### 4.3 Truncated flow equations

Approximate computations of the effective average action can be done in a variety of ways: by perturbation theory, by saddle point approximations of the functional integral, or by looking for approximate solutions of the FRGE. A nonperturbative method of the latter type consists in truncating the underlying functional space. Using an ansatz for where -independent local or nonlocal invariants are multiplied by running parameters, the FRGE (4.19) can eventually be converted into a system of ordinary differential equations for these parameters. In this section we outline how the conversion is done in principle.

A still very general truncated functional space consists of ‘all’ background invariant functionals which neglect the evolution of the ghost action. The corresponding ansatz reads [179]

To simplify the notation we wrote for in the argument of the effective actions as before. In Equation (4.29) we extracted the classical and from . The remaining functional depends on both and . It is further decomposed as where is defined in Equation (4.9) and contains the deviations for . Hence, by definition, , and can be viewed as a quantum correction to the gauge fixing term which vanishes for , too. The ansatz (4.29) satisfies the initial condition (4.21) if
Inserting the ansatz (4.29) into the exact form of the RG equation one obtains an evolution equation on the truncated space of ’s:
This equation involves
and , the Hessian of with respect to at fixed .

The truncation ansatz (4.29) is still too general for practical calculations to be easily possible. Computations simplify considerably with the choice and a local curvature polynomial for . The first specialization includes in only the wave function renormalization and gives

Here is given by Equation (4.10) with replacing ; it vanishes at .

For two choices of local curvature polynomials have been considered in detail

The truncation (4.34) will be called the Einstein–Hilbert truncation. It retains only the terms already present in the classical action, with -dependent coefficients, though. It is the one where the RG flow has been originally found in [179]. The parameter in Equation (4.33) is kept constant ( specifically; see the discussion below for generalizations) so that in this case the truncation subspace is two-dimensional: The ansatz (4.33, 4.34) contains two free functions of the scale, the running cosmological constant and or, equivalently, the running Newton constant . Here is a scale independent constant related to the fixed Newton constant in Section 1.5 by .

The truncation (4.35) will be called the -truncation. It likewise keeps the gauge fixing and ghost sector classical as in Equation (4.29) but includes a local curvature squared term in . In this case the truncated theory space is three-dimensional. Its natural (dimensionless) coordinates are , where

Even though Equation (4.35) contains only one additional invariant, the derivation of the corresponding differential equations is far more complicated than in the Einstein–Hilbert case. We shall summarize the results obtained with Equation (4.35[133131132] in Section 4.4.

We should mention that apart from familiarity and the retroactive justification through the results described later on, there is no structural reason to single out the truncations (4.34, 4.35). Even the truncated coarse graining flow in Equation (4.31) will generate all sorts of terms in , the only constraint comes from general covariance. Both local and nonlocal terms are induced. The local invariants contain monomials built from curvature tensors and their covariant derivatives, with any number of tensors and derivatives and of all possible index structures. The form of typical nonlocal terms can be motivated from a perturbative computation of ; an example is . Since approaches the ordinary effective action for it is clear that such terms must generated by the flow since they are known to be present in . For an investigation of the non-ultraviolet properties of the theory, the inclusion of such terms is very desirable but it is still beyond the calculational state of the art (see however [180]).

The main technical complication comes from evaluating the functional trace on the right-hand-side of the flow equation (4.31) to the extent that one can match the terms against those occuring on the left-hand-side. We shall now illustrate this procedure and its difficulties in the case of the Einstein–Hilbert truncation (4.34) in more detail.

Upon inserting the ansatz (4.33) into the partially truncated flow equation (4.31) it should eventually give rise to a system of two ordinary differential equations for and . Even in this simple case their derivation is rather technical, so we shall focus on matters of principle here. In order to find and it is sufficient to consider (4.31) for . In this case the left-hand-side of the flow equation becomes . The right-hand-side contains the functional derivatives of ; in their evaluation one must keep in mind that the identification can be used only after the differentiation has been performed at fixed . Upon evaluation of the functional trace the right-hand-side should then admit an expansion in terms of invariants , among them and . The projected flow equations are obtained by extracting the -dependent coefficients of these two terms and discarding all others. Equating the result to the left-hand-side and comparing the coefficients of and , the desired pair of coupled differential equations for and is obtained.

In principle the isolation of the relevant coefficients in the functional trace on the right-hand-side can be done without ever considering any specific metric . Known techniques like the derivative expansion and heat kernel asymptotics could be used for this purpose, but their application is extremely tedious usually. For example, because the operators and are typically of a complicated non-standard type so that no efficient use of the tabulated Seeley–deWitt coefficients can be made. Fortunately all that is needed to extract the desired coefficients is to get an unambiguous signal for the invariants they multiply on a suitable subclass of geometries . The subclass of geometries should be large enough to allow one to disentangle the invariants retained and small enough to really simplify the calculation.

For the Einstein–Hilbert truncation the most efficient choice is a family of -spheres , labeled by their radius . For those geometries , so they give a vanishing value on all invariants constructed from containing covariant derivatives acting on curvature tensors. What remains (among the local invariants) are terms of the form , where is a polynomial in the Riemann tensor with arbitary index contractions. To linear order in the (contractions of the) Riemann tensor the two invariants relevant for the Einstein–Hilbert truncation are discriminated by the metrics as they scale differently with the radius of the sphere: , . Thus, in order to compute the beta functions of and it is sufficient to insert an metric with arbitrary and to compare the coefficients of and . If one wants to do better and include the three quadratic invariants , , and , the family is not general enough to separate them; all scale like with the radius.

Under the trace we need the operator , the Hessian of at fixed . It is calculated by Taylor expanding the truncation ansatz, , and stripping off the two ’s from the quadratic term, . For a metric on one obtains

with
In order to partially diagonalize this quadratic form, has been decomposed into a traceless part and the trace part proportional to : , . Further, is the Laplace operator of the background geometry, and is the numerical value of the curvature scalar on .

At this point we can fix the coefficients which appear in the cutoff operators and of Equation (4.14). They should be adjusted in such a way that for every low–momentum mode the cutoff combines with the kinetic term of this mode to times a constant. Looking at Equation (4.34, 4.35) we see that the respective kinetic terms for and differ by a factor of . This suggests the following choice:

Here is the projector on the trace part of the metric. For the traceless tensor (4.39) gives , and for the different relative normalization is taken into account. Thus we obtain in the and the -sector, respectively:

From now on we may set and for simplicity we have omitted the bars from the metric and the curvature. Since we did not take into account any renormalization effects in the ghost action we set in and obtain similarly, with ,

Looking at Equation (4.37) we see that for the trace has a “wrong sign” kinetic term which corresponds to a normalization factor . The choice (4.39) complies with the rule (4.25) motivated earlier. As a result, in the -sector. The negative is a reflection of the notorious conformal factor instability in the Einstein–Hilbert action.

At this point the operator under the first trace on the right-hand-side of Equation (4.31) has become block diagonal, with the and blocks given by Equation (4.40). Both block operators are expressible in terms of the Laplacian , in the former case acting on traceless symmetric tensor fields, in the latter on scalars. The second trace in Equation (4.31) stems from the ghosts; it contains (4.41) with acting on vector fields.

It is now a matter of straightforward algebra to compute the first two terms in the derivative expansion of those traces, proportional to and . Considering the trace of an arbitrary function of the Laplacian, , the expansion up to second order derivatives of the metric is given by

The ’s are defined as
for , and for . The trace counts the number of independent field components. It equals , and , for scalars, vectors, and traceless tensors, respectively. The expansion (4.42) is derived using the heat kernel expansion
and Mellin transform techniques [179]. Using Equation (4.42) it is easy to calculate the traces in Equation (4.31) and to obtain the RG equations in the form and . We shall not display them here since it is more convenient to rewrite them in terms of the dimensionless parameters (4.36).

In terms of the dimensionless couplings and the RG equations become a system of autonomous differential equations

Here , the anomalous dimension of the operator , is explicitly given by
with the following functions of :

The beta function for is given by

The ’s and ’s appearing in Equations (4.47, 4.48) are certain integrals involving the normalized cutoff function ,
for positive integers , and .

With the derivation of the system (4.45) we managed to find an approximation to a two-dimensional projection of the FRGE flow. Its properties and the domain of applicability or reliability of the Einstein–Hilbert truncation will be discussed in Section 4.4. It will turn out that there are important qualitative features of the truncated coupling flow (4.45) which are independent of the cutoff scheme, i.e. independent of the function . The details of the flow pattern on the other hand depend on the choice of the function and hence have no intrinsic significance.

By construction the normalized cutoff function , , in Equation (C.21) describes the “shape” of in the transition region where it interpolates between the prescribed behavior for and , respectively. It is referred to as the shape function therefore.

In the literature various forms of ’s have been employed. Easy to handle, but disadvantageous for high precision calculations is the sharp cutoff  [181] defined by , where the limit is to be taken after the integration. This cutoff allows for an evaluation of the and integrals in closed form. Taking as an example, Equations (4.45) boil down to the following simple system then

(For orientation, Equation (4.50) corresponds to the sharp cutoff with in [181]). The flow described by Equation (4.50) is restricted to the halfplane since the beta functions are singular along the boundary line . When a trajectory hits this line it cannot reach the infrared () but rather terminates at a nonzero . Within the Einstein–Hilbert truncation this happens for all trajectories which approach a positive at low .

Also the cutoff with allows for an analytic evaluation of the integrals; it has been used in the Einstein–Hilbert truncation in [136]. In order to check the scheme (in)dependence of certain results it is desirable to perform the calculation, in one stroke, for a whole class of ’s. For this purpose the following one parameter family of exponential cutoffs has been used [205133131]:

The precise form of the cutoff is controlled by the “shape parameter” . For , Equation (4.51) coincides with the standard exponential cutoff (4.15). The exponential cutoffs are suitable for precision calculations, but the price to be paid is that their and integrals can be evaluated only numerically. The same is true for a one-parameter family of shape functions with compact support which was used in [133131].

The form of the expression (4.46) for the anomalous dimension illustrates the nonperturbative character of the method. For Equation (4.46) can be expanded as

which illustrates that even a simple truncation can sum up arbitrarily high powers of the couplings. It is instructive to consider the approximation where only the lowest order is retained in Equation (4.52). In terms of the dimensionful one has in and for , , so that integrating yields
The constant is -dependent via the threshold functions defined in Equation (4.49). However, for all cutoffs one finds that . An acceptable shape function must interpolate between and in a monotonic way, the ‘drop’ occuring near . This implies that and are both positive. Since typically they are of order unity this suggests that should be negative. This has been confirmed for one-parameter families of cutoffs such as Equation (4.51) or those of [133131], for the family of sharp cutoffs, and for the optimized cutoff. Using also numerical methods it seems impossible to find an acceptable which would yield a positive . The approximation (4.53) is valid for . One sees that at least in this regime is a decreasing function of . This corresponds to the antiscreening behavior discussed in Sections 1.1 and 1.5.

Above we illustrated the general ideas and constructions underlying truncated gravitational RG flows by means of the simplest example, the Einstein–Hilbert truncation (4.34). The flow equations for the truncation are likewise known in closed form but are too complicated to be displayed here. These ordinary differential equations can now be analyzed with analytical and numerical methods. Their solution reveals important evidence for asymptotic safety. Before discussing these results in Section 4.4 we comment here on two types of possible generalizations.

Concerning generalizations of the ghost sector truncation, beyond Equation (4.29) no results are available yet, but there is a partial result concerning the gauge fixing term. Even if one makes the ansatz (4.33) for in which the gauge fixing term has the classical (or more appropriately, bare) structure one should treat its prefactor as a running coupling: . After all, the actual “theory space” of functionals contains “-type” and “gauge-fixing-type” actions on a completely symmetric footing. The beta function of has not been determined yet from the FRGE, but there is a simple argument which allows us to bypass this calculation.

In nonperturbative Yang–Mills theory and in perturbative quantum gravity is known to be a fixed point for the evolution. The following heuristic argument suggests that the same should be true beyond perturbation theory for the functional integral defining the effective average action for gravity. In the standard functional integral the limit corresponds to a sharp implementation of the gauge fixing condition, i.e.  becomes proportional to . The domain of the integration consists of those ’s which satisfy the gauge fixing condition exactly, . Adding the infrared cutoff at amounts to suppressing some of the modes while retaining the others. But since all of them satisfy , a variation of cannot change the domain of the integration. The delta functional continues to be present for any value of if it was there originally. As a consequence, vanishes for all , i.e.  is a fixed point of the evolution [137].

In other words we can mimic the dynamical treatment of a running by setting the gauge fixing parameter to the constant value . The calculation for is more complicated than at , but for the Einstein–Hilbert truncation the -dependence of and , for arbitrary constant , has been found in [133]. The truncations could be analyzed only in the simple gauge, but the results from the Einstein–Hilbert truncation suggest the UV quantities of interest do not change much between and  [133131].

Up to now we considered pure gravity. As far as the general formalism is concerned, the inclusion of matter fields is straightforward. The structure of the flow equation remains unaltered, except that now and are operators on the larger space of both gravity and matter fluctuations. In practice the derivation of the projected FRG equations can be quite formidable, the main difficulty being the decoupling of the various modes (diagonalization of ) which in most calculational schemes is necessary for the computation of the functional traces.

Various matter systems, both interacting and non-interacting (apart from their interaction with gravity) have been studied in the literature. A rather detailed analysis of the fixed point has been performed by Percacci et al. In [72171170] arbitrary multiplets of free (massless) fields with spin and were included. In [170] a fully interacting scalar theory coupled to gravity in the Einstein–Hilbert approximation was analyzed, with a local potential approximation for the scalar self-interaction. A remarkable finding is that in a linearized stability analysis the marginality of the quartic self-coupling is lifted by the quantum gravitational corrections. The coupling becomes marginally irrelevant, which may offer a new perspective on the triviality issue and the ensued bounds on the Higgs mass. Making the number of matter fields large , the matter interactions dominate at all scales and the nontrivial fixed point of the expansion [216217203] is recovered [169].