Localization of Bulk Matters on a Thick Anti-de Sitter Brane
aa r X i v : . [ h e p - t h ] A ug . Localization of Gravity and Bulk Matters on a Thick Anti-deSitter Brane
Yu-Xiao Liu ∗ , Heng Guo † , Chun-E Fu ‡ , Hai-Tao Li § Institute of Theoretical Physics,Lanzhou University,Lanzhou 730000, People’s Republic of China
Abstract
In this paper, we investigate the localization and the mass spectra of gravity and various bulkmatter fields on a thick anti-de Sitter (AdS) brane, by presenting the mass-independent potentialsof the Kaluza-Klein (KK) modes in the corresponding Schr¨odinger equations. For gravity, thepotential of the KK modes tends to infinity at the boundaries of the extra dimension, which leadsto an infinite number of the bound KK modes. Although the gravity zero mode cannot be localizedon the AdS brane, the massive modes are trapped on the brane. The scalar perturbations of thethick AdS brane have been analyzed, and the brane is stable under the scalar perturbations. Forspin-0 scalar fields and spin-1 vector fields, the potentials of the KK modes also tend to infinity atthe boundaries of the extra dimension, and the characteristic of the localization is the same as thecase of gravity. For spin-1/2 fermions, by introducing the usual Yukawa coupling η ¯Ψ φ Ψ with thepositive coupling constant η , the four-dimensional massless left-chiral fermion and massive Diracfermions are obtained on the AdS thick brane. PACS numbers: 04.50.-h, 11.27.+d ∗ [email protected] † [email protected],Corresponding author ‡ [email protected] § [email protected] . INTRODUCTION The idea that our observed four-dimensional Universe might be a 3-brane, embedded ina higher dimensional space-time (the bulk), provides new insights into solving the gaugehierarchy and cosmological constant problems [1–8]. In the Randall-Sundrum (RS) braneworld model, the zero mode of gravity is localized on the brane, which reproduces thestandard Newtonian gravity on the brane [5]. But in this model, the brane is very idealbecause its thickness is neglected. In the most fundamental theory, there seems to exist aminimum scale of length, thus the thickness of a brane should be considered in more realisticfield models. For this reason, more natural thick brane scenarios have been investigated [9–41]. For some comprehensive reviews about thick branes, see Refs. [42–47].In brane world theory, only gravity is free to propagate in both the brane and bulkspace-time, however, all the matter fields (electromagnetic, Yang-Mills, etc.) in our four-dimensional Universe are confined to the 3-brane with no contradiction with present gravi-tational experiments. Hence, in order to build up the standard model, various bulk matterfields should be localized on branes by a natural mechanism. Generally, the massless scalarfields [48] could be trapped on branes of different types. Spin-1 Abelian vector fields canbe localized on the RS brane in some higher dimensional cases [49] or on the thick de Sit-ter brane and Weyl thick brane [50]. It is important to study the localization problem ofthe spin-1/2 fermions. Without introducing the scalar-fermion coupling, fermions cannotbe localized on branes in five and six dimensions [48–71]. In some cases, there may exista single bound state and a continuous gapless spectrum of massive fermion Kaluza-Klein(KK) modes [25, 50]. In some other cases, one can obtain finite discrete KK modes (massgap) and a continuous gapless spectrum starting at a positive m [28, 63, 64].Most investigations mentioned above mainly focused on flat branes. In Ref. [18], thesolutions of de Sitter (dS) and anti-de Sitter (AdS) 3-branes were presented, and the local-ization of gravity on the dS 3-brane was also studied. In this paper, in order to show the richstructures of the AdS 3-brane from the other points of view, we would like to investigate thelocalization problem of gravity and various spin matter fields (scalars, vectors and fermions)on the brane. For this AdS 3-brane solution, the behavior of the warp factor is related witha parameter δ . For δ >
1, the warp factor is divergent at z = ± z b the boundaries of theextra dimension, and the energy density has a lump at z ≈
0; i.e., the brane is located at2 ≈
0. For δ <
0, the warp factor tends to zero far away from the brane, but the energydensity has no lump at z ≈ z = ± z b , so this configuration cannot beconsidered as a thick brane. Hence, in this paper, we only consider that the parameter isconstrained as δ >
1. For gravity, the zero mode is not localized on the thick AdS brane;however, the massive KK modes can be localized on the brane, and the mass spectrum con-sists of an infinite number of discrete bound states. For free scalar fields and vector fields,all the KK modes are bound states and the massive modes, which could be trapped on thebrane. For spin-1/2 fermions coupling with the background scalar by the usual Yukawacoupling η ¯Ψ φ Ψ, the left-chiral fermion massless mode can be localized on the brane, andboth left- and right-chiral fermions are also localized on the brane, which can constitute thefour-dimensional massive Dirac fermions.The organization of this paper is as follows: In Sec. II, we first review the thick AdS3-brane in five-dimensional space-time. Then, in Sec. III, we investigate the localizationand the mass spectra of gravity on the thick AdS 3-brane. In Sec. IV, the localization andthe mass spectra of various bulk matter fields are investigated. For scalars and vectors, wegive the analytical expressions for the KK modes and the mass spectrums. For fermions,the analytical formulations of the massless modes are also obtained. Finally, our conclusionis given in Sec. V.
II. REVIEW OF THE THICK ANTI-DE SITTER 3-BRANE
We start with the following five-dimensional action of thick branes, which are generatedby a real scalar field φ with a scalar potential V ( φ ), S = Z d x √− g (cid:20) κ R − g MN ∂ M φ∂ N φ − V ( φ ) (cid:21) , (1)where R is the five-dimensional scalar curvature and κ = 8 πG with G the five-dimensionalNewton constant. The most general five-dimensional metric compatible with an AdS sym-metry can be taken as ds = g MN dx M dx N = e A ( z ) [ˆ g µν ( x ) dx µ dx ν + dz ]= e A ( z ) (cid:2) e Hx ( − dt + dx + dx ) + dx + dz (cid:3) , (2)where e A ( z ) is the warp factor, H is the AdS parameter and z stands for the extra co-ordinate. We suppose that the scalar field is a function of z only, i.e., φ = φ ( z ). The3our-dimensional cosmology constant can be expressed as Λ = − H . By considering theaction (1) and the metric (2), the filed equations reduce to the following coupled nonlineardifferential equations: φ ′ = 3 κ ( A ′ − A ′′ + H ) , (3a) V ( φ ) = − − A κ (3 H + 3 A ′ + A ′′ ) , (3b) dV ( φ ) dφ = e − A (3 A ′ φ ′ + φ ′′ ) , (3c)where the prime denotes the derivative with respect to z . These equations are not indepen-dent. The relationship between these equations was discussed in [14].Here we set κ = 1. A thick AdS brane solution in five-dimensional space-time for thepotential V ( φ ) = − δ ) H δ cosh − δ ) (cid:18) φφ (cid:19) (4)was found in Ref. [18]: A ( z ) = − δ ln | cos ¯ z | , (5) φ ( z ) = φ arcsinh(tan ¯ z ) , (6)where φ ≡ p δ ( δ − , (7)¯ z ≡ Hδ z, (8)and the parameter δ satisfies δ > δ <
0. The range of the extra dimension is − z b ≤ z ≤ z b with z b = (cid:12)(cid:12) δπ H (cid:12)(cid:12) . The metric with this choice of A ( z ) has a naked singularity at ± z b .This singularity is very similar to the one in Ref. [11] and the one encountered in the AdSflow to N = 1 super Yang-Mills theory [72]. Gremm supported that this might indicatethat it can be resolved either by lifting the five-dimensional geometry to ten dimensions orby string theory [11]. From (6), it can be seen that the background scalar field divergesat the boundaries of the extra dimension ± z b . In Ref. [11], this divergence can indicatethat the compactification manifold shrinks to zero or becomes infinitely large, so that thefive-dimensional truncation comes to be invalid. There are some examples that singularitiesin five dimensions actually correspond to nonsingular ten-dimensional geometries [73].4s is well-known, the energy-momentum tensor for a scalar field is energetically equivalentto an anisotropic fluid, T MN = − ρ ( g MN + z M z N ) + pz M z N , where z M = e A δ M and ρ ≡ (cid:2) e − A φ ′ + 2 V ( φ ) (cid:3) = − H (1 + δ ) δ cos δ − (¯ z ) , (9) p ≡ (cid:2) e − A φ ′ − V ( φ ) (cid:3) = 6 H cos δ − (¯ z ) . (10)From the above expressions, it can be shown that, for δ < δ >
1, the correspondingenergy-momentum tensor satisfies the weaker energy condition T MN ζ M ζ N ≥ ζ M anarbitrary null vector. For − < δ <
0, the weak energy condition T MN ξ M ξ N ≥ ξ M is an arbitrary future-directed timelike or null vector, and for − < δ < T MN ξ M η N ≥ η M an arbitrary future-directed timelike ornull vector is satisfied. It is clear that, for δ >
1, the energy density ρ has a lump at | z | ≈ z = ± z b , which shows that thethick 3-brane locates at | z | ≈
0. However, for δ <
0, the energy density has no lump at z ≈ ± z b , and it is difficult to regard this configuration as a thick brane,so in the following sections, we only consider the case of δ > III. LOCALIZATION OF GRAVITY ON THE THICK ANTI-DE SITTER BRANE
In this section, the localization of gravity will be investigated by presenting the mass-independent potential of the KK modes for gravitons in the corresponding Schr¨odingerequation.Let us consider the following perturbed metric: ds = ( g MN + δg MN ) dx M dx N . (11)As Ref. [5], we impose the axial gauge constraint δg M = 0, and then write the total metricin the form ds = e A ( z ) [(ˆ g µν + h µν ) dx µ dx ν + dz ] , (12)where h µν denotes the metric perturbation. Then, under the transverse-traceless gauge con-dition h µµ = ∂ µ h µν = 0, the equation for the perturbation h µν takes the following form [13][ ∂ z + 3 A ′ ( z ) ∂ z + (cid:3) + 2 H ] h µν ( x α , z ) = 0 , (13)5here (cid:3) ≡ ˆ g µν ∇ µ ∇ ν and ∇ µ denotes the covariant derivative with respect to the four-dimensional metric ˆ g µν . By making use of the KK decomposition h µν ( x α , z ) = e − A ( z ) ǫ µν ( x α ) ϕ ( z ) (14)with ǫ µν ( x α ) satisfying the transverse-traceless condition, from Eq. (13) we can get thefollowing four-dimensional equation (cid:3) ǫ µν ( x α ) + 2 H ǫ µν ( x α ) = m ǫ µν ( x α ) , (15)where m is the mass of the KK modes. The Schr¨odinger-like equation for the fifth-dimensional sector is also obtained( − ∂ z + V QM ) ϕ ( z ) = m ϕ ( z ) (16)with the potential V QM = 32 A ′′ ( z ) + 94 A ′ ( z )= 3 H δ (cid:2) (2 + 3 δ ) sec ¯ z − δ (cid:3) . (17)The shapes of the potential for the KK modes are plotted in Fig. 1. The potential V QM ( z )has minimum H δ (positive value) at the location of the brane ( z = 0), and tends to positiveinfinity at the boundaries of the fifth dimension ( z = ± z b ), so the potential has no negativevalue at the location of the brane, and it cannot trap the zero mode of gravity, but it can trapthe massive modes. Eq. (16) can be turned into the following equation with E n = m n + H : (cid:20) − ∂ z + 3 H δ (2 + 3 δ ) sec ¯ z (cid:21) ϕ n ( z ) = E n ϕ n ( z ) . (18)When the parameter δ satisfies δ >
1, the solutions of the gravity KK modes are found tobe ϕ n ( z ) ∝ F h − n, n + 3 δ, δ )2 , − sin ¯ z i × cos δ (¯ z ) , (19)where n = 1 , , · · · . Then the mass spectrum of bound states is found to be E n = Hδ (cid:18) n + 32 δ (cid:19) , (20)6r m n = Hδ p n ( n + 3 δ ) . (21)It can be found that the ground state is the massive mode with m = Hδ √ δ . All theKK modes are bound states, and localized on the thick AdS brane. The mass spectrum ofthe KK modes is discrete. The shapes of the gravity KK modes with lower mass are shownin Fig. 2. - - - V QM H z L (a) FIG. 1: The shape of the potential of the gravity KK modes V QM ( z ). The parameters are set to H = 1 and δ = 2. - - - j (a) n = 1 - - - - - j (b) n = 2 - - - - - j (c) n = 3 - - - - - - j (d) n = 4 FIG. 2: The shapes of the KK modes of the graviton ϕ n ( z ). The parameters are set to H = 1 and δ = 2. From the above discussion, m is always positive, and following the arguments given inRef. [13], it can be shown that in the present case all the corresponding perturbation modes7re stable. Now we need to further investigate the stability of this AdS brane, i.e., the scalarperturbation of the systems should be examined. The perturbed metric is given by [13] ds = e A [(1 + 2 α ) dz + (1 + 2 β )ˆ g µν dx µ dx ν ] . (22)As shown in Ref. [13], the following corresponding linearized five-dimensional Einstein-scalarequations can be obtained: δφ = 3 φ ′ ( αA ′ − β ′ ) , α + 2 β = 0 , (23) β ′′ + (cid:3) β − (cid:18) A ′ + 2 φ ′′ φ ′ (cid:19) β ′ + (cid:18) A ′ φ ′′ φ ′ − A ′′ − H (cid:19) β = 0 , (24)where δφ denotes perturbations of the background scalar field. In order to examine thestability of this system, Eq. (24) is transformed into a form of the Schr¨odinger equation:[ − ∂ z + V eff ( z )] ω ( x µ , z ) = (cid:3) ω ( x µ , z ) , (25)where ω ( x µ , z ) and V eff ( z ) are defined as ω ≡ φ ′ ( z ) e A β ( x µ , z ) , (26) V eff ≡ − A ′′ + 94 A ′ + A ′ φ ′′ φ ′ − φ ′′′ φ ′ + 2 (cid:18) φ ′′ φ ′ (cid:19) + 6 H = H δ (cid:2) (15 δ − δ −
4) + 3 δ (3 δ −
2) sec ¯ z (cid:3) . (27)From above expression (27), it is clear that because the parameter δ satisfies δ >
1, theeffective potential V eff ( z → ± z b ) → + ∞ , and has minimum H δ (3 δ − δ + 1) (positivevalue) at z = 0.We decompose ω ( x µ , z ) in Eq. (25) as ω ( x µ , z ) = f ( z ) X ( x µ ) , (28)and the following equations can be gotten (cid:3) X ( x µ ) = m X ( x µ ) , (29) − f ′′ ( z ) + V eff ( z ) f ( z ) = m f ( z ) (30)8ith m the four-dimensional mass. It is known that Eq. (29) can be solved with suitableharmonic functions. There is the Breitenlohner-Freedman bound which allows the tachyonicmass to some extent from the condition of the normalization [13, 74]. The mass m is boundedas m ≥ − , (31)which means that even when there are solutions with − ≤ m <
0, such solutions arestable in spite of the tachyonic mass.By substituting the effective potential (27) into Eq. (30), we can get (cid:20) − ∂ z + 3 H δ (3 δ −
2) sec ¯ z (cid:21) f n ( z ) = E n f n ( z ) , (32)with E n = m n − H δ (15 δ − δ − δ >
1, the solutions of the above Schr¨odinger Eq.(32) can be expressed as f n ( z ) ∝ cos δ (¯ z ) F (cid:20) − n, n + 3 δ, δ , − sin ¯ z (cid:21) , (33)where n = 0 , , , · · · . It is known that the corresponding perturbation modes tend to zeroat the boundaries of the extra dimension, i.e., they are bound states, and all the scalarmodes of the metric fluctuations can be localized on the AdS brane. The mass spectrum ofthe bound states is found to be E n = Hδ (cid:18) n + 3 δ (cid:19) , (34)or m n = Hδ p n + 6 δ + (3 n − δ − . (35)It is clear that the mass of the ground state is m = Hδ √ δ − δ −
1, and m n ≥ H δ [ n +6 δ + (3 n − δ − >
0, so it is shown that the thick AdS brane is stable under the scalarperturbations.
IV. LOCALIZATION OF VARIOUS MATTERS ON THE THICK ANTI-DE SIT-TER BRANE
In this section, we will investigate the character of the localization of the various bulkmatter fields on the thick AdS brane. Spin-0 scalars, spin-1 vectors and spin-1/2 fermions9ill be considered by means of the gravitational interaction. Certainly, we have implicitlyassumed that the various bulk matter fields considered below make little contribution tothe bulk energy so that the solutions given in the previous section remain valid even in thepresence of bulk matter fields. The mass spectra of the various matter fields on the thickbrane will also be discussed by presenting the potential of the corresponding Schr¨odingerequation for the KK modes of the various matter fields.
A. Spin-0 scalar fields
We first consider the localization of real scalar fields on the thick brane obtained in theprevious section, then turn to vectors and fermions in the next subsections. Let us start byconsidering the action of a massless real scalar coupled to gravity: S = − Z d x √− g g MN ∂ M Φ ∂ N Φ . (36)Using the metric (2), the equation of motion derived from (36) reads as1 √− ˆ g ∂ µ ( p − ˆ g ˆ g µν ∂ ν Φ) + e − A ∂ z (cid:0) e A ∂ z Φ (cid:1) = 0 . (37)Then, by using of the KK decompositionΦ( x, z ) = X n φ n ( x ) χ n ( z ) e − A/ , (38)the four-dimensional scalar fields φ n ( x ) should satisfy the four-dimensional massive Klein-Gordon equation: (cid:18) √− ˆ g ∂ µ ( p − ˆ g ˆ g µν ∂ ν ) − m n (cid:19) φ n ( x ) = 0 . (39)Therefore the equation for the scalar KK modes χ n ( z ) can be expressed as (cid:2) − ∂ z + V ( z ) (cid:3) χ n ( z ) = m n χ n ( z ) , (40)which is a Schr¨odinger equation with the effective potential given by V ( z ) = 32 A ′′ + 94 A ′ , (41)where m n is the mass of the KK excitation. It is clear that the potential V ( z ) defined in(41) is a four-dimensional mass-independent potential.10he full five-dimensional action (36) can be reduced to one four-dimensional action for amassless scalar field plus an infinite sum of massive scalar actions in four-dimension S = − X n Z d x p − ˆ g (cid:18) ˆ g µν ∂ µ φ n ∂ ν φ n + m n φ n (cid:19) (42)when integrated over the extra dimension, in which it is required that Eq. (40) is satisfiedand the following orthonormality conditions are obeyed: Z + z b − z b χ m ( z ) χ n ( z ) dz = δ mn . (43)For the thick AdS brane, because the effective potential for the scalar KK modes (41)is the same as the potential for the gravity (17), the localization of scalars is the same asthe situation of gravity. Hence, the four-dimensional massless scalar (the zero mode) is notlocalized on the thick AdS brane for δ >
1. And all the KK modes are massive bound statesand are localized on the brane. The mass spectrum of the KK modes is discrete.
B. Spin-1 vector fields
We now turn to spin-1 vector fields. We begin with the five-dimensional action of a vectorfield S = − Z d x √− g g MN g RS F MR F NS , (44)where F MN = ∂ M A N − ∂ N A M is the field tensor as usual. From this action the equations ofmotion are derived as follows 1 √− g ∂ M ( √− gg MN g RS F NS ) = 0 . (45)By using the background metric (2), the equations of motion read as1 √− ˆ g ∂ ν ( p − ˆ g ˆ g νρ ˆ g µλ F ρλ ) + ˆ g µλ e − A ∂ z (cid:0) e A F λ (cid:1) = 0 , (46) ∂ µ ( p − ˆ g ˆ g µν F ν ) = 0 . (47)Because the fourth component A has no zero mode in the effective four-dimensional theory,we assume that it is Z -odd with respect to the extra dimension z . Furthermore, in orderto be consistent with the gauge invariant equation H dzA = 0, we choose A = 0 by usinggauge freedom. Then, the action (44) can be reduced to S = − Z d x √− g h F µν F µν + 2 e − A g µν ∂ z A µ ∂ z A ν i . (48)11ith the decomposition of the vector field A µ ( x, z ) = X n a ( n ) µ ( x ) ρ n ( z ) e − A/ (49)and the orthonormality condition Z + z b − z b ρ m ( z ) ρ n ( z ) dz = δ mn , (50)the action (48) can be simplified as S = X n Z d x p − ˆ g (cid:20) −
14 ˆ g µα ˆ g νβ f ( n ) µν f ( n ) αβ − m n ˆ g µν a ( n ) µ a ( n ) ν (cid:21) , (51)where f ( n ) µν = ∂ µ a ( n ) ν − ∂ ν a ( n ) µ is the four-dimensional field strength tensor. Therefore wehave obtained a four-dimensional theory of a gauge particle (massless) and infinite towers ofmassive vector fields. In the above process, it has been required that the vector KK modes ρ n ( z ) should satisfy the following Schr¨odinger equation (cid:2) − ∂ z + V ( z ) (cid:3) ρ n ( z ) = m n ρ n ( z ) , (52)with the mass-independent potential V ( z ) given by V ( z ) = H δ (cid:2) (2 + δ ) sec ¯ z − δ (cid:3) . (53)The potential also has a minimum H δ at z = 0 and tends to positive infinity at z = z b for δ >
1. The shapes of the potential V ( z ) are plotted in Fig. 3. The potential V ( z ) arealways positive, so the zero mode is not localized on the AdS brane. However, the massivemodes can be localized on the brane. Eq. (52) with this potential can be turned into thefollowing Schr¨odinger equation: (cid:20) − ∂ z + H δ (2 + δ ) sec ¯ z (cid:21) ρ n ( z ) = E n ρ n ( z ) (54)with E n = m n + H . The solutions of Eq. (54) for δ > ρ n ∝ cos δ (¯ z ) F (cid:20) − n, n + δ, δ , − sin ¯ z (cid:21) , (55)where n = 1 , , · · · , and the discrete mass spectrum can be written as m n = Hδ p n ( n + δ ) , (56)12o all the KK modes are bound states and can also be trapped on the brane. We can findthat the ground state is a massive state with m = Hδ √ δ . The shapes of the KK modesare plotted in Fig. 4. - - - V H z L FIG. 3: The shape of the potential of the vector field V ( z ). The parameters are set to H = 1 and δ = 2. - - - Ρ (a) n = 1 - - - - - Ρ (b) n = 2 - - - - - Ρ (c) n = 3 - - - - - Ρ (d) n = 4 FIG. 4: The shapes of the KK modes of the vector fields ρ n ( z ). The parameters are set to H = 1,and δ = 2. C. Spin-1/2 fermion fields
Finally, we will study the localization of fermions on the thick AdS brane. In five-dimensional space-time, fermions are four component spinors and their Dirac structure canbe described by the curved space gamma matrices Γ M = e − A ( γ µ , γ ), where γ µ and γ arethe usual flat gamma matrices in the four-dimensional Dirac representation. The Diracaction of a massless spin-1/2 fermion coupled to the background scalar φ (6) is S / = Z d x √− g h ¯ΨΓ M ( ∂ M + ω M )Ψ − η ¯Ψ F ( φ )Ψ i , (57)13here η is a coupling constant. The nonvanishing components of the spin connection ω M for the background metric (2) are ω µ = 12 A ′ γ µ γ + ˆ ω µ , (58)with ˆ ω µ the spin connection derived from the metric ˆ g µν ( x ). Then the equation of motionis given by h γ µ ( ∂ µ + ˆ ω µ ) + γ ( ∂ z + 2 A ′ ) − η e A F ( φ ) i Ψ = 0 , (59)where γ µ ( ∂ µ + ˆ ω µ ) is the Dirac operator on the AdS brane.Now we would like to investigate the localization of the Dirac spinor on the AdS braneby studying the above five-dimensional Dirac equation. Because of the Dirac structure ofthe fifth gamma matrix γ , we expect that the left- and right-chiral projections of the four-dimensional part have different behaviors. From the equation of motion (59), we will searchfor the solutions of the general chiral decompositionΨ = X n h ψ L,n ( x ) L n ( z ) + ψ R,n ( x ) R n ( z ) i e − A , (60)where ψ L = − γ ψ and ψ R = γ ψ are the left- and right-chiral components of a four-dimensional Dirac field ψ , respectively. By demanding ψ L,R satisfy the four-dimensionalmassive Dirac equations γ µ ( ∂ µ + ˆ ω µ ) ψ L,R = mψ R,L , we obtain the following coupled equationsfor the fermion KK modes L n and R n : (cid:2) ∂ z + η e A F ( φ ) (cid:3) L n ( z ) = mR n ( z ) , (61a) (cid:2) ∂ z − η e A F ( φ ) (cid:3) R n ( z ) = − mL n ( z ) . (61b)From the above equations, we can obtain the Schr¨odinger-like equations for the KK modesof the left- and right-chiral fermions: (cid:0) − ∂ z + V L ( z ) (cid:1) L n = m L n , (62a) (cid:0) − ∂ z + V R ( z ) (cid:1) R n = m R n . (62b)where the effective potentials of Eq. (62) are given by V L ( z ) = (cid:0) η e A F ( φ ) (cid:1) − ∂ z (cid:0) η e A F ( φ ) (cid:1) , (63a) V R ( z ) = V L ( z ) | η →− η . (63b)14or the purpose of getting the standard four-dimensional action for a massless fermionand an infinite sum of the massive fermions, S = Z d x √− g ¯Ψ h Γ M ( ∂ M + ω M ) − ηF ( φ ) i Ψ= X n Z d x p − ˆ g ¯ ψ n h γ µ ( ∂ µ + ˆ ω µ ) − m n i ψ n , (64)we need the following orthonormality conditions for L n and R n : Z z b − z b L m L n dz = δ mn , (65) Z z b − z b R m R n dz = δ mn , (66) Z z b − z b L m R n dz = 0 . (67)From Eqs. (62)and (63), we can see that, in order to localize the left- and right-chiralfermions, there must be some kind of scalar-fermion coupling. This situation is similarto the one in Refs. [35, 48], in which the authors introduced the mass term mǫ ( z ) ¯ΨΨfor the localization of the fermion fields on a brane. For the thick branes arising from areal scalar field φ , the scalar-fermion coupling such as η ¯Ψ φ Ψ, η ¯Ψ φ k Ψ, η ¯Ψ tan /s ( φ )Ψ, etc.,can be introduced for fermion localization [22, 50, 53, 60]. For the thick brane generatedby two scalar φ and χ , in order to localize the fermions, the coupling terms η ¯Ψ φχ Ψ and η ¯Ψ φ Ψ + η ′ ¯Ψ χ Ψ were introduced in Ref. [70]. Moreover, if we demand that V L ( z ) and V R ( z )are Z -even with respect to the extra dimension z , F ( φ ) should be an odd function of the“kink” φ ( z ). In this paper, we choose the simplest Yukawa coupling: F ( φ ) = φ . Then theexplicit forms of the potentials (63) are V L ( z ) = η φ cos − δ (¯ z )arcsinh (tan ¯ z ) − ηHφ δ cos δ (¯ z ) h δ sin ¯ z arcsinh(tan ¯ z ) + 1 i , (68) V R ( z ) = V L ( z ) | η →− η . (69)At the location of the AdS brane, both of the potentials for the left- and right-chiral fermionshave minimum V L ( z = 0) = − ηHφ /δ and V R ( z = 0) = ηHφ /δ , and the potentials V L,R ( z → ± z b ) → + ∞ as shown in Fig. 5.From Fig. 5, only the potential of the left-chiral fermions has negative value at the locationof the thick brane for the positive coupling constant η , so only the left-chiral fermion zero15 - - V L H z L (a) V L ( z ) - - - V R H z L (b) V R ( z ) FIG. 5: The shapes of the potentials of the left- and right-chiral fermion KK modes. The parametersare set to δ = 2, H = 1 and η = 1. mode can be localized on the brane. By setting m = 0 in Eq. (61), the left-chiral fermionzero mode can be obtained: L ( z ) ∝ exp[ − ηI ( z )] , (70)where the exponential factor I ( z ) can be expressed as I ( z ) = Z z dz ′ e A ( z ′ ) φ ( z ′ )= 2 − δ φ δH ( δ − (cid:26) F (cid:16) − δ, − δ , − δ − δ , − δ − (cid:17) − ̺ − δ ( z ) (cid:20) F (cid:16) − δ, − δ , − δ − δ , − δ − ̺ ( z ) (cid:17) + ( δ − F (cid:16) − δ , − δ, − δ , − ̺ ( z ) (cid:17) ln (cid:16) ̺ ( z ) (cid:17)(cid:21)(cid:27) with ̺ ( z ) = sec ¯ z + tan ¯ z . The curve for the exponential factor is plotted in Fig. 6(a). Theexponential factor I ( z ) tends to positive infinity, so the left-chiral fermion zero mode L → z → ± z b , and is localized on the AdS brane, which is shown in Fig. 6(b).For the massive KK modes of the left- and right-chiral fermions, Eqs. (62a) and (62b)cannot be analytically solved. But we can solve them by numerical method. The KK modesof the left- and right-chiral fermions are plotted in Figs. 7 and 8, respectively. All theKK modes for both the left- and right-chiral fermions are bound states. The discrete mass16 - - H z L (a) I ( z ) - - - L H z L (b) L FIG. 6: The shape of the exponential factor I ( z ) and the left-chiral fermion zero mode L . Theparameters are set to H = 1, η = 1 and δ = 2. spectra for the KK modes are calculated as follows: m L n = { , . , . , . , . , . , . , . , . , . , . , · · · } , (71a) m R n = { . , . , . , . , . , . , . , . , . , . , · · · } , (71b)where the parameters are set to H = 1, η = 1 and δ = 2. Hence the ground state ofthe left-chiral fermions is the massless mode; however, the ground state of the right-chiralfermions is a massive mode. The mass spectra are also shown in Fig. 9. So we can obtainthe four-dimensional massless left-chiral fermion and the massive Dirac fermions consistingof the pairs of the left- and right-chiral KK modes coupling together through mass terms. - - z L (a) n = 0 - - z - - - L (b) n = 1 - - z - - L (c) n = 2 - - z - - - L (d) n = 3 FIG. 7: The shapes of the KK modes of the left-chiral fermions. The parameters are set to H = 1, η = 1 and δ = 2. - z R (a) n = 1 - - z - - - R (b) n = 2 - - z - - - R (c) n = 3 - - z - - - R (d) n = 4 FIG. 8: The shapes of the KK modes of the right-chiral fermions. The parameters are set to H = 1, η = 1 and δ = 2. Left fermions Right fermions
FIG. 9: The m L n ,R n spectra of the left- and right-chiral fermions. The parameters are set to H = 1, η = 1 and δ = 2. V. CONCLUSION AND DISCUSSION
In this paper, we first reviewed a thick AdS brane embedded in five-dimensional space-time, in which the behavior of the warp factor is related to a parameter δ . For δ <
0, thewarp factor tends to zero at the boundaries of the extra dimension, while the energy densityhas no lump at z ≈
0, which cannot be considered as a thick brane. For δ >
1, the warpfactor tends to infinity at the boundaries, and the energy density has a lump at z ≈
0, whichindicates that the AdS thick brane locates at z ≈
0. Hence, we only consider the case of δ >
1. Then we studied the mass-independent potentials of the KK modes for gravity andvarious spin fields in the corresponding Schr¨odinger equations. In this way, the localizationand mass spectra of gravity and various matters with spin-0, 1 and 1/2 on this kind of AdSbrane were investigated.For gravity, the potential of the KK modes in the corresponding Schr¨odinger equation isdivergent when far away from the brane. Such potential suggests that the mass spectrum of18he gravity KK modes consists of an infinite number of discrete bound states. Although, thezero mode of gravity is not localized on the AdS brane, the massive modes can be localizedon the brane. The scalar perturbations of the thick AdS brane has been analyzed, and as aresult, it is shown that the brane is stable under the scalar perturbations.For spin-0 scalars and spin-1 vectors, the character of localization is similar to the caseof gravity; i.e., all the scalar and vector KK modes are bound states, and the zero modesare not localized on the brane.For spin-1/2 fermions, in order to localize the left- and right-chiral fermions, we introducedthe usual Yukawa coupling η ¯Ψ φ Ψ with a positive coupling constant η . Both potentialshave the same asymptotic behavior: V L,R ( z → ± z b ) → + ∞ , and only the potential ofthe left-chiral fermion KK modes has a finite negative well at z = 0. So only the left-chiralfermion zero mode could be localized on the brane; i.e., there exists only the four-dimensionalmassless left-chiral fermion. And both the left- and right-chiral fermion KK modes have aninfinite number of bound states. Since a pair of left- and right-chiral KK modes coupletogether through a mass term to become a four-dimensional Dirac fermion, a series of four-dimensional Dirac fermions with a discrete mass spectrum could be obtained on the AdSbranes. Acknowledgement
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