Analytical Solution for Bosonic Fields in the FRW Multiply Warped Braneworld
aa r X i v : . [ h e p - t h ] J a n A NALYTICAL S OLUTION FOR B OSONIC F IELDS IN THE
FRWM
ULTIPLY W AR PED B R ANEWORLD
A P
REPRINT
A. S. Ribeiro ∗ Instituto Federal do Piauí – IFPISão Raimundo Nonato, Piauí, 64770-000, Brazil, andDepartamento de Física, Universidade Federal do CearáCampus do Pici, 60455-760, Fortaleza, Ceará, Brazil. [email protected]
G. AlencarDepartamento de Física, Universidade Federal do CearáCampus do Pici, 60455-760, Fortaleza, Ceará, Brazil. [email protected]
R. R. LandimDepartamento de Física, Universidade Federal do CearáCampus do Pici, 60455-760, Fortaleza, Ceará, Brazil. [email protected]
January 5, 2021 A BSTRACT
In this paper we find analytical solutions for the scalar and gauge fields in the Freedman-Robertson-Walker multiply warped braneworld scenario. With this we find the precise mass spectra for thesefields. We compare these spectra with that previously found in the literature for the static case. K eywords Field theories in dimensions other than four · Relativity and gravitation · Classical field theory.
In the ’s Kaluza and Klein (KK) attempted to unify the general relativity and electromagnetism theories by addingan extra compact dimension to the ordinary spacetime. This initial effort was considered as inconsistent and wasdiscarded by the scientific community. Extra dimension theories have attracted interest of researchers again in theearly eighties of the twentieth century, with the rise of the string theory as a quantum theory of the gravitational field.New approaches of extra dimension theories have been proposed in this context, with interesting consequences intheories of elementary particles and cosmology [1–4].The spacetime in these higher dimensional models is generally taken as a product of a four-dimensional spacetime with n compact extra dimensions [5]. While the gravity can propagates freely through the extra dimensions the particles inStandard Model (SM) are confined into spacetime in four dimensions. In add, the effective Planck scale is measured bythe observers in this four-dimensional boundary as M P l = M n +2 V n , where V n is the volume of the compact space. If V is sufficiently large, we can take TeV order to Planck scale. Thus, it is possible to remove the hierarchy between thePlanck scale and V n weak scale. Randall and Sundrum(RS) solved this problem proposing a new scenario with a non-factorizable geometry, and five dimensions with an exponential suppression of the Planck scale [6, 7]. This naturallygenerates the energy of the Standard Model (SM) of particles that solves the problem of the great desert between ∗ I am corresponding author.
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5, 2021the energy scales. The introduction of a cosmological evolution over the brane in the RS model was considered, forexample, in Refs. [8–10].As a natural extension of the RS model, several codimension two models were proposed [11–16]. In this directionDebajyoti Choudhury et al proposed multiply warped scenario [17]. Due to the two orbifolds, we four intersectingpoints, forming a picture that reminds a box. The walls of this box are (4 + 1) -branes, while in the intersecting pointsthere are (3 + 1) -branes. We can identify one of the intersecting points as being the Planck brane and other of thesepoints as our (3+1) universe. The authors found the mass spectrum of the Dirac field. With this a phenomenologicalconsequence of the this model, for flat branes, includes an explanation of the observed hierarchy in the masses ofstandard model fermions [17]. The mass spectrum of the scalar and gauge fields was found in [18, 19]. Whit this it ispossible to consistently describe a bulk Higgs and gauge fields with spontaneous symmetry breaking [19]. The massspectra of Refs. [18, 19] are found by considering an approximation in the equations of motion. In order to solve thisArun et al found analytical solutions for scalar and gauge fields in Refs. [20, 21].The generalization of the multiply warped scenario to include a FRW solution over the brane was obtained recently inRef. [22]. However, bosonic fields has not been considered in this background. This paper is organized as follows: insection 2 we review the static and FRW doubly warped braneworld. In section 3 we find analytical solutions to scalarand gauge fields and with this the respective mass spectra. In section 4, using the same methodology of the previoussection, we also find analytical solution for the FRW case. As it turns, in section 5 we summarize our main findingsand draw some perspectives.
Let us review the FRW doubly warped spacetime [22]. The background is given by the manifold M (1 , → (cid:2) M (1 , × S /Z (cid:3) × S /Z . Considering two consecutive orbifolds along the two extra dimensions, it is possibleto obtain a configuration of a box-like spacetime [17].Let us first define the notation. The spacetime coordinates are x M , where the index are defined as M , N , withvalues M, N = 0 , , , , , . The non-compact coordinates are x µ , with µ, ν = 0 , , , . Finally x = y and x = z are the extra dimension coordinates. The fixed points of the orbifolds, which are the -branes, are locatedin x = 0 , π and x = 0 , π . In this approach two -brane intersect to form the 3-brane at the points (cid:0) x , x (cid:1) → (0 , , (0 , π ) , ( π, , ( π, π ) .To simplify the model it is relevant to assume that our universe is homogeneous and isotropic. With this the lineelement is restricted to the form ds = B ( z ) [ A ( y ) (cid:0) − dt + V ( t ) δ ij dx i dx j (cid:1) + R dy ]+ r dz (1)where R and r are the moduli along the compact coordinates y and z , respectively. The total action of the backgroundis given by S = S + S + S ,S = Z d x √− g (cid:18) M R − Λ (cid:19) ,S = Z d xdydz {√− g [ L − V ( z )] δ ( y ) , + √− g [ L − V ( z )] δ ( y − π ) } + Z d xdydz { p − ˜ g [ L − V ( y )] δ ( z )+ p − ˜ g [ L − V ( y )] δ ( z − π ) } ,S = Z d xdydz √− g [ L − λ ] . (2)The brane tensions V i (with i = 1 , ..., ), are dependent on the coordinates of the extra dimensions. The matterlagrangian of the -branes are L i (with i = 1 , ..., ) and S is the contribution of the -brane. The matter content of thebranes is described by the perfect fluid approximation, where the energy density is ρ , and the pressure of the fluid is p .2 PREPRINT - J
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5, 2021By varying the action S , we obtain the following Einstein’s equation in six dimensions: − M √− g (cid:18) R MN − R g MN (cid:19) = Λ B √− g g MN + − h(cid:16) T γβ g αγ (cid:17) + (cid:16) T γβ g αγ (cid:17) i √− g δ αM δ βN + (3) − h(cid:16) ˜ T γβ ˜ g αγ (cid:17) + (cid:16) ˜ T γβ ˜ g αγ (cid:17) i p − ˜ g δ αM δ βN where T and ˜ T are the energy momentum tensors of the -branes.By substituting Eq. (1) in Eq. (3) it is possible to obtain the following solution V ( t ) = e H t , (4) A ( y ) = RH D sinh ( − D | y | + d ) , (5) B ( z ) = cosh ( kz )cosh( kπ ) , (6) D = Rkr cosh( kπ ) , (7) k = r r − Λ M , (8)where H and d are integration constants. In the next section we compute the mass spectrum for scalar and gaugefields in this background. As said in the introduction, the mass spectrum of the scalar field for the static case was obtained approximately inRef. [18]. Some time latter an analytical solution was found [20]. In this section we consider the FRW doubly warpedmetric given by Eq. (1) and find an analytical solution for the scalar field in this background. The action of themassless scalar field is S = Z d x √− g (cid:18) g MN ∂ M Φ ∂ N Φ (cid:19) , (9)with equation of motion given by √− g ∂ M (cid:2) √− gg MN ∂ N Φ (cid:3) = 0 . (10)Making the (KK) decomposition of the scalar field through the modes Φ ( x µ , y, z ) = X ij ˜ φ i,j ( x µ ) ξ ij ( y ) χ j ( z ) (11)and by substituting it in Eq. (10), we find V ∂ µ (cid:16) V ˜ g µν ∂ ν ˜ φ ij (cid:17) − M ij ˜ φ ij = 0 , (12) r ddz (cid:18) B dχ j dz (cid:19) = − M j B χ j , (13)and R ddy (cid:18) A dξ ij dy (cid:19) − M j A ξ ij = − M ij A ξ ij , (14)where ˜ g = − , ˜ g ij = V ( t ) δ ij is the FRW metric and M ij is the effective mass seen by an observer in the 3-brane.Let us begin by solving Eq. (13). For this we first use the warp factor B ( z ) to get d χ i dz + 5 k tanh( kz ) dχ j dz + (cid:0) sech ( kz ) r M j cosh( kπ ) (cid:1) χ j = 0 . (15)3 PREPRINT - J
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5, 2021In order to solve the above equation we define χ j ( z ) = sech / ( kz ) F j ( ω ) , ω = tanh( kz ) (16)and replace in Eq. (15) to arrive at (cid:0) − ω (cid:1) d F j dω − ω dF j dω + (cid:20) α j ( α j + 1) − − ω ) (cid:21) F j = 0 , (17)where α j = −
12 + s r M j cosh ( kπ ) k . (18)The expression (17) is Legendre’s associated equation with solution χ j ( z ) = A P j ( z ) + A Q j ( z ) , (19)where P j ( z ) = P / α j (tanh( kz )) sech / ( kz ) (20) Q j ( z ) = Q / α j (tanh( kz )) sech / ( kz ) . (21)Now we turn to solve Eq. (14). For this we replace the warp factor A ( y ) in Eq. (14) to obtain d ξ ij dy − D coth( θ ) dξ ij dy + " csch ( θ ) M ij D H − M j R ξ ij = 0 , (22)where θ = − Dy + d . (23)Applying the transformations ξ ij ( y ) = csch ( θ ) G ij ( τ ) , τ = coth( θ ) , (24)we obtain (cid:0) − τ (cid:1) d G ij dτ − τ dG ij dτ + " γ ij ( γ ij + 1) − λ j − τ ) G ij = 0 , (25)with λ j = s M j R D = α j + 12 (26)and γ ij = −
12 + s − M ij H . (27)If M ij /H > / , then γ ij = −
12 + iσ ij , (28)where σ ij = s M ij H − . (29)4 PREPRINT - J
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5, 2021Just as before, expression (25) is Legendre’s associated equation with solution ξ ij ( y ) = B P ij ( y ) + B Q ij ( y ) , (30)where P ij ( y ) = Re h P λ j γ ij (coth( − Dy + d )) i csch ( − Dy + d ) (31) Q ij ( y ) = Re h Q λ j γ ij (coth( − Dy + d )) i csch ( − Dy + d ) . (32)See that we have found a general solution and thus it is not necessary to take a fixed order. This occurs due to the factthat the α j order assumes different values for distinct j , i. e., M j is dependent of α j .With our analytical solution we can obtain the precise mass spectrum of the system. For this we must apply the fourboundary conditions in Eqs. (19) and (30). Observe that these conditions are applied taking the derivative of thefunctions that vanish at fixed points ( z = 0 , π ) , and ( y = 0 , π ) . With this we get the conditions A P ′ j (0) + A Q ′ j (0) = 0 , A P ′ j ( π ) + A Q ′ j ( π ) = 0 , (33) B P ′ ij (0) + B ′ Q ′ ij (0) = 0 , B P ij ( π ) + B Q ′ ij ( π ) = 0 . (34)Non trivial solutions of the above equations are possible only if P ′ j (0) Q ′ j ( π ) − Q ′ j (0) P ′ j ( π ) = 0 , (35) P ′ ij (0) Q ′ ij ( π ) − Q ′ ij (0) P ′ ij ( π ) = 0 . (36)From Eq. (35) we get the parameter M j and by using this in Eq. (36) we get the mass spectrum M ij of the scalar field.To clearly illustrate the dependence of the massive modes in relation to Hubble constant we produce the table 1. WeTable 1: Massive modes with Hubble constant. The M pl /H ratio of the scalar field is shown in the table. D = 11 . , k = 0 . . M ij /H M j M j M j M j M j M j M ij = H r σ ij + 94 (37)and therefore this correction is proportional to H . In this section we must study the gauge boson. The mass spectrum of this field, for the static case, was obtainedapproximately in Ref. [19]. Some time latter an analytical solution was found [20, 21]. Here we consider the FRWdoubly warped metric given by Eq. (1) and find an analytical solution for the gauge field in this background. Theaction is given by S g = Z d x √− g (cid:18) − F MN F MN (cid:19) , (38)where, g is the determinant of the metric given by Eq. (1), and F MN = ∂ M X N − ∂ N X M is the gauge field strength.By varying the action we obtain the equation of motion √− g ∂ M (cid:2) √− gg MN g LK F NL (cid:3) = 0 . (39)5 PREPRINT - J
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5, 2021We can choose a gauge in which A = A = 0 . Making the (KK) decomposition of the scalar field through the modesum X ρ ( x µ , y, z ) = X pl ˜ A plρ ( x µ ) η pl ( y ) ζ l ( z ) , (40)and by substituting it in Eq. (39), it is possible to obtain the equations V ∂ µ (cid:16) V ˜ g µν ∂ ν ˜ A plρ (cid:17) − M pl ˜ A ρ = 0 , (41) r ddz (cid:18) B dζ l dz (cid:19) = − M l Bζ l , (42) R ddy (cid:18) A dη pl dy (cid:19) − M l A η pl = − M pl η pl . (43)where M pl is the effective mass.To find M pl we solve the Eq. (42) to obtain M l , which must be used in Eq. (43) to determine M pl . Let us begin bysolving Eq. (42). First we replace the warp factor B ( z ) to obtain d ζ l dz + 3 k tanh( kz ) dζ l dz + [ sech ( kz ) r M l cosh( kπ )] ζ l = 0 . (44)Now we make the coordinate change ζ l ( y ) = sech / ( ω ) ¯ F l ( ω ) , ω = tanh( kz ) . (45)to obtain (cid:0) − ω (cid:1) d ¯ F l dω − ω d ¯ F l dω + (cid:20) µ l ( µ l + 1) − − ω ) (cid:21) ¯ F l = 0 , (46)where µ l = −
12 + s r M l cosh ( kπ ) k . (47)The above expression is Legendre’s associated equation with solution ζ l ( z ) = C P l ( z ) + C Q l ( z ) , (48)where P l ( z ) = P / µ l (tanh( kz )) sech / ( kz ) (49) Q l ( z ) = Q / µ l (tanh( kz )) sech / ( kz ) . (50)To solve Eq. (43) we use the warp factor A ( y ) and the transformations η pl ( y ) = csch ( θ ) ¯ G pl ( τ ) , τ = coth( θ ) (51)to arrive at (cid:0) − τ (cid:1) d ¯ G pl dτ − τ d ¯ G pl dτ + (cid:20) β pl ( β pl + 1) − Ω l (1 − τ ) (cid:21) ¯ G pl = 0 . (52)In the above equation we have used the definitions Ω l = r M l R D = µ l + 12 (53)6 PREPRINT - J
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5, 2021and β pl = −
12 + s − M pl H . (54)Besides, if M pl > H / , then β pl = −
12 + iσ pl (55)where σ pl = s M pl H − . (56)Again we have that Eq. (52) is the Legendre’s associated equation. We therefore arrive at the solution η lp ( y ) = D P pl ( y ) + D Q pl ( y ) , (57)where P pl ( y ) = Re h P Ω l β pl (coth( − Dy + d )) i csch ( − Dy + d ) (58) Q pl ( y ) = Re h Q Ω l β pl (coth( − Dy + d )) i csch ( − Dy + d ) . (59)From the solutions (48) and (57) we can obtain the mass spectrum by applying the boundary conditions at ( z = 0 , π ) and ( y = 0 , π ) . We get C P ′ j (0) + C Q ′ j (0) = 0 , C P ′ j ( π ) + C Q ′ j ( π ) = 0 , (60) D P ′ ij (0) + B Q ij (0) = 0 , D P ′ ij ( π ) + D Q ′ ij ( π ) = 0 . (61)Non trivial solutions of the above equations are possible only if P ′ j (0) Q ′ j ( π ) − Q ′ j (0) P ′ j ( π ) = 0 , (62) P ′ ij (0) Q ′ ij ( π ) − Q ′ ij (0) P ′ ij ( π ) = 0 . (63)As in the case of the scalar field, from Eq. (62) we get the parameter M j and by using it in Eq. (63) we get the massspectrum M ij of the gauge field. Again we find a dependence between the effective mass of the gauge boson and thebackground Hubble constant given by M pl = H r σ pl + 14 . (64)It is worthy to emphasize that due this all massive modes to vector field, taking cosmological effects, are lighter whencompared to static case.The numerical values of massive modes are shown in table 2.Table 2: Massive modes with Hubble constant. The M pl /H ratio of the gauge boson is shown in the table. D = 11 . , k = 0 . . M pl /H M l M l M l M l M l M l PREPRINT - J
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In short, we have used a FRW doubly warped spacetime in order to investigate the scalar and gauge fields. Thisproblem has been attached for the static case in Refs. [18–21]. For a FRW background (see Ref [22]), the study ofbosonic fields is lacking.For the scalar and gauge fields we find that the equations of motion can be transformed to a Legendre’s associatedequation. Therefore analytical solutions can be found and are given in Eqs. (19), (30),(48) and (57). With thesesolutions and by imposing the boundary conditions we obtain the mass spectrum of both fields. Some of the massmodes are given in tables 1 and 2. An interesting result is that in both cases the massive modes(Eqs. (37) and (64)),are given by M pl = H r σ pl + n , (65)where n = 9 or n = 1 for the scalar and gauge fields respectively. Therefore the mass tower, in the cosmologicalscenario of Ref. [22], are proportional to the parameter H .We should point that the above protocol can be used to investigate other bosonic fields in the FRW multiply warpedbackground. For example, the graviton mass spectrum has been found for the static case in Ref. [ ? ]. Another possibilityis the p -form field, which has been studied in Refs. [23–25] in other backgrounds. In the near future we must presentanalytical solutions of these fields in FRW multiply warped background. The authors would like to thanks Alexandra Elbakyan and sci-hub, for removing all barriers in the way of science.We acknowledge the financial support provided by the Conselho Nacional de Desenvolvimento Científico e Tec-nológico (CNPq) and Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) throughPRONEM PNE0112- 00085.01.00/16
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