A 6D standing-wave Braneworld with normal matter as source
AA D standing-wave Braneworld with normal matter as source.
L. J. S. Sousa ∗ Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE),Campus Canindé, 62700-000 Canindé-Ceará-Brazil
W. T. Cruz † Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE),Campus Juazeiro do Norte, 63040-000 Juazeiro do Norte-Ceará-Brazil
C. A. S. Almeida ‡ Universidade Federal do Ceará, Departamento de Física - Campus do Pici,C.P. 6030, 60455-760, Fortaleza-Ceará-Brazil
Abstract
A six dimensional standing wave braneworld model has been constructed. It consists in ananisotropic 4-brane generated by standing gravitational waves whose source is normal matter. Inthis model, the compact (on-brane) dimension is assumed to be sufficiently small in order to de-scribe our universe (hybrid compactification). The bulk geometry is non-static, unlike most of thebraneworld models in the literature. The principal feature of this model is the fact that the sourceis not a phantom like scalar field, as the original standing-wave model that was proposed in fivedimensions and its six dimensional extension recently proposed in the literature. Here, it was ob-tained a solution in the presence of normal matter what assures that the model is stable. Also, ourmodel is the first standing wave brane model in the literature which can be applied successfully tothe hierarchy problem. Additionally, we have shown that the zero-mode for the scalar and fermionicfields are localized around the brane. Particularly for the scalar field we show that it is localizedon the brane, regardless the warp factor is decreasing or increasing. This is in contrast to the caseof the local string-like defect where the scalar field is localized for a decreasing warp factor only.
PACS numbers: 04.50.-h, 11.27.+d, 12.60.-i, 11.10.Kk ∗ Electronic address: luisjose@fisica.ufc.br † Electronic address: wilami@fisica.ufc.br a r X i v : . [ h e p - t h ] F e b Electronic address: carlos@fisica.ufc.br . INTRODUCTION The so called braneworld models assume the idea that our universe is a membrane, orbrane, embedded in a higher-dimensional space-time. The success of this idea betweenthe physicists can be explained basically because these models have brought a solution forsome insoluble problems in the Standard Model (SM) physics, as the hierarchy problem.There are many theories that carry this basic idea, but the main theories in this contextare the one first proposed by Arkani-Hamed, Dimopoulos and Dvali [1–3] and the so-called,Randall-Sundrum (RS) model [4, 5].In these models, it is assumed a priori that all the matter fields are restricted to propagateonly in the brane. The gravitational field is the only one which is free to propagate in all thebulk. However, some authors have argued that this assumption is not so obvious, and it isnecessary to look for alternative theoretical mechanisms of field localization in such models[6, 7]. Accordingly, before to study the cosmology of a braneworld model, it is convenient toanalyze its capability to localize fields. Therefore, for a braneworld model to be indicatedas a potential candidate of our universe it is necessary to be able to localize the StandardModel fields.The Randall-Sundrum model was generalized to six dimensions by several interestingworks [6–29]. A high number of the works in six dimensions refer to the scenarios where thebrane has cylindrical symmetry, the so-called string-like braneworlds, which is associatedto topological defects. Some of these six dimensional models are classified as global string[6, 8, 11], the local string [12–14], thick string [16, 17, 19–22] and supersymmetric cigar-universe [23]. Also, the work proposed here is a generalization of the RS model for sixdimensional space-time. However we treat the so called standing wave brane model whichwill be discussed later.On the other hand, studies of field localization are very common in the literature in 5D[30–35] and 6D braneworld [6, 7, 12, 14, 26–28]. In general, we find strategies of localiza-tion for all the Standard Model fields but the way that this localization is possible variesin different works. In some of them, the localization is possible by means of gravitationalinteractions only [6, 7]. In other works, it is necessary to consider the existence of auxiliaryfields, like the dilaton [33, 34]. As far as we know, there is not in the literature a purelyanalytical geometry that localizes all the SM fields by means of the gravitational field inter-3ction only. Hence, looking for a model which presents the features of to be analytical andbe able to localize all the Standard Model fields is, in our point of view, an appropriatedreason to study localization of fields in different braneworld models.The search for such a model has motivated the appearance of some braneworld scenar-ios with non-standard transverse manifold. Randjbar-Daemi and Shaposhnikov obtainedtrapped massless gravitational modes and chiral fermions as well, in a model that theycalled a Ricci-flat space or a homogeneous space [36]. Kehagias proposed an interestingmodel which drains the vacuum energy, through a conical tear-drop like space which formsa transverse space with a conical singularity. In this way, it was possible to explain thesmall value of the cosmological constant [37]. Another non-trivial geometry was proposedby Gogberashvili et al . They have found three-generation for fermions on a -brane whosetransverse space has the shape of an apple [38]. It is still possible to cite other examples ofspace used like the torus [39], a space-time geometry with football-shape [40] and smoothversions of the conifold, classified as resolved conifold [41] and deformed conifold [42–44].The standing wave braneworld was first proposed in five dimensions by Gogberashvili andSingleton [45]. This is a completely anisotropic braneworld model whose source is a phantom-like scalar (a scalar field with a wrong sign in front of the kinetic term in the Lagrangian). Toavoid the problem with instability, normally presented in theory with phantom scalar, themodel is embedded in a 5D Weyl geometry in such a way that the phantom-like scalar maybe associated with the Weyl scalar [30, 31, 46], which is stable. About the Weyl scalar, wemay point out their presence also in other braneworld scenarios like the pure-gravity, whichis an extension of the RS model. In the context of field localization in the standing waveapproach, it was possible to localize several fields in five dimensions, albeit the right-handedfermions were not localized neither in increasing nor decreasing warp geometry [30, 31]. Itis worthwhile to mention that the models generated by phantom-like scalar are relevantphenomenologically since this exotic source is useful in different scenarios like cosmology[25], where the phantom scalar is used to explain dark energy theories and the acceleratedexpansion of the universe [47]. An extension for 6D of the standing wave 5D model with aphantom like scalar was first proposed by some authors of this article [48]. Additionally, thestudy of massive modes was not addressed in this model in five dimensions or even its sixdimensional version.In this article, we do not specify a priori the source or the stuff from which the brane is4one. We consider a general matter source and look for a standing wave solution. In contrastto the work of Gogberashvili and collaborators in the 5D model [30], in which the sourceis a phantom-like scalar field, here we have obtained standing gravitational waves solutionsof Einstein equations in the presence of normal matter (we are using the classification fordifferent types of matter given by M. Visser [49]). Since it is done by normal matter, themodel constructed here is stable. Our model with normal matter as a source is a first sixdimensional one, but quite recently Midodashvili et al. [50] constructed a 5D standing wavebraneworld model with a real field as a source.The model built here consists of a 6D braneworld with an anisotropic 4-brane, where thesmall, compact dimension belongs to the brane. The bulk is completely anisotropic, unlessfor some points called the AdS islands [30, 31]. Its dynamics, as in the case of the worksof Gogberashvili and collaborators and its extensions, represents a special feature in thesense that both metric and source are time dependent. We present two types of solutions:one with an isotropic cosmological constant where the source despite the fact that all itscomponents are positive do not satisfy the dominant energy condition - DEC. This sourcemay be classified as a not normal matter [49]. In the other case, we make use of a recentlyproposed approach that suggests an extension for the Randall-Sundrum model to higherdimensions in the presence of an anisotropic cosmological constant [51]. In this case, we findsolutions in the presence of normal matter.We have obtained analytical solution for the warp factor, which corresponds to a thinbrane, for both decreasing and increasing warp factor. The bulk is smooth everywhere andconverges asymptotically to an
AdS manifold. We have considered a minimally coupledscalar field, and we have shown that it is localized in this model. Here, we have obtainedresults that are more general that those encountered for the string-like defect and the 5Dand 6D versions of the standing wave approach. Indeed, here the scalar field is trapped forboth decreasing and increasing warp factor whereas in the string-like is a localized mode fora decreasing warp factor only. Also in 5D and 6D versions of the standing wave models thescalar field is localized for increasing warp factor only.Furthermore, our six dimensional standing wave braneworld with physical source is aninteresting scenario in order localize fermions fields. Indeed, we show that right-handedfermions can be localized in this brane.We organize this work as follows: in section (II) the model is described and the Einstein5quations are solved in order to obtain the general expressions for the source and the functionthat characterizes the anisotropy. In section (III), we have found the standing waves solu-tions, and we have discussed its mainly features. We still show that the energy-momentumcomponents are all positives. In the case of an anisotropic cosmological constant they obeyall the energy conditions which characterizes a normal matter source. The localization ofthe zero mode of scalar and fermionic fields have been done in the sections (IV) and (V),respectively. Some remarks and conclusions are outlined in section (VI). II. THE MODEL
Our intent is to derive a standing wave solution of the Einstein equations, by consid-ering normal matter as source. So, we consider the standard Einstein-Hilbert action insix dimensional space-time added by a matter field action which may be time dependent.Namely, S = 12 κ (cid:90) d x √− g (cid:104) ( R − ) + L m (cid:105) , (1)where κ is the six dimensional gravitational constant, Λ is the bulk cosmological constantand L m is any matter field Lagrangian.From the action (1) we derive the Einstein equations R MN − g MN R = − Λ g MN + κ T MN , (2)where M, N,... denote D-dimensional space-time indices and the T MN is the energy-momentum tensor defined as T MN = − √− g δδg MN (cid:90) d x √− gL m . (3)The general ansatz for the metric considered in this work is given as follows ds = e A (cid:0) − dt + e u dx + e u dy + e − u dz (cid:1) + dr + R e B + u dθ , (4)where the functions A ( r ) and B ( r ) depend only on r and the function u depends on r and t variables. For this metric ansatz, (4) , the Einstein equations (2) may be rewritten as G xx = G yy = (cid:18) e A + u (cid:19)(cid:16) A (cid:48) + B (cid:48) + 3 A (cid:48) B (cid:48) + 6 A (cid:48)(cid:48) + 2 B (cid:48)(cid:48) + 6( u (cid:48) − e − A ˙ u ) + 2 e − A ¨ u − A (cid:48) u (cid:48) − u (cid:48)(cid:48) (cid:17) = κ T xx − e A + u Λ , (5)6 zz = (cid:18) e A − u (cid:19)(cid:16) A (cid:48) + B (cid:48) + 3 A (cid:48) B (cid:48) + 6 A (cid:48)(cid:48) + 2 B (cid:48)(cid:48) + 6( u (cid:48) − e − A ˙ u ) − e − A ¨ u + (11 A (cid:48) + 4 B (cid:48) ) u (cid:48) + 6 u (cid:48)(cid:48) (cid:17) = κ T zz − e A − u Λ , (6) G tt = (cid:18) e A (cid:19)(cid:16) − A (cid:48) − B (cid:48) − A (cid:48) B (cid:48) − A (cid:48)(cid:48) − B (cid:48)(cid:48) − u (cid:48) + e − A ˙ u ) + ( A (cid:48) − B (cid:48) ) u (cid:48) (cid:17) = κ T tt + e A Λ , (7) G rt = 14 ˙ u ( A (cid:48) − B (cid:48) − u (cid:48) ) = κ T rt , (8) G rr = (cid:18) (cid:19)(cid:16) A (cid:48) + 4 A (cid:48) B (cid:48) − u (cid:48) + e − A ˙ u ) + ( A (cid:48) − B (cid:48) ) u (cid:48) (cid:17) = κ T rr − Λ , (9)and G θθ = (cid:18) R e B + u (cid:19)(cid:16) A (cid:48) + 8 A (cid:48)(cid:48) + 6( u (cid:48) − e − A ˙ u ) + 2 e − A ¨ u − A (cid:48) u (cid:48) − u (cid:48)(cid:48) (cid:17) = κ T θθ − R e B + u Λ . (10)The case A = B = 2 ar was treated in a previous work [48]. As a matter of fact, inthis case, it is possible to find a standing gravitational wave solution in the presence ofa phantom-like scalar field, similar to the one first found in five dimensions. Another 6Dstanding wave braneworld has been recently proposed [50]. However, in that case the metricis quite different from the one considered here as given by Eq. (4). In this last modelthe spatial metric components x, y, z are all multiplied by the same factor e ar + u , while thecompact extra dimension is multiplied by e ar − u . The solution, in this case, is similar tothe one found in Ref. [48] and the source is still a phantom-like scalar field. The two modelsare still similar in the results of field localization.As was commented above, the two 6D standing wave braneworld models recently proposedin the literature have shown interesting results in field localization, but both are generated7y exotic matter, a phantom like scalar. Here, we are interested in studying the possibilityto have a standing wave braneworld generated by normal matter whereas maintains theefficiency in localizing fields. So we will consider the case where A (cid:54) = B , A ( r ) = 2 cr , and B ( r ) = c r . In this case, the set of equations (5 - 16) will be simplified to (cid:18) (cid:19)(cid:16) c + c + 6 cc + 6( u (cid:48) − e − cr ˙ u ) + 2 e − cr ¨ u − cu (cid:48) − u (cid:48)(cid:48) (cid:17) = κ T xx − Λ , (11) (cid:18) (cid:19)(cid:16) c + c + 6 cc + 6( u (cid:48) − e − cr ˙ u ) − e − cr ¨ u + (22 c + 4 c ) u (cid:48) + 6 u (cid:48)(cid:48) (cid:17) = κ T zz − Λ , (12) − (cid:18) (cid:19)(cid:16) − c − c − cc − u (cid:48) + e − cr ˙ u ) + (2 c − c ) u (cid:48) (cid:17) = κ T tt − Λ , (13)
14 ˙ u (2 c − c − u (cid:48) ) = T rt , (14) (cid:18) (cid:19)(cid:16) c + 8 cc − u (cid:48) + e − cr ˙ u ) + (2 c − c ) u (cid:48) (cid:17) = κ T rr − Λ , (15)and (cid:18) (cid:19)(cid:16) c + 6( u (cid:48) − e − cr ˙ u ) + 2 e − cr ¨ u − cu (cid:48) − u (cid:48)(cid:48) (cid:17) = κ T θθ − Λ . (16)In order to have standing wave solution we will choose − e − cr ¨ u + 16 (22 c + 4 c ) u (cid:48) + u (cid:48)(cid:48) = 0 . (17)8n this case, the energy-momentum components have to satisfy the relations κ T xx = κ T yy = 14 (cid:18) u (cid:48) − e − cr ˙ u ) −
43 (2 c − c ) u (cid:48) + 6 cc (cid:19) , (18) κ T zz = 14 (cid:0) u (cid:48) − e − cr ˙ u )) + 6 cc (cid:1) , (19) κ T tt = − (cid:0) − u (cid:48) + e − cr ˙ u ) + (2 c − c ) u (cid:48) − cc (cid:1) , (20) κ T rr = 14 (cid:0) − u (cid:48) + e − cr ˙ u ) + (2 c − c ) u (cid:48) − c + 8 c c (cid:1) , (21)and κ T θθ = 14 (cid:18) u (cid:48) − e − cr ˙ u ) −
43 (2 c − c ) u (cid:48) + 16 c − c (cid:19) . (22)The component κ T rt must be equal to G rt . This energy-momentum component, aswill be seen, does not influence in the general results here since we will consider only timeaveraged features of the above quantities. This will be better explained later in section ( ).Finally, the bulk cosmological constant will assume the relation Λ = −
14 (24 c + c ) . (23)This will imply in Λ < which allows us to obtain relations between c , c , and | Λ | .Namely, c = ± (cid:112) | Λ | − c , (24)where c ≤ | Λ | . (25)It may be useful to highlight that for the configuration A ( r ) = 2 cr and B ( r ) = c r , themetric (4) will assume the simpler form ds = e cr (cid:0) dt − e u dx − e u dy − e − u dz (cid:1) − dr − R e c r + u dθ . (26)Here c and c ∈ R are real constants. The range of the variables r and θ are ≤ r < ∞ and ≤ θ < π , respectively. The function u = u ( r, t ) depends only on the variables r and t . The compact dimension θ , different of the string-like defect model, lives on the brane, i.e., θ is a brane coordinate for r = 0 . This particular feature is called hybrid compactification[52]. 9he metric ansatz (26) is a combination of the 6D warped braneworld model throughthe e cr and e c r terms (particularly this is similar to the global string-like defect) [6–8, 25]plus an anisotropic (r, t)-dependent warping of the brane coordinates, x, y, and z, throughthe terms e u ( t,r ) and e − u ( t,r ) . This may be seen as a six dimensional generalization of the5D standing wave braneworld model [29–31, 45, 46] and a 6D generalization of the sixdimensional standing wave braneworld [48, 50]. Still we can see our model as an extensionof the global string-like defect [6, 7]. Therefore, for u = 0 , the metric (26) is the same of thethin global string-like defects [6, 7]. As will be seen there are more than one point where u = 0 . In these points the geometry is the so-called AdS island.In addition, we can consider the exponential of the function u ( r, t ) as a correction of thestring-like models resulting in an anisotropic, time-dependent braneworld. III. STANDING WAVES SOLUTION
In order to obtain a standing wave solution we rewrite here the differential equation forthe u ( r, t ) function (17) as e − cr ¨ u ( r, t ) − au (cid:48) ( r, t ) − u (cid:48)(cid:48) ( r, t ) = 0 , (27)where prime and dots mean differentiation with respect to r and t , respectively, and a = (22 c + 4 c ) . In order to solve equation (27) we proceed as in Ref. [45] by choosing u ( r, t ) =sin( ωt ) ρ ( r ) . The general solution to the equation for the variable ρ ( r ) is given by ρ ( r ) = D e − a r J − a c ( ωc e − cr ) + D e − a r J a c ( ωc e − cr ) , (28)where D = C ( ω/ c ) a/ c Γ(1 − a/ c ) and D = C ( ω/ c ) a/ c Γ(1 + a/ c ) . C and C areintegration constants. J − a c and J a c are the first kind Bessel functions of orders − a c and a c ,respectively, and Γ represents the Gamma function. Now that we found the solution (28)we have the so called standing waves solution which generalizes the 5D work [45], and the6D works [48, 50]. Depending on the values of c and a one can obtain solutions similar tothat in six dimensions. If one has a = 5 c and D = 0 the solution will depends on the Besselfunction J which is the case in the works in six dimensions. So these present solutions aremore general then those obtained in the works cited above.Some features of the function u ( r, t ) can be derived from the above solution. The firstone is the fact that both functions J − a c and J a c are regular at the origin and at infinity10 r → ∞ ), given the possibility to maintain the general solution (28). Depending on therelation between ω , c and a the functions J − a c and J a c converge for both c > or c < enabling solutions with decreasing and increasing warp factor. Furthermore, we require thatthe function u is zero on the brane, i.e., at r = 0 [45]. This assumption may be expressedby ωc = X ± a c ,n , (29)where X ± a c ,n represents the n-th zero of J − a c or J a c depending if we do C or C equal tozero in (28) . The boundary condition (29) quantizes the ω frequency.By this consideration the u function will assume the value zero in some specific r values,namely r m . For these r m values our model may be identified with other 6D braneworldmodels [6–8, 12–14, 23, 25] as one can see in the metric (26). For c > the convergence ofthe function (28) for C = 0 or C = 0 , will depends essentially on the value of the ratio ω/c . The quantity of zeros, or AdS island, will depend on the value of c and mainly on thevalue of this ratio. For the case discussed here we have a finite number of zero. For c < (with either C = 0 or C = 0) the solution will present infinite zeros.Once we know u we may obtain the components of the energy-momentum tensor. Thiswill be done for the cases where a and c have the same sign and for the case where theyhave opposite sign. A. Case A: the same sign for a and c
In this case we will choose a = 4 c which will imply in c = c . Here we will consider onlythe time average of the energy-momentum tensor components. This option will be betterexplained in the section about field localization. In the case a = 4 c and D = 0 , the solution(28) will depends on J , so our energy-momentum components will be done in terms of thisfunction.In the figures below we plot the quantities (cid:104) T xx (cid:105) = (cid:104) T yy (cid:105) = (cid:104) T zz (cid:105) , (cid:104) T tt (cid:105) , (cid:104) T rr (cid:105) and (cid:104) T θθ (cid:105) for D = κ = c = 1 ; ω = 5 . . In figure (1) the dot-dashed line represents (cid:104) T xx (cid:105) = (cid:104) T yy (cid:105) = (cid:104) T zz (cid:105) ,the doted one represents (cid:104) T rr (cid:105) , the dashed line represents (cid:104) T θθ (cid:105) and finally, the filled linerepresents the energy density (cid:104) T tt (cid:105) . As one can see all these quantities are positive (part of T rr is negative but |(cid:104) T rr (cid:105)| < |(cid:104) T rr (cid:105)| ), what is an advantage when one compares it with theother works in this context [45, 48, 50]. But it is not possible to say that this is a normal11atter once the dominant energy condition (DEC) is violated. However it is not an exoticsource once the null (NEC), strong (SEC) and weak (WEC) energy conditions are satisfied.By following the matter classification given in [49] this is a not normal matter. (cid:45) T MN (cid:64) r (cid:68) Figure 1: (cid:104) T MN (cid:105) profile. But the we are interested in a solution generated by normal matter. In order to have anormal matter solution it is necessary that ρ ≥ p . In order to treat this unique possibilitywe have to consider an anisotropic cosmological constant. As a matter of fact, recently,a higher dimensional Randall-Sundrum toy model was proposed by Archer and Huber [51]which contains a bulk with anisotropic cosmological constant given by Λ = Λ η µν Λ Λ ,where η µν is the metric of the brane.Following this procedure it is possible to find our solution in the presence of normalmatter. For an anisotropic cosmological constant where its brane part is given by Λ = − ( c +6 cc ) , Λ = − (8 cc ) , and Λ = − (16 c ) , the components of the energy-momentumtensor (18 - 22) will assume the form κ (cid:104) T xx (cid:105) = κ (cid:104) T yy (cid:105) = κ (cid:104) T zz (cid:105) = κ (cid:104) T θθ (cid:105) = 14 (cid:0) u (cid:48) − e − cr ˙ u ) + 24 c (cid:1) , (30) κ T tt = − (cid:0) − u (cid:48) + e − cr ˙ u ) − c (cid:1) , (31)and κ (cid:104) T rr (cid:105) = 14 (cid:0) − u (cid:48) + e − cr ˙ u ) + 24 c (cid:1) . (32)12e plot in figure (2) these quantities as in figure (1), with the same values for theconstants. The dotted line represents the spatial components, except the r componentwhich is represented by the shaded line. The filled line represents the temporal componentof the energy-momentum tensor. As one can see all these quantities are positive and allthe energy conditions (particularly DEC) are satisfied. This is sufficient to assure that oursource is a normal matter and that our model is stable. Of course it is possible to choosethe value of the cosmological constant in a different way and still keep the normal mattersolution. T MN (cid:64) r (cid:68) Figure 2: (cid:104) T MN (cid:105) profile. B. Case B: a and c have opposite signs
For this case we will consider a = − c which will give c = − c . As in the othercase if the cosmological constant is isotropic it is possible to find solution with all energy-momentum tensor components positive, but it would not possible to obey the dominantenergy condition, as in the case above. In fact once we choose Λ = − (24 c + 6 cc ) , whichis positive for a = − c (meaning that the bulk is asymptotically dS), than we obtain notnormal matter as in case A above. But the principal interest consists in a source whichcorresponds to normal matter. So we will once again look for a solution with an anisotropiccosmological constant. There are several ways to choose the energy-momentum componentsand cosmological constant in order to have a solution in the presence of normal matter.Here we assume Λ = − (24 c + 6 cc ) , Λ = − (24 c + 8 cc − c ) and Λ θ = − (40 c − c ) .Since we know the relation between c and c it is easy to see that the components of theanisotropic cosmological constant are all positive. The time average components of the13nergy-momentum tensor are κ (cid:104) T xx (cid:105) = κ (cid:104) T yy (cid:105) = κ (cid:104) T zz (cid:105) = κ (cid:104) T θθ (cid:105) = 14 (cid:0) u (cid:48) − e − cr ˙ u ) + c (cid:1) , (33) κ T tt = − (cid:0) − u (cid:48) + e − cr ˙ u ) − c (cid:1) , (34)and κ (cid:104) T rr (cid:105) = 14 (cid:0) − u (cid:48) + e − cr ˙ u ) + c (cid:1) . (35)For D = 0 and a = − c in (28) we plot the time averaged components of the energy-momentum tensor (34 - 35) in figure (3). As in figure (2) the filled line represents the energydensity, the dotted one gives κ (cid:104) T xx (cid:105) = κ (cid:104) T yy (cid:105) = κ (cid:104) T zz (cid:105) = κ (cid:104) T θθ (cid:105) and the dashed linerepresents the (cid:104) T rr (cid:105) component. As one can see all these quantities are positive and ρ ≥ p which assures the dominant energy condition. Therefore, again we obtained a standing wavesolution generated by normal matter. T MN (cid:64) r (cid:68) Figure 3: (cid:104) T MN (cid:105) profile. On the other hand, one important feature of the braneworld models is the possibility tosolve the hierarchy problem. In others standing wave braneworld works this feature was notexplored. Here we are interested in showing that it is possible to readdress this solutionin this context. The condition to solve the hierarchy problem in this context is that theintegral below be convergent, namely M = 2 πM (cid:90) ∞ dre (2 c + c ) r . (36)For c = c = 2 a , where a is a positive constant, as in the two 6D standing wave braneworldmodels cited above, the integral is not convergent and the hierarchy problem is not solvable.The same is valid for the first case presented in this work where c = c and c > . But in14his second case where c = − c and c is positive we obtain the hierarchy problem solution.So this is another advantage of the model presented here in relation to the other ones donein this same context.After we obtain the solutions for the standing wave braneworld and after we demonstratethat one of our solutions is able to solve the hierarchy problem, we are now interest in thepotential of our model in order to localize the Standard Model (SM) fields.In the other 6D standing wave braneworld [48, 50] the localization issues of scalar, vectorand fermion fields were already exhaustively studied. Therefore, since from our model wemay obtain these other 6D standing wave braneworld, here it is sufficient to assure that thescenario presented here is convenient to localizes standard model fields. However, we willbriefly present the study of localization for the scalar and fermion fields. This last one isinteresting because, in five dimension, it was not possible to localize the right fermion. IV. SCALAR FIELD LOCALIZATION
This section is devoted to the study of the localization of the bulk scalar field. We willfollow again the proceedings given in Refs. [30, 31, 45]. Then, considering the general metric(4) we have that √− g = R e A + B/ . Therefore, the equation for the scalar field may bewritten as (cid:20) ∂ t − e − u (cid:0) ∂ x + ∂ y (cid:1) − e u ∂ z − e − u R ∂ θ (cid:21) Φ = e − A − B/ (cid:16) e A + B/ Φ (cid:48) (cid:17) (cid:48) . (37)Next, we consider a solution of the form Φ( x M ) = Ψ( r, t ) χ ( x, y ) ζ ( z ) e ilθ . (38)If one separates the variables r and t by making Ψ( r, t ) = e iEt ¯ ρ ( r ) , the equation for the r variable will assume the form (cid:16) e A + B/ ¯ ρ ( r ) (cid:48) (cid:17) (cid:48) − e A + B/ G ( r ) ¯ ρ ( r ) = 0 , (39)where G ( r ) = (cid:0) p x + p y (cid:1) (cid:0) e − u − (cid:1) + p z (cid:0) e u − (cid:1) + l R e − u . (40)It will be convenient to write (39) as an analogue non-relativistic quantum mechanicproblem. So we will assume the change of variable ¯ ρ ( r ) = e − ( A + B/ ¯Ψ( r ) . With this change15e will find ¯Ψ (cid:48)(cid:48) ( r ) − V ( r ) ¯Ψ( r ) = 0 , (41)where V ( r ) = 12 (2 A (cid:48)(cid:48) + B (cid:48)(cid:48) A (cid:48) + B (cid:48) + e − A G ( r ) . (42)From now on, we will consider A = 2 cr , B = c r and the simplified metric (26). Next,we will obtain the rt-dependent function Ψ , in order to analyze the localization of the scalarfield. In other words it is necessary to solve equation (41), but this will be done onlyfor the zero mode scalar and s-wave. This case is obtained when we assume ( l = 0) and E = p x + p y + p z . Additionally it is considered that ω >> E , which justifies to perform thetime-averaging of V ( r ) reducing the number of independent variables to one, namely r . Byapplying this simplification we will find the following expansion (cid:10) e bu (cid:11) = 1 + + ∞ (cid:88) n =1 ( b ) n n ( n !) [ D e − a r J − a c ( ωc e − cr ) + D e − a r J a c ( ωc e − cr )] n , (43)or (cid:10) e bu (cid:11) = I ( bρ ( r )) , (44)where I is the modified Bessel function of the first kind. It is evident from the expressionabove that our problem is still very complex. As can be seen from the expression (43),the approach in order to solve analytically the equation (41) is a hard work. Our strategyconsists in consider simplification and asymptotic approximations for the above expression.Let us begin the approximations by making D = 0 in (28). Once we do this, the u ( r, t ) will depends on first kind Bessel function J a c . The expansion (43) will be given by (cid:10) e bu (cid:11) = 1 + + ∞ (cid:88) n =1 ( bD ) n e − anr n ( n !) [ J a c ( ωc e − cr )] n . (45)It is evident that our solution is still very general since the order of the function J is a/ c . This give us the advantage to choose the order of the function J more convenientfor our interest, since our choosing is in accordance with the relations (24) and (25). If onechoose a = 4 c the order of J will be 2, as in the case A discussed above. This solution isvery similar to the one first proposed in five dimensions for the localization of the scalarfield, with the difference that there the authors considered the Bessel function of the secondkind, Y , rather than J [45]. 16fter applying what it was discussed above, let us study (41) by considering asymptoticapproximation far from and near the brane. For the first case, r → + ∞ , the expression J a c = J goes to zero ( ( ω/c ) e − cr → ) and the relation (45) will be approximated as (cid:10) e bu (cid:11) ≈ . This will result in the following simpler form for equation (41), namely ¯Ψ (cid:48)(cid:48) ( r ) − c ¯Ψ( r ) = 0 , (46)whose solution is e ± cr . We choose ¯Ψ = e − cr and c > which is convergent for all r values. This solution is similar to that one found in 5D standing wave context in the caseof scalar field localization, for this same asymptotic limit assumed here [31].The other case to be considered here for asymptotic approximation is the case where r → . In this case the equation (41) may be approximated as ¯Ψ (cid:48)(cid:48) ( r ) − (cid:16) dc r − dcr + d (cid:48) (cid:17) ¯Ψ( r ) = 0 . (47)This equation is more general that the equivalent equation considered in five dimension[31], once there it was considered only first order approximation. The constants d and d (cid:48) are given, respectively, by d = (cid:18) D (cid:19) (cid:16) ωc (cid:17) ( p x + p y + 9 p z ) , (48)and d (cid:48) = 94 c + d. (49)The solution of the equation (47) is given by ¯Ψ( r ) = E D µ − / (cid:115) √ dc + 2 (cid:18)(cid:113) c √ d (cid:19) r + E D ν − i / (cid:115) √ dc + 2 i (cid:18)(cid:113) c √ d (cid:19) r , (50)where D is the parabolic cylinder function, and E and E are integration constants. We seethat E must be zero in order to have a real solution. The µ , ν indexes are given respectivelyby µ = − c + 16 √ dc − d √ dc , (51)17nd ν = 18 √ c − √ dc − √ d √ dc . (52)For E = 0 and ω/a = 5 . , which corresponds to the first zero of J , and requiring µ = 0 it is possible to show that this solution is convergent for either c > or c < , as can beseen in the figures below. We see that the extra part of the scalar zero-mode wave function ¯ ρ ( r ) has a minimum at r = 0 , increases and then fall off, for the case c = 1 , as can be seenin figure (4). For c = − the function has a maximum at r = 0 and it rapidly falls off as wemove away from the brane, as can be seen in figure (5). On the other hand, for r → ∞ , itassumes the asymptotic form e − (17 / cr which is in accordance with [31] only for c > . Ingeneral, however, for other relations between a and c , it is possible to have localization for c positive or negative, i.e., for increasing or decreasing warp factor. Ρ (cid:64) r (cid:68) Figure 4: ¯ ρ profile. c = 1 Ρ (cid:64) r (cid:68) Figure 5: ¯ ρ profile. c = − The results of this section show that we have the localization of the zero-mode scalarfield in the model considered in this work. This is an expected result since the study oflocalization of the scalar field was performed in simpler 5D and 6D models than the oneconsidered here. It is relevant to stress the fact that the localization here is possible for bothincreasing or decreasing warp factor whereas in the thin string-like brane the localizationof the zero-mode scalar field is obtained only for a decreasing warp factor, and in otherstanding wave braneworld this result was obtained only for the case of an increasing warpfactor [6, 7, 48, 50]. 18 . LOCALIZATION OF SPIN / FERMIONIC ZERO MODE
The study of localization of zero mode spin / fermion is interesting in this context since,in 5D standing wave braneworld, it was not possible to localize the zero mode right fermion.However in the six dimensional models cited above it was demonstrated that this field islocalized. Once the model presented here is more general than that, it is natural that we findthe same results here. As a matter of fact, our results, as will be seen in this section is verysimilar to the one found in Refs. [48, 50], except that there, the Bessel function consideredis J , and here we are using J . Therefore we begin by the action for the massless spin / fermion in six dimensions which may be written as S = (cid:90) d x √− g ¯Ψ i Γ M D M Ψ . (53)From this action we derive the respective equation of motion, namely (cid:0) Γ µ D µ + Γ r D r + Γ θ D θ (cid:1) Ψ( x M ) = 0 . (54)In this expression Γ M represents the curved gamma matrices which relate to the flat ones as Γ M = h M ¯ M γ ¯ M , (55)where the vielbein h M ¯ M is defined as follows g MN = η ¯ M ¯ N h ¯ MM h ¯ NN . (56)The covariant derivative assumes the classical form D M = ∂ M + 14 Ω ¯ M ¯ NM γ ¯ M γ ¯ N . (57)The spin connection Ω ¯ M ¯ NM in this case is defined as Ω ¯ M ¯ NM = 12 h N ¯ M (cid:16) ∂ M h ¯ NN − ∂ N h ¯ NM (cid:17) + − h N ¯ N (cid:16) ∂ M h ¯ MN − ∂ N h ¯ MM (cid:17) − h P ¯ M h Q ¯ N h ¯ RM (cid:0) ∂ P h Q ¯ R − ∂ Q h P ¯ R (cid:1) . (58)In order to find the relations between the curved gamma matrices and the flat gammamatrices we refer to the metric ansatz (26) and use the relation (55), which will give us thenon zero results Γ t = e − cr γ ¯ t ; Γ x = e − cr − u γ ¯ x ; Γ y = e − cr − u γ ¯ y ;Γ z = e − cr + u γ ¯ z ; Γ r = γ ¯ r ; Γ θ = R − e − c r − u γ ¯ θ . (59)19t is still necessary to put in evidence the non-vanishing components of the spin connection(58), namely Ω ¯ t ¯ xx = Ω ¯ t ¯ yy = 1 R Ω ¯ t ¯ θθ = − ˙ u e u/ ; Ω ¯ t ¯ zz = 3 ˙ u e − u/ ;Ω ¯ r ¯ xx = Ω ¯ r ¯ yy = (cid:18) c + u (cid:48) (cid:19) e cr + u/ ; Ω ¯ r ¯ zz = (cid:18) c − u (cid:48) (cid:19) e cr − u/ ;1 R Ω ¯ r ¯ θθ = (cid:18) c u (cid:48) (cid:19) e c r + u/ ; Ω ¯ r ¯ tt = ce cr . (60)On the other hand, in order to solve Eq. (54) it will be necessary to consider the casewhere ω >> E , as in the last section. In addition, a time average of the equation of motionmust be performed. Furthermore, we assume the decomposition Ψ( x A ) = ψ ( x µ ) ρ ( r ) e ilθ forthe wave function. This will allow us to write the equation of motion (54) as (cid:20) D + γ r (cid:18) c + c ∂ r (cid:19) − e − cr R − l (cid:10) e − u/ (cid:11)(cid:21) ψ ( x µ ) ρ ( r ) = 0 , (61)where the operator D is given as D = e − ar (cid:2)(cid:0)(cid:10) e − u/ (cid:11) − (cid:1) ( γ x ∂ x + γ y ∂ y ) + (cid:0)(cid:10) e u/ (cid:11) − (cid:1) γ z ∂ z (cid:3) . (62)Once again it is not possible to analytically solve the equation above and then we have tostudy this equation in the limits r → and r → ∞ . It is relevant to note that the operator D may be approximated as D ≈ in this two distinct regions. This is a consequence ofthe fact that (cid:10) e bu (cid:11) ≈ for r → . This result it is shown schematically in figure (6) below.The constant b assumes the values b = − . and b = 1 . , represented by filled and dottedline respectively. As one can see from this figure the quantity (cid:10) e bu (cid:11) is approximately onefor both values of the constant b . Therefore, in both cases, the equation (61) for the s-wave ( l = 0) , may be simplified as (cid:18) c + c ∂ r (cid:19) ρ ( r ) = 0 . (63)It is very easy to solve this equation resulting in ρ ( r ) ∝ e − c + c r . This solution showsthat for r → the function ρ has a maximum at the origin and that it decays as e − c + c r for a > when r → ∞ . To show the localization, we insert this solution in the action (53).By doing this, the resultant integral in variable r will assume the form I ∝ (cid:82) ∞ dre − ar . It isevident that this integral is convergent for a > . This is sufficient to assure that the spin20 e bu (cid:72) r (cid:76) Figure 6: (cid:104) e bu (cid:105) profile. The filled line represents b = − . and the dotted line represents b = 1 . / fermion zero mode is localized in this model. Therefore, similar to the other results insix dimension, we show that the geometry is important to localize fields in contrast to 5Dstanding wave braneworld where is not possible to find localization for the fermion. This stillshows that the model presented here is more general that the other six dimensional standingwave braneworld model and allows the localization of fields for different Bessel function. VI. REMARKS AND CONCLUSIONS
In this work, we have obtained standing gravitational waves solution for the six dimen-sional Einstein equation in the presence of an anisotropic brane generated by normal matter.The compact dimension belongs to the brane and is small enough to assure that our modelis realistic. Our metric ansatz is anisotropic and non-static, unlike most models consideredin the braneworld literature. We find a solution for the warp factor which represents a thinbrane, and in this case the bulk may be seen as a generalization of the string-like defect[6, 7]. Apart from having as source a physical field, our model is more general than otherssix dimensional standing wave braneworld recently considered in the literature [48, 50] sincetheir solutions may be derived from our solution as special cases. Our metric ansatz has twodifferent warp factors e cr and e c r , similar to the global string-like defect, and it has alsogeneral warp factors e u ( r,t ) and e − u ( r,t ) similar to the 6D standing wave braneworld withan exotic source. The different warp factors permit us to find a solution for the function u for increasing or decreasing warp factor e ± cr which is an advantage when compared with itssimilar model.We find standing wave solution for the function u ( r, t ) and we show that this solution ismore general and comprises the others six dimensional standing wave braneworld solutions21hich have recently appeared in the literature [48, 50]. In fact, our solutions depend onthe Bessel function J ± a c which for the special case a = ± c coincides with the models citedabove.From the u ( r, t ) function, it was possible to choose the type of matter that generates thebrane. We have found two types of matter depending if we consider isotropic or anisotropiccosmological constant. In the first case, for a negative cosmological constant, and a = 4 c ,we have demonstrated that the energy density and pressure components are all positive andthe energy conditions NEC, WEC and SEC are satisfied, although the dominant energycondition is violated in this case. If one consider a = − c , similar results are found forthe matter but, in this case, the cosmological constant is positive which means that in thiscase the bulk is asymptotically dS, while in the first case we have an 6D AdS space-time.It is interesting to note that an asymptotically AdS space-time has been considered in theothers 5D and 6D standing wave braneworld, but a dS geometry was found for the first timehere. For the case of anisotropic cosmological constant, we have considered the situationrecently suggested in the literature for higher dimensional Randall-Sundrum model [51]. Sowe consider that the cosmological constant on the brane has a fixed value λ , but the extracomponents Λ and Λ may assume different values. This is reasonable in our case sincewe are dealing with an anisotropic bulk. Therefore, for a = 4 c all the "components" ofthe cosmological constant are negative and for a = − c they are positive, in line with thefindings for the case of isotropic cosmological constant discussed above. In the two cases,we have obtained the components of the energy-momentum tensor and showed that all ofthem are positive and that all the energy conditions are satisfied. This means that wehave constructed a 6D standing wave braneworld which is generated by normal matter and,therefore, stable.An important feature of some braneworld models is their ability to solve the so-calledhierarchy problem, but in the context of standing wave braneworld this problem was nevertreated. Here, we have demonstrated that it is possible to solve the hierarchy problem in thiscontext. This is another advantage of the model proposed in this work over other standingwave braneworld models.Finally, we have considered the localization of fields in our model. Since it generalizes ourprevious work in this subject [48], it is reasonable to expect that the localization of the fieldswhich was studied there, it is also possible here. Indeed, here we have considered a = 4 c and22e studied only the zero mode scalar and fermion fields localization and as was expectedwe have shown that there is a zero mode localized for both scalar and fermion fields. Thesolution found here for the scalar field is in accordance with the ones that were encounteredin five dimensions [31] and six dimensions [48, 50].It is known that quantum effects may play important roles in braneworld models. Indeed,as an example, the mechanism of generating 4D newtonian gravity in static 3-brane [53, 54].Moreover, quantum corrections in warped backgrounds may lead to gravity delocalization[55]. Therefore, although we are dealing with normal matter, it is interesting to study howquantum fluctuation could interfere in the stability of our ansatz metric. This also will beleft for a future work. Acknowledgments
The authors thank the Fundação Cearense de apoio ao Desenvolvimento Científico eTecnológico (FUNCAP), the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior(CAPES), and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)for financial support. [1] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B , 257(1998) [hep-ph/9804398].[2] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B , 263 (1998) [hep-ph/9803315].[3] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Rev. D , 086004 (1999) [hep-ph/9807344].[4] L. Randall and R. Sundrum, Phys. Rev. Lett. , 3370 (1999) [arXiv:hep-ph/9905221].[5] L. Randall and R. Sundrum, Phys. Rev. Lett. , 4690 (1999) [arXiv:hep-th/9906064].[6] I. Oda, Phys. Lett. B , 113 (2000) [arXiv:hep-th/0006203].[7] I. Oda, Phys. Rev. D , 126009 (2000) [hep-th/0008012].[8] R. Gregory, Phys. Rev. Lett. , 2564 (2000) [arXiv:hep-th/9911015].[9]
9] J. W. Chen, M. A. Luty and E. Ponton, JHEP , 012 (2000) [arXiv:hep-th/0003067].[10] A. G. Cohen and D. B. Kaplan, Phys. Lett. B , 52 (1999) [arXiv:hep-th/9910132].[11] I. Olasagasti and A. Vilenkin, Phys. Rev. D , 044014 (2000) [hep-th/0003300].[12] T. Gherghetta and M. E. Shaposhnikov, Phys. Rev. Lett. , 240 (2000) [arXiv:hep-th/0004014].[13] E. Ponton and E. Poppitz, JHEP , 042 (2001) [hep-th/0012033].[14] M. Giovannini, H. Meyer and M. E. Shaposhnikov, Nucl. Phys. B , 615 (2001) [hep-th/0104118].[15] P. Tinyakov and K. Zuleta, Phys. Rev. D , 025022 (2001) [hep-th/0103062].[16] S. Kanno and J. Soda, JCAP , 002 (2004) [hep-th/0404207].[17] J. Vinet and J. M. Cline, Phys. Rev. D , 083514 (2004) [hep-th/0406141].[18] J. M. Cline, J. Descheneau, M. Giovannini and J. Vinet, JHEP , 048 (2003) [hep-th/0304147].[19] E. Papantonopoulos, A. Papazoglou and V. Zamarias, Nucl. Phys. B , 520 (2008)[arXiv:0707.1396 [hep-th]].[20] I. Navarro, JCAP , 004 (2003) [hep-th/0302129].[21] I. Navarro and J. Santiago, JHEP , 007 (2005) [hep-th/0411250].[22] E. Papantonopoulos and A. Papazoglou, JCAP , 004 (2005) [hep-th/0501112].[23] B. de Carlos and J. M. Moreno, JHEP , 040 (2003) [arXiv:hep-th/0309259].[24] M. Gogberashvili and P. Midodashvili, Europhys. Lett. , 308 (2003) [hep-th/0111132].[25] R. Koley and S. Kar, Class. Quant. Grav. , 79 (2007) [hep-th/0611074].[26] R. S. Torrealba, Phys. Rev. D , 024034 (2010) [arXiv:1003.4199 [hep-th]].[27] J. E. G. Silva and C. A. S. Almeida, Phys. Rev. D , 085027 (2011) [arXiv:1110.1597 [hep-th]].[28] L. J. S. Sousa, W. T. Cruz and C. A. S. Almeida, Phys. Lett. B , 97 (2012) [arXiv:1203.5149[hep-th]].[29] Localization of Matter Fields in the 5D Standing Wave Braneworld, M. Gogberashvili,arXiv:1204.2448 [hep-th].[30] M. Gogberashvili, P. Midodashvili and L. Midodashvili, Phys. Lett. B , 169 (2012)[arXiv:1105.1866 [hep-th]].[31] M. Gogberashvili, P. Midodashvili and L. Midodashvili, Phys. Lett. B , 276 (2011)[arXiv:1105.1701 [hep-th]].
32] A. Kehagias and K. Tamvakis, Phys. Lett. B , 38 (2001) [hep-th/0010112].[33] W. T. Cruz, M. O. Tahim and C. A. S. Almeida, Europhys. Lett. , 41001 (2009)[arXiv:0912.1029 [hep-th]].[34] W. T. Cruz, A. R. Gomes and C. A. S. Almeida, Europhys. Lett. , 31001 (2011).[35] M. O. Tahim, W. T. Cruz and C. A. S. Almeida, Phys. Rev. D , 085022 (2009)[arXiv:0808.2199 [hep-th]].[36] S. Randjbar-Daemi and M. E. Shaposhnikov, Phys. Lett. B , 329 (2000) [hep-th/0008087].[37] A. Kehagias, Phys. Lett. B , 133 (2004) [arXiv:hep-th/0406025].[38] M. Gogberashvili, P. Midodashvili and D. Singleton, JHEP , 033 (2007) [arXiv:0706.0676[hep-th]].[39] Y. -S. Duan, Y. -X. Liu and Y. -Q. Wang, Mod. Phys. Lett. A , 2019 (2006) [hep-th/0602157].[40] J. Garriga and M. Porrati, JHEP , 028 (2004) [hep-th/0406158].[41] J. F. Vazquez-Poritz, JHEP , 001 (2002) [arXiv:hep-th/0111229].[42] H. Firouzjahi and S. H. Tye, JHEP , 136 (2006) [arXiv:hep-th/0512076].[43] T. Noguchi, M. Yamaguchi and M. Yamashita, Phys. Lett. B , 221 (2006) [arXiv:hep-th/0512249].[44] F. Brummer, A. Hebecker and E. Trincherini, Nucl. Phys. B , 283 (2006) [arXiv:hep-th/0510113].[45] M. Gogberashvili and D. Singleton, Mod. Phys. Lett. A , 2131 (2010) [arXiv:0904.2828[hep-th]].[46] M. Gogberashvili, A. Herrera-Aguilar and D. Malagon-Morejon, Class. Quant. Grav. ,025007 (2012) [arXiv:1012.4534 [hep-th]].[47] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003); S.M. Carroll, M Hoffman and M. Trodden, Phys. Rev. D68, 023509 (2003); P. Frampton, PhysLett. B555 139 (2003); V. Sahni and Y. Shtanov, JCAP 0311, 014 (2003).[48] A 6D standing wave Braneworld, L. J. S. Sousa, J. E. G. Silva and C. A. S. Almeida,arXiv:1209.2727 [hep-th].[49] M. Visser, Phys. Rev. D 56, 7578 (1997).[50] Localization of Matter Fields in the 6D Standing Wave Braneworld, P. Midodashvili,arXiv:1211.0206v1 [hep-th].[51] P. R. Archer and S. J. Huber JHEP 1103:018,2011.
52] B. Carter, A. Nielsen and D. Wiltshire, JHEP (2006) 34 [arxiv:0602086 [hep-th]].[53] G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B485 (2000) 208.[54] G. Dvali, G. Gabadadze, Phys. Rev. D63 (2001) 065007.[55] Z. Kakushadze, Phys. Lett. B497 (2001) 125.(2006) 34 [arxiv:0602086 [hep-th]].[53] G. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B485 (2000) 208.[54] G. Dvali, G. Gabadadze, Phys. Rev. D63 (2001) 065007.[55] Z. Kakushadze, Phys. Lett. B497 (2001) 125.