MZ-TH/10-19

Bimetric Renormalization Group Flows in

[1.5ex] Quantum Einstein 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 formulation of an exact functional renormalization group equation for Quantum Einstein Gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of “background independence” is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which are purely background, and those involving a mixture of both metrics. This work initiates the first study of the full-fledged gravitational RG flow, which explicitly accounts for this bimetric structure, by considering an ansatz for the effective average action which includes all three classes of interactions. It is shown that the non-trivial gravitational RG fixed point central to the Asymptotic Safety program persists upon disentangling the dynamical and background terms. Moreover, upon including the mixed terms, a second non-trivial fixed point emerges, which may control the theory’s IR behavior.

## 1 Introduction

Background independence constitutes one of the central guiding principles in the quest for a viable quantum theory of gravity. This requirement is central in loop quantum gravity [1, 2, 3] and also implemented in lattice approaches towards quantum gravity [4]-[7]. Loosely speaking, it implies that the spacetime structure realized in Nature should not be part of the theory’s definition, but rather emerge from a dynamical principle. This strict background invariance, referring to no background structure whatsoever, is, however, very hard to implement. In particular, without an ab initio metric the notions of causality and equal time commutation relations are not defined, so that the usual quantization procedures underlying ordinary quantum field theories cannot be applied straightforwardly.

A milder, but nevertheless equally admissible road towards a viable quantum gravity theory is the requirement of “background covariance”. This allows to introduce a background metric as an auxiliary tool, as long as none of the theory’s basic rules and assumptions, calculational methods, and predictions, depend on this special metric. In other words, all metrics of physical relevance are obtained from the dynamics of the theory. This is the viewpoint adopted in many continuum field theory approaches to quantum gravity, in particular by the functional renormalization group approach initiated in [8].

The latter is based on a functional renormalization group equation (FRGE) which encodes a kind of (continuous) Wilsonian RG flow on the space of diffeomorphism invariant action functionals. These actions naturally depend on the expectation value of the quantum metric . In addition, the coarse graining operation requires a background structure, which can be used to define volumes over which the quantum fluctuations are averaged. This structure is conveniently provided by the background field method [9] which also ensures the background covariance of the approach. Here the quantum metric is split according to

(1.1) |

where is a fixed, but unspecified, background metric and are the quantum fluctuations around this background which are not necessarily small. This allows the formal construction of the gauge-fixed (Euclidean) gravitational path integral

(1.2) |

Here is a generic action, which depends on only, while the background gauge fixing and ghost contribution contain and in such a way that they do not combine into a full . They have an “extra -dependence” and are not invariant under split-symmetry , which is respected by the combination (1.1).

The key ingredient in the construction of the FRGE is the coarse graining term . It is quadratic in the fluctuation field , , plus a similar term for the ghosts. The kernel provides a -dependent mass term which separates the fluctuations into high momentum modes and low momentum modes with respect to the scale set by the covariant Laplacian of the background metric. The profile of ensures that the high momentum modes are integrated out unsuppressed while the contribution of the low momentum modes to the path integral is suppressed by the -dependent mass term. Varying then naturally realizes Wilson’s idea of coarse graining by integrating out the quantum fluctuations shell by shell.

Taking the formal -derivative, eq. (1.2) provides the starting point for the construction of the functional renormalization group equation for the effective average action [10, 11].(See [12] for reviews.) For gravity this flow equation takes the form [8]

(1.3) |

Here , STr is a functional supertrace which includes a minus sign for the ghosts , is the matrix valued (in field space) IR cutoff introduced above, and is the second variation of with respect to the fluctuation fields. Notably, depends on two metrics, the background metric and the expectation value field

(1.4) |

The explicit dependence on the two metrics is essential for being able to write down the exact flow equation (1.3), as the Hessian is the variation of with respect to the fluctuation fields at fixed . In this sense, and its flow is of an intrinsically bimetric nature. In particular, the construction of involves the terms and where the -dependence does not combine with into the full averaged metric . These terms therefore provide a source for the extra background field dependence of . To stress this point, it may be illustrative to write

(1.5) |

where now depends on two full fledged metrics, and .

One way to extract physics information from the FRGE is by applying perturbation theory [13, 14, 15]. The main virtue of the flow equation is, however, that its use is not limited to perturbation theory and can also be employed to obtain non-perturbative information. Here the most common approximation scheme consists of truncating the space of functionals to a finite-dimensional subspace and projecting the flow equation onto this subspace. Studying the gravitational RG flow, within these truncations, the most exciting result obtained to date is a substantial body of evidence [8],[13]-[50] in support of Weinberg’s asymptotic safety scenario for gravity [16, 17, 18]. All truncations of the FRGE have displayed a non-Gaussian fixed point (NGFP) of the gravitational RG flow and there is also mounting evidence [13, 14, 32, 33] that its number of relevant couplings is actually finite. This fixed point may thus provide a fundamental and predictive UV completion of gravity within Wilson’s generalized framework of renormalization. (See [51]-[54] for cosmological applications of this framework.)

While already impressive, a serious caveat in this body of evidence is that all computations carried out to date are essentially “single-metric” and do not properly reflect the bimetric nature of the flow equation. Typically the ansatz made for falls into the class

(1.6) |

where are interaction monomials built from the expectation value metric only. The split-symmetry violating interactions built from both are encoded in which, by construction, vanishes for . Finally, and are taken as the classical gauge-fixing and ghost terms. The single-metric computations then proceed by taking the second variation of with respect to the fluctuation fields and setting afterwards. This suffices to extract the running of the coupling constants contained in .

The potentially problematic feature of these computations is that the -functions encoding the running of the coupling constants multiplying interactions built from the “genuine” metric are tainted by contributions originating from pure background terms. A single-metric truncation does not distinguish between the running of, say,

(1.7) |

It determines the running of a linear combination of the couplings and their background analogs only.

The bimetric nature of the gravitational average action has been appreciated only very recently, by carrying out preliminary studies in conformally reduced gravity [43], and studying the bimetric terms induced by quantum effects in the matter sector [44]. Currently there are no results on full-fledged gravity available. There are, however, several good reasons why disentangling between the and contributions is of central importance. Firstly, the single-metric computations do not account for the bimetric nature of the gauge-, ghost-, and cutoff-terms which inject an extra -dependence into the path integral. They are sources of split-symmetry breaking action monomials, which will inevitably be “switched on” along the RG flow, leading to new interactions which are either constructed from the background metric only or a mixture of background and expectation value metric. The preliminary results obtained in [43] and [44] suggest that disentangling these interactions may lead to a significant alteration of the results obtained in the single-metric case. In particular, separating the and pieces in may destroy the NGFP underlying Weinberg’s Asymptotic Safety idea. Secondly, identifying does not probe the direction of the IR cutoff in theory space which may give rise to a important contribution to the RG flow in the UV. Thirdly, employing the background field method, the counterterms found in perturbation theory are constructed solely from the background fields [55], so that isolating their effect requires careful distinction between the and field monomials.

Based on this motivation, our work initiates the first study of the full-fledged gravitational RG flow in a fully bimetric setting. Concretely, we study the RG flow in the bimetric Einstein-Hilbert truncation which distinguishes the four action monomials in eq. (1.7), i.e.

The rest of the paper is organized as follows. In Section 2 we describe the details of the setup and state our main new result: the -functions of the double-Einstein-Hilbert truncation in four dimensions. The properties of these -functions are analyzed in Section 3 and we discuss our findings in Section 4. A brief summary of the heat-kernel techniques employed in the paper and the rather lengthy -functions for the bimetric Einstein-Hilbert truncation valid for any spacetime dimension are relegated to the Appendices A and B, respectively.

## 2 -functions of the double-Einstein-Hilbert truncation

In this section we derive the -functions of the double-Einstein-Hilbert truncation.
Besides the Einstein-Hilbert action constructed from , known from previous single-metric
truncations, the corresponding truncation ansatz also encompasses a Einstein-Hilbert
action constructed from the background metric and a simple class
of interaction monomials including both and . This prototypical setup accounts for
the bimetric nature of the FRGE (1.3), for the first time disentangling the quantum
gravity effects in and in a full gravity computation.^{1}^{1}1For a related analysis
in the framework of conformally reduced and matter induced gravity, see ref. [43] and [44], respectively.

### 2.1 The truncation ansatz

Our ansatz for the double-Einstein-Hilbert truncation takes the form

(2.1) |

where is the metric part of the effective action (built from both and ) which we supplement by the classical gauge-fixing and ghost action and , respectively. Explicitly, we consider the following one-parameter class of gravitational actions

(2.2) |

Here the unbared (bared) quantities are constructed from the expectation value metric (background metric ). Furthermore, and denote the Newton’s constants and cosmological constants, with the superscript and indicating that the corresponding interaction term is constructed from and respectively. The form of the bimetric term appearing in the last line is motivated by the structure of the flow equation encountered in [44], where it is precisely the ratio that naturally appears on its right-hand-side. In the following, we will consider integer exponents only. The terms , give rise to the monomials multiplying or , respectively, and are already included in the Einstein-Hilbert actions constructed from the “genuine” and background metric.

In the sequel, we will work with the geometric gauge-fixing, setting

(2.3) |

and subsequently taking the Landau gauge limit . As it was shown in [13, 32], this gauge choice is perfectly adapted to the transverse-traceless decomposition utilized in Section 2.3 below, where it leads to significant simplifications. The ghost action exponentiating the resulting Faddeev-Popov determinant takes the form

(2.4) |

with

(2.5) |

This completes the specification of our truncation ansatz.

### 2.2 The conformal projection technique

Our next task is to project the gravitational RG flow onto the subspace spanned by (2.1), so that we can compute the -functions for the -dependent couplings contained in . Obviously, this cannot be achieved by evaluating the flow equation setting , which underlies the single-metric computations. Instead, we will resort to the conformal projection technique introduced in [44] which identifies and up to a constant conformal factor:

(2.6) |

Substituting this identification into (2.2) and performing a double-expansion in and the background curvature yields

(2.7) |

where the dots indicate higher powers in the expansion. Plugging this expansion into (1.3), the left-hand-side of the equation indicates that the running of the coupling constants contained in the ansatz (2.2) is captured by the coefficients

(2.8) |

of this double expansion. Thus, by extracting the corresponding contributions from the right-hand-side of the flow equation, we are able to disentangle the running of and together with .

At this stage, we feel obliged to add the following word of caution. While the conformal projection technique employed here is capable of distinguishing between the running coupling constants associated with monomials built from , the background metric or a mixture of the two, it has only limited power for resolving different tensorial structures. As an illustrative example, we consider the following three mixed terms in :

(2.9) |

Under the conformal identification (2.6) all three invariants are projected onto the same structure, ,
and are thus indistinguishable.^{2}^{2}2The situation is completely analogous to the projection of the three -couplings on a spherical background [13, 22, 23, 24, 33], which also determines the -functions for one particular linear combination of the three couplings only. Resolving this ambiguity will require a much more sophisticated computational technique, like the -expansion advocated in [44]. Owed to the increased technical complexity of the -expansion, however, we will refrain from resolving this ambiguity and resort to the conformal projection scheme in the sequel. In any case, we expect that the latter is sufficiently elaborate to give some first insights into the properties of the gravitational RG flow taking the bimetric nature of the flow equation into account.

### 2.3 Hessian , cutoff implementation, and the flow equation

In the next step, it is convenient to first compute the quadratic forms arising at second order in the -expansion of . These forms will considerably simplify the computation of the Hessian later on. We start by constructing the Taylor series of around the background (2.6):

(2.10) |

To simplify the notation it is useful to abbreviate the interaction monomials in by

(2.11) |

The interaction term multiplying and are special cases of , corresponding to .

Expanding the curvature invariants up to second order in and setting afterwards, the second variations become

(2.12) |

Here , , and we have freely integrated by parts. All indices are raised and lowered with the background metric. Furthermore, we have specified the background metric as the one of the -dimensional sphere, satisfying

(2.13) |

which suffices to keep track of the expansion in the background scalar curvature terms, cf. eq. (2.8). All terms are generalized homogeneous in the conformal factor and scale with exponents

(2.14) |

for the cosmological constant, , and , respectively.

In order to diagonalize the Hessian we implement the transverse-traceless (TT)-decomposition [56] of the fluctuations fields with respect to the spherical background, according to

(2.15) |

for the metric fluctuations and

(2.16) |

for the ghost fields, respectively. The component fields are subject to the (differential) constraints

(2.17) |

The resulting Jacobian determinants resulting from the TT-decomposition are exponentiated by introducing suitable auxiliary fields along the lines of the Faddeev-Popov trick. For this purpose, we introduce the transverse vector ghosts , the transverse vector , the scalar ghosts , the real scalar , and a complex scalar , which enter into the auxiliary action (see [13, 32] for more details). On the spherical background (2.13) it reads:

(2.18) |

Substituting the decomposition (2.15), it is now straightforward to obtain the component field representation of (2.12)

(2.19) |

where we abbreviated

(2.20) |

Analogously, the quadratic form arising from the gauge-fixing term reads

(2.21) |

while the ghost action gives

(2.22) |

Reinstalling the -dependent coupling constants multiplying the action monomials, it is then straightforward to compute the Hessian

(2.23) |

where is the multiplet of all fluctuation fields, and takes values 0 or 1 for Grassmann-even or odd, respectively. The matrix elements of in field space are then summarized in the second column of Table 1. In order to uniformize the expressions in the gravitational sector, we introduced the -dependent constants

(2.24) |

Fields | Hessian | Kernel |
---|---|---|

Notably, the entries of in the and sector contain the contribution from the gauge-fixing term only, and omit the terms originating from . The latter are subleading in and can be shown to drop out of the flow equation once the Landau limit is taken. Anticipating this result, Table 1 gives only the leading -terms.

The next step in obtaining the -functions is the construction of the matrix-valued IR-cutoff operator . This operator provides a -dependent mass term for the fluctuation fields which is built from the background metric only. This implies that cannot depend on . The -dependence of then enforces a modification of the cutoff schemes used in previous single-metric computations. Focusing on the cutoff of Type I [14] the rule for determining in a single-metric truncation adjusts in such a way that all covariant Laplacians are dressed by a -dependent mass-term according to

(2.25) |

Here and is a shape function interpolating monotonously between and . For the bimetric setup of this paper, we generalize this rule in the minimal sense

(2.26) |

i.e. provides a -dependent mass term at zeroth order in the -expansion. This definition reduces to the standard Type I cutoff implementation for the single-metric case.

Applying (2.26) to the matrix entries then determines the entries of uniquely. The result is displayed in the third column of Table 1, with the explicit form of being

(2.27) |

With this result, we now have all ingredients for the explicit construction of the operator trace appearing on the right-hand-side of the flow equation resulting from our ansatz. Utilizing the block diagonal form of in field-space, this trace decomposes as

(2.28) |

Here the left-hand-side is given by the -derivative of the double expansion (2.7), while , , , and are the operator traces constructed from the transverse-traceless -fluctuations, the metric scalar , the gauge-fixing sector in the second block, and the auxiliary field contribution given in the third block of Table 1, respectively. By first carrying out a double expansion of the trace-arguments with respect to , retaining all the terms indicated in (2.8), the traces can be evaluated using standard early-time heat-kernel techniques. Since the corresponding computation is rather technical, it has been relegated to Appendix B, where we also give the explicit expressions for the -dimensional -functions. For the rest of the paper we restrict ourselves to the case for which the explicit -functions are given in the next subsection.

We close this subsection with a remark on the unphysical exceptional modes arising from working with the TT-decomposition on a spherical background. Performing a spectral decomposition of the component fields in terms of -eigenmodes, one finds that the two lowest scalar eigenmodes (the constant mode and the lowest non-trivial eigenfunction satisfying the conformal Killing equation) and the lowest vector-eigenmode (satisfying the Killing equation) do not contribute to and therefore require special care when evaluating the operator traces (2.28). One finds that their contribution to the flow equation enters only at , however, so that this subtlety can safely be disregarded in the present computation.

### 2.4 The four-dimensional -functions

Based on eq. (2.28), the -functions

(2.29) |

arising in the -dimensional double-Einstein-Hilbert truncation are computed in Appendix B. To simplify our notation we will set in the following and work with the -dimensional version of the dimensionless coupling constants (B.1)

(2.30) |

Furthermore, we define the anomalous dimensions of the two Newton constants as

(2.31) |

The -functions are most conveniently expressed in terms of the dimensionless threshold functions (A.5). In this context, it turns out to be convenient to introduce a short-hand notation for the - and -independent terms appearing in the square brackets in the first and second line of Table 1. The arguments of the threshold functions are then given by the -limit of these terms. For the and -contributions they read

(2.32) |

Furthermore, we denote the first and second derivative of these terms with respect to by

(2.33) |

These expressions can also be obtained by setting in (B.2) and (B.3), respectively.

Using these notations the -functions in four spacetime dimensions can be summarized as follows. First we explicitly solve (B.11) from Appendix B for . This yields the anomalous dimension

(2.34) |

The and are obtained by splitting into the terms independent and linear in :

(2.35) |

Here the threshold functions and without explicit argument are understood to be evaluated at . Utilizing (2.34) the -functions for and can be obtained from equation (B.13):

(2.36a) | ||||

(2.36b) | ||||

(2.36c) |

The running of the background couplings is governed by

(2.37a) | ||||