of Lorentzian gravity are not an issue in the FRG approach. For example if is of the type, the running inverse propagator when expanded around flat space has ghosts similar to those in perturbation theory. For the FRG flow this is irrelevant, however, since in the derivation of the beta functions no background needs to be specified explicitly. All one needs is that the RG trajectories are well defined down to . This requires only that is a positive operator for all . In the exact theory this is believed to be the case.
A rather encouraging first result in this direction comes from the truncation [131]. In the FRG formalism the problem of the higher derivative ghosts is to some extent related to the negative factors discussed in Section 2.1. It was found that, contrary to the Einstein–Hilbert truncation, the truncation has only positive factors in the fixed point regime . Hence in this truncation the existence of ‘safe’ couplings appears to be compatible with the absence of unphysical propagating modes, as required by the scenario.
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