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4.4 Einstein–Hilbert and 2 R truncations

In this section we review the main results obtained in the effective average action framework via the truncated flow equations of the previous Section 4.3. To facilitate comparison with the original papers we write here G = g ∕(16π ) k N for the running Newton constant; unless stated otherwise the results refer to d = 4.

4.4.1 Phase portrait of the Einstein–Hilbert truncation

In [181Jump To The Next Citation Point] the coupling flow (4.45View Equation) 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 gλ plane are shown in Figure 2View Image. The trajectories were obtained by numerically solving the system (4.50View Equation) for a sharp cutoff; using a smooth one all qualitative features remain unchanged.

View Image

Figure 2: Flow pattern in the gλ plane with a sharp mode cut-off. The arrows point in the direction of the coarse graining, i.e. of decreasing k. (From [181Jump To The Next Citation Point].)

The RG flow is dominated by two fixed points (g∗,λ∗): a Gaussian fixed point (GFP) at g∗ = λ ∗ = 0, and a non-Gaussian fixed point (NGFP) with g∗ > 0 and λ∗ > 0. 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 k → 0. The short–distance properties of Quantum Einstein Gravity are governed by the NGFP; for k → ∞, in Figure 2View Image all RG trajectories on the half–plane g > 0 run into this fixed point – its unstable manifold is two-dimensional. Note that near the NGFP the dimensionful Newton constant vanishes for k → ∞ according to Gk = gk∕k2 ≈ g∗∕k2 → 0. 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 [227Jump To The Next Citation Point].

The trajectories of Type IIIa cannot be continued all the way down to the infrared (k = 0) but rather terminate at a finite scale kterm > 0. (This feature is not resolved in Figure 2View Image.) At this scale the β-functions diverge. As a result, the flow equations cannot be integrated beyond this point. The value of kterm depends on the trajectory considered. The trajectory terminates when the dimensionless cosmological constant reaches the value λ = 1∕2. This is due to the fact that the functions Φp (w ) n and ^ p Φ n(w) – for any admissible choice of (0) ℛ – have a singularity at w = − 1, and because w = − 2λ in all terms of the β–functions. In Equations (4.50View Equation) the divergence at λ = 1 ∕2 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 k.

4.4.2 Evidence for asymptotic safety – Survey

Here we collect the evidence for asymptotic safety obtained from the Einstein–Hilbert and R2 truncations, Equation (4.34View Equation) and Equation (4.35View Equation), respectively, of the flow equations in Section 4.2 [133Jump To The Next Citation Point131Jump To The Next Citation Point].

The details of the flow pattern depend on a number of ad-hoc 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 (0) ℛ in Equation (4.31View Equation) can be varied, the gauge parameter α in Equation (4.33View Equation) can be varied, and the vector and transversal parts in the traceless tensor modes can be treated differently (type A and B cutoffs).

Picking a specific value for the gauge parameter has a somewhat different status than the other two choices. The truncations are actually one-parameter families of truncations labelled by α; in a more refined treatment α = αk would be a running parameter itself determined by the FRGE.

In practice the shape function (0) ℛ was varied within the class (4.51View Equation) of exponential cutoffs and a similar one-parameter class of cutoffs with compact support [133Jump To The Next Citation Point131Jump To The Next Citation Point]. Changing the cutoff function Ck at fixed k 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:

  1. Existence of a non-Gaussian fixed point: The NGFP exists no matter how ℛ (0) and α are chosen, both for type A and B cutoffs.
  2. Positive Newton constant: While the position of the fixed point is scheme dependent (see below), all cutoffs yield positive values of g ∗ and λ∗. A negative g∗ would have been problematic for stability reasons, but there is no mechanism in the flow equation which would exclude it on general grounds. This feature is preserved in the R2 truncation.
  3. Unstable manifold of maximal dimension: The existence of a nontrivial unstable manifold is crucial for the asymptotic safety scenario. The fact that the unstable manifold has (for d = 4) its maximal dimension (at least in the vicinity of the fixed point) indicates that the set of curvature invariants retained is dynamically natural. Again this is (in d = 4) a bonus feature not built into the flow equations. It holds for both the Einstein–Hilbert and the R2 truncation.
  4. Smallness of 2 R coupling: Also with the generalized truncation the fixed point is found to exist for all admissible cutoffs. It is quite remarkable that ν∗ is always significantly smaller than λ ∗ and g∗. Within the limited precision of our calculation this means that in the three-dimensional parameter space the fixed point practically lies on the (λ,g)-plane with ν = 0, i.e. on the parameter space of the pure Einstein–Hilbert truncation.

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.

4.4.3 Structure of the unstable manifold

This can be studied in the vicinity of the fixed point by a standard linearized stability analysis. We summarize the results for the non-Gaussian fixed point, first in the Einstein–Hilbert truncation and then in the more general R2 truncation. To set the notation recall that for a flow equation of the form k∂ g = β (g ,g ,...) k i i 1 2 the linearized flow near the fixed point is governed by the stability matrix with components Θij := ∂βi∕∂gj |g=g∗,

∑ ∗ k ∂k gi(k) = Θij (gj(k) − gj). (4.54 ) j
The general solution to this equation reads
( ) ∗ ∑ I k0- −ϑI gi(k) = gi + CI Vi k , (4.55 ) I
where the VI’s are the right-eigenvectors of Θ with eigenvalues ϑI, i.e. ∑ Θij VjI= ϑI V Ii j. The C I’s are constants of integration, k 0 is a reference scale, and decreasing k is the direction of coarse graining. Since Θ is not symmetric in general the ϑI’s are not guaranteed to be real. In principle Θ 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 Θ is non-degenerate for both the Einstein–Hilbert truncation and for the R2-truncation. In such a situation, the eigendirections with Re ϑI > 0 are irrelevant; they die out upon coarse graining and span the tangent space of the fixed point’s stable manifold. The remaining eigendirections with Re ϑI < 0 are relevant perturbations which span the tangent space of the fixed point’s unstable manifold. The eigenvalues ϑI 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 Ck.

As explained in Section 2.1 it is often convenient to set t := lnk0 ∕k (which is to be read as lnΛ ∕k − lnk ∕Λ 0 in the presence of an ultraviolet cutoff Λ) and ask “where a coarse graining trajectory comes from” by formally sending t to − ∞ (while the coarse graining flow is in the direction of increasing t). The tangent space to the unstable manifold has its maximal dimension if all the essential couplings taken into account hit the fixed point as t 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 2 R truncation, as we shall describe now in more detail.

Linearizing the flow equation (4.45View Equation) according to Equation (4.54View Equation) we obtain a pair of complex conjugate eigenvalues ϑ1 = ϑ∗ 2 with negative real part ϑ ′ and imaginary parts ±ϑ ′′. In terms of t = ln(k0∕k ) the general solution to the linearized flow equations reads

(λk, gk)T = (λ∗,g∗)T + 2{[Re C cos(ϑ′′t) + Im C sin(ϑ ′′t)]Re V + [Re C sin (ϑ′′t) − Im C cos(ϑ′′t)]Im V}e −ϑ′t, (4.56 )
with C := C1 = (C2)∗ being an arbitrary complex number and V := V 1 = (V 2)∗ the right-eigenvector of Θ with eigenvalue ϑ1 = ϑ∗2. Equation (4.54View Equation) implies that, due to the positivity of − ϑ′, all trajectories hit the fixed point as t is sent to − ∞. The nonvanishing imaginary part ϑ′′ has no impact on the stability. However, it influences the shape of the trajectories which spiral into the fixed point for t → − ∞. In summary, for any mode-cutoff employed the non-Gaussian fixed point is found to be ultraviolet attractive in both directions of the (λ,g)-plane.

Solving the full, nonlinear flow equations numerically [181Jump To The Next Citation Point] shows that the asymptotic scaling region where the linearization (4.56View Equation) is valid extends from k = ∞ down to about k ≈ mPl with the Planck mass defined as −1∕2 mPl = G 0. Here mPl marks the lower boundary of the asymptotic scaling region. We set k0 := mPl so that the asymptotic scaling regime extends from about t = 0 to t = − ∞.

The non-Gaussian fixed point of the 2 R-truncation likewise proves to be ultraviolet attractive in any of the three directions of the (λ,g,ν ) tangent space for all cutoffs used. The linearized flow in its vicinity is always governed by a pair of complex conjugate eigenvalues ϑ1 = ϑ′ + iϑ′′ = ϑ∗2 with ϑ ′ < 0, and a real negative one ϑ3 < 0. The linearized solution may be expressed as

(λk,gk,νk)T = (λ∗,g∗,ν∗)T + 2{[Re C cos(ϑ ′′t) + Im C sin(ϑ′′t)]Re V ′′ ′′ −ϑ′t + [Re C sin(ϑ t) − Im C cos(ϑ t)]Im V }e +C3V 3e −ϑ3t, (4.57 )
with arbitrary complex ∗ C := C1 = (C2 ), arbitrary real C3. Here 1 2∗ V := V = (V ) and 3 V are the right-eigenvectors of the stability matrix (Θij)i,j∈{λ,g,ν} with eigenvalues ∗ ϑ1 = ϑ2 and ϑ3, respectively. Clearly the conditions for ultraviolet stability are ϑ′ < 0 and ϑ3 < 0. They are indeed satisfied for all cutoffs. For the exponential shape function with s = 1, for instance, we find − ϑ′ = 2.15, − ϑ′′ = 3.79, − ϑ = 28.8 3, and Re V = (− 0.164,0.753, − 0.008)T, Im V = (0.64,0,− 0.01)T, 3 T V = − (0.92, 0.39, 0.04 ). (The vectors are normalized such that 3 ∥V ∥ = ∥V ∥ = 1.) The trajectories (4.57View Equation) comprise three independent normal modes with amplitudes proportional to Re C, Im C, and C3, 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 non-Gaussian fixed point, the 2 R flow is rather well approximated by the Einstein–Hilbert truncation:

  1. The ν-components of Re V and Im V are tiny. Hence these two vectors span a plane which virtually coincides with the (λ, g) subspace at ν = 0, i.e. with the parameter space of the Einstein–Hilbert truncation. As a consequence, the Re C- and Im C-normal modes are essentially the same trajectories as the “old” normal modes already found without the 2 R-term. Also the corresponding ′ ϑ- and ′′ ϑ-values coincide within the scheme dependence.
  2. The new eigenvalue ϑ3 introduced by the R2-term is significantly larger in modulus than ′ ϑ. When a trajectory approaches the fixed point from below (t → − ∞ ), the “old” normal modes ∝ Re C, Im C are proportional to ′ exp (− ϑ t), but the new one is proportional to exp(− ϑ3t), so that it decays much more quickly. For every trajectory running into the fixed point, i.e. for every set of constants (Re C,Im C, C3 ), we find therefore that once − t is sufficiently large the trajectory lies entirely in the Re V-Im V subspace, i.e. in the ν = 0-plane practically.

    Due to the large value of − ϑ3, the new scaling field is ‘very relevant’. However, when we start at the fixed point (t = − ∞ ) and raise t it is only at the low energy(!) scale k ≈ m Pl (t ≈ 0) that exp(− ϑ3t) reaches unity, and only then, i.e. far away from the fixed point, the new scaling field starts growing rapidly.

  3. Since the matrix Θ is not symmetric its eigenvectors have no reason to be orthogonal. In fact, one finds that V3 lies almost in the Re VIm V-plane. For the angles between the eigenvectors given above we obtain ∘ ∢(Re V, Im V) = 102.3, 3 ∘ ∢ (Re V,V ) = 100.7, 3 ∘ ∢(Im V, V ) = 156.7. Their sum is ∘ 359.7 which confirms that Re V, Im V, and 3 V are almost coplanar. Therefore as one raises t and moves away from the fixed point so that the V 3 scaling field starts growing, it is again predominantly the ∫ dx √g-- and ∫ dx √gR- (g) invariants which get excited, but not ∫ dx√gR- (g)2.

Summarizing the three points above we can say that very close to the fixed point the R2 flow seems to be essentially two-dimensional, and that this two-dimensional flow is well approximated by the coupling flow of the Einstein–Hilbert truncation. In Figure 3View Image we show a typical trajectory which has all three normal modes excited with equal strength √ -- (Re C = Im C = 1∕ 2, C3 = 1). All its way down from k = ∞ to about k = mPl it is confined to a very thin box surrounding the ν = 0-plane.

View Image

Figure 3: Trajectory of the linearized flow equation obtained from the R2-truncation for − 1 ≥ t = ln(k0∕k) > − ∞. In the right panel we depict the eigendirections and the “box” to which the trajectory is confined. (From [132Jump To The Next Citation Point].)

4.4.4 Robustness of qualitative features

As explained before the details of the coupling flow produced by the various truncations of Equation (4.19View Equation) depend on the choice of the cutoff action ((0) ℛ, type A vs. B) and the gauge parameter α. 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 non-Gaussian fixed point. Importantly, both for type A and type B cutoffs the non-Gaussian fixed point is found to exists for all shape functions ℛ (s0). This generalizes earlier results in [205]. Indeed, it seems impossible to find an admissible mode-cutoff which destroys the fixed point in d = 4. This is nontrivial since in higher dimensions (d ≳ 5) the fixed point exists for some but does not exist for other mode-cutoffs [181Jump To The Next Citation Point] (see however [79Jump To The Next Citation Point]).

Within the Einstein–Hilbert truncation also a RG formalism different from (and in fact much simpler than) that of the average action was used [39Jump To The Next Citation Point]. The fixed point was found to exist already in a simple RG improved 1-loop calculation with a proper time cutoff.

We take this as an indication that the fixed point seen in the Einstein–Hilbert [204133Jump To The Next Citation Point13639] and the 2 R truncations [131Jump To The Next Citation Point] 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 g∗λ∗ of the fixed point coordinates. Recall from Section 2.3.2 that the product g(k)λ(k) 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 4View Image we show the fixed point coordinates (λ ∗,g ∗,ν ∗) for the family of shape functions (4.51View Equation) and the type B cutoff. For every shape parameter s, the values of λ∗ and g ∗ are almost the same as those obtained with the Einstein–Hilbert truncation. Despite the rather strong scheme dependence of g∗ and λ∗ separately, their product has almost no visible s-dependence for not too small values of s! For s = 1, for instance, one obtains (λ∗,g∗) = (0.348, 0.272 ) from the Einstein–Hilbert truncation and (λ∗,g∗,ν∗) = (0.330,0.292,0.005) from the generalized truncation. One can also see that the R2 coupling ν at the fixed point is uniformly small throughout the family of exponential shape functions (4.51View Equation).

View Image

Figure 4: g∗, λ ∗, and g∗λ∗ as functions of s for 1 ≤ s ≤ 5 (left panel) and ν∗ as a function of s for 1 ≤ s ≤ 30 (right panel), using the family of exponential shape functions (4.51View Equation). (From [132Jump To The Next Citation Point].)

A similar situation is found upon variation of the gauge parameter α. Within the Einstein–Hilbert truncation the analysis has been performed in ref. [133Jump To The Next Citation Point] for an arbitary constant gauge parameter α, including the ‘physical’ value α = 0. For example one finds

{ g λ ≈ 0.12 for α = 1, (4.58 ) ∗ ∗ 0.14 for α = 0.
The differences between the ‘physical’ (fixed point) value of the gauge parameter, α = 0, and the technically more convenient α = 1 are at the level of about 10 to 20 per-cent. In view of this the much more involved analysis in the R2 truncation has been performed in the simpler α = 1 gauge only [131Jump To The Next Citation Point]. The product g λ ∗ ∗ with α = 1 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 α = 0 and the α = 1 results of the Einstein–Hilbert truncation. Taken together the analysis suggests the universal value g ∗λ ∗ ≈ 0.14.

Next we consider the ℛ (0) (in)dependence of the “critical exponents” ϑ′, ϑ ′′ in Equation (4.56View Equation, 4.57View Equation). Within the Einstein–Hilbert truncation the eigenvalues are found to be reasonably constant within about a factor of 2. For α = 1 and α = 0, for instance, they assume values in the ranges ′ 1.4 ≲ − ϑ ≲ 1.8, ′′ 2.3 ≲ − ϑ ≲ 4 and ′ 1.7 ≲ − ϑ ≲ 2.1, ′′ 2.5 ≲ − ϑ ≲ 5, respectively. The corresponding results for the R2 truncation are shown in Figure 5View Image. It presents the ℛ (0) dependence of the critical exponents, using the family of shape functions (4.51View Equation). For the cutoffs employed − ϑ′ and − ϑ′′ assume values in the ranges 2.1 ≲ − ϑ′ ≲ 3.4 and 3.1 ≲ − ϑ′′ ≲ 4.3, respectively. While the scheme dependence of ′′ ϑ is weaker than in the case of the Einstein–Hilbert truncation one finds that it is slightly larger for ′ ϑ. The exponent − ϑ3 suffers from relatively strong variations as the cutoff is changed, 8.4 ≲ − ϑ3 ≲ 28.8, but it is always significantly larger in modulus than ϑ ′.

View Image

Figure 5: θ′ = − Re ϑ1 and θ′′ = − Im ϑ1 (left panel) and θ3 = − ϑ3 (right panel) as functions of s, using the family of exponential shape functions (4.51View Equation).

In summary, the qualitative properties listed above (′ ϑ ,ϑ3 < 0, ′ − ϑ3 ≫ − ϑ, etc.) hold for all cutoffs. The ϑ’s have a stronger scheme dependence than g∗λ∗, however. This is most probably due to having neglected further relevant operators in the truncation so that the Θ 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 d and it is instructive to analyze them for d ⁄= 4. This was done for the Einstein–Hilbert truncation in [181Jump To The Next Citation Point79Jump To The Next Citation Point], with the result that the coupling flow is quantitatively similar to the 4-dimensional one for not too large d. The robustness features have been explored with varios cutoffs with the result that the sensitity on the cutoff parameters increases with increasing d. In [181] a strong cutoff dependence was found for d larger than approximately 6, for two versions of the sharp cutoff (with s = 1,30) and for the exponential cutoff with s = 1. In [79] a number of different cutoffs were employed and no sharp increase in sensitivity to the cutoff parameters was reported for d ≤ 10.

Close to d = 2 the results of the ε-expansion are recovered. Indeed, the fixed point of Section 1 originally found in the ε-expansion is recovered in the present framework [179Jump To The Next Citation Point],

g = 3-ε, λ = − 3-Φ1 (0)ε. (4.59 ) ∗ 38 ∗ 38 1
The coefficient for g ∗ coincides with the one found in the ε-expansion using the volume operator ∫ √ -- g as a reference. In the expression for λ∗, 1 Φ 1(0) is a scheme dependent positive constant. Of course here ε = d − 2 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. [133Jump To The Next Citation Point131Jump To The Next Citation Point132]. On the basis of these robustness properties we believe that the non-Gaussian fixed point seen in the Einstein–Hilbert and 2 R 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 R2-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 4-dimensional Quantum Einstein Gravity possesses a RG fixed point with precisely the properties needed for its asymptotic safety.

4.4.5 Comments on the full FRGE dynamics

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 (4.29View Equation, 4.33View Equation) in which the bi-metric 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 Γ k[g,¯g,σ,¯σ]. Within the restricted functional space (4.29View Equation, 4.33View Equation) only the ansatz for ¯Γ [g] k can be successively generalized. A generic finite-dimensional truncation ansatz for ¯Γ [g] k has the form

N ¯Γ [g] = ∑ g (k)kdiI [g], (4.60 ) k i i i=0
where g = ⟨g⟩ is the averaged metric and the Ii are ‘well-chosen’ 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 ℓ loop contribution is always local and thus can be added as a counter term to a local bare action ∑ S[g] = iuiPi[g], where the sum is over local curvature invariants Pi[g]. The scaling pattern of the monomials Pi[g] with respect to the perturbative Gaussian fixed point will thus reflect those of the Ii[g] in the effective action and vice versa. As explained in Section 2.3 the short-distance behavior of the perturbatively defined theory will be dominated by the Pi’s with the largest number of derivatives acting upon gαβ. In a local invariant containing the Riemann tensor to the pth power and q covariant derivatives acting on it, the number of derivatives acting on gαβ is 2p + q. If one starts with just a few Pi’s and performs loop calculations one discovers that higher Pi’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 blow-up 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 ∑ Γ¯k[g ] = iui(k )Ii[g], 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 g (k) = k−diu (k) i i approach constant values g∗ i for k → ∞. As a consequence, the dimensionful couplings have the following k-dependence in the fixed point regime:

∗ di ui(k) ≈ gi k . (4.61 )
Obviously ui(k → ∞ ) = 0 for any di < 0. The traces on the right-hand-side of the exact flow equation (4.31View Equation) are a compact representation of the beta functions for all g i’s. They contain the Hessian ¯ (2) ∑ (2) Γ k [g ] = iui(k )Ii [g]. Let us perform the traces in the eigenbasis of 2 − ∇, denoting its eigenvalues by p2. At least when all Hessians I(i2)[g] can be expressed in terms of ∇2 they will be diagonal in this basis, with eigenvalues of the form ci(p2)−di∕2, where the constants ci’s are of order unity. Now, as explained previously, due to the k ∂kℛk factor the functional traces receive significant contributions only from a small band of eigenvalues near 2 2 p = k. Hence (p2)−di∕2 is effectively the same as (k2)− di∕2 under the trace, and the corresponding Hessian is
¯Γ (2)(− ∇2 ≈ k2) ≈ ∑ u (k)c k− di. (4.62 ) k i i i
If the coefficients ui(k ) were constant, the high energy limit k → ∞ of Equation (4.62View Equation) would indeed be dominated by the higher derivative invariants, their importance growing as k− di. However the couplings u (k) i are k-dependent by themselves and near a NGFP they run according to Equation (4.61View Equation). As a result the growing factor −di k is compensated by the di k which stems from the fixed point running. Therefore (2) ¯Γ k is essentially a sum of the form ∑ ig∗ici 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 (4.31View Equation) by quantities which could have a well-defined limit as k → ∞.

Clearly the above argument can be generalized to action functionals depending on all the fields gαβ, ¯gαβ, α σ, ¯σα. 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 S ∗[g] = limk → ∞ ¯Γ k[g] = limk→ ∞ Γ k[g,g,0, 0] which is well-defined 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 (4.57View Equation) arguably only a finite number should be relevant. This is because the power counting dimensions d < 0 i 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 ∫ 4 √-- 2 I2[g] := d x gR (g) invariant. It is power counting marginal (d2 = 0) but with respect to the NGFP the scaling dimension of the associated dimensionless coupling is shifted to a large positive value dNGFP = − ϑ3 2 of O(10 ). 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 F (I [g ]) 2 of the above I [g] 2 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

1 ∫ √ -- 1 ¯Γ k[g] = −----- dx gR (g) + ----Fk(V [g]) (4.63 ) 16 πG 8πG
was considered in [180182]. Here ∫ V [g] = dx √g-- is the volume of the Riemannian manifold, Fk is an arbitrary scale dependent function, and G 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 infinite-dimensional 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 higher-derivative theories of Lorentzian gravity are not an issue in the FRG approach. For example if Γ k is of the 2 R + R type, the running inverse propagator (2) Γk 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 k = 0. This requires only that (2) Γ k + ℛk is a positive operator for all k. In the exact theory this is believed to be the case.

A rather encouraging first result in this direction comes from the R2 truncation [131Jump To The Next Citation Point]. In the FRG formalism the problem of the higher derivative ghosts is to some extent related to the negative Z φ k factors discussed in Section 2.1. It was found that, contrary to the Einstein–Hilbert truncation, the R2 truncation has only positive φ Z k factors in the fixed point regime k ≥ mPl. 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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