PHYSICAL REVIEW D 101, 065001 (2020)
Functional renormalization group approach and gauge dependence in gravity theories
† ‡ Vítor F. Barra ,1,* Peter M. Lavrov,2,3, Eduardo Antonio dos Reis,1, ∥ Tib´erio de Paula Netto,4,§ and Ilya L. Shapiro 1,2,3, 1Departamento de Física, ICE, Universidade Federal de Juiz de Fora, 36036-330 Juiz de Fora, MG, Brazil 2Department of Theoretical Physics, Tomsk State Pedagogical University, 634061 Tomsk, Russia 3National Research Tomsk State University, 634050 Tomsk, Russia 4Departament of Physics, Southern University of Science and Technology, 518055 Shenzhen, China
(Received 21 October 2019; accepted 11 February 2020; published 4 March 2020)
We investigate the gauge symmetry and gauge-fixing dependence properties of the effective average action for quantum gravity models of general form. Using the background field formalism and the standard Becchi-Rouet-Stora-Tyutin (BRST)-based arguments, one can establish the special class of regulator functions that preserves the background field symmetry of the effective average action. Unfortunately, regardless if the gauge symmetry is preserved at the quantum level, the noninvariance of the regulator action under the global BRST transformations leads to the gauge-fixing dependence even under the use of the on-shell conditions.
DOI: 10.1103/PhysRevD.101.065001
I. INTRODUCTION The perturbative renormalization group in this model is – The interest in the nonperturbative formulation in quan- well explored [7 9], but it is not conclusive for the tum gravity has two strong motivations. First, there are discussion of the dressed propagator. In general, the long-standing expectations that even the perturbatively attempts to explore this possibility in the semiclassical nonrenormalizable models such as the simplest quantum and perturbative quantum level [10,11] has been proved gravity based on general relativity may be quantum mechan- nonconclusive [12], and hence the main hope is related to ically consistent due to the asymptotic safety scenario [1] the nonperturbative calculations in the framework of the (see [2,3] for comprehensive reviews). On the other hand, functional renormalization group approach [13] (see [14] there is a possibility that the nonperturbative effects may for an extensive review). provide unitarity in the fourth derivative theory by trans- Thus, in both cases the consistency of the results forming the massive unphysical pole, which spoils the obtained within the functional renormalization group spectrum of this renormalizable theory [4]. The presence approach is of the utmost importance. In this respect there of such a massive ghost violates the stability of classical are two main dangers. For the quantum gravity models solutions (see e.g., discussion and further references in [5]). based on general relativity, the running of a Newton At the quantum level, the perturbative information is constant in four-dimensional spacetime is always obtained insufficient to conclude whether in the dressed propagator on the basis of quadratic divergences. These divergences the ghost pole does transform into a nonoffensive pair of are known to have strong regularization dependence. In complex conjugate poles [6]. particular, they are absent in dimensional regularization and can be freely modified in all known cutoff schemes by changing the regularization parameter. This part of the problem does not exist for the functional renormalization *[email protected] † group applied to the fourth derivative quantum gravity. [email protected] ‡ [email protected] However, in this case there is yet another serious problem, §[email protected] related to the gauge-fixing dependence of the effective ∥ [email protected] average action. This problem is the subject of the present work. In previous publications [15–17] we explored the Published by the American Physical Society under the terms of gauge-fixing dependence in Yang-Mills theories and it was the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to shown that such dependence for the effective average the author(s) and the published article’s title, journal citation, action does not vanish on-shell, except maybe in the fixed and DOI. Funded by SCOAP3. point where this object becomes identical with the usual
2470-0010=2020=101(6)=065001(12) 065001-1 Published by the American Physical Society BARRA, LAVROV, REIS, NETTO, and SHAPIRO PHYS. REV. D 101, 065001 (2020) effective action in quantum field theory. Except this special theories to study the gauge dependence problem in this point there is uncontrollable dependence on the set of kind of theory. This method allows us to work with the arbitrary gauge-fixing parameters, and thus one can expect effective action which is invariant under the gauge trans- a strong arbitrariness in the renormalization group flows formations of the background fields. which lead to the fixed point and, in fact, define its position Despite the numerous aspects of quantum properties and proper existence. The main purpose of the present work successfully studied with the background field method is to extend this conclusion to quantum gravity. It is [51–60], the gauge dependence problem remains important remarkable that one can complete this task for the quantum [16] and needs to be considered in more details. We obtain gravity theory of an arbitrary form, without using the restrictions on the tensor structure of the regulator functions concrete form of the action. One can use this consideration, which allows us to construct a gauge invariant effective e.g., for the superrenormalizable models of quantum average action. Nevertheless, the effective average action gravity, when the perturbative renormalization group remains dependent of the gauge choice at on-shell level. may be exact and, moreover, completely independent on The paper is organized as follows. In Sec. II we introduce the gauge fixing [18,19]. This example is somehow the general considerations of quantum gravity theories through most explicit one, since it shows that the transition from the background field method. In Sec. III we consider standard quantum field theory to the functional renormal- the FRG approach for quantum gravity theories and ization group (FRG) may actually spoil the “perfect” find conditions that we must impose in the regulator situation, namely exact and universal renormalization functions to maintain the gauge invariance of background group flow. effective average action. Based on this, we also present some In Yang-Mills theories [20], the gauge symmetry of the interesting candidates to the regulator functions. In Sec. IV initial action is broken on a quantum level due to the gauge- we prove the gauge dependence of vacuum functional for the fixing procedure in the process of quantization. In turn, the model under consideration. Finally, our conclusions and effective potential depends on gauge [21–24]. This depend- remarks are presented in Sec. V. ence occurs in a special way, such that it disappears on In the paper, DeWitt notations [61] are used.R TheR short shell [25,26], which means that it is possible to give a notation for integration in D dimension is dDx ¼ dx. physical interpretation to the results obtained at the All the derivatives with respect to fields are left derivative quantum level. unless otherwise stated. The Grassmann parity of a quantity One of the well-developed nonperturbative methods in A is denoted as εðAÞ. quantum field theory to study quantum properties of physical models beyond the perturbation theory is the FRG approach II. QUANTUM GRAVITY IN THE – [27,28] (see also the review papers [29 34]). As we have BACKGROUND FIELD FORMALISM already mentioned above, when applied to gauge theories, this method leads to obstacles related to the on-shell gauge Let us consider an arbitrary action S0 ¼ S0ðgÞ, where dependence of the effective average action. g ¼ gαβðxÞ is the metric tensor of an arbitrary Riemann There are some efforts to solve this problems. One of manifold. We assume that the action is invariant under them consists from reformulation of Yang-Mills theory general coordinates transformations with the application of a gauge-invariant cutoff dependent μ μ μ regulator function that is introduced as a covariant form x → x ¼ x ðx0Þ; ð1Þ factor into the action of Yang-Mills fields, which leads to an invariant regulator action [35,36]. As a consequence, which leads to the metric transformations the effective average action is gauge invariant on shell. ∂xμ ∂xν A second approach [37,38] is based on the use of the 0 0 gαβðx Þ¼gμνðxÞ α β ð2Þ Vilkovisky-DeWitt covariant effective action [39,40]. This ∂x0 ∂x0 technique provides gauge independence even off shell, but and consider the infinitesimal form of these transforma- it introduce other types of ambiguities. An alternative σ σ σ tions, x0 ¼ x þ ξ ðxÞ, when formulation was presented in [15]; it consists of an alternative way of introducing the regulator function as a σ σ σ δgαβ ¼ −ξ ðxÞ∂σgαβðxÞ − gασðxÞ∂βξ ðxÞ − gσβðxÞ∂αξ ðxÞ: composite operator. When applied in gauge theories, this approach leads to the on-shell gauge invariance of the ð3Þ effective average action. In the present work, we apply the background field The diffeomorphism (3) can be considered as the gauge method [41–43] (recent advances for the gauge theories can transformation for gαβðxÞ be found in [16,44–49] and discussion for the quantum Z gravities case in [50]) in the FRG approach as a reformu- σ δgαβðxÞ¼ dyRαβσðx; y; gÞξ ðyÞ; ð4Þ lation to the quantization procedure for quantum gravity
065001-2 FUNCTIONAL RENORMALIZATION GROUP APPROACH AND … PHYS. REV. D 101, 065001 (2020)
Z pffiffiffiffiffiffiffiffiffiffiffiffi where ¯ ¯ ¯ α βγ ¯ Sghðϕ; gÞ¼ dxdydz −gðxÞC ðxÞHα ðx; y; g; hÞ
Rαβσðx; y; gÞ¼−δðx − yÞ∂σgαβðxÞ − gασðxÞ∂βδðx − yÞ σ × Rβγσðy; z; g¯ þ hÞC ðzÞ; ð11Þ − gσβðxÞ∂αδðx − yÞð5Þ Z pffiffiffiffiffiffiffiffiffiffiffiffi α S ϕ; g¯ dx −g¯ x B x χα x; g;¯ h ; 12 are the generators of the gauge transformations of the gfð Þ¼ ð Þ ð Þ ð Þ ð Þ metric tensor, and ξσðyÞ are the gauge parameters. The algebra of generators is closed, namely where Z δRαβσðx;y;gÞ δRαβγðx;z;gÞ βγ δχαðx; g;¯ hÞ du Rμνγðu;z;gÞ− Rμνσðu;y;gÞ Hα ðx; y; g;¯ hÞ¼ ð13Þ δgμνðuÞ δgμνðuÞ δhβγðyÞ Z − ; λ i ¼ duRαβλðx;u gÞFσγðu;y;zÞ; ð6Þ and χαðx; g;¯ hÞ are the gauge-fixing functions, ϕ ðxÞ¼ α α ¯ α fhαβðxÞ;B ðxÞ;C ðxÞ; C ðxÞg is the set of quantum fields, where CαðxÞ and C¯ αðxÞ are the ghost and antighost fields, respectively, and BαðxÞ are the Nakanishi-Lautrup auxiliary λ λ λ Fμνðx;y;zÞ¼δðx − yÞδν∂μδðx − zÞ − δðx − zÞδμ∂νδðx − yÞ; fields. The Grassmann parity of all quantum fields are as follows: ð7Þ α ¯ α α i λ λ εðhαβÞ¼εðB Þ¼0; εðC Þ¼εðC Þ¼1; εðϕ Þ¼εi: with Fμνðx; y; zÞ¼−Fνμðx; z; yÞð8Þ The ghost numbers are are structure functions of the algebra which does not depend on the metric tensor. Let us stress that the Bα h 0; Cα 1; C¯ α −1: mentioned features are valid for an arbitrary action of ghð Þ¼ghð αβÞ¼ ghð Þ¼ ghð Þ¼ gravity, since the algebra presented above is independent on the initial action. Therefore, any theory of gravity is a Independent of gauge-fixing function choice, the action gauge theory and the structure functions are independent of (10) is invariant under a global supersymmetry trans- the fields, that means quantum gravity is similar to the formation, known as Becchi-Rouet-Stora-Tyutin symmetry Yang-Mills theory. [63,64]. The gravitational BRST transformations were A useful procedure to quantize gauge theories is the so- introduced in [4,65,66] and can be presented as called background field formalism. In what follows, we δ − σ ∂ ∂ σ shall perform the quantization of gravity on an arbitrary BhαβðxÞ¼ ðC ðxÞ σgαβðxÞþgασðxÞ βC ðxÞ σ external background metric g¯αβðxÞ. The standard references þ gσβðxÞ∂αC ðxÞÞλ; on the background field formalism in quantum field theory α σ α δ C x C x ∂σC x λ; are [41–43] (see also recent advances for the gauge theories B ð Þ¼ ð Þ ð Þ ¯ α α in [16] and [48] for the discussion of quantum gravity). δBC ðxÞ¼B ðxÞλ; In the background field formalism, the metric tensor α δBB ðxÞ¼0; ð14Þ gαβðxÞ is replaced by the sum where λ is a constant Grassmann parameter. In condensed gαβðxÞ¼g¯αβðxÞþhαβðxÞ; ð9Þ notation, we can write the BRST transformations as where g¯αβðxÞ is an external (background) metric field and i i i hαβðxÞ is the variable of integration, also called quantum δBϕ ðxÞ¼R ðx; ϕ; g¯Þλ; εðR Þ¼εi þ 1; ð15Þ metric. Thus, the initial action is replaced by Ri RðhÞ;Rα ;Rα ;Rα where ¼f αβ ðBÞ ðCÞ ðC¯ Þg and S0ðgÞ → S0ðg¯ þ hÞ: ϕ ¯ ðhÞ σ σ The Faddeev-Popov SFPð ; gÞ action is constructed in Rαβ ðx; ϕ; g¯Þ¼−C ðxÞ∂σgαβðxÞ − gασðxÞ∂βC ðxÞ the standard way [62] σ − gσβðxÞ∂αC ðxÞ; α σ α S ðϕ; g¯Þ¼S0ðg¯ þ hÞþS ðϕ; g¯ÞþS ðϕ; g¯Þ; ð10Þ ; ϕ ¯ ∂ FP gh gf RðCÞðx ; gÞ¼C ðxÞ σC ðxÞ; α α R ¯ ðx; ϕ; g¯Þ¼B ðxÞ; where Sghðϕ; g¯Þ is the ghost action and Sgfðϕ; g¯Þ is the ðCÞ gauge-fixing action. In the presence of external metric α ; ϕ ¯ 0 RðBÞðx ; gÞ¼ : ð16Þ g¯αβðxÞ, they can be written as
065001-3 BARRA, LAVROV, REIS, NETTO, and SHAPIRO PHYS. REV. D 101, 065001 (2020)
The generating functional of Green functions is δωSFPðϕ; g¯Þ¼0: ð23Þ defined as Z A consequence of (23) is the gauge invariance of (17). ¯ ϕ i ϕ ¯ ϕ Namely, ZðJ; gÞ¼ d exp ℏ ½SFPð ; gÞþJ δωZðJ; g¯Þ¼0: ð24Þ i ¯ ¼ exp ℏ WðJ; gÞ ; ð17Þ From this expression it is possible to prove that ΓðΦ; g¯Þ is where WðJ; g¯Þ is the generating functional of connected also gauge invariant Green functions. In Eq. (17) the product of the sources i δ Γ Φ ¯ 0 JiðxÞ and the fields ϕ ðxÞ was written in the condensed ω ð ; gÞ¼ : ð25Þ notation of DeWitt [61]. Explicitly, Z In the next sections we will discuss this and other Jϕ ¼ dxJ ðxÞϕiðxÞ; features for quantum gravity theories in the framework i of the FRG approach. μν ðBÞ ¯ where JiðxÞ¼fJ ;Jα ðxÞ; JαðxÞ;JαðxÞg ð18Þ III. FRG APPROACH FOR QUANTUM ε ε ϕi with the Grassmann parities ðJiÞ¼ ð Þ and ghost GRAVITY THEORIES i numbers ghðJiÞ¼ghðϕ Þ. The effective action ΓðΦ; g¯Þ is defined in terms of the The main idea of a functional renormalization group is to Γ Γ Legendre transformation use instead of an effective average action, k, where k is a momentum-shell parameter [27], in a way that ΓðΦ; g¯Þ¼WðJ; g¯Þ − JΦ; ð19Þ ¯ ¯ limΓkðϕ; gÞ¼Γðϕ; gÞ: ð26Þ i k→0 where Φ ¼fΦ g are the mean fields and the Ji are the solution of the equation In order to obtain Γkðϕ; g¯Þ, we introduce the average δWðJ; g¯Þ action ¼ Φi: ð20Þ δJi SkFPðϕ; g¯Þ¼SFPðϕ; g¯ÞþSkðϕ; g¯Þ; ð27Þ It is well known [25,26] that the effective action is gauge ¯ independent on shell, where Skðϕ; gÞ is the scale-dependent regulator action defined in curved spacetime δΓ Φ ¯ 0 ð ; gÞjδΓðΦ;g¯Þ 0 ¼ : ð21Þ Z δΦ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi Skðϕ; g¯Þ¼ dx −g¯ðxÞLkðϕ; g¯Þð28Þ At this moment we have considered only the trans- formations of the quantum fields. However, the background metric also transforms together with the quantum fields and the Lagrangian density is written as in the so-called background field transformations. The 1 rules of such transformation can be written, in the local ð1Þαβγδ Lkðϕ; g¯Þ¼ hαβðxÞR ðx; g¯ÞhγδðxÞ formulation, as 2 k ¯ α ð2Þ ; ¯ β σ σ σ þ C ðxÞRkαβðx gÞC ðxÞ; ð29Þ δωg¯αβðxÞ¼−∂σg¯αβðxÞω − g¯ασðxÞ∂βω − g¯σβðxÞ∂αω ; σ σ σ 1 αβγδ 2 δωhαβðxÞ¼−∂σhαβðxÞω − hασðxÞ∂βω − hσβðxÞ∂αω ; ð Þ ; ¯ ð Þ ; ¯ where the regulator functions Rk ðx gÞ and Rkαβðx gÞ ¯ α σ ¯ α ¯ σ α ¯ δωC ðxÞ¼−ω ∂σC ðxÞþC ðxÞ∂σω ; are dependent on the external fields gαβðxÞ. The regulator α σ α σ α functions obey the properties δωC ðxÞ¼−ω ∂σC ðxÞþC ðxÞ∂σω ; α σ α σ α 1 αβγδ 2 δωB ðxÞ¼−ω ∂σB ðxÞþB ðxÞ∂σω ; ð22Þ ð Þ ; ¯ 0 ð Þ ; ¯ 0 limRk ðx gÞ¼ and limRkαβðx gÞ¼ ; ð30Þ k→0 k→0 where ωσ ¼ ωσðxÞ are arbitrary functions. The background field transformations have the same structure of tensor which means that the average action recovers the Faddeev- → 0 transformations for tensors of types (0,2) and (1,0). Poppov action (10) in the limit when k . The regulator ð1Þαβγδ ; ¯ The background invariance of Faddeev-Popov action for functions Rk ðx gÞ also obey, by construction, the quantum gravity is known [48] and reads symmetry properties
065001-4 FUNCTIONAL RENORMALIZATION GROUP APPROACH AND … PHYS. REV. D 101, 065001 (2020) Z n ð1Þαβγδ ð1Þβαγδ ð1Þαβδγ pffiffiffiffiffiffiffiffiffiffiffiffi R ðx; g¯Þ¼R ðx; g¯Þ¼R ðx; g¯Þ ¯ ¯ ¯ k k k δωSkðϕ; gÞ¼ dx δω −gðxÞLkðϕ; gÞ ð1Þγδαβ ; ¯ pffiffiffiffiffiffiffiffiffiffiffiffi o ¼ Rk ðx gÞ: ð31Þ þ −g¯ðxÞδωLkðϕ; g¯Þ ¼ 0: ð33Þ
We want to solve the problem of average action For the first term in (33) we have invariance under background field transformations, Z pffiffiffiffiffiffiffiffiffiffiffiffi namely dxδω −g¯ðxÞLkðϕ; g¯Þ Z pffiffiffiffiffiffiffiffiffiffiffiffi ¯ σ ¯ ¼ − dx∂σð −gðxÞω ÞLkðϕ; gÞ δωSkFPðϕ; g¯Þ¼δωSkðϕ; g¯Þ¼0; ð32Þ Z pffiffiffiffiffiffiffiffiffiffiffiffi σ ¼ dx −g¯ðxÞω ∂σLkðϕ; g¯Þ; ð34Þ where the relation (23) is used. In what follows, we present explicit calculation where integration by parts was used. of variation of action (28). For this purpose, we write The variation of Lkðϕ; g¯Þ in the second term of Eq. (33) (32) as can be presented as
1 ð1Þαβγδ 1 ð1Þαβγδ δωL ϕ; g¯ δωhαβ x R x; g¯ hγδ x hαβ x δωR x; g¯ hγδ x kð Þ¼2 ð Þ k ð Þ ð Þþ2 ð Þ k ð Þ ð Þ 1 ð1Þαβγδ ¯ α ð2Þ β hαβ x R x; g¯ δωhγδ x δωC x R x; g¯ C x þ 2 ð Þ k ð Þ ð Þþ ð Þ kαβð Þ ð Þ ¯ α δ ð2Þ ; ¯ β ¯ α ð2Þ ; ¯ δ β þ C ðxÞ ωRkαβðx gÞC ðxÞþC ðxÞRkαβðx gÞ ωC ðxÞ: ð35Þ
In terms of transformations (22), the above expression reads
1 σ σ σ ð1Þαβγδ δωL ϕ; g¯ − ∂σhαβ x ω hασ x ∂βω hσβ x ∂αω R x; g¯ hγδ x kð Þ¼ 2 ð ð Þ þ ð Þ þ ð Þ Þ k ð Þ ð Þ 1 1 αβγδ − h Rð Þ x; g¯ ∂ h x ωσ h x ∂ ωσ h x ∂ ωσ 2 αβ k ð Þð σ γδð Þ þ γσð Þ δ þ σδð Þ γ Þ 1 ð1Þαβγδ ¯ α ð2Þ β hαβ x δωR x; g¯ hγδ x C x δωR x; g¯ C x þ 2 ð Þ k ð Þ ð Þþ ð Þ kαβð Þ ð Þ −ωσ∂ ¯ α ¯ σ ∂ ωα ð2Þ ; ¯ β þð σC ðxÞþC ðxÞ σ ÞRkαβðx gÞC ðxÞ ¯ α ð2Þ ; ¯ −ωσ∂ β σ ∂ ωβ þ C ðxÞRkαβðx gÞð σC ðxÞþC ðxÞ σ Þ: ð36Þ
Thus, by means of Eqs. (34) and (36) the variation of the action can be written in the compact way Z pffiffiffiffiffiffiffiffiffiffiffiffi 1 ð1Þαβγδ ¯ α ð2Þ β δωS ϕ; g¯ dx −g¯ x hαβ x M x; g¯ hγδ x C x M x; g¯ C x ; kð Þ¼ ð Þ 2 ð Þ ωk ð Þ ð Þþ ð Þ ωkαβð Þ ð Þ ð37Þ where
ð1Þαβγδ ; ¯ δ ð1Þαβγδ ; ¯ ωσ∂ ð1Þαβγδ ; ¯ − ∂ ωα ð1Þσβγδ ; ¯ Mωk ðx gÞ¼ ωRk ðx gÞþ σRk ðx gÞ σ Rk ðx gÞ − ∂ ωβ ð1Þασγδ ; ¯ − ∂ ωγ ð1Þαβσδ ; ¯ − ∂ ωδ ð1Þαβγσ ; ¯ σ Rk ðx gÞ σ Rk ðx gÞ σ Rk ðx gÞ and
ð2Þ ; ¯ δ ð2Þ ; ¯ ωσ∂ ð2Þ ; ¯ ð2Þ ; ¯ ∂ ωσ ð2Þ ; ¯ ∂ ωσ Mωkαβðx gÞ¼ ωRkαβðx gÞþ σRkαβðx gÞþRkσβðx gÞ α þ Rkασðx gÞ β : ð38Þ
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ð1Þαβγδ αβ γδ ð1Þ ¯ In order to ensure the invariance of (28), it is necessary R ðx; g¯Þ¼g¯ ðxÞg¯ ðxÞR ð□Þ that the following conditions are satisfied: k k αγ βδ αδ βγ ¯ þðg¯ ðxÞg¯ ðxÞþg¯ ðxÞg¯ ðxÞÞQkð□Þ ð1Þαβγδ ; ¯ 0 ð2Þ ; ¯ 0 42 Mωk ðx gÞ¼ and Mωkαβðx gÞ¼ : ð39Þ ð Þ
As a result, we obtain expressions for the variation of the and regulator functions, ð2Þ ; ¯ ¯ ð2Þ □¯ Rkαβðx gÞ¼gαβRk ð Þ; ð43Þ δ ð1Þαβγδ ; ¯ −ωσ∂ ð1Þαβγδ ; ¯ ωRk ðx gÞ¼ σRk ðx gÞ ð1;2Þ ¯ ¯ ¯ ð1Þσβγδ α ð1Þασγδ β where R ð□Þ and Qkð□Þ are scalar functions and □ is þ R ðx;g¯Þ∂σω þ R ðx;g¯Þ∂σω k k k the d’Alembertian operator defined in terms of the covar- ð1Þαβσδ ; ¯ ∂ ωγ ð1Þαβγσ ; ¯ ∂ ωδ þ Rk ðx gÞ σ þ Rk ðx gÞ σ iant derivatives of the background metric: ð40Þ ¯ μν ¯ ¯ □ ¼ g¯ ∇μ∇ν; ð44Þ and with the metricity property
δ ð2Þ ; ¯ −ωσ∂ ð2Þ ; ¯ − ð2Þ ; ¯ ∂ ωσ ∇¯ ¯ 0 ωRkαβðx gÞ¼ σRkαβðx gÞ Rkσβðx gÞ α τgμν ¼ : ð45Þ − ð2Þ ; ¯ ∂ ωσ Rkασðx gÞ β : ð41Þ It is possible to show that (42) and (43) present the same variational structure of (40) and (41), respectively. By using If the relations (40) and (41) are fulfilled, then the action the inverse background metric variation Skðϕ; g¯Þ is invariant under background field transforma- μν σ μν μσ ν σν μ tions. Therefore, the regulator functions should have a δωg¯ ðxÞ¼−ω ∂σg¯ ðxÞþg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω ; tensor structure in order to ensure the invariance. Thus, taking into account the symmetry properties presented in ð46Þ (31) we can propose the following solutions for the regulator functions: we have
δ ð1Þαβγδ ; ¯ −ωσ∂ ¯αβ ¯ασ ∂ ωβ ¯σβ ∂ ωα ¯γδ ð1Þ □¯ ωRk ðx gÞ¼ð σg ðxÞþg ðxÞ σ þ g ðxÞ σ Þg ðxÞRk ð Þ ¯αβ −ωσ∂ ¯γδ ¯γσ ∂ ωδ ¯σδ ∂ ωγ ð1Þ □¯ þ g ðxÞð σg ðxÞþg ðxÞ σ þ g ðxÞ σ ÞRk ð Þ σ αγ ασ γ σγ α βδ ¯ þð−ω ∂σg¯ ðxÞþg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω Þg¯ ðxÞQkð□Þ αγ σ βδ βσ δ σδ β ¯ þ g¯ ðxÞð−ω ∂σg¯ ðxÞþg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω ÞQkð□Þ σ αδ ασ δ σδ α βγ ¯ þð−ω ∂σg¯ ðxÞþg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω Þg¯ ðxÞQkð□Þ αδ σ βγ βσ γ σγ β ¯ þ g¯ ðxÞð−ω ∂σg¯ ðxÞþg¯ ðxÞ∂σω þ g¯ ðxÞδσω ÞQkð□Þ − ¯αβ ¯γδ ωσ∂ ð1Þ □¯ − ¯αγ ¯βδ ωσ∂ □¯ g ðxÞg ðxÞ σRk ð Þ g ðxÞg ðxÞ σQkð Þ αδ βγ σ ¯ − g¯ ðxÞg¯ ðxÞω ∂σQkð□Þ: ð47Þ ð1Þ □¯ □¯ The derivatives in metric tensor and in functions Rk ð Þ and Qkð Þ can be combined to obtain
δ ð1Þαβγδ ; ¯ ¯ασ ∂ ωβ ¯σβ ∂ ωα ¯γδ ð1Þ □¯ ¯αβ ¯γσ ∂ ωδ ¯σδ ∂ ωγ ð1Þ □¯ ωRk ðx gÞ¼ðg ðxÞ σ þ g ðxÞ σ Þg ðxÞRk ð Þþg ðxÞðg ðxÞ σ þ g ðxÞ σ ÞRk ð Þ ασ γ σγ α βδ ¯ αγ βσ δ σδ β ¯ þðg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω Þg¯ ðxÞQkð□Þþg¯ ðxÞðg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω ÞQkð□Þ ασ δ σδ α βγ ¯ αδ βσ δ σγ β ¯ þðg¯ ðxÞ∂σω þ g¯ ðxÞ∂σω Þg¯ ðxÞQkð□Þþg¯ ðxÞðþg¯ ðxÞ∂σω þ g¯ ðxÞδσω ÞQkð□Þ − ωσ∂ ¯αβ ¯γδ ð1Þ □¯ − ωσ∂ ¯αγ ¯βδ □¯ − ωσ∂ ¯αδ ¯βγ □¯ σðg ðxÞg ðxÞRk ð ÞÞ σðg ðxÞg ðxÞQkð ÞÞ σðg ðxÞg ðxÞQkð ÞÞ: ð48Þ
As a result, it is possible to see that (40) is satisfied.
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For the second function its variation can be expressed, where after some algebra, as Z pffiffiffiffiffiffiffiffiffiffiffiffi ¯ α Ψðϕ; g¯Þ¼ dx −g¯ðxÞC ðxÞχαðx; h; g¯Þð53Þ δ ð2Þ ; ¯ −ωσ∂ ¯ ð2Þ □¯ − ∂ ωσ ¯ ð2Þ □¯ ωRkαβðx gÞ¼ σgαβRk ð Þ α gσβðxÞRk ð Þ − ∂ ωσ ¯ ð2Þ □¯ β gασðxÞRk ð Þ is the fermionic gauge-fixing functional and Z σ ð2Þ ¯ − ω g¯αβðxÞ∂σR ð□Þ: ð49Þ δ k Rˆ ðϕ; g¯Þ¼ dx r Riðx; ϕ; g¯Þð54Þ δϕiðxÞ The combination of derivatives in the metric tensor and in the scalar function leads to is the generator of BRST transformations (15). As far as we saw in the previous section, the regulator ð2Þ σ ð2Þ ¯ action (28) does not depend on the gauge Ψðϕ; g¯Þ.Now,we δωR αβðx; g¯Þ¼−ω ∂σðg¯αβðxÞR ð□ÞÞ k k shall consider another choice of gauge-fixing functional − ∂ ωσ ¯ ð2Þ □¯ Ψ → Ψ δΨ 0 α gσβðxÞRk ð Þ þ and set J ¼ in (52). Thus, 2 Z − ∂ ωσ ¯ ð Þ □¯ i β gασðxÞRk ð Þ; ð50Þ ¯ ϕ ϕ ¯ ˆ ϕ ¯ δΨ ϕ ¯ ZkΨþδΨðgÞ¼ d exp ℏ½SkFPð ;gÞþRð ;gÞ ð ;gÞ which has the same structure as (41). i Finally, the scale-dependent regulator Lagrangian den- ¯ ¼ exp ℏWkΨþδΨðgÞ ; ð55Þ sity (29) in terms of (42) and (43) reads 1 where αβ γδ ð1Þ ¯ L ϕ; g¯ hαβ x g¯ x g¯ x R □ Z kð Þ¼2 ð Þ½ ð Þ ð Þ k ð Þ pffiffiffiffiffiffiffiffiffiffiffiffi δΨ δΨ ϕ ¯ −¯ ¯ α δχ ¯ αγ βδ αδ βγ ¯ ¼ ð ; gÞ¼ dx gðxÞC ðxÞ αðh; gÞ: ð56Þ þðg¯ ðxÞg¯ ðxÞþg¯ ðxÞg¯ ðxÞÞQkð□Þ hγδðxÞ ¯ α ¯ ð2Þ □¯ β We will try to compensate the additional term Rˆ δΨ in þ C ðxÞgαβðxÞRk ð ÞC ðxÞ; ð51Þ (55). To do this, we change the variables in the functional ϕ ¯ which maintains the background field symmetry, integral related to the symmetries of action SFPð ; gÞ, δωSkðϕ; g¯Þ¼0. namely the BRST symmetry and the background gauge invariance. First, we shall consider the BRST symmetry IV. GAUGE DEPENDENCE OF EFFECTIVE (14), but trading the constant parameter λ by a functional AVERAGE ACTION Λ ¼ Λðϕ; g¯Þ. The variation of (28) under such transforma- tion is the following: In order to understand the gauge invariance and gauge Z dependence problems in the background field method, we pffiffiffiffiffiffiffiffiffiffiffiffi δ ϕ ¯ 4 −¯ δ L ϕ ¯ shall consider the generating functionals of Green functions BSkð ; gÞ¼ d x gðxÞf B kð ; gÞg; ð57Þ Z ¯ ϕ i ¯ ˆ ϕ ¯ Ψ ϕ ¯ where ZkΨðJ;gÞ¼ d exp ℏ ½S0ðh þ gÞþRð ; gÞ ð ; gÞ 1 ð1Þαβγδ δBLk ¼ δBhαβðxÞR ðx;g¯ÞhγδðxÞ þ Skðϕ; g¯ÞþJϕ 2 k
Z 1 ð1Þαβγδ hαβ x R x;g¯ δ hγδ x ϕ i ϕ ¯ ϕ þ2 ð Þ k ð Þ B ð Þ ¼ d exp ℏ ½SkFPð ; gÞþJ δ ¯ α ð2Þ ; ¯ β ¯ α ð2Þ ; ¯ δ β þ BC ðxÞRkαβðx gÞC ðxÞþC ðxÞRkαβðx gÞ BC ðxÞ: i ¯ ¼ exp ℏ WkΨðJ; gÞ ; ð52Þ After some algebra, δBLkðϕ; g¯Þ reads as
1 σ σ σ ð1Þαβγδ δ L − C x ∂σgαβ x gασ x ∂βC x gσβ x ∂αC x ΛR x; g¯ hγδ x B k ¼ 2 ð ð Þ ð Þþ ð Þ ð Þþ ð Þ ð ÞÞ k ð Þ ð Þ
1 ð1Þαβγδ σ σ σ − hαβ x R x; g¯ C x ∂σgγδ x gγσ x ∂δC x gσδ x ∂γC x Λ 2 ð Þ k ð Þð ð Þ ð Þþ ð Þ ð Þþ ð Þ ð ÞÞ α Λ ð2Þ ; ¯ β ¯ α ð2Þ ; ¯ σ ∂ β Λ þ B ðxÞ Rkαβðx gÞC ðxÞþC ðxÞRkαβðx gÞC ðxÞ σC ðxÞ :
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From the above expression, it is clear that the action Skðϕ; g¯Þ is not invariant under BRST transformations δBSkðϕ; g¯Þ ≠ 0. The Jacobian J1 of such transformation can be obtained in the standard way Z α ¯ α δ δ hαβ x δ δ δ δ ð B ð ÞÞ − ð BC ðxÞÞ − ð BC ðxÞÞ J1 ¼ exp dx α ¯ α ; ð58Þ δhαβðxÞ δC ðxÞ δC ðxÞ where the functional derivatives are δ δ ð BhαβðxÞÞ DðD þ 1Þ σ ðD þ 1ÞðD − 2Þ σ ¼ − δð0ÞC ðxÞ∂σΛðϕ; g¯Þ − δð0Þ∂σC ðxÞΛðϕ; g¯Þ δhαβðxÞ 2 2
σ σ σ δΛðϕ; g¯Þ − ½CðxÞ ∂σgαβðxÞþgασðxÞ∂βC ðxÞþgσβðxÞ∂αC ðxÞ ; ð59Þ δhαβðxÞ
α δðδBC ðxÞÞ σ σ σ α δΛðϕ; g¯Þ ¼ðD þ 1Þδð0Þ∂σC ðxÞΛðϕ; g¯ÞþDδð0ÞC ðxÞ∂σΛðϕ; g¯ÞþC ðxÞ∂σC ðxÞ ; ð60Þ δCαðxÞ δCαðxÞ
δðδ C¯ αðxÞÞ δΛðϕ; g¯Þ B ¼ BαðxÞ : ð61Þ δC¯ αðxÞ δC¯ αðxÞ
It is possible to choose a regularization scheme such that δð0Þ¼0. As a result, the Jacobian for BRST transformations is Z δΛ ϕ ¯ δΛ ϕ ¯ δΛ ϕ ¯ ðhÞ ; ϕ ¯ ð ; gÞ − α ; ϕ ¯ ð ; gÞ − α ; ϕ ¯ ð ; gÞ J1 ¼ exp dx Rαβ ðx ; gÞ RðCÞðx ; gÞ α R C¯ ðx ; gÞ ¯ α ; ð62Þ δhαβðxÞ δC ðxÞ ð Þ δC ðxÞ where (15) is used. Jacobian of this transformation can be obtained as It is also interesting to consider the background before Z gauge transformation related to expressions (22).Asfar α ¯ α δðδΩhαβðxÞÞ δðδΩC ðxÞÞ δðδΩC ðxÞÞ as the regulator functions transform as (40) and (41), J2 dx − − ; ¼ exp δ δ α δ ¯ α the action Skðϕ; g¯Þ is invariant under such transforma- hαβðxÞ C ðxÞ C ðxÞ ω ωσ tion. But now, instead of functions ¼ ðxÞ we ð63Þ shall consider the functional Ωσ ¼ Ωσðx; ϕ; g¯Þ.The action (28) remains invariant, and the corresponding with the following functional derivatives:
δ δ σ ð ΩhαβðxÞÞ ðD þ 1ÞðD − 2Þ σ δΩ ðx; ϕ; g¯Þ ¼ δð0Þ∂σΩ ðx; ϕ; g¯Þ − ð∂σhαβðxÞþhασðxÞ∂β þ hσβðxÞ∂αÞ ; ð64Þ δhαβðxÞ 2 δhαβðxÞ
δ δ ¯ α δΩσ ϕ ¯ δΩα ϕ ¯ ð ΩC ðxÞÞ σ ðx; ; gÞ ¯ α ¯ σ ðx; ; gÞ ¼ðD þ 1Þδð0Þ∂σΩ ðx; ϕ; g¯Þ − ∂σC ðxÞ − C ðxÞ∂σ ; ð65Þ δC¯ αðxÞ δC¯ αðxÞ δC¯ αðxÞ
α σ α δðδΩC ðxÞÞ σ δΩ ðx; ϕ; g¯Þ α σ δΩ ðx; ϕ; g¯Þ ¼ðD þ 1Þδð0Þ∂σΩ ðx; ϕ; g¯Þ − ∂σC ðxÞ − C ðxÞ∂σ : ð66Þ δCαðxÞ δCαðxÞ δCαðxÞ
As before, δð0Þ¼0. Thus, the Jacobian for background gauge transformations reads Z δΩσ ϕ ¯ δΩσ ϕ ¯ δΩα ϕ ¯ − ∂ ∂ ∂ ðx; ; gÞ ðx; ; gÞ ∂ ¯ α ¯ σ ∂ ðx; ; gÞ J2 ¼ exp dx ð σhαβðxÞþhασðxÞ β þ hσβðxÞ αÞ þ ¯ α σC ðxÞþC ðxÞ σ ¯ α δhαβðxÞ δC ðxÞ δC ðxÞ σ α δΩ ðx; ϕ; g¯Þ α σ δΩ ðx; ϕ; g¯Þ þ ∂σC ðxÞþC ðxÞ∂σ : ð67Þ δCαðxÞ δCαðxÞ
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If it is possible to fulfill the condition σð2Þ i ð1Þ ð1Þτσγδ ∂σωαβ ðx; h; g¯Þ¼ ½∂βgτσðxÞλα ðh; g¯ÞR ðx; g¯ÞhγδðxÞ Z 2ℏ i ð1Þτσγδ ð1Þ ˆ ϕ ¯ δΨ ϕ ¯ δ ϕ ¯ 1 þ hτσðxÞR ðx; g¯Þ∂βgγδðxÞλα ðh; g¯Þ : J1J2 exp ℏ dx½Rð ; gÞ ð ; gÞþ BSkð ; gÞ ¼ ð78Þ ð68Þ From Eq. (74) since Ωσð0Þðx; h; g¯Þ is an arbitrary function the generating vacuum functional ZkΨðg¯Þ does not depend we can not have just one particular solution. In addition, the on the gauge fixing functional Ψ. In order to verify that, let ð3Þ λαβγ relation in (76) creates a nonlocal term of structure us expand the functionals Λ and Ω, with Grassmann parity BC¯ CCC¯ which can be only eliminated if we consider a εðΛÞ¼1 and εðΩÞ¼0 and ghost numbers ghðΛÞ¼−1 new functional Λ of higher orders in ghost fields. Even so, and ghðΩÞ¼0 in the lower power of ghost fields this process would repeat endlessly. The only case left for Ωσ 0 ð1Þ ð3Þ σ σð0Þ σð2Þ us is to consider the simple solution when ¼ and Λ ¼ Λ þ Λ ; Ω ¼ Ω þ Ω ; ð69Þ 1 Λ ¼ Λð Þ; we have the result Z where ¯ ϕ i ϕ ¯ δ ϕ Z ZkΨþδΨðgÞ¼ d exp ℏ ½SkFPð ; gÞþ SkFPð Þ ; 1 ¯ α ð1Þ Λð Þ ¼ dxC ðxÞλα ðx; h; g¯Þ; ð70Þ ¯ ≠ ¯ ZkΨðgÞ ZkΨþδΨðgÞ: ð79Þ Z 1 As a final result, the vacuum functional in the FRG Λð3Þ ¯ α ¯ β λð3Þ ¯ γ ¼ dx2C ðxÞC ðxÞ αβγðx;h;gÞC ðxÞ; ð71Þ approach for gravity theories depends on the gauge fixing even on shell, which leads to a gauge-dependent S-matrix.
Ωσð0Þ Ωσð0Þ ¯ ðxÞ¼ ðx; h; gÞ; ð72Þ V. CONCLUSIONS AND PERSPECTIVES We explored the problem of gauge invariance and the Ωσð2Þ ¯ ¯ α ωσð2Þ ¯ β ðx; h; gÞ¼C ðxÞ αβ ðx; h; gÞC ðxÞ: ð73Þ gauge-fixing dependence using the background field for- malism, for gravity theories in the FRG framework. It was The terms that vanish in (68) and do not depend on ghost shown that the background field invariance is achieved fields lead to when the regulator functions are chosen to have the tensor structure. Nevertheless, even in this case the on-shell gauge δΩσð0Þðx;h;g¯Þ dependence cannot be cured in the standard FRG approach ð∂σhαβðxÞþhασðxÞ∂β þhσβðxÞ∂αÞ ¼0: ð74Þ δhαβðxÞ which we dealt with. In this respect the situation is qualitatively the same as in the Yang-Mills theories, as Analyzing the terms which are linear in the antighost fields, was discussed in [15]. which contains the auxiliary fields BðxÞ, we obtain The on-shell gauge dependence takes place due to the fact that the regulator action (28) is not BRST invariant. It turns out that this is a fundamentally important feature, that λð1Þ ¯ i δχ ¯ α ðx; h; gÞ¼ℏ αðx; h; gÞð75Þ can be changed only by trading the standard and conven- tional FRG framework to an alternative one, which is based and on the use of composite operators for constructing the regulator action. Unfortunately, until now there is no way to perform practical calculations in this alternative formu- λð3Þ ¯ λð1Þ ¯ ð2Þ ; ¯ αβγðx; h; gÞ¼ α ðh; gÞRkβγðx gÞ lation. For this reason, taking our present results into 1 2 − λð Þðh; g¯ÞRð Þ ðx; g¯Þ; ð76Þ account, it remains unclear whether the quantum gravity β kαγ results obtained within the FRG formalism can have a reasonable physical interpretation. One can expect that all where predictions of this formalism will depend on an arbitrary Z choice of the gauge fixing. Thus, one can, in principle, ð1Þ ð1Þ λα ðh; g¯Þ¼ dxλα ðx; h; g¯Þ: ð77Þ provide any desirable result, but the value of this output is not clear. Alternatively, there should be found some ¯ physical reason to claim that one special gauge fixing is Now, the vanishing terms with structure CðxÞCðxÞ can be “correct” or “preferred” for some reason, but at the moment Ωσ ¯ related to the second order of ðx; h; gÞ functional leading it is unclear what this reason could be, since the original σð2Þ to a differential equation for ωαβ ðx; h; g¯Þ, theory is gauge (diffeomorphism) invariant.
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It is worthwhile to comment on the effect of the on-shell making it completely ambiguous. Thus, in our opinion, it is gauge-fixing dependence on the nonuniversality of the difficult to take the results extracted from A1 as a physical renormalization group flow in the framework of the func- prediction, even without gauge-fixing ambiguity. tional renormalization group approach. First of all, the The theories of quantum gravity of the second kind are complete understanding of this issue requires a detailed those with higher derivatives, being fourth derivative [4] analysis and, in principle, explicit calculations in the models, the super-renormalizable models with six or more sufficiently general model of quantum gravity. This calcu- derivatives [18], or the nonlocal super-renormalizable mod- lation is a separate problem that is beyond the scope of the els. In all these cases the renormalization group flow in the present work. At the same time, we can give some strong functional renormalization group has a well-defined pertur- arguments in favor of such a gauge-fixing dependence for bative limit (see e.g., [3]), and hence it should be supposed both renormalization group flow and the fixed point, which that the invariance or noninvariance of the renormalization is obtained from this flow. group flow should be the same in both usual perturbative and One can classify the models of quantum gravity into two functional renormalization group approaches. On the other classes. The first class is the quantum general relativity, hand, without the on-shell invariance condition there is where the renormalization group equation for the Newton absolutely no way to consider the renormalization group constant G does not exist at the perturbative level, at least flows in any one of these models in a consistent way. This without the cosmological constant term. In the presence of issue was analyzed in detail in [7,18,68,69] and [19], thus the cosmological constant there is a perturbative renorm- there is no point to repeat these considerations here. Thus, alization group equation for the Newton constant, but it is we can conclude that in those versions of functional strongly gauge-fixing dependent, as discussed e.g., in [7] renormalization group where one can expect a minimally and more recently, with very general calculations, in [67] reasonable interpretation of the application to quantum (see more references therein). Let us add that in the gravity, it is difficult to expect the gauge-invariant flow in functional renormalization group approach the renormali- the framework of the functional renormalization group, if zation group flow for G is extracted from the quadratic this approach is not reformulated in a new invariant way. divergences, and for this reason the corresponding results Finally, we mention two recent papers. In [70] it was do not have the perturbative limit. On the other hand, the shown that the renormalization group flow in quantum quadratic divergences are also gauge-fixing dependent [67] general relativity should be compatible with the Ward and, furthermore, suffer from the total ambiguity in defin- identities related to the possibility of changing the conformal ing the cutoff parameter Ω. For instance, if we start from frame. Certainly, this is not the symmetry that we discussed in the total expression for the one-loop divergences in the the previous sections. Also, even if the mentioned compat- Schwinger-DeWitt formalism, ibility is achieved, this does not make the whole approach Z sufficiently unambiguous, as we explained above. However, 1 ffiffiffi 1 Ω2 ð1Þ 4 p 4 2 qualitatively, the importance of the Ward identities to the Γ ¼ − d x g A0Ω þ A1Ω þ A2 log ; div 2 2 μ2 consistency of the renormalization group flow is certainly ð80Þ coherent with our analysis. On the other hand, Ref. [71] proposed a new formulation of the functional renormaliza- where A0, A1, and A2 are the algebraic sums of the tion group for quantum gravity. In this new version, there is contributions of quantum metric and ghosts, given by not one scale parameter, but two: one for regularizing the UV the functional traces of the coincidence limits of the sector and another one for regularizing the IR sector. The idea corresponding Schwinger-DeWitt coefficients. Without is certainly interesting for quantum gravity since the two the cosmological constant only A1 has the term linear in limits in all known models are very much different. However, the scalar curvature R, and hence only this term can since the UV part is essentially the same as in the standard contribute to the renormalization group flow for G.Now, formulation, it is difficult to expect that the gauge-fixing changing the cutoff as dependence on shell in this new formulation will be a smaller problem than it is in the standard versions with one scale Ω2 → Ω2 þ αR; ð81Þ parameter. We can state that the preliminary analysis of this model confirms this qualitative conclusion. The full consid- with an arbitrary coefficient α, we observe that the quartic eration is possible but would be quite involved and is beyond and logarithmic divergences do not change, while the the scope of the present work. quadratic ones modify according to1 The results of the considerations which we described above make more interesting the discussion of the possible → α A1 A1 þ A0R; ð82Þ ways to solve the problem of on-shell gauge-fixing dependence, such as the ones suggested in [34–36] or in 1One of the authors (I.S.) is grateful to M. Asorey for [15]. In our opinion, the last approach is more transparent explaining this point. and physically motivated, but (as we have already
065001-10 FUNCTIONAL RENORMALIZATION GROUP APPROACH AND … PHYS. REV. D 101, 065001 (2020) mentioned above) there is no well-developed technique of during his long-term visit, when this work was initiated. The using it for practical calculations. work of P. M. L. was partially supported by the Ministry of Science and Higher Education of the Russian Federation, ACKNOWLEDGMENTS Grant No. 3.1386.2017 and by the RFBR Grant No. 18-02- 00153. I. L. S. was partially supported by Conselho Nacional E. A. R. and V.F. B. are grateful to Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) de Desenvolvimento Científico e Tecnológico (CNPq) under for supporting their Ph.D. and Ms. projects. P. M. L. is the Grant No. 303635/2018-5 and Fundação de Amparoa ` grateful to the Department of Physics of the Federal Pesquisa de Minas Gerais (FAPEMIG) under the Project University of Juiz de Fora (MG, Brazil) for warm hospitality No. APQ-01205-16.
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