#
Mz-Th/09-17

Bimetric Truncations for

Quantum Einstein Gravity and

Asymptotic Safety

###### Abstract

In the average action approach to the quantization of gravity the fundamental requirement of “background independence” is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper we analyze a number of conceptual issues related to the bimetric character of the gravitational average action and explore a first nontrivial bimetric truncation in the simplified setting of conformally reduced gravity. Possible implications for the Asymptotic Safety program and the cosmological constant problem are discussed in detail.

## 1 Introduction and motivation

Unifying the principles of quantum mechanics and general relativity is perhaps still the most challenging open problem in fundamental physics [1]. While the various approaches that are currently being developed, such as string theory, loop quantum gravity [2, 3, 4], or asymptotic safety [5]-[33], for instance, are based upon rather different physical ideas and are formulated in correspondingly different mathematical frameworks, they all must cope with the problem of “background independence” in one way or another. Whatever the ultimate theory of quantum gravity will look like, a central requirement we impose on it is that it should be “background independent” in the same sense as general relativity is background independent. Loosely speaking, this means that the spacetime structure actually realized in Nature should not be part of the theory’s definition but rather emerge as a solution to certain dynamical equations. In classical general relativity the spacetime structure is encoded in a Lorentzian metric on a smooth manifold, and this metric, via Einstein’s equation, is a dynamical consequence of the matter present in the Universe.

### 1.1 The requirement of “background independence”

In the following we shall explore the possibility of constructing a quantum field theory of gravity in which the metric carries the dynamical degrees of freedom. Even though this property is taken over from classical general relativity the fundamental dynamics of those metric degrees of freedom, henceforth denoted , is allowed to be different from that in classical general relativity.

Furthermore, the theory of quantum gravity we are searching for will be required to respect the principle of “background independence”: In the formulation of the theory no special metric should play any distinguished role. The actual metric of spacetime should arise as the expectation value of the quantum field (operator) with respect to some state: . This is in sharp contradistinction to the traditional quantum field theory on Minkowski space whose conceptual foundations heavily rely on the availability of a non-dynamical (rigid) Minkowski spacetime as a background structure.

Trying to set up a similar quantum field theory of the metric itself, let us assume that we are given some candidate for a microscopic interaction which is described by a diffeomorphism invariant classical (i.e. bare) action functional . Whatever this action is (Einstein-Hilbert, , etc.), well before one encounters the notorious problems related to the ultraviolet (UV) divergences, profound conceptual problems arise. In absence of a rigid background when the metric is dynamical, there is no preferred time direction, for instance, hence no notion of equal time commutators, and clearly the usual rules of quantization cannot be applied straightforwardly. Many more problems arise when one tries to apply the familiar methods of quantum field theory to the metric itself without introducing a rigid background structure. Some of them are conceptually deep, others are of a more technical nature. Here we only mention one type of difficulties which later on will become central in our discussion.

In conventional field theory on a rigid background typical regulator schemes (by higher derivatives, for example) which are used to make the theory well behaved, both in the infrared (IR) and the UV, employ the background metric. As a result, it is not obvious if and how such schemes can be applied to quantum gravity. This problem is particularly acute for approaches based upon some sort of functional renormalization group equation (FRGE) which is supposed to implement a Wilson-like renormalization group (RG) flow by a continuous coarse graining [34]-[40]. In conventional Euclidean field theory and statistical mechanics every such coarse graining is described by an associated length scale which measures the size of the spacetime blocks within which the microscopic degrees of freedom got averaged over. When the metric is dynamical and no rigid background is available, this concept becomes highly problematic since it is not clear a priori in terms of which metric one should express the physical, i.e. proper diameter of a spacetime block.

The principle of ‘‘background independence”^{1}^{1}1Here and in the following we use quotation marks when “background independence” is
supposed to stand for this principle (rather than for the independence of some quantity of the background field). which we would like to
implement in the quantum field theory of the metric we are aiming at can be summarized as the requirement that
none of the theory’s basic rules and assumptions, calculational methods, and none of its predictions, therefore, may depend on any
special metric fixed a priori. All metrics of physical relevance must result from the intrinsic quantum gravitational dynamics.

A possible objection^{2}^{2}2This argument is due to D. Giulini [63]. We thank him and A. Ashtekar for a discussion of this point.
against this working definition could be as follows: A theory can be made “background independent” in the above sense, but
nevertheless has a distinguished rigid background if the latter arises as the unique solution to some field equation which is
made part of the “basic rules”. For instance, rather than introducing a Minkowski background directly one simply
imposes the field equation . However, this objection can apply only in a setting where the dynamics,
the field equations, can be chosen freely. In asymptotically safe gravity, the case we are actually interested in, this
is impossible as the dynamics is dictated by the fixed point action.

There are two quite different strategies for complying with the requirement of “background independence”:
(i) One can try to define the theory, and work out its implications, without ever employing a background metric or a similar
nondynamical structure. While this is the path taken in loop quantum gravity [2, 3, 4] and the discrete approaches
[41],[42]-[44], for instance, it seems very hard , if not impossible to realize it in a continuum field
theory.^{3}^{3}3The typical difficulties are reminiscent of those encountered in the quantization of topological Yang-Mills theories.
Even when the classical action can be written down without the need of a metric, the gauge fixing and quantization
of the theory usually requires one.
Hence the only way of proving the topological character of some result is to show its independence of the metric chosen.
(ii) One employs an arbitrarily chosen background metric at the intermediate steps of the quantization,
but verifies at the end that no physical prediction depends on which metric was chosen. This is the route taken in the
gravitational average action approach [6] which we are going to use in this paper.

### 1.2 The bimetric solution to the “background independence” problem

In the average action approach one decomposes the quantum metric as and quantizes the (non-linear) fluctuation in essentially the same way one would quantize a matter field in a classical spacetime with metric . In this way all of the conceptual problems alluded to above, in particular the difficulties related to the construction of regulators, disappear. Technically the quantization of gravity proceeds then almost as in standard field theory on a rigid classical spacetime, with one essential difference, though: In the latter, one concretely fixes the background typically as or as in the Euclidean case. In “background independent” quantum gravity instead, the metric is never concretely chosen. All objects that one has to compute in this setting, generating functionals, say, are functionals of the variable . An example is the effective action which depends on the background and the fluctuation expectation value . In a sense, the “background independent” quantization of gravity amounts to its quantization on all possible backgrounds at a time.

There are two metrics now which are almost equally important: the background and the expectation value metric

(1.1) |

Alternatively we may regard the effective action as a functional of the two metrics, defining

(1.2) |

In this language both arguments of can be varied freely over the same space of tensor valued functions. Because of the symmetric status enjoyed by the two metrics we can characterize this setting as a “bimetric” approach.

### 1.3 Background-covariant coarse graining

The effective average action (EAA) is a scale dependent version of the ordinary effective action, with built-in IR cutoff at a variable momentum scale [34, 39, 40]. In the case of gravity [6] its formal definition starts out from the modified gauge fixed path integral over and the Faddeev-Popov ghosts and :

(1.3) |

Here includes, besides the bare action , the gauge fixing and the ghost terms. The new ingredient is the cutoff action which suppresses the IR modes. It has the structure

(1.4) |

plus a similar term for the ghosts. The coarse graining kernel is a functional of . In fact, the background field is fundamental for a covariant (in the sense explained below) coarse graining and for giving a proper (as opposed to a pure “coordinate”) meaning to the momentum scale . When the integration variable is expanded in eigenmodes of the covariant Laplacian constructed from , the eigenvalue of the lowest mode integrated out unsuppressed has eigenvalue .

For the discussion in the present paper it is important to note that is a functional of the two independent fields and . Contrary to a classical action it is not just a functional of their sum . We say that has an extra background field dependence, that is, a dependence on which does not combine with to form the full metric . (The same is also true for .) For a detailed discussion of this point, and the corresponding background-quantum field split symmetry, we refer to [25] and [26].

### 1.4 Properties of the gravitational average action

With a modified path integral (1.3) as the starting point, the remaining steps in the construction of the gravitational EAA proceed almost as in the case of the ordinary effective action: one introduces source terms for and for the ghosts, defines a connected generating functional , and introduces its Legendre transform . It depends on the expectation values of and the Faddeev-Popov ghosts, denoted and , respectively. Finally, the EAA is defined as the difference where the expectation value fields are inserted into .

Let us list the main properties of the gravitational EAA, to the extent they are relevant to the present discussion. For further details we refer to [6].

(A) At , coincides with the ordinary effective action .

(B) Originally, depends, besides the background metric, on the expectation value fields . This presentation of the EAA fits with the intuition that we are quantizing the three “matter” fields in the classical -background spacetime. Sometimes the bimetric point of view is more convenient. Then one replaces by as the independent argument and defines

(1.5) |

This notation allows us to interpret the second, i.e. the -argument of as an extra background dependence, since it is this -dependence that does not combine with a corresponding -dependence to a full metric . The extra -dependence of has (at least) two sources, namely the one of and , respectively. The former disappears at , the latter does not.

(C) In the construction of [6] a gauge fixing condition which is invariant under the so called background gauge transformations [45] has been employed. As a result, is invariant under a simultaneous general coordinate transformation of all its arguments, including :

(1.6) |

Here denotes the Lie derivative with respect to a generating vector field . At the standard discussion of the background gauge technique applies [45]. Hence the functional restricted to or is sufficient to generate all on-shell graviton Green’s functions. They can be obtained in a comparatively simply way by first setting and afterwards differentiating with respect to .

(D) For vanishing ghosts^{4}^{4}4In the general case there are two more equations similar to (1.7), involving derivatives
w.r.t. and . It follows from ghost number neutrality that those latter equations always admit the solution
., the EAA implies an effective Einstein equation whose (-dependent!) solution determines
the expectation value of the metric as a functional of :

(1.7) |

Note that (1.7) involves , not itself. In the matter field interpretation we may regard (1.7) as an equation for . We call a selfconsistent background, , if it gives rise to a vanishing fluctuation average, that is, if there are no quantum corrections to the background metric:

(1.8) |

As is bilinear in , the condition for a selfconsistent background can be written directly in terms of itself:

(1.9) |

Note that in eq.(1.9) we may insert only after the functional differentiation. In order to set up the “tadpole equation” (1.9) we need to know at least to first order in in an expansion about .

(E) On the theory space spanned by functionals of the type , restricted by the condition (1.6), the gravitational average action satisfies an exact FRGE. This FRGE was derived in [6] and up to now all applications of the EAA concept in gravity focused on finding approximate solutions to this equation [6]-[31] and to explore their physics contents [46]-[57]. In particular the EAA-based investigations of the Asymptotic Safety scenario used this equation. For our present purposes a second type of exact functional equation is equally relevant to which we turn next.

(F) The EAA satisfies an exact functional BRS Ward identity. To formulate it, one has to enlarge the theory space to functionals of the form where and are sources of the BRS variation of and , respectively. Then, abbreviating , the path integral-based definition of implies

(1.10) |

with the trace functional

(1.11) |

where denotes the Hessian of , and is summed over. Furthermore, and are the coarse graining kernels for the graviton and the ghosts, and denotes their direct sum [6]. The standard BRS Ward identities have the structure of (1.10) with . The nonzero contributions to stem from the cutoff term which is not BRS invariant. Since vanishes for it follows that so that is BRS invariant in the usual way.

(G) The dependence of on the background metric is governed by a similar exact functional
equation^{5}^{5}5For the derivation of an analogous relation in Yang-Mills theory see Appendix A of [37].:

(1.12) |

Obviously (1.12) measures the degree possesses an “extra” background dependence. The functional on the RHS is similar to (1.4); it consists of various traces involving itself. The action appearing under the path integral, , contains various sources of contributions to , in particular the extra background dependences of and , respectively. The former is nonzero even for , the latter vanishes in this limit.

Indeed, all coarse graining kernels have the structure

(1.13) |

where interpolates between zero and unity for large and small arguments, respectively. Therefore vanishes for , and as a consequence no longer provides an extra background dependence. However, a crucial observation, and in fact one of the motivations for the present work, is that the contributions stemming from are likely to become large in the UV limit . After all, itself behaves like a divergent mass term in this limit.

(H) Exact solutions to the FRGE automatically satisfy the BRS Ward identity and the -equation (1.12). For approximate solutions to the flow equation this is not necessarily the case. One can then evaluate the Ward identity and/or the -equation for the approximate RG trajectory and check how well these relations are satisfied. In principle this is a useful tool in order to judge the reliability of approximations, truncations of theory space in particular. Because of the extreme complexity of these equations this has not been done so far for gravity. In the present paper we shall use a simplified version of (1.12) for this purpose, however.

### 1.5 QEG on truncated theory spaces

The FRGE of the gravitational average action has been used in many investigations of the nonperturbative RG flow of Quantum Einstein Gravity (QEG), in particular in the context of the Asymptotic Safety conjecture. In all of those investigations[6]-[31] the RG flow had been projected onto a truncated theory space which can be described by the ansatz

(1.14) |

In this ansatz the classical gauge fixing and ghost terms were pulled out of , and also the coupling to the BRS variations is taken to have the same form as in the bare action. The remaining functional depends on and . It is further decomposed as where is defined by putting and equal,

(1.15) |

and is the remainder. Hence, by definition, it vanishes when the metrics are equal: .
Furthermore, it was argued [6] that setting should be a good first approximation,
and in fact in all calculations performed so far has been neglected essentially^{6}^{6}6At most the
effect of the running -wave function normalization on the gauge fixing term has been taken into account
[8, 10]. This amounts to setting ,
where is given by the running of Newton’s constant.. Then, what remains to be determined from the FRGE is the
-dependence of , a functional of one metric variable only.

This brings us to a subtle, but important issue which will be the main topic of the present paper.

The general situation we are confronted with can be described as follows. We are given an exact RG equation on the full theory space consisting of “all” action functionals of a given (symmetry, etc.) type. Somewhat symbolically, the FRGE has the structure , where encodes the beta functions of all running couplings. These beta functions are the components of a vector field which the FRGE defines on theory space, and the corresponding integral curves are the RG trajectories.

Now we make a truncation ansatz for the EAA which specifies a certain subspace of this full theory space. The idea is to study an RG flow on the subspace which is induced by the flow (vector field) on the full space. The problem is that in general the vector field on the full space will not be tangent to the subspace and hence it does not give rise to a flow on the latter. Stated in more practical terms, when we insert an action from the subspace into the RHS of the FRGE, the calculation of the functional traces will produce terms different from those present in the truncation ansatz: the RG trajectories try to “leave” the subspace.

In order to obtain RG trajectories we must invoke a kind of generalized projection which maps the full vector field, restricted to the subspace, onto a vector field tangent to the subspace. The result of merely restricting the full vector field to the subspace will not be tangent to it; it has normal components corresponding to terms in the actions we would like to discard. Therefore the specification of a truncation involves two items: a truncation ansatz for the action, to define the subspace, and a description for mapping the full vector field on the subspace onto a new one tangent to it. Clearly a truncation approximates the exact flow the better the smaller the normal components of the vector field are. Ideally one would like to find a subspace such that at least in some domain it is automatically tangent to it, without any projection.

Now let us return to the truncation ansatz (1.5) and discuss the related projection. If we insert
the of (1.5), with a possibly nontrivial , into the exact form of the FRGE^{7}^{7}7
See eq.(2.32) of ref.[6]., we obtain a simpler flow equation for the functional

(1.16) |

It reads

(1.17) |

where

(1.18) |

Here denotes the Hessian of with respect to at fixed , and
is the Faddeev-Popov kinetic operator.^{8}^{8}8See eq.(2.11) of ref.[6].

Let us first assume the subspace consists of all actions of the type (1.5), that is, is an arbitrary functional of two metrics, vanishing at . Then the vector field in (1.5) happens to be tangent to the subspace. In fact, the RHS of (1.17) is an arbitrary, diffeomorphically invariant functional of and , as is its LHS. Since is -independent, the RG equation (1.17) with (1.16) reads

(1.19) |

Setting in (1.19) we obtain an equation for alone, and upon subtracting it from (1.19) we get a flow equation for :

(1.20) | ||||

(1.21) |

Given appropriate initial conditions, the eqs.(1.20) and (1.21) suffice to determine the scale dependence of and , i.e. of all actions of the type (1.5).

We refer to truncations of this type, involving a running functional of two metrics, and , as bimetric truncations.

All truncations worked out so far in the literature [6]-[20] are single metric truncations. They set in the general ansatz (1.5), hence discard the second flow equation (1.21), and set on the RHS of the first one, eq.(1.20). In this way, the latter assumes the form, symbolically,

(1.22) |

This is a closed equation for , or stated differently, the vector field it defines is tangent to the subspace.

What makes the truncations potentially dangerous is that they cannot discriminate background field monomials in the action, such as , from similar ones containing the dynamical metric, , say. Hence the RG running of the coefficient is combined with that of into a single beta function.

The truncation should provide a good approximation if the flow on the larger space is approximately tangent to the smaller subspace defined by the additional constraint . This is the case if the RHS of (1.21) is small so that the “ -directions” in theory space do not get “turned on”: . This condition is met precisely if the extra background dependence is small.

Note, however, that in setting on the RHS of (1.21) we replace with under the functional traces, and one might wonder about the impact this has on the beta functions of pure monomials, for instance. This is in fact the topic of the present paper.

### 1.6 Aim of the present paper

The above discussion suggests that the degree of reliability of the class of truncations is intimately related to the extra background dependence of the EAA which in turn is at the very heart of the bimetric solution to the “background independence” problem.

In this paper we are therefore going to analyze a first bimetric truncation, and we asses how stable the predictions of the corresponding approximation are under this generalization of the truncation. Indeed, we shall focus on the non-trivial UV fixed point that is known to exist in all truncations investigated so far. The regime is particularly susceptible to “ contaminations” since diverges for and could possibly give rise to a large extra background dependence therefore.

For this reason it is even the more gratifying that we shall find a non-Gaussian fixed point (NGFP) in the RG flow of the bimetric truncation, too. However, our analysis will not be performed within full fledged Quantum Einstein Gravity but rather a toy model which shares many features with full QEG, in particular the existence of a NGFP in the truncation. This toy model is the “conformally reduced gravity” studied in [25] and [26], a caricature of QEG in which the conformal factor of the metric is quantized in its own right, rather than the real metric degrees of freedom. In [25] and [26] a number of conceptual issues related to the Asymptotic Safety program, in particular on the role of “background independence”, has been investigated within this comparatively simple theoretical laboratory. A generalization of the model including higher derivatives has been considered in [27].

For the time being the corresponding bimetric analysis of the Asymptotic Safety program within full QEG is beyond the technical state of art. It would require the evaluation of for kept different from , by a derivative expansion, say. We would have to calculate traces of the form where is a function of two different, in general noncommuting covariant Laplacians which involve , and , respectively. There exist no standard heat kernel techniques that could be applied here.

The remaining sections of this paper are organized as follows. In Section 2 we briefly review the discussion of conformally reduced gravity and extend it in various directions. Using this model as our main theoretical laboratory we shall then, in Section 3, introduce bimetric truncations and obtain the corresponding RG flow. Section 4 is devoted to a detailed discussion of this flow and of the general lessons it teaches us about full fledged QEG. We summarize our main results in Section 5.

Several discussions of a more technical nature are relegated to three appendices. Appendix A is dedicated to the evaluation of various beta functions, in Appendix B we use the exact -equation (1.12) in order to test the quality of the truncations used, and in Appendix C we describe the relation between the effective and the bare fixed point action in the bimetric setting.

##
2 Conformally reduced gravity as a

theoretical laboratory

Our toy model is inspired by the observation that the (Euclidean) Einstein-Hilbert action,

(2.1) |

when evaluated for metrics , assumes the form of a standard action:

(2.2) |

Here is a “reference metric” which is fixed once and for all; in the following we usually assume it flat, whence . The corresponding classical equation of motion reads then

(2.3) |

In [25, 26] the scalar-like theory defined by (2.2) was considered in its own right, detached from the original quantum field theory of metrics, and the FRGE approach has been used to quantize it. As compared to a conventional scalar theory crucial differences arise since the background value of itself, denoted , determines the proper cutoff momentum a given value of corresponds to.

### 2.1 The EAA setting for the toy model

Let us explain the quantization and the FRGE of the toy model from a more general perspective. We start from a formal path integral

(2.4) |

representing the partition function of the scalar with a bare action , not necessarily related to of (2.2). The “microscopic” conformal factor , the analogue of in full gravity, is decomposed as , and the -integral is replaced by an integral over . Here is an arbitrary but fixed background field, and the dynamical fluctuation field. The corresponding expectation values and are the counterpart of and , respectively.

By now we arrived at the path integral . We think of as being expanded in the eigenmodes of the covariant Laplacian pertaining to the background metric , whereby the measure corresponds to an integration over the expansion coefficients. The IR cutoff responsable for the coarse graining is now implemented by introducing a smooth cutoff in the spectrum of , i.e. by suppressing the contribution of all eigenmodes with eigenvalues below a given value . In practice one replaces the path integral by with a cutoff action which is quadratic in the fluctuation,

and contains a -dependent integral kernel . Upon adding a source term to the action the path integral equals with the coarse grained generating functional of connected Green’s functions. Denoting its Legendre transform by the definition of the effective average action for the toy model reads

(2.5) |

As in full gravity, we may alternatively regard as a functional of two complete metrics rather than a fluctuation and a background. Hence we define

(2.6) |

Using the notation , the second argument stands for an extra background dependence, in the sense that it does not appear combined with as . Furthermore, in analogy with full gravity, we introduce the “diagonal” functional with and identified,

(2.7) |

and the remainder . Thus every functional has a unique decomposition of the form

(2.8) |

whereby vanishes for equal fields,

(2.9) |

From the path integral-based definition (2.5) of the EAA its flow equation can be derived in the usual way [25]:

(2.10) |

The Hessian operator reads, in the position representation,

(2.11) |

In [25, 26] the coarse graining kernel has been constructed in such a way that, when added to ,
it effects the replacement , with an arbitrary shape function
interpolating between and . For the generalized truncations considered in the present
paper this requirement is met if we choose^{9}^{9}9This choice of generalizes the one used in earlier investigations for the
case of a position dependent . If , the operator (2.12) reduces to the one employed in [25, 26].

(2.12) |

Note the explicit -dependence of besides the one implicit in , the Laplace-Beltrami operator related to .

By a similar derivation one obtains the following exact equation for the extra -dependence of the EAA:

(2.13) |

Here involves derivatives with respect to at fixed . Eq.(2.13) is a simplified version of (1.12) in full QEG.

By inserting the decomposition (2.8) into the FRGE we obtain the following coupled system of two flow equations which is still fully equivalent to (2.10):

(2.14) | ||||

(2.15) |

Here we introduced

(2.16) |

Up to this point all equations are exact.

In the case at hand the truncations discard the second flow equation, (2.15), and neglect the contribution in the on the RHS of the first one. Eq.(2.14) becomes a closed equation for then:

(2.17) |

Here we see quite explicitly why the truncations are potentially dangerous: The coarse graining kernel under the trace, originally , has now become . Hence the cutoff terms generate contributions to the beta functions which mix with those from the true -terms in !

### 2.2 Examples of single-metric truncations ( )

In [25] and [26] the FRGE of conformally reduced gravity was solved in various truncations of the type. The simplest one is the “conformally reduced Einstein-Hilbert” (CREH) truncation; here the ansatz for the EAA has exactly the structure of the classical action (2.2), with a running Newton and cosmological constant, though:

(2.18) |

In [26] a generalization motivated by the “local potential approximation” frequently used in standard scalar theories [39] was employed; it retains the classical kinetic term of (2.18) but allows for an arbitrary potential:

(2.19) |

The crucial feature of the functionals (2.18) and (2.19) is that actually they have no extra dependence on , that is, the background field always appears combined with the fluctuation to form a complete field . Hence we have and .

In [25] the flow corresponding to the CREH truncation (2.18) has been worked out whereby the coarse graining operator (2.12) was used. This operator is designed in such a way that the cutoff scale is proper with respect to the background metric . It was found that, with this , the RG flow is qualitatively very similar to the one in full QEG; in particular both a Gaussian and a non Gaussian fixed point were found to exist in this truncation.

If instead is tailored in such a way that becomes proper with respect to the metric , the RG flow is that of a conventional scalar theory, and no NGFP exists.

In [25] it was argued that the first choice of is the correct one to be used in gravity since only this choice respects the principle of “background independence”, while the second makes use of a rigid structure, the reference metric , which does not even have an analogue in full QEG.

## 3 Bimetric truncations

### 3.1 Generalized local potential approximation

In the following we employ a generalized truncation ansatz which will allow us to disentangle the - and -dependencies
of the EAA. We no longer identify the dynamical metric with the background metric
as in (2.17). The ansatz has a nontrivial extra dependence now. It
reads^{10}^{10}10In ref.[58], Floreanini and Percacci have performed a similar calculation with two independent conformal factors
in a perturbatively renormalizable gravity model.

(3.1) |

This ansatz differs from (2.19) by a separate kinetic term for the background field and an extra dependence of the potential . Clearly the functional related to (3.1) is nonvanishing for generic fields. This ansatz may be regarded a generalized local potential ansatz for two “scalars”; if we require to be separately invariant under and there is no cross term .

Note that that there exist two versions of Newton’s constant now: the prefactor of involves the ordinary Newton constant associated with the (self-) couplings of the dynamical gravitational field , while the prefactor of contains a kind of background Newton constant . (In the potential term we pulled out a factor of to facilitate the comparison with (2.19).) The scale dependence of the two Newton constants is governed by their respective anomalous dimension, defined as

(3.2) |

In order to project out the term we must allow for a -dependent background field in the following; in [25, 26] a constant one had been sufficient.

In the following we assume to be a flat metric on a manifold with -topology. We shall set where convenient.

### 3.2 The 3-parameter potential ansatz

For an explicit solution of the differential equations we shall impose a further truncation on the ansatz (3.1). We assume that involves only three running couplings, multiplying the monomials , and , respectively:

(3.3) |

This ansatz allows us to disentangle the - and the -contributions, respectively, to the cosmological constant term in .

The actions and contain the potentials and with and , respectively.

The single metric potential contains a cosmological constant term ; this is the one whose running has been computed in earlier studies. All three monomials in the ansatz (3.3) contribute to this term upon equating the fields:

The 3-parameter potential (3.3) is further motivated by the fact that there is a natural way of projecting the flow on the corresponding truncation subspace (see below and Appendix A), and by its natural interpretation in the language of conventional scalar field theory. In fact, besides the “true” cosmological constant related to the dynamical field and the “background” one , analogously related to , the potential contains the mixed term . As we shall see the later it is closely related to a conventional mass term.

The functional (3.1) with the special potential (3.3) can be written in the following suggestive form:

(3.4) |

Here it is understood that the RHS of this equation is evaluated for the metrics and . Obviously the above functional consists of two separate Einstein-Hilbert actions for and , respectively, plus a novel non-derivative term which couples the two metrics. Unusual as it looks, it is precisely the kind of terms that is expected to arise in the effective average action of full quantum gravity. In fact, the RHS of (3.2) is invariant under simultaneous diffeomorphisms of and , as it should.

It is instructive to write down the tadpole equation for the toymodel. The analogue of eq.(1.9) for a selfconsistent background reads, with the general truncation (3.1)

(3.5) |

Here the prime denotes a derivative with respect to the -argument before is set. For the 3-parameter potential (3.3) we find, for instance,

(3.6) |

with the selfconsistent background (scb) parameter

(3.7) |

While the equation (3.6) which governs consistent background fields has the same structure as the classical field equation (2.3), or the one following from the