Matter induced bimetric actions for gravity

Annals of Physics 326 (2011) 440–462

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Matter induced bimetric actions for gravity Elisa Manrique ⇑, Martin Reuter, Frank Saueressig Institute of Physics, University of Mainz, Staudingerweg 7, D-55099 Mainz, Germany

a r t i c l e

i n f o

Article history: Received 25 June 2010 Accepted 3 November 2010 Available online 11 November 2010

a b s t r a c t 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 N expansion. Ó 2010 Elsevier Inc. All rights reserved.

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 [2–4]. 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 [1–32], see [33,34] for reviews.1 A complete and everywhere regular RG trajectory of the effective average action [39–42] then defines a fundamental and predictive quantum field theory of gravity. One of the key requirements every future fundamental quantum theory of gravity must meet is that of ‘‘background independence’’ [43]. 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. (See [27,44] for a more detailed discussion of this point.) While in

⇑ Corresponding author. E-mail addresses: [email protected] (E. Manrique), [email protected] (M. Reuter), [email protected] (F. Saueressig). 1 For related results in conformally reduced gravity see [35–37]. 0003-4916/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2010.11.003

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loop quantum gravity [45–47] and in the discrete approaches [48–51] the requirement of ‘‘background independence’’2 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 glm at the intermediate steps of the quantization, but verifies at the end that no physical prediction depends on which glm 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 glm and quantize the nonlinear fluctuations of the dynamical metric, hlm, in the ‘‘arena’’ provided by glm , and we repeat this quantization process for any choice of glm . In this manner we arrive at an infinite family of quantum theories for hlm, whereby the family members are labeled by a classical (pseudo-) Riemannian metric glm .  lm ; glm  which deThe dynamical content of this family is fully described by an effective action C½h lm  hhlm i, and the background pends on two arguments3: the expectation value of the fluctuation, h lm gives rise to a metric. If the background-quantum field split is chosen linear, the expectation value h corresponding expectation value

lm g lm ¼ glm þ h

ð1:1Þ

 g entails an effective lm þ hlm . The action C½h; of the metric operator clm, i.e., glm = hclmi where clm ¼ g   field equation which governs the dynamics of hlm ðxÞ  hlm ½gðxÞ in dependence on the background metric:

d lm ðxÞ dh

 g ¼ 0: C½h;

ð1:2Þ

For special, so-called ‘‘self-consistent’’ backgrounds glm  gselfcon it happens that Eq. (1.2) is solved by lm lm ½gselfcon ðxÞ  0. Then the expectation value of an identically vanishing fluctuation expectation value h lm  equals exactly the background metric, g lm ¼ glm . The defining condithe quantum metric g lm  g lm ½g tion for a self-consistent background,

   selfcon  C½h; g    dhlm ðxÞ d

¼ 0;

ð1:3Þ

h¼0

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

     C ½ h; g      : lm ðx1 Þ dh  qr ðxn Þ  dh h¼0 d

d

ð1:4Þ

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. 3 For simplicity we ignore the Faddeev–Popov ghosts and possible matter fields here.

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It is a well known ‘‘magic’’ of the background formalism that by using an appropriate gauge fixing condition [52] a set of on-shell  equivalent Green’s functions is generated by differentiating the re g duced functional C½g  C½h; with respect to glm :  h¼0

d d C½g:  dglm ðx1 Þ dgqr ðxn Þ

ð1:5Þ

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 glm as a background plus a fluctuation. In the linear case the corresponding symmetry transformation are

lm ¼ lm ; dh

dglm ¼ lm

ð1:6Þ

for any lm. If the split symmetry is appropriately implemented at the quantum level, the reduced  which depends on one argument only has exactly the same physical contents as the functional C½g  g. more complicated C½h; Up to now we tacitly assumed that the microscopic (bare) action governing the dynamics of clm 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  g, to search for nontrivial fixed points of this flow and if (diffeomorphism invariant) functionals C½h; 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 [1,33]. Here one defines a coarse grained counterpart of the ordinary effective action, the effective average action  g, in terms of a functional integral containing an additional mode suppression factor exp(D S) Ck ½h; k quadratic in h, and performs the coarse graining by suppressing the contributions of the hlm-modes with a covariant momentum smaller than the variable IR cutoff scale k [39–41]. The k-dependence  g is governed by a functional RG equation (FRGE) which defines of the effective average action Ck ½h; a vector field (a ‘‘flow’’) on theory space. At k = 0, the average action Ck equals the ordinary effective  g does not action C, and only the gauge fixing term breaks the split symmetry. Hence, for k = 0, Ck ½h;  g contain more gauge invariant information than the reduced functional Ck ½g  Ck ½h; does.  h¼0 It is crucial to realize that this (on-shell) equivalence of the two functionals at k = 0 does not generalize to k > 0. At every nonzero scale k the coarse graining operation unavoidably leads to an addi lm and glm tional violation of the split symmetry since the action DkS is not invariant. It contains h     separately, not only in the split invariant combination hlm þ g lm . In a sense, Ck ½h; g  contains more information than Ck ½g if k > 0. An immediate consequence is the well known fact [40,1] that it is impossible to write down a FRGE in terms of the reduced functional Ck ½g alone. The actual theory space is more complicated, consisting of functionals with two metric arguments and, to be precise,  lm ; C l ; C l ; g lm . Often it is convenient to replace also ghost arguments Cl and C l , respectively: Ck ½h   hlm with g lm  glm þ hlm as the independent argument

Ck ½g; g; C; C  Ck ½h ¼ g  g; C; C; g:

ð1:7Þ

In the Ck ½g; C; C; g-notation the second argument parametrizes the so-called extra background field dependence, i.e., that part of the glm -dependence which does not combine with a corresponding  lm -dependence to a full metric g lm  g . If the split symmetry was intact we had lm þ h h lm d  C ½g; g ; C; C ¼ 0. In general Ck has a nontrivial dependence on glm though, and so we can rightfully dg k call Ck a bimetric action. Besides the FRGE, the effective average action satisfies two important functional identities which, however, are ‘‘instantaneous’’ in the sense that they contain no k-derivative but constrain the form of Ck at every fixed value of k. The first one is the modified BRST Ward identity

Z

1 dC0 dC0k dC0k dC0k þ d x pffiffiffi  k g dhlm dblm dC l dsl d

!

¼ Y k ½Ck :

ð1:8Þ

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Here C0k  Ck  Sgf , and blm and sl are sources for the BRS variation of clm and Cl, respectively. The RHS of Eq. (1.8), the functional Yk, is given by a set of complicated traces involving Rk , see [1] for their explicit form. The standard BRS Ward identities have the structure of (1.8) with Yk ? 0. The nonzero contributions to Yk stem from the cutoff term DkS which is not BRS invariant. Since Rk vanishes for k ? 0 it follows that limk?0Yk = 0 so that limk?0Ck  C is BRS invariant in the usual way. The dependence of Ck on the background metric glm is governed by a similar exact functional equation4

d Ck ½g; g; n; n ¼ Ye k ½Ck lm ðxÞ: dglm

ð1:9Þ

The RHS of Eq. (1.9) measures the degree Ck too possesses an ‘‘extra’’ background dependence. The e k consists on various traces involving Rk and Ck itself. The gauge fixed action appearing functional Y under the path integral contains various sources of contributions to Yk, in particular the extra background dependence of the gauge fixing and ghost terms, as well as of DkS, respectively. The former is nonzero even for k ? 0, the latter vanishes in this limit. Indeed, all coarse graining kernels have 2 2 the structure Rk / k Rð0Þ ðD2 =k Þ where R(0) interpolates between zero and unity for large and small R arguments, respectively. Therefore Rk vanishes for k ? 0, and as a consequence Dk S / hRk h no longer provides an extra background dependence. However, the contributions stemming from DkS possibly become significant in the UV limit k ? 1 since Rk behaves like a divergent mass term /k2 in this limit. Exact solutions to the FRGE automatically satisfy the BRS Ward identity and the d=dg-equation (1.9). For approximate solutions to the flow equation this is not necessarily the case. One can then evaluate the Ward identity and/or the d=dg-equation for the approximate RG trajectory and check how well these relations are satisfied. 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 full fledged gravity. However in Ref. [27] the d=dg-equation has been analyzed for conformally reduced gravity and its predictions were compared to those of the flow equation. The enlarged theory space is the price one has to pay for the ‘‘background independence’’ of the average action approach. Including a matter field A(x) its ‘‘points’’ are functionals C½A; g; g; C; C which are invariant under arbitrary diffeomorphisms acting on all arguments simultaneously. If vl is a generating vector field and Lv the corresponding Lie derivative we have, to first order in vl, C½A þ Lv A; g þ Lv g; g þ Lv g; . . . ¼ C½A; g; g; . . .. For later use we write down the resulting Ward identity, for simplicity at C ¼ C ¼ 0 and for a scalar matter field:

2g qm Dl

dC½A; g; g dC½A; g; g dC½A; g; g ¼ 0: þ 2gqm Dl  ð@ q AÞ dg lm ðxÞ dglm ðxÞ dAðxÞ

ð1:10Þ

Here and in the following the covariant derivatives Dl and Dl refer to the Levi-Cività connections of glm and glm , respectively. Up to now almost all applications of the gravitational average action employed a truncated theory space with functionals of the form C½g; g; C; C ¼ Ck ½g + classical gauge fixing and ghost terms, where Ck ½g  Ck ½g; g; 0; 0. Hence in practice one had to deal with the RG evolution of a single metric functional only, Ck ½g. In Ref. [27] 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 [27] 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 [53]. 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 Ck ½A; g; g in certain truncations. The quantum effects originating from the gravitational sector are neglected, which allows to avoid the additional technical complications linked to the

4

For the derivation of an analogous relation in Yang-Mills theory see Appendix A of [38].

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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 [54,55]. As a spin-off, we re-examine the Asymptotic Safety conjecture within a large-N approximation [14–16,18], which gives the first insights on how important the nontrivial bimetric dependence of Ck actually is. We stress that the objective behind the bimetric truncations studied in this paper is a more precise description of the theory’s RG flow, irrespective of the precise physical interpretation of the new running coupling constants at intermediate scales. 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, pffiffiffi lm ¼ g  glm is not required to be small. In Section 3 we employ lm Þ. Here the fluctuation h g Y k ðg lm ; g lm  lm -expansion which includes all interaction a complementary truncation of Ck, based on a systematic h lm . We derive the RG flow on the corterms built from up to two derivatives and up to first order in h responding five-dimensional theory space and analyze its fixed point structure. The technical details underlying this calculation are relegated to the Appendix A. As an application, the resulting k-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 ns quantized scalar fields (AI)  A, I = 1, . . . , ns, 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 Ck ½A; g; g. Even in this comparatively simple setting where gravity itself is not quantized this action is ‘‘bimetric’’: it depends on the full metric glm since the scalars couple directly to it, and it also depends on glm because it is the backR d pffiffiffi ground metric which enters the scalar mode suppression term Dk S½A; g  12 d x gAðxÞRk ½gAðxÞ where Rk ½g is the coarse graining operator [39,40,1]. 2.1. The functional RG equation The FRGE for the quantized scalars A(x) interacting with classical gravity reads

@ t Ck ½A; g; g ¼

  1 1 ð2Þ Tr Ck ½A; g; g þ R½g @ t Rk ½g : 2

ð2:1Þ

As gravity is not quantized here, the trace is over the fluctuations of the scalars only, and the Hessian Cð2Þ k involves functional derivatives with respect to A only. As usual, t = ln k denotes the ‘‘RG time’’. pffiffiffi pffiffiffi In the following it will be important to carefully distinguish the volume elements g and g, respectively. The matrix elements of the Hessian operator contain two factors of the latter,



Ckð2Þ

 xy

 E D  1 d2  ð2Þ   xCk y ¼ pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi Ck ½A; g; g: gðxÞ gðyÞ dAðxÞdAðyÞ ð2Þ

ð2:2Þ

Products of operators such as Ck are defined in terms of their matrix elements as R d pffiffiffiffiffiffiffiffiffi ðABÞxy ¼ d z gðzÞAxz Bzy , and the position representation of the trace in (2.1) reads R d pffiffiffiffiffiffiffiffiffi ðxÞAxx . Furthermore, if some operator has matrix elements Axy , we define the associTrðAÞ ¼ d x g R d pffiffiffiffiffiffiffiffiffi ated (pseudo) differential operator Adiff-op by ðAf Þx  d y gðyÞAxy fy ¼ ðAdiff-op f ÞðxÞ for every ‘‘column vector’’ fx  f(x). We shall write h  glmDlDm and   glm Dl Dm for the Laplace–Beltrami operators belonging to glm and glm , respectively.

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

Ck ½A; g; g ¼

Z

 Z 1 2 2 1 d pffiffiffi 1 lm d pffiffiffi  kA þ g @ l A@ m A þ m d x g d x g Y k ðg lm ðxÞ; glm ðxÞÞ: 2 2 8p G

ð2:3Þ

Here A always stands for (AI), I = 1, . . . , ns. The ansatz (2.3) is taken to be O(ns) invariant; appropriate sums over the index I are understood (A2  AIAI, etc.). The first term in (2.3) is the action of a standard scalar coupled to the full metric glm, 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 pffiffiffi glm and glm . The normalization and the explicit factor of g in the second term on the RHS of (2.3) are chosen such that the constant term of the function Yk equals the cosmological constant:

Y k ðg lm ; glm Þ ¼ Kk þ \more":

ð2:4Þ

For the time being we neglect the dynamics of the gravitational field. Within the truncation considered this approximation becomes exact in the large ns-limit, where the leading quantum corrections to the running of Yk arise from the matter fields while the gravitational corrections are suppressed by a factor 1/ns. We do not claim that our result is exact to order 1/n in the full theory space, i.e. without the a priori assumption of a truncated action functional. It would be substantially harder to prove a result of this kind (if true). In order to project out the beta function of Yk it is sufficient to insert x-independent metrics glm and glm into the FRGE. In this case the Hessian resulting from the ansatz (2.3) is

pffiffiffi g

g



  2k ; pffiffiffi  þ m Cð2Þ k ½A; g; g  ¼

ð2:5Þ

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 Rk may depend on the background metric only, we define it by the requirement that upon adding Rk to the Hessian and setting g ¼ g the operator ðÞ which then ð2Þ  must get replaced by  þ k2 Rð0Þ ð=k2 Þ. Here R(0) is an arbitrary shape funcappears in Ck ½A; g; g tion [39,41] with the standard properties R(0)(0) = 1 and limz?1R(0)(z) = 0. The condition



 

  Cð2Þ k ½A; g; g  þ Rk ½g  

g¼g

  ð2Þ ¼ Ck ½A; g; g

ð2:6Þ

!þk2 Rð0Þ ð=k2 Þ

leads to an operator which at first sight appears familiar, 2 2 Rk ½g  Rk ðÞ ¼ k Rð0Þ ð=k Þ

ð2:7Þ

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

pffiffiffi g

g



 2k þ k2 Rð0Þ ð=k2 Þ:   pffiffiffi  þ m Cð2Þ k ½A; g; g  þ Rk ½g  ¼

ð2:8Þ

Most of the unfamiliar features we are going to find in the following are due to the interplay of the pffiffiffi pffiffiffi standard cutoff operator Rk with a Hessian containing the ratio of the volume elements g = g. 2.4. The RG equation for Yk The trace of the flow equation is easily evaluated in a standard plane wave basis now. With  k =k we have: mm

pffiffiffi d g @ t Y k ðg lm ; glm Þ ¼ 8pGns k

Z

0

Rð0Þ ðglm ql qm Þ  ðglm ql qm ÞRð0Þ ðglm ql qm Þ

pffiffiffi pffiffiffi :

p ffiffiffi p ffiffiffi g = g ðg lm ql qm Þ þ Rð0Þ ðglm ql qm Þ þ m2 g = g ð2pÞd d

d q

ð2:9Þ

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The integral on the RHS of this flow equation represents a scalar density which depends on two constant matrices, glm and glm . It cannot be evaluated in closed form. In the following we calculate it for two special cases, namely for glm and glm both proportional to the unit matrix, and by expanding in lm . lm ¼ h their difference g lm  g 2.5. The volume element truncation Now we make a further truncation and restrict Yk to depend on the metrics via their volume ele pffiffiffi pffiffiffi ments only: Y k ðg lm ; glm Þ  P k g ; g . In this case, the beta function of P k can be found by inserting two conformally flat metrics

g lm ¼ L2 dlm ;

glm ¼ L2 dlm

ð2:10Þ

into the flow equation (2.9) and keeping track of the constants L and L. One then observes that @ t P k pffiffiffi pffiffiffi = g ¼ ðL=LÞd  qd only. Thus, setting Y k ðg lm ; glm Þ  Qk ðqd Þ with actually depends on the ratio g q  L=L, we obtain d

@ t Qk ðqd Þ ¼ 16pv d Gns k qd

Z

1

dyyd=21

0

Here,

v d  ½2dþ1 pd=2 Cðd=2Þ1 ;

b ð0Þ ðyÞ  y R b ð0Þ0 ðyÞ R : b ð0Þ ðyÞ þ ðm=qÞ2 yþR

ð2:11Þ

y ¼ L2 dlm ql qm , and

b ð0Þ ðyÞ  qd2 Rð0Þ ðyÞ: R

ð2:12Þ

From now on we employ the ‘‘optimized’’ shape function [42] R(0)(y) = (1  y)h(1  y), whence d

@ t Qk ðqd Þ ¼ 16pv d Gns k qd J d ða; lÞ:

ð2:13Þ

a  q2d  1  ðL=LÞ2d  1; l  m2 qd ;

ð2:14Þ

Here

and

J d ða; lÞ 

Z

1

dyyd=21 ½1 þ l þ ay1 :

ð2:15Þ

0

The integrals (2.15) can easily be computed explicitly. Let us specialize for d = 4 now. Then

J 4 ða; lÞ ¼

1

a

   1þl ln 1 þ 1

a

a



1þl

ð2:16Þ

and the flow equation reads

@ t Qk ðq4 Þ ¼

G 4 ns k q4 J 4 ða; lÞ: 2p

ð2:17Þ

At first, let us neglect the scalar mass, setting m = 0 and l = 0 therefore. Then the RG equation (2.17) 4 can be integrated trivially: Qk ¼ Qk¼0 þ 8Gp ns k q4 I4 ðaÞ, where we abbreviate I4 ðaÞ  J 4 ða; 0Þ. As for the constant of integration, Qk¼0 , it is important to recall that the glm -dependence of Ck is entirely due to the cutoff Rk ½g in the case at hand as we have no gauge fixing terms. As a result, Ck and in particular Yk must become independent of g at k = 0 since Rk ½g vanishes there. This implies that Qk¼0 may not have any q-dependence. In fact, it equals the ‘‘ordinary’’ cosmological constant K0 which multiplies pffiffiffi the volume element g in the effective action C  C0:

Qk ðq4 Þ ¼ K0 þ

1 4 Gns k q4 I4 ðaÞ: 8p

ð2:18Þ

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Using (2.18) in the truncation ansatz (2.3) we obtain the following explicit representation for the nonderivative terms in the average action: " pffiffiffi pffiffiffi # Z Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffi 1 g 1 ns 4 1 g 4 pffiffiffi 4 non-deriv   q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g= g d x g K0  d x Ck ½A; g; g  ¼ k ln pffiffiffi : pffiffiffi pffiffiffi 1 þ 2 1  8pG 64p2 g 1 g = g

ð2:19Þ This is one of the our main results, and several comments are in order here. pffiffiffi pffiffiffi (A) Obviously the induced non-derivative terms are neither proportional to g nor to g but rather pffiffiffi to a complicated function of their ratio, times an extra factor of g. (The explicit factor of q4 on R pffiffiffi pffiffiffi pffiffiffi g Y k to a g.) the RHS of (2.18) has converted the g in the ansatz (B) The induced term is regular for any ratio g=g 2 ð0; 1Þ. In fact, writing (2.19) as

Cnon-deriv ½A; g; g ¼ k

K0 8pG

Z

ns 4 4 pffiffiffi k d x gþ 64p2

Z

pffiffiffi 4 d x gI4 ða ¼ ðg=gÞ1=4  1Þ;

ð2:20Þ

and using the expansion

I4 ðaÞ ¼

1 1 1  a þ a2 þ Oða3 Þ; 2 3 4

ð2:21Þ

. we see that actually Cnon-deriv ½A; g lm ; glm  has no singularity at a = 0, i.e., at g ¼ g k (C) Let us specialize the result for the regime g  g. This amounts to expanding in the fluctuation lm  g  glm or, equivalently, in the variable a since variable h lm

  1 1  l 2 1  lm 3 : hl  hlm h þ O h a  ðg=gÞ1=4  1 ¼ hll þ lm 4

32

ð2:22Þ

8

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffi pffiffiffi pffiffiffipffiffiffiffi pffiffiffi pffiffiffi 1 pffiffiffi 1 1 pffiffiffi g  g g  g þ g g þ g þ Oða3 Þ: gI4 ðaÞ ¼ g2 2 3 4

pffiffiffi gI4 ðaÞ. The

ð2:23Þ

In the g  g regime there is clearly no preference for g-monomials over g-monomials or vice versa, so the truncation ansatz cannot be simplified accordingly: any meaningful truncation is genqffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffi g g which was individually included in the uinely ‘‘bimetric’’! Note also that the monomial truncation studied in [27]5 never is generated in isolation. According to (2.23) it is always accompffiffiffi pffiffiffi panied by other, a priori equally important terms involving g and g.   (D) It is instructive to rewrite the expansion about g ¼ g in terms of hlm . Up to linear order the rel lm ; g lm   Ck ½A; h lm  are evant terms in Ck ½A; g lm ; g

 g ¼ Cnon-deriv ½A; h; k

Kð0Þ k 8pG

Z

  pffiffiffi Kð1Þ 1 Z 4 pffiffiffi lm 4 lm þ O h 2 : d x gg h d x g þ k  lm 8p G 2

ð2:24Þ

R pffiffiffi R pffiffiffi ð0Þ ð1Þ g , and Kk is an Here Kk is a background cosmological constant multiplying g rather than  lm . Explicitly, analogous, but numerically different running coupling in the (only) term linear in h

Kð0Þ k ¼ K0 þ

ns 4 Gk ; 16p

Kð1Þ k ¼ K0 

ns 4 Gk : 48p

ð2:25Þ ð0Þ

ð1Þ

Note the different signs on the RHS of the equations: Kk increases, but Kk decreases for growing k. Note also that when k – 0 the RHS of (2.24) cannot be written as a functional of lm  g  þh the single metric g lm alone as this would require it to be proportional to pffiffiffi pffiffiffi 1 pffiffiffi lm  lm  2 Þ. But this is not the case just because Kð0Þ – Kð1Þ . g ¼ g þ 2 gg hlm þ Oðh lm k k qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffi Note, however, that g g plays a distinguished role in the conformally reduced gravity setting of [27]. It amounts to a mass term of the scalar field appearing there. 5

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(E) Computations within the setting of single metric truncations retain only the terms of zeroth lm . They equate the two metrics g and g lm , and traditionally denote the one metric order in h lm which is left over then by glm. In this setting, Eq. (2.24) would boil down to metric Cnon-deriv;single ½A; g; g ¼ k

Kð0Þ k 8pG

Z

4 pffiffiffi d x g:

ð2:26Þ ð0Þ

Thus, in a single metric truncation, it is the parameter Kk 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 misleading. We shall discuss this point in detail in Section 4. pffiffiffi pffiffiffi pffiffiffi (F) We saw that, in the regime g  g, the true cosmological constant monomial g plays no dispffiffiffi pffiffiffi tinguished role. Next let us see whether this could be the case when g  g. Let us try an qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi g = g  1. In a r2-expansion of (2.20) with a  r2  1 there would expansion in r2  pffiffiffi indeed appear a g monomial in isolation, the first few terms of the power series being pffiffiffi 2 0 pffiffiffi pffiffiffi 2 1 qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffi pffiffiffi 2 2 pffiffiffi pffiffiffi 2 3 ; gðr Þ ¼ g g; gðr Þ ¼ g ; gðr Þ ¼ g 3=4 g1=4 ; . . . However, taking gðr Þ ¼ g the explicit form of I4 into account one finds that an expansion of this type actually does not exist, since I4(a = r2  1)  (1 + ln r2) when r  1. Hence I4 is not analytic at r2 = 0. So the pffiffiffi conclusion is that in this regime, too, the g -monomial plays no distinguished role in the trunqffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffipffiffiffiffi  term considered in [27].) g g cation. (The same is true for the (G) Up to now, the scalars A were If we allow for a non-zero dimensionless mass  assumed  pmassless. ffiffiffi pffiffiffid  k =k so that now l ¼ m  , the RG equation (2.17) no longer can be inte 2k =k2 mm g= g  k , the matter field grated trivially. The general properties of the flow are clear though: if k  m k is still approximately massless and the above discussion applies. If, on the other hand, k  m the scalar decouples and its contribution to the running of Qk becomes tiny. If we neglect the  k M the parameter l diverges /1/k2 below the threshold running of the dimensionful mass m 1 at k = M. Expanding J ða; lÞ ¼ 2l þ O l12 we obtain the leading term below the threshold:

Cnon-deriv ¼ k

1 8pG

Z

4

d x

( " !#) pffiffiffi 2 4 pffiffiffi pffiffiffi G k g 4 k : g K0 þ g pffiffiffi ns k þ O g 24p M2 M4

ð2:27Þ

In the massive regime an extra factor k2/M2  1 suppresses the running of the non-derivative pffiffiffi pffiffiffi term. Here, again, it is not of the standard g -form, but rather proportional to g= g . (H) So far, the discussion was based upon an evaluation of the integral formula (2.9) for metrics glm and glm which are conformal to dlm, see (2.10). If one wants to know the tensorial structure of the terms in Yk one must go beyond this special case. A systematic strategy for doing this is an  lm ¼ g  glm , whereby g and glm are still constant matrices, but not necessarily expansion in h lm lm  lm we find a result of the proportional to dlm. Carrying out this expansion up to linear order in h form (2.24), generalized for arbitrary d, with

ns 2 1 Ud=2 ð0ÞGkd ; ð4pÞd=21 d ns ðd  2Þ 2 ¼ K0  Ud=2þ1 ð0ÞGkd : d ð4pÞd=21

Kkð0Þ ¼ K0 þ Kð1Þ k

ð2:28Þ

Here the U’s are the usual R(0)-dependent threshold function of Ref. [1], see also Eq. (A.18) in Appendix A. If one specializes for d = 4 and the ‘‘optimized’’ R(0), the result (2.28) accidentally coincides with the one obtained from the conformally flat ansatz (2.25). As (2.28) contains different U-functions, they depend on R(0) in a different way. This implies that any special relationð0Þ ð1Þ ship between the two cosmological constants Kk and Kk which one might invoke, for instance in order to restore split symmetry, cannot have a universal (scheme independent) meaning at k – 0.

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 lm -expansion 3. A systematic derivative- and h 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  g in powers of h  lm and the number of derivatives. Ck ½A; g; g  Ck ½A; h; 3.1. The truncation ansatz In the following, we will consider an average action of the form

 g ¼  Ck ½A; h;

Z

1 ð0Þ

 pffiffiffi d ð0Þ d x g R  2Kk þ

Z

1 ð1Þ

16pGk 16pGk   Z  d pffiffiffi 1 lm g @ l A@ m A þ U k ðAÞ  þ d x g : 2  lm g lm ¼glm þh

 pffiffiffi 1 d ð1Þ lm d x g Glm  Ek glm R þ Kk glm h 2 ð3:1Þ

mr Gqr , and Here R is the curvature scalar built from the background metric, Glm  glq g Glm  Rlm  12 glm R denotes the background Einstein tensor. The functional (3.1) is obviously invariant lm ; A, and g lm , respectively. The gravitational part of under diffeomorphisms acting simultaneously on h lm and this ansatz is complete in the sense that it contains all possible terms with no or one factor of h lm , and diffeomorphism invariat most two derivatives. The latter can always be arranged to act on g ance 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 glm is to be read as an abbreviation for  lm , though. glm þ h  -independent field monomials with zero and two derivatives, respectively, There exist two h R pffiffiffi R plffiffiffim ð0Þ ð0Þ namely g and gR. In Eq. (3.1) the corresponding prefactors are proportional to Kk =Gk and ð0Þ  1=Gk , respectively. In the sector with one power of hlm there are three possible tensors structures for the corresponding one-point function, namely Glm and glm R with two, and glm with no derivatives. ð1Þ ð1Þ ð1Þ ð1Þ Their prefactors define new running couplings 1=Gk ; Ek =Gk , and Kk =Gk , respectively. (The super lm -order in which the coupling in question occurs.) scripts (0), (1), . . . indicate the h  lm and g lm ; hence Ck has an ‘‘exThe functional (3.1) is defined for completely independent fields h   lm . Nev tra’’ g lm -dependence in general which does not combine with hlm to a full metric g lm ¼ glm þ h ertheless it is instructive to consider (3.1) for the special case of no extra glm -dependence. The  lm via their background-quantum field split symmetry is intact then and Ck depends on glm and h sum only. The resulting gravitational part is obtained by expanding the Einstein–Hilbert action,

  CEH k ½g þ h  

1 16p

ð0Þ Gk

Z

  d pffiffiffi ð0Þ d x g RðgÞ  2Kk 

;

ð3:2Þ

 g lm ¼g lm þh lm

lm . The expansion is of the form (3.1) with special coefficients, however: up to terms of first order in h ð1Þ

ð0Þ

Gk ¼ Gk ; ð1Þ k

K ¼K Ek ¼ 0:

ð0Þ k ;

ð3:3aÞ ð3:3bÞ ð3:3cÞ

It needs to be stressed that the relations (3.3) are not satisfied in general. Since the cutoff term DkS breaks the split symmetry, Ck does have an extra background dependence, and so the terms with one  lm are not the linearization of any functional depending on the sum glm þ h  lm only. As a conpower of h ð0Þ ð1Þ sequence, we encounter two running couplings, Gk and Gk , both of which are related to the classical Newton constant, but have a different conceptual status and numerical value. The same remark ð0Þ ð1Þ applies to the running cosmological constants Kk and Kk . Furthermore, we emphasize that when the split symmetry is broken there is no symmetry principle lm factor to be proportional to the Einstein that would force the second derivative terms with one h  lm and Rglm h lm , respectively; they tensor. There are actually two independent tensor structures, Rlm h  lm together with Rg  lm . Hence E – 0 in general. (The implications lm h can equivalently be chosen as Glm h k of a non-vanishing Ek for the effective field equations will be discussed below.)

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If, as in the system discussed in the present paper, the cutoff term DkS is the only source of split symmetry violation the average action looses its extra g-dependence in the ‘‘physical limit’’ k ? 0. We expect the relations (3.3) to be satisfied then. In non-gauge theories the ordinary effective action limk!0 Ck ½A; g; g  C½A; g has no ‘‘extra’’ g-dependence. 3.2. The five-dimensional RG flow In order to obtain the beta functions for the running couplings in the truncation ansatz we must insert (3.1) into the flow equation (2.1), compute the trace on its RHS, and project it on the various monomials. For bimetric truncations computations of this kind are considerably more involved than lm the in the single metric case. In the present situation where one must retain only one power of h calculation still can be done in a comparatively elegant way. Some details, in particular the new strat lm -dependence are given in Appendix A. Here we only present and analyze the egies to cope with the h results. As we are not interested in the renormalization of the matter sector we only include a mass term in  2k A2 . The dimensionful mass m  k has the same value for all components of A, the potential: U k ðAÞ ¼ 12 m  k  m.  The RG equations are most conveniently written down and we neglect its scale dependence, m  in terms of the dimensionless quantities mk  m=k; Ek ; and d2

ð0Þ

gk  k

ð0Þ

Gk ;

ð0Þ

2

ð0Þ

ð1Þ

k k  k Kk ;

gk  k

d2

ð1Þ

Gk ;

ð1Þ

2

ð1Þ

kk  k Kk :

ð3:4Þ

Furthermore, it is convenient to introduce the following anomalous dimensions for the two running Newton constants: ð0Þ

gð0Þ N 

@ t Gk

ð0Þ Gk

ð1Þ

;

gð1Þ N 

@ t Gk

ð3:5Þ

:

ð1Þ Gk

 lm , the RG equations assume the following form: For the couplings of order zero in h

h i ð0Þ ð0Þ ð0Þ @ t g k ¼ d  2 þ gN g k ;

ð3:6aÞ

h i

 ð0Þ ð0Þ ð0Þ ð0Þ @ t kk ¼ gN  2 kk þ 2ns ð4pÞ1d=2 g k U1d=2 m2k :

ð3:6bÞ

Similarly, one finds at order one:

h i ð1Þ ð1Þ ð1Þ @ t g k ¼ d  2 þ gN g k ;

ð3:7aÞ

h i h



i ð1Þ ð1Þ ð1Þ ð1Þ @ t kk ¼ gN  2 kk  ns ð4pÞ1d=2 g k ðd  2ÞU2d=2þ1 m2k þ 2m2k U2d=2 m2k ;

ð3:7bÞ

 @ t Ek ¼



 1 2 ð1Þ ð1Þ ðd  2Þ þ Ek gN þ ns ð4pÞ1d=2 m2k g k U2d=21 m2k : 2 3

ð3:7cÞ

The anomalous dimensions are explicitly given by



2 3



ð0Þ ns ð4pÞ1d=2 g k U1d=21 m2k ; gð0Þ N ¼

2 3





2 2 gNð1Þ ¼ ns ð4pÞ1d=2 g ð1Þ k Ud=2 mk :

ð3:8aÞ ð3:8bÞ

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Various remarks are in order here: ð0;1Þ

(A) The above five ordinary differential equations for g k completely decoupled: the order zero quantities ð1Þ

ð0Þ gk

ð0;1Þ

; kk

and

, and Ek form two sets which are

ð0Þ kk

enter only the coupled system

ð1Þ

(3.6), while the order-one couplings g k ; kk , and Ek occur only in the three-dimensional system (3.7).6 (B) The two-dimensional subsystem for the order zero quantities, Eqs. (3.6a) and (3.6b), is exactly ð0Þ ð1Þ ð0Þ ð1Þ what one obtains in a conventional single metric truncation, with g k ¼ g k and kk ¼ kk interpreted as ‘‘the’’ Newton and cosmological constant, respectively, [18]. (C) The above equations make it manifest that the RG flow does indeed generate split symmetry violating terms: the beta function of Ek is nonzero, and the beta functions for the zeroth and first order Newton and cosmological constants are different. (D) As we want split symmetry to be restored at k = 0, the constants of integration contained in solutions to the above RG equations must be chosen in accord with (3.3). This does not fix them completely, of course, but reduces the set of undetermined coupling constants to the one arising in the single metric computation. 3.3. Nontrivial fixed points in the large N approximation We close this section by studying the fixed point structure of the RG flow defined by the above equations. In the large N limit N  ns ? 1 the renormalization effects due to the matter fields overwhelm those stemming from the gravitational field which have been neglected here. Hence it makes sense to ask whether the above RG equations admit ‘‘asymptotically safe’’ solutions, i.e. RG trajectories which run into a non-Gaussian fixed point (NGFP) when k ? 1. For the sake of simplicity we restrict the discussion to d = 4 and m = 0. The system for the orderzero quantities (3.6) then becomes

h i ð0Þ ð0Þ ð0Þ @ t g k ¼ 2 þ gN g k ; ð0Þ

h

ð0Þ

i

ð0Þ

@ t kk ¼ gN  2 kk þ

gð0Þ N ¼

ns ð0Þ 1 g U ð0Þ; 6p k 1

ns ð0Þ 1 g U ð0Þ: 2p k 2

ð3:9aÞ ð3:9bÞ

The three coupled equations for the order-one couplings (3.7) simplify to:

h i ð1Þ ð1Þ ð1Þ @ t g k ¼ 2 þ gN g k ;

gNð1Þ ¼

ns ð1Þ 2 g U ð0Þ; 6p k 2

h i ns ð1Þ 2 ð1Þ ð1Þ ð1Þ @ t kk ¼ gN  2 kk  g U ð0Þ; 2p k 3

ð3:10aÞ ð3:10bÞ

ð1Þ

@ t Ek ¼ ½1 þ Ek gN :

ð3:10cÞ

Owed to their decoupling, the two sets of equations can then be analyzed independently. It is easy to see that both the order zero and the order one subsystem has a Gaussian fixed point (GFP) as well as a non-Gaussian fixed point (NGFP). From (3.9) we read off that the beta functions ð0Þ ð0Þ of g k and kk vanish at the following two points:

GFPð0Þ : g ð0Þ ¼ 0; NGFPð0Þ : g ð0Þ ¼ 

kð0Þ ¼ 0; 12p ns U11 ð0Þ

;

ð3:11aÞ k ð0Þ ¼ 

1 2 ð0Þ : 1 1 ð0Þ

3U 2U

ð3:11bÞ

6 The complete decoupling of the two sets of gravitational flow equations is accidental, however, and owed to the simplicity of the model. In a more elaborate framework, which also includes the quantum effects in the gravitational sector, the coupling  constants appearing at higher orders in the h-expansion will, in general, ‘‘feed back’’ into the beta functions capturing the running of the coupling constants at lower orders [27,54].

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The zeros of the order one beta functions are:

GFPð1Þ : g ð1Þ ¼ 0; NGFP

ð1Þ

kð1Þ ¼ 0;

: g ð1Þ ¼ 

12p ns U22 ð0Þ

;

E arbitrary; k ð1Þ

3 U23 ð0Þ ; ¼ 2 U22 ð0Þ

ð3:12aÞ E ¼ 1:

ð3:12bÞ

The following points should be noted here. (A) Any fixed point of the order zero system may be combined with any fixed point of the order one equations. Hence we find four fixed points of the total system; symbolically:

GFPð0Þ GFPð1Þ ;

GFPð0Þ NGFPð1Þ ;

NGFPð0Þ GFPð1Þ ;

NGFPð0Þ NGFPð1Þ :

ð3:13Þ

Most likely, these fixed points constitute projections of fixed points on the full bimetric theory space, which, in addition, should contain contributions associated with higher derivative and higher-order hlm-terms not captured by the truncation ansatz. Within our projection, the fixed points are exact, up to corrections in 1/ns originating from the dynamics of the gravitational field. (B) The constants Upn ð0Þ are cutoff scheme, i.e., R(0)-dependent, but their signs are universal. At both non-Gaussian fixed points the respective Newton constants assume negative values. (In the case of g ð0Þ this was known already [18].) Inserting the ‘‘optimized’’ shape function as an example one finds:

12p ; ns 24p ¼ ; ns

NGFPð0Þ : g ð0Þ ¼  NGFPð1Þ : g ð1Þ

3 k ð0Þ ¼  ; 4 1 ð1Þ k ¼ ; E ¼ 1: 2

ð3:14aÞ ð3:14bÞ

ð1Þ Note also the different signs of kð0Þ and k . (C) While we knew already from the single metric truncations that there exists a NGFP in the order zero sector, the result concerning a nontrivial fixed point at order one is new. Clearly this is encouraging news for the Asymptotic Safety program. It requires a fixed point on the ‘‘big’’ the g, and they comprise terms of all orders in ory space spanned by functionals of the type C½A; h; lm , of course. h (D) If there was no split symmetry breaking the fixed points in the zero and first order sectors were related, with (3.3) implying

g ð1Þ ¼ g ð0Þ ;

ð3:15aÞ

kð1Þ

ð3:15bÞ ð3:15cÞ

kð0Þ ;

¼ E ¼ 0:

Clearly the explicit results (3.14) are far from satisfying these relations, and we must conclude that the violation of the split symmetry due to the mode cutoff has a significant impact on the fixed point structure. It needs to be emphasized that it is quite nontrivial that the known single metric fixed point did not get destroyed within the more general bimetric truncation. We rather find that it splits into two different ones, one at the zeroth and another one at the first level of lm -expansion. the h (E) The running coupling Ek enters the action (3.1) via the combination

1 1 Glm  Ek glm R ¼ Rlm  ð1 þ Ek Þglm R: 2 2

ð3:16Þ

The NGFP(1) fixed point value E* = 1 is quite special therefore. It is precisely such that the Einstein tensor is converted to a pure Ricci tensor. Note that also that at the GFP(1) the value ð1Þ of E* is arbitrary: if g k ¼ 0, the RHS of (3.10c) vanishes whatever is the value of Ek. So strictly speaking we encounter a whole line of fixed points. Imposing split symmetry leads to E* = 0 though.

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(F) Imposing the restauration of split symmetry at k = 0, it is possible to solve the RG equation ð1Þ (3.10c) explicitly for Ek in terms of Gk , for all values of k ð1Þ

Ek ¼ Gk =G0  1:

ð3:17Þ ð0Þ Gk

ð1Þ Gk

Here G0 is the commonh value of andi at k = 0. In the semiclassical regime we have 2 2 ð1Þ approximately Gk ¼ G0 1 þ 12nsp U22 ð0ÞG0 k and Ek ¼ 12nsp U22 ð0ÞG0 k , hence Ek is a tiny number 2 of order k =m2Pl there. ð1Þ (G) The negative sign of g ð0Þ and g is of no relevance to the formal considerations of the present ð0Þ paper; actually g is known to be positive for fermionic matter fields [18]. 4. Tadpole equation and self-consistent backgrounds One of the perhaps somewhat unusual features of the effective average action at k – 0 is that its field-source relations (‘‘effective field equations’’) which govern the expectation value fields involve e k  Ck þ Dk S rather than Ck. In an arbitrary theory with dynamical (i.e., non-background) fields Ui C these relations read, for vanishing sources [39],

e k ½U dCk ½U dDk S½U dC  þ ¼ 0: dUi dUi dUi

ð4:1Þ

R

Since, symbolically, Dk S / URk U, the last term in Eq. (4.1) equals essentially Rk Ui . It vanishes at k = 0 since R0 ¼ 0 by construction.  lm . The effective For the gravity-scalar system of the present paper the set (Ui) comprises A and h field equation for the scalar reads

 g pffiffiffi dCk ½A; h; þ gRk ½gAðxÞ ¼ 0: dAðxÞ

ð4:2Þ

lm , holds for the metric fluctuation. This latter equation simA similar equation, involving a term Rk h lm ¼ 0 so that the plifies when we search for self-consistent backgrounds, since then we impose h lm term is absent. Hence the generalization of the tadpole equation (1.3) for k – 0 reads simply Rk h

   selfcon   C ½A; h; g   k lm ðxÞ  dh d

¼ 0:

ð4:3Þ

h¼0

We call gselfcon a self-consistent background metric for the scale k if there exists a scalar field configuration A(x) such that the coupled system of Eqs. (4.2) and (4.3) is satisfied. For the truncation ansatz (3.1) the scale dependent equation (4.3) reads explicitly, omitting the selfcon , superscript from g

1 ð1Þ ð1Þ Glm  Ek glm R þ Kk glm ¼ 8pGk T lm ½A; g: 2

ð4:4Þ

Here Tlm denotes the Euclidean energy–momentum tensor

  1 d  M   T ½A; g   2 pffiffiffi Ck ½A; g; g  :  g dg lm g¼g lm

ð4:5Þ

 Within the truncation studied above the matter part of the average action, CM k ½A; g; g , is given by the last line in Eq. (3.1). In this section we are slightly more general and allow for an arbitrary matter action which may have an ‘‘extra’’ g-dependence. Above, such a dependence was excluded by the form of the ansatz, but in more general truncations an ‘‘extra’’ g-dependence can be induced by the RG  but also by split running. At k – 0 gravity interacts with matter not only via the full metric g ¼ g þ h, symmetry violating ‘‘extra’’ g-couplings. The condition for a self-consistent background (4.4) differs from Einstein’s equation by the Ek-term on its LHS. This term is not forbidden by any general principle since (4.4) is not the variation of a diffeomorphism invariant functional F[A, g], evaluated at g ¼ g. If it was, the pure gravity part in the

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R pffiffiffi g RðgÞ, which produces a pure Einstein tensor with no Ek2-derivative sector of F[A, g] could only be correction. In fact, the Ek-term owes its existence to the violated split symmetry or, stated differently,  g. What goes beyond the familiar variational principle   Ck ½A; h; to the bimetric character of Ck ½A; g; g . that would lead to the standard Einstein equation is precisely the ‘‘extra’’ g-dependence of Ck ½A; g; g Interestingly, the tadpole equation (4.4) can be written as an ordinary Einstein equation, with no Ek-term on its LHS, but with a modified energy momentum tensor on its RHS. To see this we take the trace7 of (4.4)

R¼

  1 ð1Þ ð1Þ 4Kk  8pGk glm T lm : 1 þ 2Ek

ð4:6Þ

If we substitute this formula for the curvature scalar into the Ek-term of (4.4) we arrive at a conventionally-looking Einstein equation

e ð1Þ glm þ 8pGð1Þ Te lm ½A; g: Glm ¼  K k k

ð4:7Þ

Yet, this equation contains a rescaled cosmological constant,

e ð1Þ  K k

Kð1Þ k ; 1 þ 2Ek

ð4:8Þ

and a modified energy–momentum tensor

Te lm ½A; g  T lm ½A; g 

Ek glm gab T ab ½A; g: 2ð1 þ 2Ek Þ

ð4:9Þ

The ordinary Einstein equation comes with a nontrivial integrability condition which is of central importance, both from the mathematical and the physical point of view. As a consequence of Bianchi’s identity, the equation is consistent only when the energy–momentum tensor is covariantly conserved. For the tadpole equation (4.4) the situation is more complicated. Applying Dl to it we obtain

Dl T lm ½A; g ¼ 

1 1 ð1Þ 8pGk Ek glm @ l R: 2

ð4:10Þ

We see that if Ek – 0, the existence of a self-consistent background requires that the energy–momentum tensor is not conserved, Dl T lm – 0, unless @ l R ¼ 0. Using (4.6) we can re-express (4.10) as

Dl T lm ½A; g ¼

Ek glm Dl ðgab T ab ½A; gÞ: 2ð1 þ 2Ek Þ

ð4:11Þ

ab T ab , and therefore R, happens to be Again, we see that Tlm can be conserved only if the trace g constant. Equivalently, we could have started from the form (4.7) of the tadpole equation. Then the integrae lm ¼ 0 which, when rewritten in terms of Tlm, likewise leads to Eq. (4.11). bility condition is Dl T So, given this new kind of integrability condition at finite k, can we actually expect the tadpole equation to have solutions? First of all it is clear that for Tlm = 0 the effective Einstein equation is of the standard type, so for pure gravity8 the situation is the same as in classical general relativity. As for gravity coupled to matter, in classical relativity a standard argument shows that the energy– momentum tensor is conserved if the matter action is invariant under diffeomorphisms and the matter fields satisfy their equations of motion. Remarkably, the same argument does not go through for the tad pole equation at k – 0, for two independent reasons. Indeed, as CM k ½A; g; g  is diffeomorphism invariant  it satisfies a Ward identity of the form (1.10) which, with (4.5), can be seen to imply at g ¼ g

Dl T lm ½A; g ¼ 

7 8

   glm @ l A dCM 2 dCM ½A; g; g k ½A; g ; g  pffiffiffi þ pffiffiffi Dl k   dA dglm g g

:

ð4:12Þ

g¼g

For simplicity we set d = 4 here. While this case is not considered in the present paper, we expect that in pure quantum gravity, too, an Ek-term is generated.

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Note that the first term on the RHS of (4.12) is not in general zero when A is on shell, the reason being pffiffiffi M   that (4.12) involves dCM k =dA while the equation of motion is (4.2), or dCk =dA ¼  g Rk ½g A. Thus, this lm ð@ l AÞRk ½g A and so it is nonzero even when A is on-shell. However, as it should be, it term equals g vanishes in the limit k ? 0.  The second term on the RHS of (4.12) is nonzero precisely when CM k has an ‘‘extra’’ g -dependence. As we discussed already, this dependence must disappear for k ? 0, but at finite k there is no reason why the RG evolution should not generate such a g-dependence. So we can summarize the situation with matter by saying that, on the one hand, the integrability of the tadpole equation generically requires a non-conserved Tlm at k – 0, and that on the other hand the very structure of the average action entails that the coarse graining and the RG running actually do produce a non-conservation of a certain type. Whether this generic non-conservation of Tlm is such that it renders the tadpole equation integrable cannot be assessed by general arguments but only in special examples; we shall come back to this issue elsewhere. It should also be emphasized that for the existence of an asymptotically safe fundamental theory it is by no means necessary that there are self-consistent backgrounds for all physical situations. Such backgrounds are merely a convenient tool for the visualization of special properties and predictions of the theory, but they are not needed for its construction in terms of an RG flow on theory space. It is quite conceivable that in a regime where the quantum effects are strong no self-consistent background will exist, indicating that the concept of a classical mean field metric breaks down there. 5. Conclusion The gravitational average action approach to quantum gravity solves the ‘‘background independence’’ problem by giving an extra g-dependence to Ck. The average action Ck ½g lm ; glm ; . . . is inherently of a bimetric nature. In particular a functional flow equation which is exact in Wilson’s sense can be formulated only on a theory space of functionals depending on two metrics. Almost all investigations within this framework performed to date employ truncations which retain only the minimum extra gdependence due to the gauge fixing term. A comprehensive analysis of the extra background field dependence is highly desirable, however, both for improving the quality and precision of the predictions, and for gaining insights into various structural issues and problems which are obscured in a single metric approximation. In particular in the Asymptotic Safety context bimetric truncations are likely to be unavoidable in order to improve upon the precision that has been achieved already. In fact, in a first investigation based upon a bimetric ansatz [27] utilizing the conformally reduced gravity framework, it was found that the modifications caused by the generalization of the truncation are probably larger than the typical scheme dependence within the original single metric truncation. Generalizing these investigations to full fledged gravity is difficult because of the complicated functional traces appearing in the FRGE. In the present paper we took the first step in this direction by computing the RG flow of a bimetric action Ck ½A; g; g for a set of scalar fields interacting with all irreducible components of the metric. Yet, to simplify matters and to make the structural issues as transparent as possible, we neglected the contribution of the metric fluctuations to the corresponding beta functions. This system was known to display a non-Gaussian fixed point when analyzed within a single metric truncation [14–16,18]. As for the Asymptotic Safety program, our most important finding is that the bimetric truncation, too, gives rise to such a fixed point, a very encouraging result indeed. We stress that for Asymptotic Safety to work, it is presumably not strictly necessary that all coupling constants coordinatizing the bimetric theory space approach fixed point values. For physical divergencies to be absent, it might suffice that some subset of couplings (i.e., the ones entering into physical observables, like scattering amplitudes) remain finite in the UV. However given the notorious difficulty of finding such observables, it is not possible to identify any subset of this kind. On the conceptual side, the discussions in this paper illustrates that defining ‘‘the’’ Newton’s constant or ‘‘the’’ cosmological constant in the RG context requires some care. At scales k – 0 the split  and g is broken by the cutoff, and therefore an expansion of symmetry which ties together h    Ck ½A; h; g  in powers of h contains an infinite set of different field monomials, each being accompanied

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 by its own running coupling. For instance, the expansion of the zero-derivative terms in powers of h ð0Þ ð1Þ ð2Þ  induces a set of couplings Kk ; Kk ; Kk ; . . .. In the h-expansion of Ck they are the coefficients of the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi  lm Þ with respect to h  lm . Those monomials monomials which arise when we expand g  detðglm þ h  are characterized not only by the number of h’s they contain, but also by their tensor structure. At or2 , for instance, the two basis monomials are ðg lm Þ2 and h qr . Imposing the restoration lm glq g lm h mr h der h of split symmetry at k = 0, all these monomials degenerate to the same quantity.  and/or g . If Similar remarks apply to the sectors with a nonzero number of derivatives acting on h    I½g ¼ I½g þ h is any invariant built from a single metric we can expand it in h and arrive at a represen g. Again, the expansion generates infinitely many diffeomorphically  ¼ P1 uðnÞ IðnÞ ½h;  þ h tation I½g n¼0 class ðnÞ   invariant monomials in h; I ½h; g. We assume that they are normalized in some canonical way (‘‘unit ðnÞ

prefactor’’). Hence the Taylor series gives rise to a well defined set of coefficients uclass multiplying them. It is convenient to take the I(n)’s as a subset of the basis elements spanning theory space. Then a single invariant I[g], depending on one metric, is seen to supply infinitely many terms to be included  and g independently. As split symmetry is broken in general they in Ck, and these terms depend on h P ðnÞ ðnÞ   with k½h; g evolve differently under the RG flow. Hence the corresponding part of Ck reads 1 n¼0 uk I ðnÞ

ðnÞ

ðnÞ

dependent coefficients uk . Only if split symmetry happens to be intact we have uk ¼ uclass for all n. A priori, this appears as a substantial proliferation of coupling constants determining a given RG trajectory. Imposing the restoration of split symmetry at k = 0, however, puts sever constraints on ðnÞ ðnÞ these couplings by fixing the initial conditions u0 ¼ uclass for all n, while coupling constants not asso have to vanish. These relations reduce the number of undetermined ciated to an expansion of I½g þ h couplings governing an ‘‘asymptotically safe’’ bimetric RG trajectory to precisely the ones present in the single-metric case. In other words, while the bimetric setting contains significantly more coupling constants, it does not result in additional initial conditions for the resulting RG trajectories, as one could have expected initially. In Sections 3 and 4 we made this structure explicit, working at the lowest nontrivial order of both  and the derivative-expansion. We derived the RG flow of the terms linear in h,  investigated its the hfixed point properties and, as an application, set up the tadpole equation for self-consistent background  does not exactly have geometries. Even at the two-derivative level this k-dependent field equation for g the structure of the classical Einstein equation. As it cannot be obtained as the variation of any single metric action, further tensor structures are possible. As demonstrated in Section 4, the two-derivative approximation of the tadpole equation can be rewritten in the style of Einstein’s equation, but with a e ð1Þ . non-standard source term on its RHS. It also contains still another running cosmological constant, K k lm only, but In Section 2 we performed a complementary analysis, for constant metrics glm and g  ¼ g  g is small. Studying the RG flow of all possible non-derivative terms without assuming that h pffiffiffi we saw that, contrary to common belief, it is not predominantly a g term which is induced by the pffiffiffi vacuum fluctuations of the matter fields, but rather g and more complicated invariants involving both metrics. Even though this observation by itself is not a solution to the cosmological constant problem it suggests that the possibility of vacuum fluctuations curving spacetime should be investigated with much more care. At least for simple physical settings in flat space, the effective average action gives rise to an effective field theory valid at the scale k. By analogy, this property of Ck ½g; g ¼ g has been exploited in some phenomenological investigations of asymptotically safe gravity [56–66], for instance in inflationary cosmology [60], see also [67]. For the time being, it remains an open question if and how the new couplings contained in the bimetric setup affect such renormalization group improvements. We expect, however, that single-metric approximations of the RG flow suffice to capture the leading quantum gravity effects in this framework. Acknowledgements We thank G. ’t Hooft and J. Pawlowski for interesting discussions. The research of E.M. and F.S. is supported by the Deutsche Forschungsgemeinschaft (DFG) within the Emmy-Noether program (Grant SA/1975 1-1).

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457

 Appendix A. Beta functions for the derivative- and h-expansion In this appendix we derive the RG equation (3.6) and (3.7) resulting from the truncation ansatz (3.1). A.1. The master equation Using Eq. (3.1) as the LHS of the FRGE (2.1), we obtain the following ‘‘master equation’’

( ! ! ! ! pffiffiffi Kkð0Þ Kkð1Þ lm  1 1 1 1 1  lm   þ @t R þ @t Glm h d x g @t g hl m  @ t ð0Þ ð1Þ ð0Þ ð1Þ 2 2 2 Gk Gk Gk Gk ! )   1 1 Ek 1 ð2Þ l m   g þ R ½g    ¼ h Tr R g C ½A; g ¼ g þ h; @ R ½ g  ;  @t l m t k k k ð1Þ 4 2 Gk

1 8p

Z

d

ðA:1Þ

from which all beta-functions for the running coupling constants can be read off, by matching the  and derivative expansion on both sides. Thus we must expand its RHS in h  lm coefficients of the h  lm appears only inside the Hessian and retain all terms of order zero and one. The field h





ð2Þ ð2Þ 2       Cð2Þ k ½A; g ¼ g þ h; g  ¼ Ck ½A; g ; g  þ dCk ½g  þ O hlm :

ðA:2Þ

 lm . Upon inserting (A.2) into the functional Here, by definition, dC stands for the term linear in h trace of (A.1) and expanding the denominator we see that the RHS of the master equation (A.1) is given by the sum of the following two traces, containing the order zero and order one contributions, respectively ð2Þ  k ½g 

T0 ¼

  1 1 ð2Þ @ t Rk ½g ; Tr Ck ½A; g; g þ Rk ½g 2

ðA:3Þ

  2 1 ð2Þ ð2Þ @ t Rk ½gdCk ½g : T 1 ¼  Tr Ck ½A; g; g þ Rk ½g 2

ðA:4Þ

In the sequel, we will evaluate these traces up to first order in the background curvature, employing early-time heat-kernel techniques. ð2Þ

ð2Þ

A.2. Matrix elements of Ck and dCk

 – 0, We start by computing the position space matrix elements (2.2) for the functional (3.1). For h  i.e., g – g



 Cð2Þ k ½A; g; g 

 xy

hpffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffi 1 1 pffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffi gðxÞx dðx  yÞ þ gðyÞy dðx  yÞ 2 gðxÞ gðyÞ pffiffiffiffiffiffiffiffiffi gðxÞ 00 þ U ðAðxÞÞdðx  yÞ: gðxÞ k

¼

ðA:5Þ

Note the occurrence of both g and g in (A.5). As always, h  glmDlDm stands for the Laplace–Beltrami lm we obtain operator built from glm. Further expanding the matrix elements (A.5) up to first order in h  lm , the linear term, with the abbreviation c  12 glm h

" #   1 cðxÞ cðyÞ ð2Þ dCk ¼  pffiffiffiffiffiffiffiffiffi x dðx  yÞ þ pffiffiffiffiffiffiffiffiffi y dðx  yÞ xy 2 gðyÞ gðxÞ " #  1 1 1  pffiffiffiffiffiffiffiffiffi dx dðx  yÞ þ pffiffiffiffiffiffiffiffiffi dy dðx  yÞ    2 gðyÞ gðxÞ

cðxÞ þ pffiffiffiffiffiffiffiffiffi U 00k ðAðxÞÞdðx  yÞ: gðxÞ g¼g

ðA:6Þ

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lm to (A.5) and setting g ¼ g afterwards. Formally this result is obtained by applying a variation dg lm ¼ h The variation of the Laplacian, djg¼g , is left unevaluated for the time being. In fact we shall see that there is a very elegant way of calculating this type of traces without relying on explicit expressions. A.3. Associated differential operators and IR-cutoff Applying the rule described in  the paragraph following Eq. (2.2) we can associate a differential ð2Þ operator to the matrix elements Ck . For (A.5) this yields

C

ð2Þ  k ½A; g; g 

xy pffiffiffiffiffiffiffiffiffi  gðxÞ  00 ¼ pffiffiffiffiffiffiffiffiffi  þ U k ðAÞ : gðxÞ

ðA:7Þ

In particular, for g ¼ g,

Ckð2Þ ½A; g; g ¼  þ U 00k ðAÞ:

ðA:8Þ

Analogously, the differential operator resulting from (A.6) is ð2Þ

d Ck ¼ 

i 1h cðxÞ þ cðxÞ þ d þ ðdÞy þ cðxÞU 00k ðAÞ: 2

ðA:9Þ

Here the hermitian adjoint operator (dh)  is defined with respect to the inner product given by R d pffiffiffiffiffiffiffiffiffi ðA1 ; A2 Þ  d x gðxÞA1 ðxÞA2 ðxÞ. ð2Þ At this point a word of caution might be appropriate. While the matrix elements of dCk can be obð2Þ tained by applying a formal variation ‘‘d’’ to those of Ck , the same is not true for the associated differential operators. If we naively apply a ‘‘d’’ to Eq. (A.7) the result is not the (correct) Eq. (A.9); in place of the hermitian combinations c þ c and dh + (dh) , respectively, we would obtain the non-hermitian operators 2c and 2dh. Considering (A.8), we can see that the correctly adjusted cutoff operator is given by

  2 Rk ½g ¼ k Rð0Þ  2  Rk ðÞ; k

ðA:10Þ

since then

Ckð2Þ ½A; g; g þ Rk ½g ¼  þ Rk ½g þ U 00k ðAÞ;

ðA:11Þ 2

so that the inverse propagator of the low momentum modes becomes  þ k þ U 00k , as it should be. A.4. The relevant traces Let V : R ! R be an arbitrary real valued function. Then V(D2(g))  = V(D2(g)) is hermitian with respect to (, ), whence Tr[V(D2(g))] is real for any metric glm. As a consequence, dTr½VðD2 ðgÞÞ ¼   Tr½VðD2 ðg ÞÞdD2   Tr½VðD2 ðg  þ hÞÞ ÞÞ is real, too. Furthermore, since Tr½Ay  ¼ Tr½A for Tr½V 0 ðD2 ðg any matrix A, we may write

Tr½V 0 ðD2 ðgÞÞdD2  ¼ Tr½fV 0 ðD2 ðgÞÞdD2 gy  ¼ Tr½fdD2 gy fV 0 ðD2 ðgÞÞgy  ¼ Tr½V 0 ðD2 ðgÞÞfdD2 gy : This relation implies, for an arbitrary function F  V0 ,

1 Tr½FðÞðd þ ðdÞy Þ ¼ Tr½FðÞd: 2

ðA:12Þ

Likewise exploiting the cyclicity of the trace we have

1 Tr½FðÞðc þ cÞ ¼ Tr½FðÞc: 2

ðA:13Þ

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The identities (A.12) and (A.13)  imply that under the trace (A.4) we may effectively replace the operator dC(2) of Eq. (A.9) with  þ U 00k ðAÞ c  d. As a consequence of this, and also by using (A.9) we may rewrite T0 and T 1  T A1 þ T B1 according to

i 1 1 h

Tr  þ Rk ðÞ þ U 00k ðAÞ @ t Rk ðÞ ; 2 2

  i 1 h

T A1 ¼  Tr  þ Rk ðÞ þ U 00k ðAÞ @ t Rk ðÞ  þ U 00k ðAÞ glm h lm ; 4 h i

 1 2 T B1 ¼  Tr  þ Rk ðÞ þ U 00k ðAÞ @ t Rk ðÞdðÞ : 2

T0 ¼

ðA:14Þ ðA:15Þ ðA:16Þ

Thus the RHS of the master formula is given by the sum T 0 þ T A1 þ T B1 . A.5. Explicit evaluation of the operator traces

U 00k

We are now in the position to compute the traces T0, T A1 , and T B1 explicitly. From now on we set  2 ¼ const, as in the main text. ¼m (A) Evaluation of T0 The trace T0 of (A.14) provides the order zero contribution that occurred already in the single metric truncation. It is straightforwardly evaluated using the familiar heat kernel based expansion [1]



Z Z 1 d pffiffiffi d pffiffiffi Tr½WðD2 Þ ¼ ð4pÞd=2 trðIÞ Q d=2 ½W d x g þ Q d=21 ½W d x g R þ OðR2 Þ 6

ðA:17Þ

R1 1 with Q n ½W  CðnÞ dz zn1 WðzÞ and tr(I) denoting the unit-trace with respect to the internal 0 indices which, in our case, just counts the number of scalars. The result is most conveniently expressed in terms of the standard threshold functions [1]

Upn ðwÞ 

1 CðnÞ

Z

0

1

dz zn1

0

Rð0Þ ðzÞ  zRð0Þ ðzÞ ½z þ Rð0Þ ðzÞ þ wp

:

ðA:18Þ

lm , Applying (A.17) to (A.14) yields, retaining terms with at most two derivatives of g

T0 ¼



Z Z pffiffiffi 1 d2 1 pffiffiffi d 1 d d 2 2   k k U ðm Þ d x U ðm Þ d x R : g þ g d=2 d=21 6 ð4pÞd=2 ns

ðA:19Þ

(B) Evaluation of T A1 The trace T A1 is also fairly simple to compute since the local form of (A.17) suffices for this purpose. In fact, when we evaluate the trace (A.15) in the position space representation we may  lm ð^ pull out the position dependent operator glm ð^ xÞh xÞ from the matrix element by virtue of the eigenvalue equation of ^ x:

T A1 ¼ ¼

Z

Z

d

lm ð^xÞjxi d xhxjf  gglm ð^xÞh d

lm ðxÞ: d xhxjf  gjxiglm ðxÞh

ðA:20Þ

Using the local analog of Eq. (A.17) for the diagonal matrix element hxj{  }jxi we arrive at

T A1 ¼  

ns 4ð4pÞ ns

d=2

24ð4pÞ

h iZ pffiffiffi d d lm k dU2d=2þ1 ðm2 Þ þ 2m2 U2d=2 ðm2 Þ d x gglm h

k d=2

d2

h iZ pffiffiffi d  lm : ðd  2ÞU2d=2 ðm2 Þ þ 2m2 U2d=21 ðm2 Þ d x gRglm h

ðA:21Þ

(C) Evaluation of T B1 In general the evaluation of traces involving factors of dh requires very lengthy computations  lm in the present and this is in fact the reason why we restrict ourselves to the first order in h paper. The trace (A.16) has the structure Tr[{  }dh] with a single dh insertion only, and such traces can be calculated very elegantly by means of the following trick.

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The expansion (A.17) is identically satisfied for all metrics glm. We may therefore perform the variation glm ? glm + dglm on both sides of this expansion. After a short calculation the result is found to be

Z 1 1 1 d pffiffiffi Tr½WðD2 ÞdðD2 Þ ¼ trðIÞ d x g  Q d=2þ1 ½Wg lm þ Q d=2 ½WðGlm  Dl Dm þ g lm D2 Þ dg lm : d=2 2 6 ð4pÞ

ðA:22Þ With this identity in our hands it is now straightforward to calculate lm to (A.16) we find, discarding surface terms, g lm ¼ glm and dg lm ¼ h

T B1 ¼

ns

d

k U2d=2þ1 ðm2 Þ 2ð4pÞd=2

Z

pffiffiffi d  lm  d x gglm h

ns 6ð4pÞ

k d=2

d2

U2d=2 ðm2 Þ

Z

T B1 .

Applying (A.22) with

pffiffiffi d  lm : d x gGlm h

ðA:23Þ

2 and higher are considerably more involved [55]. The analogous computations at order h lm (D) The beta functions The RHS of Eq. (A.1) is explicitly known at this point; it is given by the sum of (A.19), (A.21) and (A.23). By equating the coefficients of like field monomials on both sides of (A.1) we can now read off the beta functions for the various combinations of coupling constants. Expressing the result in terms of dimensionless variables we thus arrive at the final results given in Eqs. (3.6) and (3.7) of the main text.

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