We set up a nonperturbative gravitational coarse graining flow and the corresponding functional renormalization group equation on the as to yet unexplored “tetrad only” theory space. It comprises action functionals which depend on the tetrad field (along with the related background and ghost fields) and are invariant under the semi-direct product of spacetime diffeomorphisms and local Lorentz transformations. This theory space differs from that of Quantum Einstein Gravity (QEG) in that the tetrad rather than the metric constitutes the fundamental variable and because of the additional symmetry requirement of local Lorentz invariance. It also differs from “Quantum Einstein Cartan Gravity” (QECG) investigated recently since the spin connection is not an independent field variable now. We explicitly compute the renormalization group flow on this theory space within the tetrad version of the Einstein-Hilbert truncation. A detailed comparison with analog results in QEG and QECG is performed in order to assess the impact the choice of a fundamental field variable has on the renormalization behavior of the gravitational average action, and the possibility of an asymptotically safe infinite cutoff limit is investigated. Implications for nonperturbative studies of fermionic matter coupled to quantum gravity are also discussed. It turns out that, in the context of functional flow equations, the “hybrid calculations” proposed in the literature (using the tetrad for fermionic diagrams only, and the metric in all others) are unlikely to be quantitatively reliable. Moreover we find that, unlike in perturbation theory, the non-propagating Faddeev-Popov ghosts related to the local Lorentz transformations may not be discarded but rather contribute quite significantly to the beta functions of Newton’s constant and the cosmological constant.

## 1 Introduction

In classical General Relativity there exists a remarkably rich variety of different variational principles which give rise to Einstein’s equation, or equations equivalent to it but expressed in terms of different field variables. The best known examples are the Einstein-Hilbert action expressed in terms of the metric, , or the tetrad, respectively, . The latter action functional is obtained by inserting the representation of the metric in terms of vielbeins into the former: .

Another classically equivalent formulation, at least in absence of spinning matter, is provided by the first order Hilbert-Palatini action which, besides the tetrad, depends on the spin connection assuming values in the Lie algebra of . Variation of with respect to leads, in vacuo, to an equation of motion which expresses that this connection has vanishing torsion. It can be solved algebraically as which, when inserted into , brings us back to .

Another equivalent formulation is based upon the self-dual Hilbert-Palatini action which depends only on the (complex, in the Lorentzian case) self-dual projection of the spin connection, [1, 2, 3, 4]. This action in turn is closely related to the Plebanski action [5], containing additional 2-form fields, and to the Capovilla-Dell-Jacobson action [6] which involves essentially only a self-dual connection. Similarly, Krasnov’s diffeomorphism invariant Yang-Mills theories [7] allow for a “pure connection” reformulation of General Relativity as well as deformations thereof.

The above variational principles are Lagrangian in nature; the fields employed provide a parametrization of configuration space. The corresponding Legendre transformation yields a Hamiltonian description in which the “carrier fields” of the gravitational interaction parametrize a phase-space now. In this way the ADM-Hamiltonian [8] and Ashtekar’s Hamiltonian [9], for instance, make their appearance.

Regarding the ongoing search for a quantum (field) theory of gravity this multitude of classical formalisms offers many equally plausible possibilities to explore. A priori it is not clear which one of the above hamiltonian systems, if any, is linked to the as to yet unknown fundamental quantum theory in the simplest or most easy to guess way.

In the traditional approach of “quantizing” a known classical system more input than the field equations, such as the Lagrangian is needed, so that classically equivalent theories might possibly give rise to inequivalent quantum theories. Among those, at most one can be “correct”, in the sense of being realized in Nature. Of course, given the limitations of the available observational and experimental data it is not clear whether the most natural and/or simplest presentation of the classical limit emerging from this “correct” theory is in the above list, or even close to it. Up to now, because of the many well known conceptual and technical problems [4] we are not in a position to discriminate the various classical gravity theories on the basis of the quantum properties they imply.

Rather than trying to quantize a given classical dynamical system, there is another strategy one can adopt in order to search for a quantum theory consistent with the observed classical limit, the Asymptotic Safety program [10, 11, 12, 13, 14]. One of its advantages as compared to a “quantization” is that it depends on the classical input data to a lesser extent. The idea is to fix a certain theory space of action functionals, a coarse graining flow of it, and then search for a renormalization group (RG) fixed point (FP) on it at which the infinite ultraviolet (UV) cutoff limit can be taken in a “safe” way.

While originally motivated by the possibility of sidestepping the problem of perturbative nonrenormalizability, this search strategy in principle can predict the theory’s fundamental action. The only input needed is theory space. Once it is chosen one can “turn the crank” and, in case a suitable fixed point is found, construct a UV-regularized functional integral representation of the resulting theory [15]. Only at this very last stage we can identify the hamiltonian system which, implicitly, was quantized by taking the continuum limit at the respective fixed point.

To characterize a theory space we must pick a certain set of fields, collectively denoted , a space of action functionals , and a group of symmetry transformations under which they are required to be invariant. In this sense, the above classical gravity theories motivate us to explore, for instance, the case where is the metric and the diffeomorphism group, or, as in Einstein-Cartan theory, where is the semidirect product of local Lorentz transformations and spacetime diffeomorphisms.

We emphasize that these spaces and symmetries are the only “inspiration” drawn from the classical examples. Their dynamics, i. e. the specific classical action they postulate, plays no special rôle in the Asymptotic Safety program. It is just one special point in the pertinent theory space, and usually not the sought for fixed point of the RG flow.

Most of the work on Asymptotic Safety has been done in “Einstein” gravity which, by definition, is based upon a theory space of functionals111The dots stand for the background fields and Faddeev-Popov ghosts to be introduced later. invariant under , the diffeomorphisms of the spacetime manifold . Recently also first investigations of the “Einstein-Cartan” choice

 TEC={A[ea μ,ωab  μ,⋯];G=Diff(M)⋉O(4)loc} (1.1)

were published [16]. Here plays the rôle of the Euclidean Lorentz group222We shall consider the case of Euclidean signature throughout., and denotes the group of the corresponding local gauge transformations.

The present paper instead is devoted to the “tetrad only” theory space pertaining to a dimensional spacetime :

 Ttet={A[ea μ,⋯];G=Diff(M)⋉O(d)loc}. (1.2)

With actions depending on the vielbein only, this space is intermediate between and : Coming from the “Einstein” side it generalizes by declaring the fundamental field and the metric a composite thereof, . Conversely, coming from the “Einstein-Cartan” side, every implies a certain upon inserting , where is the torsion-free (Levi-Civita) connection the vielbein gives rise to.

There are various independent motivations for this investigation.

(A) The first functional RG based results obtained on the Einstein-Cartan theory space , in a truncation with a scale dependent Hilbert-Palatini action (including a running Immirzi term), show certain characteristic differences in comparison with the familiar case of truncated with a running Einstein-Hilbert action; in particular, the results show a stronger RG scheme and gauge fixing dependence than the older ones on the “Einstein” case [16]. It would be interesting to know whether these differences are mainly due to the use of the different truncations, different field variables, or both. In the present paper we shall change only the field variable (and the group G correspondingly), but not the truncation, and so it should be possible to disentangle the two sources of deviations to some extent.

We note here that, like most settings of quantum field theory, the flow equation of the average action is not invariant under diffeomorphisms in field space, . Thus, at intermediate steps, as long as one does not compute observables, there is no reason to expect any field parametrization independence. Moreover, and perhaps this is even more important, the gauge fixing and ghost sectors are quite different for and , respectively. Therefore the -functions for the running Newton constant or cosmological constant , for instance, may well depend on whether the functional renormalization group equation (FRGE) is formulated in terms of the metric or tetrad. Similar remarks apply also to a recent study of the perturbative RG running of and the Immirzi parameter [17].

(B) On theory spaces involving fermions coupled to gravity introducing vielbeins is compulsory. Besides the pure gravity couplings, such as , etc. the average action will then depend on additional couplings related to the matter field monomials. If we collectively denote these couplings by and , respectively, their -functions are of the form

 βgrav =βgravgrav(ugrav)+β% matgrav(ugrav,umat) (1.3) βmat =βmatmat(umat)+β% gravmat(ugrav,umat) (1.4)

Diagrammatically speaking, the two parts and of the pure gravity -functions stem from the graviton and matter loops, respectively. Conversely, the running of the matter couplings has a part due to pure matter loops, , plus mixed matter-gravity contributions, .

In order to get a first impression of the impact the fermions have on the gravitational RG flow one might neglect the running of the matter couplings, and try to compute only. While the evaluation of from the fermion loops clearly requires a vielbein and a spin connection, the pure gravity part does not obviously do so. From a pragmatic point of view it is therefore tempting to take the part from a (much simpler, and already available) computation in the metric formalism. The invariants occurring in the latter one would interpret as . For some (but not all) field monomials in this establishes a correspondence to monomials in , and one can try to identify their running prefactors; for instance, , when and denote the determinants of and , respectively.

Thus it seems that only the fermion loops, , need to be calculated. This requires fixing a Lorentz gauge in order to associate a unique to a given , and for one might take the unique Levi-Civita connection associated to this vielbein, .

We shall refer to this procedure as a hybrid calculation. Clearly it can be meaningful at most within a truncation of and that allows an identification of monomials; an example is the Einstein-Hilbert action regarded as a functional of and , respectively, with the same two couplings and occurring in both cases. At the exact level there exists certainly no such one-to-one correspondence between action monomials in and . Nevertheless, if it was possible to establish the “hybrid” scheme as a reliable approximation, this would be of considerable importance for the feasibility of practical calculations.

As to yet, all investigations for the gravity + fermions theory space, in particular in the Asymptotic Safety context, are, in fact, hybrid computations of this form [18, 19, 20]. They combine the metric-formalism -functions for and in the Einstein-Hilbert truncation with certain matter contributions , solve for , and insert the result into (1.4) to obtain the running of the matter couplings. In ref. [20] the gravity corrections to certain 4-fermion couplings were studied in this way.

A necessary condition for the consistency of the hybrid approach is that the pure gravity part does not change much when we switch from to as the fundamental field variable in the Einstein-Hilbert truncation. In the present paper we shall be able to explicitly test whether or not this is actually the case. It will be one of our main results that the hybrid scheme is very hard, if not impossible to justify, at least at the quantitative level. We shall demonstrate in detail that if one aims at some degree of numerical precision, one should consistently work with the vielbein and its corresponding ghost system already at the pure gravity level.

(C) Picking the vielbein as the fundamental field variable requires fixing a gauge. In perturbation theory, a popular choice is the Deser-van Nieuwenhuizen algebraic gauge fixing condition where the antisymmetric part of the matrix is required to vanish [21, 22]. As has parameters, this reduces the independent components of to , which is precisely the number of independent fields has in dimensions.

It was shown that, for this gauge, and in perturbation theory, no Faddeev-Popov ghosts need to be introduced for the factor of G, and that it allows to explicitly express vielbein fluctuations purely in terms of metric fluctuations [23]. Therefore the point of view was advocated that even in presence of fermions the vielbein can be eliminated in favor of the metric.

While this method was proven to be correct in a well defined perturbative context, recently it has been proposed to use this same procedure, in particular the omission of the ghosts, also in the context of a nonperturbative flow equation for the gravity–fermion system [19, 20]. If applicable, it would provide a very economic framework for hybrid computations of the type sketched above.

However, as we are going to discuss in detail there are reasons to doubt that the perturbative arguments justifying the omission of the ghosts carry over to the nonperturbative setting of the FRGE. In fact, in perturbation theory the ghosts are omitted since their inverse propagator contains no derivatives, they are non-propagating, leading to a trivial Faddeev-Popov determinant. In the FRGE, instead, a straightforward evaluation of the functional traces cuts off all field modes in a uniform fashion, no matter if their kinetic term contains 2, or more, or no derivatives at all.

In the present paper we shall explicitly evaluate the contributions to from the non-propagating ghosts pertaining to the symmetric vielbein gauge, and we shall analyze whether they really can be discarded in setting up the flow equation for the average action.

Fortunately, the particular fermionic -functions computed in [20] happen to be independent on whether the ghosts are retained or not. However, in future extensions of such studies it will be important to know how to treat them correctly.

The remaining sections of this article are organized as follows. In Section 2 we summarize various preliminaries on the gravitational average action and its FRGE which will be needed later on. In Section 3 we focus on the “tetrad only” theory space in the Einstein-Hilbert truncation, and calculate the corresponding -functions. The resulting RG flow is analyzed with numerical methods in Section 4 then. Our results, in particular on the issues (A)–(C) raised above, are summarized in Section 5.

## 2 The average action approach to quantum gravity

Introducing the scale-dependent effective average action it has been possible to construct a functional RG flow for quantum gravity [11]. This “running action” can be considered the generating functional of the 1 PI correlation functions that take into account quantum fluctuation of all scales between the UV and an infrared cutoff scale . For it is closely related to the bare action and for vanishing cutoff it coincides with the usual effective action . Its scale-dependence is governed by an exact renormalization group equation:

 ∂tΓk=12STr[∂tˆRkΓ(2)k+ˆRk]. (2.1)

Here , and denotes the matrix of the second functional derivative of with respect to the dynamical fields. Furthermore, is an operator that implements the infrared cutoff in the path integral by replacing the bare action with where is quadratic in the fluctuations, . Finally, the supertrace in (2.1) comprises a trace over all internal indices as well as an integral/sum over all modes of ; for fermionic fields it contains an additional minus sign [24].

As is a generic point in “theory space”, i. e. a functional of a given set of fields restricted only by the required symmetries, solving this exact equation is usually a formidable task. For this reason one has to resort to truncations of theory space in order to find approximate solutions to eq. (2.1). This is done by expanding in a basis of integrated field monomials , i. e. and restricting the sum to a finite number of terms. The scale-dependence of is then described by a finite number of running couplings . If we project the RHS of (2.1) onto this subspace of theory space the functional equation reduces to a coupled system of ordinary differential equations in these couplings.

If we describe pure gravity with the metric as field variable, the simplest truncation is the Einstein-Hilbert truncation with only two running couplings: Newton’s constant and the cosmological constant . As gravity is a gauge theory we also have to add a gauge fixing and a ghost term to the truncation ansatz; its running shall be ignored in our approximation. Our ansatz for can therefore be decomposed into a “bosonic part” and the classical ghost contribution :

 Γk=ΓEHk+Γgf+Sgh≡˘Γk+Sgh (2.2)

Using this decomposition the FRGE can be written in the following form:

 ∂tΓk=12Tr⎡⎢⎣∂tˆRk˘Γ(2)k+ˆRk⎤⎥⎦−Tr⎡⎢ ⎢⎣∂tˆRghkS(2)gh+ˆRghk⎤⎥ ⎥⎦. (2.3)

The gravitational average action heavily relies on the background field method [25]. The field chosen to represent gravity is split arbitrarily into a background part and a fluctuation: . is constructed as a background gauge invariant functional of both fields, , i. e. it is invariant under a simultaneous action of on both and . As we only deal with a so-called single metric truncation [26] in this paper, we will set the fluctuations to zero after the second derivative with respect to the fluctuations has been taken. At the end we therefore arrive at a system of differential equations for the running couplings parametrizing .

For example, in the “tetrad only” case the average action is a curve in the theory space which, to be precise now, consists of invariant functionals of the type ; besides the vielbein and its background, they depend on the diffeomorphism ghosts and ghosts . Instead of we shall often consider the vielbein fluctuation the independent argument of the action.

## 3 Tetrad theory space in Einstein-Hilbert truncation

At this point we take two decisions. One of them refers to the deeper level of the exact theory, the other to the practical (computational) level of concrete approximations.

First, we fix the theory space to be the “tetrad only” one, , so that all actions to be considered depend only on , along with the corresponding background and ghost fields.

Second, to be able to perform practical calculations we decide to truncate by an ansatz for which is essentially a -dependent version of the Einstein-Hilbert action reexpressed in terms of the tetrad, .

### 3.1 The FRGE on Ttet

In this subsection we derive the RG flow of tetrad gravity in the Einstein-Hilbert truncation

 ˘Γk[e,¯e]=−116πGk∫ddx√g(e)(R(g(e))−2¯λk)+Γgf[e,¯e]. (3.1)

This action involves two running couplings, the cosmological constant and Newton’s constant ; the latter is frequently expressed in terms of the dimensionless function according to with a constant .

To be as general as possible we re-express the metric in terms of the new field variable in the following way:

 gμν=ξ−1ea μeb νηab. (3.2)

This representation resembles the usual vielbein decomposition of the metric, except for the additional free parameter . For this reason we will refer to the field as a generalized vielbein for a given . Treating as the independent variable we assume that the basis 1-forms indeed form a non-degenerate co-frame. The parameter is merely a mathematical tool that enables us to study a continuous class of field redefinitions at a time.

As for the usual vielbein this generalized decomposition of the metric is not unique, but there exists an manifold of vielbein fields corresponding to the same metric. We will treat this arbitrariness as an additional gauge freedom, such that the total group of gauge transformations is given by . Compared to the metric formulation we therefore have to add a second gauge fixing term; the corresponding background gauge invariant ghost-action can be constructed using the formalism introduced in [27].

If we decompose both the metric and the vielbein into background fields and fluctuations, we find

 ¯gμν+¯hμν=gμν=ξ−1(¯ea μ+¯εa μ)(¯eb ν+¯εb ν)ηab=ξ−1¯ea μ¯eb νηab+2ξ−1¯ε(μν)+O(¯ε2). (3.3)

Here and in the following we use the background vielbein to change the type of the first (i. e., frame) index of the vielbein fluctuation: . We see that the symmetric part of the vielbein fluctuations, , is proportional to the metric fluctuations in lowest order, while we can relate the additional gauge degrees of freedom carried by to the antisymmetric part of the fluctuations, .

This observation motivates the following choice of gauge conditions. For the diffeomorphisms we choose the usual harmonic gauge fixing function for metric fluctuations, replacing , with :

 Fμ=2√2κξ−1(¯Dν¯ε(μν)−12¯Dμ¯εν ν). (3.4)

The transformations are gauge fixed using

 Gab=2ξ−12¯gμν¯ε[a μ¯eb] ν=2ξ−12¯ε[ab], (3.5)

corresponding to a suppression of the antisymmetric vielbein fluctuations.

With these gauge conditions the gauge fixing term in the effective average action assumes the usual form, involving parameters and :

 Γgf,k[e,¯e]=12αD∫ddx√¯g¯gμνFμFν+12αL∫% ddx√¯gGabGab. (3.6)

In the following we fix the diffeomorphism gauge parameter to which leads to the same cancelation in the kinetic operator as in metric gravity [11].

In order to obtain a background G-invariant ghost action with respect to both transformations and diffeomorphisms, we can make use of the Faddeev-Popov construction only if we first reparametrize the gauge transformations in such a way, that the new generators of diffeomorphisms and transformations commute. This corresponds to an covariantization of the Lie derivative. Following this procedure, described in detail in [16, 27], while treating the ghost sector classically (i. e. we can set already at the level of the ghost action) we arrive at

 (3.7)

Here represent the diffeomorphism ghosts and the ghost fields.

As the infinitesimal transformation under diffeomorphisms contains a derivative, while the corresponding transformation does not, the diffeomorphism ghosts have a canonical mass dimension of one unit less compared to the ghosts. In order to obtain a Hessian operator of a well-defined mass dimension we have rescaled the fields , with an arbitrary mass parameter ; consequently the Hessian operator obtains a mass dimension of 2.

### 3.2 Structure of the vielbein sector

After having presented the details of our truncation we can now pass on to the evaluation of the FRGE (2.3) in this truncation. On the LHS of the equation, after setting , we obtain the same result as in the metric version of the Einstein-Hilbert truncation [11]:

 ∂tΓk[e,e]=2κ2∫ddx√g(e)[−R(g(e))∂tZNk+2∂t(ZNk¯λk)] (3.8)

On the RHS of the FRGE, however, we find two types of additional contributions to the supertrace as compared to those already present in the metric description. While the second type of contributions is due to the extended gauge group of the theory, the first type is closely linked to the off-shell character of the FRGE. This can be seen as follows.

In order to obtain we expand to second order in the vielbein fluctuations and read off the operator from the quadratic term . As is already quadratic in the fluctuations we only have to expand . For

we find

Here we have used the chain rule for functional derivatives. Obviously, the first term on the RHS of (3.10) corresponds exactly to the one known from the metric calculation, while the second term is due to the field redefinition. We note that those two terms come with different powers of , which enables us to keep track of their respective origin during the entire calculation and in the final result. This was in fact our main motivation for introducing this book-keeping device.

Note also that in (3.10) the term due to the field redefinition is proportional to the first variation . So it would vanish if we were to go “on shell”, i. e. to insert a special metric or vielbein which happens to be a stationary point of . We emphasize that in the process of computing -functions this would be a severe mistake. To see this, consider an (exact) average action expanded as

 Γk[ϕ,¯ϕ]=∑αcα(k)Iα[ϕ,¯ϕ], (3.11)

where denote the running couplings and the ’s are -invariant basis functionals (integrated field monomials, say) independent of . When represented in this fashion one may think of as a “generating function” for the set of running couplings, , which are “projected out” by expanding in the basis . In this picture the fields , have a subordinate status only. They serve as arguments of the ’s, and their only rôle is that of a dummy variable needed in order to define the basis functionals . Therefore, in order for the set to remain complete it is in general not possible to narrow down the function space , are drawn from in any way, for instance by stationary point conditions or the like. In this sense, the average action and its associated FRGE are intrinsically “off shell” in nature.

At most at the level of truncations where the set is incomplete anyhow we may opt for special choices of the fields (e. g. satisfying convenient symmetry conditions) as long as the invariants in the truncation ansatz when calculated for these fields can still be distinguished from all other invariants and from each other. This is an often used computational trick that simplifies practical calculations without affecting the result in any way.

For the total quadratic part of the action we obtain, with ,

 ˘Γquadk[¯ε;¯e] =4ZNkκ2ξ2∫ddx√¯g¯ε(μν)[−Kμν  ρσ¯D2+Uμν  ρσ]¯ε(ρσ) (3.12) +2ZNkκ2ξ∫ddx√¯g(¯Rμν+Λ¯gμν−¯R2¯gμν)¯εa(ν¯εa μ)+12αL4ξ∫ddx√¯g¯ε[ab]¯ε[ab]

where

 Kμν  ρσ≡14(δμρδνσ+δμσδνρ−¯gμν¯gρσ) (3.13)

and

 (3.14)

We observe that the first term on the RHS of (3.12) is exactly the contribution known from the metric computation [11]; in particular thanks to all non-minimal terms in the differential operator canceled. The second and third terms in (3.12) correspond to the already mentioned first and second type of new contributions, respectively.

In a next step we decompose the vielbein fluctuations into their symmetric traceless part , antisymmetric part , and trace part , according to

 ¯εμν=ˆεμν+~εμν+1d¯gμνϕε (3.15)

with , and . In addition we specify the background spacetime to be a maximally symmetric Einstein space with

 ¯Rμνρσ=1d(d−1)[¯gμρ¯gνσ−¯gμσ¯gνρ]¯Rand¯Rμν=1d¯gμν¯R. (3.16)

This spacetime is still sufficiently general to identify the contributions to the relevant invariants and unambiguously. Within the present truncation it is thus a permissable restriction of the function space of the metric; it does not affect the generality of the calculation and so is an example of the computational trick mentioned above.

Using the relations (3.15) and (3.16) the quadratic part of the action reads

with the constants

 CT(ξ)≡d(d−3)+4d(d−1)−d−22dξ,CS(ξ)≡d−4+ξd. (3.18)

Note that whereas the symmetric tensor has a standard positive definite kinetic term, its antisymmetric counterpart is non-propagating; the -bilinear contains no derivatives at all, but only a (gauge dependent) mass term. Note also that in the trace part has a “wrong sign” kinetic term, reflecting the well known conformal factor instability [11].

Let us now fix the precise form of the cutoff operator in the various sectors of field space. Generically it has the structure

 ˆRk=Zkk2R(0)(−¯D2/k2), (3.19)

where is a matrix in field space, and is a dimensionless “shape function” that interpolates smoothly between and . At least in simple matter field theories on a rigid background spacetime, there is a simple rule for finding a suitable , and this rule has also been used in the metric calculation in [11]: If a certain field mode has a kinetic operator of the form , the is fixed in such a way that in the sum this operator gets replaced by .

In the case at hand it is straightforward to implement this rule for and . In the different sectors we choose

 (Zk)ˆεˆε=2ZNkξ−2κ2,(Zk)~ε~ε=2ξ−1ZNkκ2,(Zk)ϕεϕε=−2ξ−2ZNkκ2d−22d. (3.20)

As for the antisymmetric tensor , we fixed the corresponding in such a way that, taking the overall prefactor into account, the addition of to the inverse propagator replaces the square brackets in the -bilinear of (3.17) by

 [k2R(0)(−¯D2/k2)+1ZNkκ2αL+¯λk−d−22d¯R]. (3.21)

Now we have specified all ingredients entering the supertrace on the RHS of (2.3) in the different sectors.

First of all we note that the contributions of the antisymmetric sector vanish in the limit of , as this part of the trace is given by

 12Tr~ε~ε[∂t(ZNkk2R(0)(−¯D2/k2))ZNk(k2R(0)(−¯D2/k2)+¯λk+1/(ZNkκ2αL)−¯R(d−2)/(2d))]=αL2Tr~ε~ε[∂t(ZNkk2R(0)(−¯D2/k2))ZNk(αLk2R(0)(−¯D2/k2)+αL¯λk+1/(ZNkκ2)−αL¯R(d−2)/(2d))]αL→0−−−→0. (3.22)

This behavior is easy to understand as the limit corresponds to a sharp implementation of the gauge condition that introduces a delta functional into the path integral. Since the domain of tensors with is invariant under the coarse graining operation it is obvious that the antisymmetric fluctuations should not contribute to any RG running in this limit. From now on we will choose the gauge in order to simplify the discussion.

In this particularly simple gauge the quadratic form (3.17) is structurally similar to the corresponding equation in the metric formalism, see eq. (4.12) in [11]. However, the prefactors of in the various terms of and the now -dependent coefficients , of the curvature scalar are different and this will have a rather significant impact on the resulting RG flow. Replacing these constants appropriately in the original metric calculation we can obtain the “bosonic” contributions to the -functions without a new calculation from those of [11].

### 3.3 Propagating and non-propagating ghosts

Let us move on and discuss the ghost sector. Here we choose the cutoff operator to be

 ˆRghk=⎛⎝√2ξ−1δμ ρk2R(0)(−¯D2/k2)00ZghLkδ[μρδν]σk2R(0)(−¯D2/k2)⎞⎠. (3.23)

In the diffeomorphism-ghost sector we have adjusted to the kinetic term according to the above rule.

In the ghost sector, however, there is no kinetic term; the ghosts do not propagate. Nevertheless, a consistent application of the FRGE requires us not to ignore, but to systematically integrate out these non-propagating modes in the same way as all the others, i. e. ordered, and eventually cut off according to their -eigenvalue. Therefore we introduce a cutoff-operator (with a prefactor unrelated to the couplings in , denoted by ) in this sector as well.333Recall that ideally, at the exact level, the cutoff action would be independent of the running couplings present in [24].

In the gauge chosen, the inverse ghost propagator is a triangular matrix, such that the contributions of the different sectors to the trace decouple.

For any constant choice of we obtain contributions of the ghost sector of the form

 (3.24)

with the abbreviation . Introducing the dimensionless mass parameter , and then neglecting any further running of , we observe that the trace (3.24) depends only on the -independent dimensionless quantity

 −M2ZghLk2≡−2μ2ZghLξ1/2. (3.25)

In order to avoid divergences due to a vanishing denominator in (3.24) we have to choose a negative value for , as known from the conformal sector. Since both parameters, and , occur only in the combination (3.25) we can mimic any choice of by choosing a suitable . (In particular , upon replacing .)

In the following we will discuss three distinguished choices of :

(i) : the cutoff term is unrelated to , the ghost contribution will therefore depend on and .

(ii) : the cutoff is optimally adapted to the form of leading to a cancelation of the parameters and . This procedure is closest to the above rule for usual kinetic term adaptation and we therefore expect the most reliable results for this choice.

(iii) : no cutoff term introduced. This choice corresponds to neglecting the ghost modes completely, the trace (3.24) vanishes.

As explained above, these three choices are equivalent to using and setting equal to , , and , respectively. We shall refer to them as the ghost adaptation schemes (i)–(iii) from now on.

### 3.4 The interpolating beta functions

The remaining part of the calculation consists of projecting out the invariants and from the supertrace in order to find the -functions for and ; it follows exactly the metric calculation in [11].

If we turn over to dimensionless couplings

 gk=kd−232πZNkκ2=kd−2Gk,λk=k−2¯λk (3.26)

the resulting system of coupled RG equations is autonomous and has the structure

 ∂tgk =βg(gk,λk)≡[d−2+ηN(gk,λk)]gk, (3.27) ∂tλk =βλ(gk,λk) (3.28)

with the anomalous dimension . We shall employ the standard threshold functions , of [11] along with a new type of threshold function, , defined according to

 Φpn(w) =1Γ(n)∫∞0dzzn−1R(0)(z)−zR(0)′(z)[z+R(0)(z)+w]p (3.29) ~Φpn(w) =1Γ(n)∫∞0dzzn−1R(0)(z)[z+R(0)(z)+w]pforn>0 (3.30) ~~Φpn(w) =1Γ(n)∫∞0dzzn−1R(0)(z)−zR(0)′(z)[R(0)(z)+w]p (3.31)

and . We can write down an explicit expression for in terms of the couplings then:

 ηN(g,λ)=g¯B1(λ)1−g¯B2(λ) (3.32)

The functions and are -dependent generalizations of similar ones occurring in [11]:

 ¯B1(λ)=13(4π)1−d/2[(d−1)(d+2)Φ1d/2−1((ξ−2)λ)+2Φ1d/2−1(−2λd−2+ξd−2)−4dΦ1d/2−1(0)−2d(d−1)~~Φ1d/2−1(2μ2√ξ)−6(d−1)(d+2)CT(ξ)Φ2d/2((ξ−2)λ)−12CS(ξ)Φ2d/2(−2λd−2+ξd−2)−24Φ2d/2(0)] (3.33)

and

 (3.34)

For the -function of the cosmological constant we obtain

 βλ=−(2−ηN)λ+12gk(4π)1−d/2[2(d−1)(d+2)Φ1d/2((ξ−2)λ)+4Φ1d/2(−2λd−2+ξd−2)−8dΦ1d/2(0)−4d(d−1)~~Φ1d/2(2μ2√ξ)−(d−1)(d+2)ηN~Φ1d/2((ξ−2)λ)−2η
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