MZ-TH/10-08

Matter Induced Bimetric Actions

[1.2ex] for Gravity

[10mm] Elisa Manrique, Martin Reuter and Frank Saueressig

[3mm] Institute of Physics, University of Mainz

Staudingerweg 7, D-55099 Mainz, Germany

[1.1ex]

Abstract

The gravitational effective average action is studied in a bimetric truncation with a nontrivial background field dependence, and its renormalization group flow due to a scalar multiplet coupled to gravity is derived. Neglecting the metric contributions to the corresponding beta functions, the analysis of its fixed points reveals that, even on the new enlarged theory space which includes bimetric action functionals, the theory is asymptotically safe in the large expansion.

## 1 Introduction

The gravitational average action [1] is a universal tool for investigating the scale dependence of the quantum gravitational dynamics. It can be used in both effective and fundamental field theories of gravity. In particular it has played an important role in the Asymptotic Safety program [2, 3, 4]. In fact, the effective average action seems to evolve along renormalization group (RG) trajectories which have exactly the properties postulated by Weinberg [2], that is, in the ultraviolet (UV) they run into a nontrivial fixed point with a finite dimensional UV-critical surface [1]-[36]. A complete and everywhere regular RG trajectory then defines a fundamental and predictive quantum field theory of gravity. At every fixed coarse graining scale the effective average action [37]-[40] gives rise to an effective field theory valid at that scale. This property has been exploited in some first phenomenological investigations of asymptotically safe gravity [41]-[51], for instance in inflationary cosmology [45], see also [52].

One of the key requirements every future fundamental quantum theory of gravity must meet is that of “background independence” [53].
Loosely speaking this means that none of the theory’s basic rules and assumptions, calculational methods, and none of its predictions, therefore,
may depend on any special metric that is fixed a priori. All metrics of physical relevance must result from the intrinsic quantum gravitational
dynamics^{1}^{1}1See [30] and [54] for a more detailed discussion of this point..
While in loop quantum gravity [55, 56, 57] and in the discrete approaches [58]-[61] the requirement of
‘‘background independence”^{2}^{2}2 Here and in the following we write “background independence” in quotation marks when it is supposed to stand for the above general principle,
rather than for the independence of a background field.
is met in the obvious way by completely avoiding the use of any background metric or a similar non-dynamical structure, this seems very hard to
do in a continuum field theory. In fact, in the gravitational average action approach [1] “background independence” is implemented by quite
a different strategy: one introduces an arbitrarily chosen background metric at the intermediate steps of the quantization, but verifies at the
end that no physical prediction depends on which was chosen. In this way one may take advantage of the entire arsenal of techniques
developed for quantizing fields in a fixed curved background. However, what complicates matters as compared to the usual situation, is that the background spacetime
is never concretely specified; hence there is no way of exploiting the simplifications that would arise for special, highly symmetric backgrounds such as
Minkowski or de Sitter space, say. In a sense, one is always dealing with the “worst case” as far as the complexity of the background structures is concerned.
On the other hand, the crucial advantage of this approach is that it sidesteps all the profound conceptual difficulties and the resulting technical
problems that emerge when one tries to set up a quantum theory without any metric at the fundamental level.
The difficulties one faces in a program of this type are comparable to those encountered when one tries to quantize a topological field theory
on a manifold which carries only a smooth but no Riemannian structure.

Thus, from now on we assume that the gravitational degrees of freedom can be encoded in a metric tensor field. We fix a background metric and quantize the nonlinear fluctuations of the dynamical metric, , in the “arena” provided by , and we repeat this quantization process for any choice of . In this manner we arrive at an infinite family of quantum theories for , whereby the family members are labeled by a classical (pseudo-) Riemannian metric .

The dynamical content of this family is fully described by
an effective action which depends on two arguments^{3}^{3}3
For simplicity we ignore the Faddeev-Popov ghosts and possible matter fields here.:
the expectation value of the fluctuation, , and the background metric. If the background-quantum field
split is chosen linear, the expectation value gives rise to a corresponding expectation value

(1.1) |

of the metric operator , i.e., where . The action entails an effective field equation which governs the dynamics of in dependence on the background metric:

(1.2) |

For special, so-called “self-consistent” backgrounds it happens that eq. (1.2) is solved by an identically vanishing fluctuation expectation value . Then the expectation value of the quantum metric equals exactly the background metric, . The defining condition for a self-consistent background,

(1.3) |

is referred to as the tadpole equation since it expresses the vanishing of the fluctuation 1-point function. In fact, the corresponding -point 1PI Green’s functions for generic are given by

(1.4) |

It is a well known “magic” of the background formalism that by using an appropriate gauge fixing condition [62] a set of on-shell equivalent Green’s functions is generated by differentiating the reduced functional with respect to :

(1.5) |

The Green’s functions (1.4) and (1.5) are equivalent on-shell only if the quantization scheme employed respects the background-quantum field split symmetry in the physical sector. It expresses the arbitrariness of the decomposition of as a background plus a fluctuation. In the linear case the corresponding symmetry transformation are

(1.6) |

for any . If the split symmetry is appropriately implemented at the quantum level, the reduced functional which depends on one argument only has exactly the same physical contents as the more complicated .

Up to now we tacitly assumed that the microscopic (bare) action governing the dynamics of is known a priori, as this would be the case in an ordinary quantum field theory. In the Asymptotic Safety program [2], the situation is different: the pertinent bare action is not an input but rather a prediction of the theory. More precisely, the idea is to set up a functional coarse graining flow on the space of all (diffeomorphism invariant) functionals , to search for nontrivial fixed points of this flow and if there are any, to identify the bare action with one of them. This construction yields a quantum field theory with a well behaved UV-limit.

This idea has been implemented in the framework of the gravitational average action [1, 14]. Here one defines a coarse grained counterpart of the ordinary effective action, the effective average action , in terms of a functional integral containing an additional mode suppression factor quadratic in , and performs the coarse graining by suppressing the contributions of the -modes with a covariant momentum smaller than the variable IR cutoff scale [37, 38, 39]. The -dependence of the effective average action is governed by a functional RG equation (FRGE) which defines a vector field (a “flow”) on theory space. At , the average action equals the ordinary effective action , and only the gauge fixing term breaks the split symmetry. Hence, for , does not contain more gauge invariant information than the reduced functional does.

It is crucial to realize that this (on-shell) equivalence of the two functionals at does not generalize to . At every nonzero scale the coarse graining operation unavoidably leads to an additional violation of the split symmetry since the action is not invariant. It contains and separately, not only in the split invariant combination . In a sense, contains more information than if . An immediate consequence is the well known fact [38, 1] that it is impossible to write down a FRGE in terms of the reduced functional alone. The actual theory space is more complicated, consisting of functionals with two metric arguments and, to be precise, also ghost arguments and , respectively: . Often it is convenient to replace with as the independent argument

(1.7) |

In the -notation the second argument parametrizes the so-called extra background field dependence, i.e., that part of the -dependence which does not combine with a corresponding -dependence to a full metric . If the split symmetry was intact we had . In general has a nontrivial dependence on though, and so we can rightfully call a bimetric action.

The enlarged theory space is the price one has to pay for the “background independence” of the average action approach. Including a matter field its “points” are functionals which are invariant under arbitrary diffeomorphisms acting on all arguments simultaneously. If is a generating vector field and the corresponding Lie derivative we have, to first order in , . For later use we write down the resulting Ward identity, for simplicity at and for a scalar matter field:

(1.8) |

Here and in the following the covariant derivatives and refer to the Levi-Cività connections of and , respectively.

Up to now almost all applications of the gravitational average action employed a truncated theory space with functionals of the form +classical gauge fixing and ghost terms, where . Hence in practice one had to deal with the RG evolution of a single metric functional only, . In ref. [30] a first example of an RG flow with a nontrivial bimetric truncation was analyzed, albeit only in conformally reduced gravity rather than the full fledged theory. In [30] also a number of conceptual issues related to the bimetric character of the average action approach have been explained; we refer the reader to this discussion for further details. A similar calculation in a different gravity theory has been performed in [63].

The purpose of the present paper is to perform the first investigation of a bimetric RG flow involving the full fledged gravitational field. To be precise, we compute the contribution of scalar matter fields to the beta functions of various Newton- and cosmological constant-like running couplings which parametrize in certain truncations. The quantum effects originating from the gravitational sector are neglected, which allows to avoid the additional technical complications linked to the gauge-fixing of the theory. The resulting RG flow is simple enough to be investigated analytically in a completely explicit way. This enables a detailed study of the conceptual points underlying the bimetric truncations by identifying and highlighting the general aspects, which will be central to more sophisticated computations in the future [64, 65]. As a spin-off, we re-examine the Asymptotic Safety conjecture within a large- approximation [15, 17], which gives the first insights on how important the nontrivial bimetric dependence of actually is.

The remaining sections of this paper are organized as follows. In Section 2 we analyze the RG behavior of the most general bimetric non-derivative term contained in the average action, . Here the fluctuation is not required to be small. In Section 3 we employ a complementary truncation of , based on a systematic -expansion which includes all interaction terms built from up to two derivatives and up to first order in . We derive the RG flow on the corresponding five-dimensional theory space and analyze its fixed point structure. The technical details underlying this calculation are relegated to the Appendix. As an application, the resulting -dependent tadpole equation for self-consistent background geometries is discussed in Section 4. Finally, Section 5 contains our conclusions.

## 2 The Running Cosmological Constants

In the following we consider a multiplet of quantized scalar fields coupled to classical gravity. We quantize the scalars by means of a (truncated) functional flow equation, being particularly interested in the gravitational interaction terms induced by the quantum effects in the matter sector. In most parts of the analysis we take the scalars free, except for their interaction with gravity. The system is described by an effective average action . Even in this comparatively simple setting where gravity itself is not quantized this action is “bimetric”: it depends on the full metric since the scalars couple directly to it, and it also depends on because it is the background metric which enters the scalar mode suppression term

### 2.1 The functional RG equation

The FRGE for the quantized scalars interacting with classical gravity reads

(2.1) |

As gravity is not quantized here, the trace is over the fluctuations of the scalars only, and the Hessian involves functional derivatives with respect to only. As usual, denotes the “RG time”.

In the following it will be important to carefully distinguish the volume elements and , respectively. The matrix elements of the Hessian operator contain two factors of the latter,

(2.2) |

Products of operators such as are defined in terms of their matrix elements as , and the position representation of the trace in (2.1) reads . Furthermore, if some operator has matrix elements , we define the associated (pseudo) differential operator by for every “column vector” . We shall write and for the Laplace-Beltrami operators belonging to and , respectively.

### 2.2 The truncation ansatz

To start with, we explore the contents of the FRGE (2.1) with the following ansatz:

(2.3) |

Here always stands for . The ansatz (2.3) is taken to be invariant; appropriate sums over the index are understood (, etc.). The first term in (2.3) is the action of a standard scalar coupled to the full metric , the last represents a generalization of the running cosmological constant induced by the scalars. It is assumed to contain all possible non-derivative terms built from and . For the time being we discard induced derivative terms. In particular the running of Newton’s constant is neglected. The normalization and the explicit factor of in the second term on the RHS of (2.3) are chosen such that the constant term of the function equals the cosmological constant:

(2.4) |

In order to project out the beta function of it is sufficient to insert -independent metrics and into the FRGE. In this case the Hessian resulting from the ansatz (2.3) is

(2.5) |

where we employed the differential operator notation.

### 2.3 The cutoff operator

Now we come to a crucial step which highlights the role of the background metric: the construction of the cutoff operator. Recalling that may depend on the background metric only, we define it by the requirement that upon adding to the Hessian and setting the operator which then appears in must get replaced by . Here is an arbitrary shape function [37, 39] with the standard properties and . The condition

(2.6) |

leads to an operator which at first sight appears familiar,

(2.7) |

which, however, appears in the FRGE combined with the Hessian for different from :

(2.8) |

Most of the unfamiliar features we are going to find in the following are due to the interplay of the standard cutoff operator with a Hessian containing the ratio of the volume elements .

### 2.4 The RG equation for

The trace of the flow equation is easily evaluated in a standard plane wave basis now. With we have:

(2.9) |

The integral on the RHS of this flow equation represents a scalar density which depends on two constant matrices, and . It cannot be evaluated in closed form. In the following we calculate it for two special cases, namely for and both proportional to the unit matrix, and by expanding in their difference .

### 2.5 The volume element truncation

Now we make a further truncation and restrict to depend on the metrics via their volume elements only: . In this case, the beta function of can be found by inserting two conformally flat metrics

(2.10) |

into the flow equation (2.9) and keeping track of the constants and . One then observes that actually depends on the ratio only. Thus, setting with , we obtain

(2.11) |

Here, , , and

(2.12) |

From now on we employ the “optimized” shape function [40] , whence

(2.13) |

Here

(2.14) |

and

(2.15) |

The integrals (2.15) can easily be computed explicitly. Let us specialize for now. Then

(2.16) |

and the flow equation reads

(2.17) |

At first, let us neglect the scalar mass, setting and therefore. Then the RG equation (2.17) can be integrated trivially:

(2.18) |

Using (2.18) in the truncation ansatz (2.3) we obtain the following explicit representation for the non-derivative terms in the average action:

(2.19) |

This is one of the our main results, and several comments are in order here.

(A) Obviously the induced non-derivative terms are neither proportional to nor to but rather to a complicated function of their ratio, times an extra factor of . (The explicit factor of on the RHS of (2.18) has converted the in the ansatz to a .)

(B) The induced term is regular for any ratio . In fact, writing (2.19) as

(2.20) |

and using the expansion

(2.21) |

we see that actually has no singularity at , i.e., at .

(C) Let us specialize the result for the regime . This amounts to expanding in the fluctuation variable or, equivalently, in the variable since

(2.22) |

(Here indices are raised and lowered with .) The essential quantity in (2.20) is . The first few terms of its expansion read

(2.23) |

In the regime there is clearly no preference for -monomials over -monomials or vice versa, so the truncation ansatz cannot be simplified
accordingly: any meaningful truncation is genuinely “bimetric”! Note also that the monomial which was individually included in
the truncation studied in [30]^{4}^{4}4
Note, however, that plays a distinguished role in the conformally reduced gravity setting of [30]. It amounts to a mass term
of the scalar field appearing there.
never is generated in isolation. According to (2.23) it is always accompanied by other,
a priori equally important terms involving and .

(D) It is instructive to rewrite the expansion about in terms of . Up to linear order the relevant terms in are

(2.24) |

Here is a background cosmological constant multiplying rather than , and is an analogous, but numerically different running coupling in the (only) term linear in . Explicitly,

(2.25) |

Note the different signs on the RHS of the equations: increases, but decreases for growing . Note also that when the RHS of (2.24) cannot be written as a functional of the single metric alone as this would require it to be proportional to . But this is not the case just because .

(E) Computations within the setting of single metric truncations retain only the terms of zeroth order in . They equate the two metrics and , and traditionally denote the one metric which is left over then by . In this setting, eq. (2.24) would boil down to

(2.26) |

Thus, in a single metric truncation, it is the parameter which would be interpreted as “the” cosmological constant, the one, and only one, responsible for the curvature of spacetime. From the more general perspective of the bimetric truncation we understand that this is actually missleading. We shall discuss this point in detail in Section 4.

(F) We saw that, in the regime , the true cosmological constant monomial plays no distinguished role. Next let us see whether this could be the case when . Let us try an expansion in . In a -expansion of (2.20) with there would indeed appear a monomial in isolation, the first few terms of the power series being . However, taking the explicit form of into account one finds that an expansion of this type actually does not exist, since when . Hence is not analytic at . So the conclusion is that in this regime, too, the -monomial plays no distinguished role in the truncation. (The same is true for the term considered in [30].)

(G) Up to now, the scalars were assumed massless. If we allow for a non-zero dimensionless mass so that now , the RG equation (2.17) no longer can be integrated trivially. The general properties of the flow are clear though: if , the matter field is still approximately massless and the above discussion applies. If, on the other hand, the scalar decouples and its contribution to the running of becomes tiny. If we neglect the running of the dimensionful mass the parameter diverges below the threshold at . Expanding we obtain the leading term below the threshold:

(2.27) |

In the massive regime an extra factor suppresses the running of the non-derivative term. Here, again, it is not of the standard -form, but rather proportional to .

(H) So far, the discussion was based upon an evaluation of the integral formula (2.9) for metrics and which are conformal to , see (2.10). If one wants to know the tensorial structure of the terms in one must go beyond this special case. A systematic strategy for doing this is an expansion in , whereby and are still constant matrices, but not necessarily proportional to . Carrying out this expansion up to linear order in we find a result of the form (2.24), generalized for arbitrary , with

(2.28) |

Here the ’s are the usual -dependent threshold function of ref. [1], see also eq. (A.18) in the Appendix. If one specializes for and the “optimized” , the result (2.28) accidentally coincides with the one obtained from the conformally flat ansatz (2.25). As (2.28) contains different -functions, they depend on in a different way. This implies that any special relationship between the two cosmological constants and which one might invoke, for instance in order to restore split symmetry, cannot have a universal (scheme independent) meaning at .

## 3 A systematic derivative- and -expansion

In this section we explore the matter induced gravitational coupling constants in a different parameterization of the average action. It comprises the first terms of a systematic expansion of in powers of and the number of derivatives.

### 3.1 The truncation ansatz

In the following, we will consider an average action of the form

(3.1) |

Here is the curvature scalar built from the background metric, , and denotes the background Einstein tensor. The functional (3.1) is obviously invariant under diffeomorphisms acting simultaneously on , and , respectively. The gravitational part of this ansatz is complete in the sense that it contains all possible terms with no or one factor of and at most two derivatives. The latter can always be arranged to act on , and diffeomorphism invariance then implies that they occur as contractions of the background Riemann tensor. The matter part of (3.1) has the same structure as in the previous section; now is to be read as an abbreviation for , though.

There exist two -independent field monomials with zero and two derivatives, respectively, namely and . In eq. (3.1) the corresponding prefactors are proportional to and , respectively. In the sector with one power of there are three possible tensors structures for the corresponding one-point function, namely and with two, and with no derivatives. Their prefactors define new running couplings , , and , respectively. (The superscripts indicate the -order in which the coupling in question occurs.)

The functional (3.1) is defined for completely independent fields and ; hence has an “extra” -dependence in general which does not combine with to a full metric . Nevertheless it is instructive to consider (3.1) for the special case of no extra -dependence. The background-quantum field split symmetry is intact then and depends on and via their sum only. The resulting gravitational part is obtained by expanding the Einstein-Hilbert action,

(3.2) |

up to terms of first order in . The expansion is of the form (3.1) with special coefficients, however:

(3.3a) | ||||

(3.3b) | ||||

(3.3c) |

It needs to be stressed that the relations (3.3) are not satisfied in general. Since the cutoff term breaks the split symmetry, does have an extra background dependence, and so the terms with one power of are not the linearization of any functional depending on the sum only. As a consequence, we encounter two running couplings, and , both of which are related to the classical Newton constant, but have a different conceptual status and numerical value. The same remark applies to the running cosmological constants and .

Furthermore, we emphasize that when the split symmetry is broken there is no symmetry principle that would force the second derivative terms with one factor to be proportional to the Einstein tensor. There are actually two independent tensor structures,