Bulk Matters on Symmetric and Asymmetric de Sitter Thick Branes
aa r X i v : . [ h e p - t h ] F e b Preprint typeset in JHEP style - HYPER VERSION
Bulk Matters on Symmetric and Asymmetric de SitterThick Branes
Yu-Xiao Liu ∗ , Zhen-Hua Zhao , , Shao-Wen Wei , Yi-Shi Duan Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, P. R. China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, P. R.ChinaE-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
An asymmetric thick domain wall solution with de Sitter ( dS ) expansion infive dimensions can be constructed from a symmetric one by using a same scalar (kink)with different potentials. In this paper, by presenting the mass-independent potentialsof Kaluza–Klein (KK) modes in the corresponding Schr¨odinger equations, we investigatethe localization and mass spectra of various bulk matter fields on the symmetric andasymmetric dS thick branes. For spin 0 scalars and spin 1 vectors, the potentials of KKmodes in the corresponding Schr¨odinger equations are the modified P¨oschl-Teller potentials,and there exist a mass gap and a series of continuous spectrum. It is shown that thespectrum of scalar KK modes on the symmetric dS brane contains only one bound mode(the massless mode). However, for the asymmetric dS brane with a large asymmetricfactor, there are two bound scalar KK modes: a zero mode and a massive mode. For spin1 vectors, the spectra of KK modes on both dS branes consist of a bound massless modeand a set of continuous ones, i.e., the asymmetric factor does not change the number ofthe bound vector KK modes. For spin 1/2 fermions, two types of kink-fermion couplingsare investigated in detail. For the usual Yukawa coupling η ¯Ψ φ Ψ, there exists no massgap but a continuous gapless spectrum of KK states. For the scalar-fermion coupling η ¯Ψ sin( φφ ) cos − δ ( φφ )Ψ with a positive coupling constant η , there exist some discrete boundKK modes and a series of continuous ones. The total number of bound states increaseswith the coupling constant η . For the case of the symmetric dS brane and positive η , thereare N L ( N L ≥
1) left chiral fermion bound states (including zero mode and massive KKmodes) and N L − dS brane scenario, the asymmetric factor a reduces the number of thebound fermion KK modes. For large enough a , there would not be any right chiral fermionbound mode, but at least one left chiral fermion zero mode. Keywords:
Extra Dimensions, Brane world. ∗ Corresponding author. ontents
1. Introduction 12. Review of the symmetric and asymmetric thick branes 23. Localization and mass gaps of various matters on the thick branes 4
4. Conclusion and discussion 20
1. Introduction
The idea of embedding our universe in a higher dimensional space has received a great ofrenewed attention. The suggestion that extra dimensions may not be compact [1, 2, 3, 4, 5]or large [6, 7] can provide new insights for solving gauge hierarchy problem [7], i.e., the largedifference in magnitude between the Planck and electroweak scales, and the long-standingcosmological constant problem [1, 3, 8]. According to the brane scenarios, gravity is freeto propagate in all dimensions, while all the matter fields (electromagnetic, Yang-Millsetc.) are confined to a 3–brane in a high-dimensional space. In Ref. [4], an alternativescenario of the compactification had been proposed. In this scenario, the internal manifolddoes not need to be compactified to the Planck scale any more, it can be large, or eveninfinite non-compact, which is one of reasons why this new compactification scenario hasattracted so much attention. Among all of the brane world models, there is an interestingand important model in which extra dimensions comprise a compact hyperbolic manifold[9]. The model is known to be free of usual problems that plague the original ADD modelsand share many common features with Randall-Sundrum (RS) models.Recently, an increasing interest has been focused on the study of thick brane scenarioin higher dimensional space-time [10, 11, 12, 13, 14, 15, 16], since in more realistic modelsthe thickness of the brane should be taken into account. A virtue of these models is that thebranes can be obtained naturally rather than introduced by hand. In this scenario the scalarfield configuration is usually a kink, which provides a thick brane realization of the braneworld as a domain wall in the bulk. However, the inclusion of the gravitational evolutioninto a dynamic thick wall is a highly non-trivial problem because of the non-linearity of theEinstein equations. For this reason, there are not so many analytic solutions of a dynamic– 1 –hick domain wall. The symmetric de Sitter ( dS ) branes have been studied in five andhigher dimensional spacetimes, for examples in [14, 17, 18]. Ref. [19] presented a methodto construct asymmetric thick dS brane solutions from known ones, where the spacetimesassociated to them are physically different. With the method, asymmetric brane worldswith dS expansion were obtained. These branes interpolate between two spacetimes withdifferent cosmological constants, and the vacua correspond to dS and AdS geometry. Itwas shown that gravity is localized on such branes.In brane world scenarios, an important and complex question is localization of variousbulk fields on a brane by a natural mechanism. It is well known that massless scalar fields[20] and graviton [4] can be localized on branes of different types. However, spin 1 Abelianvector fields can not be localized on the RS brane in five dimensions, but can be localized onthe RS brane in some higher-dimensional cases [21] or on the thick dS brane and Weyl thickbrane [22]. The localization problem of spin 1/2 fermions on thick branes is interestingand important. Fermions do not have normalizable zero modes in five and six dimensionswithout the scalar-fermion coupling [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. InRef. [30], the authors obtained trapped discrete massive fermion states on the brane, whichin fact are quasi-bound and have a finite probability of escaping into the bulk. In fact,fermions can escape into the bulk by tunnelling, and the rate depends on the parameters ofthe scalar potential [31]. In five dimensions, with the scalar–fermion coupling, there mayexist a single bound state and a continuous gapless spectrum of massive fermion Kaluza–Klein (KK) states [22, 33]. While in some other brane models, there exist finite discreteKK states (mass gap) and a continuous gapless spectrum starting at a positive m [34, 35].Since a physically different asymmetric thick dS brane solution can be constructed froma known symmetric one by including an asymmetric factor, we will address the localizationand mass spectrum problems of various bulk matters on the symmetric and asymmetric dS branes, and investigate the influence of the asymmetric factor on the mass spectra of bulkmatters in this paper. We will show that all bulk matters (scalars, vectors and fermions)can be localized on these branes and the corresponding mass spectra have a mass gap (forspin 1/2 fermions the scalar-fermion coupling should not be the usual Yukawa coupling η ¯Ψ φ Ψ in order to trap the zero mode). The large asymmetric factor increases the numberof the scalar bound states but reduces that of the fermion ones, and does not change thenumber of the vector bound states.The organization of the paper is as follows: In section 2, we first review the symmetricand asymmetric dS thick branes in 5-dimensional space-time. Then, in section 3, we studythe localization and mass spectra of various bulk fields on the symmetric and asymmetricthick branes by presenting the shapes of the potentials of the corresponding Schr¨odingerproblem. For spin 1/2 fermions, we consider two different types of scalar-fermion interac-tions. Finally, the conclusion and summary are given.
2. Review of the symmetric and asymmetric thick branes
Let us consider thick branes arising from a real scalar field φ with a scalar potential V ( φ ).– 2 –he action for such a system is given by S = Z d x √− g (cid:20) κ R − g MN ∂ M φ∂ N φ − V ( φ ) (cid:21) , (2.1)where R is the scalar curvature and κ = 8 πG with G the 5-dimensional Newton constant.Here we set κ = 1. The line-element for a 5-dimensional spacetime with planar-paralellsymmetry is assumed as ds = e A ( z ) (cid:0) ˆ g µν ( x ) dx µ dx ν + dz (cid:1) = e A ( z ) (cid:0) − dt + e βt dx i dx i + dz (cid:1) , (2.2)where e A ( z ) is the warp factor and z stands for the extra coordinate. For the positiveconstant β >
0, we will have dynamic solutions. The scalar field is considered to be afunction of z only, i.e., φ = φ ( z ). In the model, the potential could provide a realizationof a thick brane, and the soliton configuration of the scalar field dynamically generate thedomain wall configuration with warped geometry. The field equations generated from theaction (2.1) with the ansatz (2.2) reduce to the following coupled nonlinear differentialequations φ ′ = 3( A ′ − A ′′ − β ) , (2.3) V ( φ ) = 32 e − A ( − A ′ − A ′′ + 3 β ) , (2.4) dV ( φ ) dφ = e − A (3 A ′ φ ′ + φ ′′ ) , (2.5)where the prime denotes derivative with respect to z . For positive and vanishing β we willobtain dynamic and static solutions, respectively.A symmetric thick domain wall with dS expansion in five dimensions for the potential V ( φ ) = 1 + 3 δ δ β (cid:18) cos φφ (cid:19) − δ ) , (2.6)was found in Refs. [36, 37]: e A = cosh − δ (cid:18) βzδ (cid:19) , (2.7) φ = φ arctan (cid:18) sinh βzδ (cid:19) , (2.8)where φ = p δ (1 − δ ), 0 < δ < β >
0. In this system, The scalar field takesvalues ± φ π/ z → ±∞ , corresponding to two consecutive minima of the potential withcosmological constant Λ = 0. The scalar configuration in fact is a kink, which provides athick brane realization of the brane world as a domain wall in the bulk. δ plays the roleof the wall’s thickness. The thick brane has a well-defined distributional thin wall limitwhen δ → / < δ <
1, the hypersurfaces | z | = ∞ represent non-scalar spacetime singularities [14].– 3 –n asymmetric thick domain wall solution with dS expansion in five dimensions forthe same kink configuration φ in (2.8) was found in Ref. [19]: e − A = cosh δ (cid:18) βzδ (cid:19) + iaδβ − βδ cosh − δ (cid:18) βzδ (cid:19) coth (cid:18) βzδ (cid:19) × (cid:12)(cid:12)(cid:12)(cid:12) sinh (cid:18) βzδ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:18) − δ, , − δ, cosh (cid:18) βzδ (cid:19)(cid:19) , (2.9)where F is the hypergeometric function. Here we will consider the case δ = 1 / e A = β sech2 βz [ β + a arctan(tanh βz )] , (2.10) φ = √
32 arctan (sinh 2 βz ) , (2.11) V ( φ ) = 32 (cid:12)(cid:12)(cid:12)(cid:12) cos 2 φ √ (cid:12)(cid:12)(cid:12)(cid:12) (cid:26) − a + 5 β − aβ tan 2 φ √ a arctan (cid:20) tanh (cid:18)
12 arcsinh (cid:18) tan 2 φ √ (cid:19)(cid:19)(cid:21) +2 a arctan (cid:20) tanh (cid:18)
12 arcsinh (cid:18) tan 2 φ √ (cid:19)(cid:19)(cid:21) (cid:18) β − a tan 2 φ √ (cid:19) (cid:27) , (2.12)where | a | < β/π in order to prevent singularities in the metric tensor. The parameter a decides the asymmetry of the solution. For a = 0, we recover the symmetric domain wallsolution. For positive (negative) a , the spacetime for z → + ∞ is asymptotically AdS ( dS )with cosmological constant − a (4 β + aπ ) and for z → −∞ is asymptotically dS ( AdS )with cosmological constant 3 a (4 β − aπ ). The scalar curvature R and the energy density ρ for the dS brane are calculated as follows: R = 4sech2 βz (cid:2) β − a + 14 aβ arctan(tanh βz )+7 a arctan (tanh βz ) − a ( β + a arctan(tanh βz )) sinh 2 βz (cid:3) , (2.13) ρ = sech2 βz (cid:2) β − a + 6 aβ arctan(tanh βz )+3 a arctan (tanh βz ) − a ( β + a arctan(tanh βz )) sinh 2 βz (cid:3) . (2.14)The shapes for the metric factor e A , the potential V ( φ ), the scalar curvature R , and thedensity energy ρ are shown in Fig. 1.
3. Localization and mass gaps of various matters on the thick branes
In this section let us investigate whether various bulk mater fields such as spin 0 scalars,spin 1 vectors and spin 1/2 fermions can be localized on the thick branes by means ofonly the gravitational interaction. Of course, we have implicitly assumed that various bulkfields considered below make little contribution to the bulk energy so that the solutionsgiven in previous section remain valid even in the presence of bulk fields. We will analyzethe spectra of various mater fields for the thick brane by presenting the potential of the– 4 – ã A -1 -0.5 0 0.5 1-150-100-50050 Φ V H Φ L -1 -0.5 0 0.5 1-600-400-2000200 zR -1 -0.5 0 0.5 1-150-100-50050 z Ρ Figure 1:
The shapes of the metric factor e A , scalar potential V ( φ ), the scalar curvature R , andthe energy density ρ for the dS branes with β = 3. The parameter a is set to a = 0 for dashedlines, a = 2 for thin lines, and a = 3 for thick lines. corresponding Schr¨odinger equation. It can be seen from the following calculations that themass-independent potential can be obtained conveniently with the conformally flat metric(2.2). We first study localization of a real scalar field on the branes obtained in previous section.Let us start by considering the action of a massless real scalar coupled to gravity S = − Z d x √− g g MN ∂ M Φ ∂ N Φ . (3.1)By considering the conformally flat metric (2.2) the equation of motion derived from (3.1)is read 1 √− ˆ g ∂ µ ( p − ˆ g ˆ g µν ∂ ν Φ) + e − A ∂ z (cid:0) e A ∂ z Φ (cid:1) = 0 . (3.2)Then, by decomposing Φ( x, z ) = P n φ n ( x ) χ n ( z ) e A/ and demanding φ n ( x ) satisfiesthe 4-dimensional massive Klein–Gordon equation (cid:16) √− ˆ g ∂ µ ( √− ˆ g ˆ g µν ∂ ν ) − µ n (cid:17) φ n ( x ) = 0,we obtain the equation for χ n ( z ): (cid:2) − ∂ z + V ( z ) (cid:3) χ n ( z ) = µ n χ n ( z ) , (3.3)– 5 –hich is a Schr¨odinger equation with the effective potential given by V ( z ) = 32 A ′′ + 94 A ′ , (3.4)where µ n is the mass of the KK excitations. It is clear that V ( z ) defined in (3.4) is amass-independent potential.The full 5-dimensional action (3.1) reduces to the standard 4-dimensional action forthe massive scalars S = − X n Z d x p − ˆ g (cid:18) ˆ g µν ∂ µ φ n ∂ ν φ n + µ n φ n (cid:19) , (3.5)when integrated over the extra dimension, in which it is required that Eq. (3.3) is satisfiedand the following orthonormality condition is obeyed: Z ∞−∞ dz χ m ( z ) χ n ( z ) = δ mn . (3.6)For the symmetric and asymmetric dS brane world solutions (2.7) and (2.10), thepotentials corresponding to (3.4) are V S ( z ) = 3 β δ (cid:0) δ − (2 + 3 δ )sech ( βz/δ ) (cid:1) (3.7)and V A ( z ) = 9 β aβ sech2 βz tanh 2 βz β + a arctan tanh βz ) − β (cid:0) β − a + 7 a arctan tanh βz (2 β + a arctan tanh βz ) (cid:1) β + a arctan tanh βz ) cosh (2 βz ) , ( δ = 12 ) (3.8)respectively. For the case a = 0, the potential (3.8) is reduced to (3.7) with δ = 1 / V S ( z ) = 34 β (cid:0) − (2 βz ) (cid:1) . ( δ = 12 ) (3.9)We first investigate the potential (3.7) for the symmetric dS brane. It has a minimum(negative value) − β δ at z = 0 and a maximum (positive value) β at z = ±∞ . Let p = β/δ and q = 1 + 3 δ/
2, Eq. (3.3) with the potential (3.7) turns into the well-knownSchr¨odinger equation with E n = µ n − δ p : h − ∂ z − q ( q − p sech ( pz ) i χ n = E n χ n . (3.10)For this equation with a modified P¨oschl-Teller potential, the energy spectrum of boundstates is found to be E n = − p ( q − − n ) (3.11)or µ n = n (3 δ − n ) β δ , (3.12)– 6 – Figure 2:
The shapes of the potential V S ( z ) for the symmetric dS brane. The parameters are setto β = 3, δ = 1 / δ = 2 / δ → where n is an integer and satisfies 0 ≤ n < δ . It is clear that the energy for n = 0 or µ = 0 always belongs to the spectrum of the potential (3.7) for δ >
0. For 0 < δ ≤ , thereis only one bound state, i.e., the ground state χ ( z ) = s β Γ( + δ ) δ √ π Γ( δ ) sech δ/ ( βz/δ ) (3.13)with µ = 0, which is just the normalized zero-mass mode and also shows that there is notachyonic scalar mode. The continuous spectrum starts with µ = β and asymptoticallyturn into plane waves, which represent delocalized KK massive scalars. For < δ < χ ( z ) ∝ sech δ/ ( βz/δ ) sinh z (3.14)with mass µ = (3 δ − β /δ . The continuous spectrum also start with µ = β . Fromabove analysis, we come to the conclusion: for 0 < δ ≤ , there is only one bound state (ismassless mode) for the symmetric potential (3.7).Next we turn to the potential (3.8) for asymmetric dS brane. It has a negative valueat some z ( z < a > z > a <
0) and the asymptotic behavior: V A ( z = ±∞ ) = β , which implies that there is also a mass gap. For the massless mode χ ( z ) with µ = 0, the Schr¨odinger equation (3.3) with the potential (3.8) can be solvedanalytically, and the normalizable eigenfunction is found to be χ ( z ) ∝ (cid:18) β sech2 βz [ β + a arctan(tanh βz )] (cid:19) / . (3.15)This zero mode is the ground state since it has no node. For the limit a →
0, the masslessmode (3.15) is reduced to (3.13) but with δ = 1 /
2. Now, we ask an interesting question:are there other bound states except the zero mode for the asymmetric potential (3.8)?This is very important for producing 4-dimensional massive scalars. If the answer is yes,we will get massive scalars on the asymmetric dS brane. We have known that there– 7 – , Χ n , Μ n2 (a) δ = 1 / -1.5 -1 -0.5 0 0.5 1 1.5-30-20-1001020 zV , Χ n , Μ n2 (b) δ = 0 . Figure 3:
The shapes of the potential V S ( z ) (thick lines), KK modes χ n ( z ) (dashed lines for χ ( z )and thin lines for χ ( z )) and the mass spectrum (thick gray lines) for symmetric dS brane with β = 3, δ = 1 / δ = 0 . -2 -1 0 1-30-20-10010203040 zV Figure 4:
The shapes of the potential V A ( z ) for the asymmetric dS brane. The parameters areset to β = 3, δ = 1 / a = 0 for dashed line, and a = 2, 3, 3.78, 3.818 for solid lines with thicknessincreases with a . is no any massive bound state for the symmetric potential (3.9), the limit case of thecurrent asymmetric one. Hence we can extrapolate that the answer should be no for smallasymmetric factor a . However, what will happen for large a ? We note that the asymmetricpotential (3.8) has the same asymptotic behavior as the symmetric case: V A ( ±∞ ) = β ,but a different minimum V A min , which is larger than that of the symmetric potential. Theabsolute value of the minimum of the asymmetric potential decreases with the increase ofthe asymmetry (see Fig. 4). This leads to the increase of the relative depth of the potentialwell V A ( ±∞ ) / | V A min | , which indicates that the potential well may trap more bound stats.By numerical method, we do get a massive bound state with µ = 18 .
11 at a = 3 .
818 (seeFig. 5). – 8 – , Χ n , Μ n2 (a) a = 2 -3 -2 -1 0 1 2-20-10010203040 zV , Χ n , Μ n2 (b) a = 3 . Figure 5:
The shapes of the potential V A ( z ) (thick lines), KK modes χ n (dashed lines and thinlines) and the mass spectrum µ n (thick gray lines) for dS brane with β = 3, δ = 1 /
2, and a = 2,3.818. Next we turn to spin 1 vector fields. We begin with the 5D action of a vector field S = − Z d x √− gg MN g RS F MR F NS , (3.16)where F MN = ∂ M A N − ∂ N A M as usual. From this action and the background geometry(2.2), the equations of motion √− g ∂ M ( √− gg MN g RS F NS ) = 0 are reduced to1 √− 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 , (3.17) ∂ µ ( p − ˆ g ˆ g µν F ν ) = 0 . (3.18)We assume that A is Z -odd with respect to the extra dimension z , which results inthat A has no zero mode in the effective 4D theory. Furthermore, in order to consistentwith the gauge invariant equation H dzA = 0, we use gauge freedom to choose A = 0.Under the assumption, the action (3.16) is reduced to S = − Z d x √− g (cid:26) g µα g νβ F µν F αβ + 2 e − A g µν ∂ z A µ ∂ z A ν (cid:27) . (3.19)Then, with the decomposition of the vector field A µ ( x, z ) = P n a ( n ) µ ( x ) ρ n ( z ) e A/ , andimporting the orthonormality condition Z ∞−∞ dz ρ m ( z ) ρ n ( z ) = δ mn , (3.20)the action (3.19) is read S = X n Z d x p − ˆ g (cid:18) −
14 ˆ g µα ˆ g νβ f ( n ) µν f ( n ) αβ − µ n ˆ g µν a ( n ) µ a ( n ) ν (cid:19) , (3.21)– 9 –here f ( n ) µν = ∂ µ a ( n ) ν − ∂ ν a ( n ) µ is the 4-dimensional field strength tensor, and it has beenrequired that the ρ n ( z ) satisfies the following Schr¨odinger equation (cid:2) − ∂ z + V ( z ) (cid:3) ρ n ( z ) = µ n ρ n ( z ) , (3.22)where the mass-independent potential is given by V S ( z ) = β δ (cid:0) δ − (2 + δ )sech ( βz/δ ) (cid:1) (3.23)and V A ( z ) = β aβ sech2 βz tanh 2 βz β + a arctan tanh βz ) − β (cid:0) β − a + 5 a (2 β + a arctan tanh βz ) arctan tanh βz (cid:1) β + a arctan tanh βz ) cosh (2 βz ) , ( δ = 12 ) (3.24)for the symmetric and asymmetric dS brane world solutions given in previous section, re-spectively. The asymmetric potential at the limit a → δ = 1 / V S ( z ) = β (cid:0) − (2 βz ) (cid:1) . (3.25)The symmetric potential (3.23) for arbitrary 0 < δ < − β δ at z = 0and a maximum β / z = ±∞ . Eq. (3.22) with this potential can be turned into thefollowing Schr¨odinger equation with a modified P¨oschl-Teller potential: h − ∂ z − q ( q − p sech ( pz ) i χ n = E n χ n , (3.26)where p = β/δ , q = 1 + δ/ E n = µ n − δ p . The energy spectrum of bound statesis found to be E n = − p ( q − − n ) or µ n = n ( δ − n ) β δ , n ∈ Z , ≤ n < δ. (3.27)For 0 < δ <
1, we get only one bound state, i.e., the normalized zero mode ρ ( z ) = s β Γ( + δ ) δ √ π Γ( δ ) sech δ/ ( βz/δ ) (3.28)with µ = 0. There is a mass gap between the zero mode and the first excited mode. Thecontinuous spectrum starts with µ = β and asymptotically turn into plane waves, whichrepresent delocalized KK massive vectors.For the asymmetric dS brane, the asymmetric potential (3.24) has a negative minimumvalue at some z and the asymptotic behavior: V A ( z = ±∞ ) = β , which implies thatthere is also a mass gap. The normalizable massless mode ρ ( z ) with µ = 0 is found to be ρ ( z ) ∝ (cid:18) β sech2 βz [ β + a arctan(tanh βz )] (cid:19) / . (3.29)– 10 – Figure 6:
The shapes of the potential V A ( z ) for the asymmetric dS brane. The parameters areset to β = 3, δ = 1 / a = 0 for dashed line, and a = 2, 3, 3.78, 3.818 for solid lines with thicknessincreases with a . This zero mode is the ground state since it has no node. For the limit a →
0, the masslessmode (3.29) is reduced to (3.28) but with δ = 1 /
2. Now, we also ask the question: are thereother bound states except the zero mode? Since there is no massive bound state for thesymmetric potential (3.25), we can conclude that the answer is also no for small asymmetricfactor a . Just as the case of scalar, the absolute value of the minimum asymmetric potentialdecreases with the increase of the asymmetry (see Fig. 6). However, we do not find massivebound states by numerical method even for large a . -4 -3 -2 -1 0 1 2-4-2024 zV , Ρ Μ Figure 7:
The shapes of the potential V A ( z ) (thick line), zero mode (dashed line) and the massspectrum (thick gray line) for dS brane with β = 3, δ = 1 / a = 3 . It was shown in the RS model in
AdS space that a spin 1 vector field is not localizedneither on a brane with positive tension nor on a brane with negative tension so the Dvali-Shifman mechanism [39] must be considered for the vector field localization [20]. Here, it isturned out that a vector field can be localized on the dS thick branes and we do not needto introduce additional mechanism for the vector field localization in the case at hand. For0 < δ <
1, we get only one bound state which is the massless mode. Furthermore, thereexists a mass gap between the bound ground state and the first exited state.– 11 – .3 Spin 1/2 fermion fields
In five dimensions, fermions are four component spinors and their Dirac structure is de-scribed by Γ M = e M ¯ M Γ ¯ M with e M ¯ M being the vielbein and { Γ M , Γ N } = 2 g MN . In thispaper, ¯ M , ¯ N , · · · = 0 , , , , µ, ¯ ν, · · · = 0 , , , ¯ M are the flat gamma matrices in five dimensions. In our set-up,the vielbein is given by e ¯ MM = e A ˆ e ¯ νµ
00 e A ! , (3.30)Γ M = e − A (ˆ e µ ¯ ν γ ¯ ν , γ ) = e − A ( γ µ , γ ), where γ µ = ˆ e µ ¯ ν γ ¯ ν , γ ¯ ν and γ are the usual flat gammamatrices in the 4D Dirac representation. The Dirac action of a massless spin 1/2 fermioncoupled to the scalar is S / = Z d x √− g (cid:0) ¯ΨΓ M ( ∂ M + ω M )Ψ − η ¯Ψ F ( φ )Ψ (cid:1) , (3.31)where the spin connection is defined as ω M = ω ¯ M ¯ NM Γ ¯ M Γ ¯ N and ω ¯ M ¯ NM = 12 e N ¯ M ( ∂ M e ¯ NN − ∂ N e ¯ NM ) − e N ¯ N ( ∂ M e ¯ MN − ∂ N e ¯ MM ) − e P ¯ M e Q ¯ N ( ∂ P e Q ¯ R − ∂ Q e P ¯ R ) e ¯ RM . (3.32)The non-vanishing components of the spin connection ω M for the background metric (2.2)are ω µ = 12 ( ∂ z A ) γ µ γ + ˆ ω µ , (3.33)where µ = 0 , , , ω µ = ¯ ω ¯ µ ¯ νµ Γ ¯ µ Γ ¯ ν is the spin connection derived from the metricˆ g µν ( x ) = ˆ e ¯ µµ ˆ e ¯ νν η ¯ µ ¯ ν . Then the equation of motion is given by (cid:8) γ µ ( ∂ µ + ˆ ω µ ) + γ ( ∂ z + 2 ∂ z A ) − η e A F ( φ ) (cid:9) Ψ = 0 , (3.34)where γ µ ( ∂ µ + ˆ ω µ ) is the Dirac operator on the brane.Now we study the above 5-dimensional Dirac equation, and write the spinor in termsof 4-dimensional effective fields. Because of the Dirac structure of the fifth gamma matrix γ , we expect that the left- and right-handed projections of the four dimensional part tobehave differently. From the equation of motion (3.34), we will search for the solutions ofthe general chiral decompositionΨ( x, z ) = e − A X n ψ Ln ( x ) α Ln ( z ) + X n ψ Rn ( x ) α Rn ( z ) ! (3.35)with ψ Ln ( x ) = − γ ψ Ln ( x ) and ψ Rn ( x ) = γ ψ Rn ( x ) the left-handed and right-handedcomponents of a 4D Dirac field. Here, to obtain the equations for the basis functions ψ Ln ( x )– 12 –nd ψ Rn ( x ), we assume that ψ L ( x ) and ψ R ( x ) satisfy the 4D massive Dirac equations γ µ ( ∂ µ + ˆ ω µ ) ψ Ln ( x ) = µ n ψ R n ( x ) and γ µ ( ∂ µ + ˆ ω µ ) ψ Rn ( x ) = µ n ψ L n ( x ). Then the KK modes α Ln ( z ) and α Rn ( z ) satisfy the following coupled equations (cid:2) ∂ z + η e A F ( φ ) (cid:3) α Ln ( z ) = µ n α Rn ( z ) , (3.36a) (cid:2) ∂ z − η e A F ( φ ) (cid:3) α Rn ( z ) = − µ n α Ln ( z ) , (3.36b)i.e., (cid:2) ∂ z − η e A F ( φ ) (cid:3) (cid:2) ∂ z + η e A F ( φ ) (cid:3) α Ln ( z ) = − µ n α Ln ( z ) , (3.37a) (cid:2) ∂ z + η e A F ( φ ) (cid:3) (cid:2) ∂ z − η e A F ( φ ) (cid:3) α Rn ( z ) = − µ n α Rn ( z ) . (3.37b)Hence, we get the Schr¨odinger-like equations for the left and right chiral fermions (cid:0) − ∂ z + V L ( z ) (cid:1) α Ln = µ n α Ln , (3.38) (cid:0) − ∂ z + V R ( z ) (cid:1) α Rn = µ n α Rn , (3.39)where the mass-independent potentials are given by V L ( z ) = e A η F ( φ ) − e A η ∂ z F ( φ ) − ( ∂ z A )e A ηF ( φ ) , (3.40a) V R ( z ) = V L ( z ) | η →− η . (3.40b)In order to obtain the standard 4D action for the massive chiral fermions: S / = Z d x √− g ¯Ψ (cid:0) Γ M ( ∂ M + ω M ) − ηF ( φ ) (cid:1) Ψ= X n Z d x p − ˆ g (cid:8) ¯ ψ Rn γ µ ( ∂ µ + ˆ ω µ ) ψ Rn − ¯ ψ Rn µ n ψ Ln (cid:9) + X n Z d x p − ˆ g (cid:8) ¯ ψ Ln γ µ ( ∂ µ + ˆ ω µ ) ψ Ln − ¯ ψ Ln µ n ψ Rn (cid:9) = X n Z d x p − ˆ g ¯ ψ n [ γ µ ( ∂ µ + ˆ ω µ ) − µ n ] ψ n , (3.41)we need the following orthonormality conditions for α L n and α R n : Z ∞−∞ α Lm α Ln dz = δ mn , (3.42) Z ∞−∞ α Rm α Rn dz = δ mn , (3.43) Z ∞−∞ α Lm α Rn dz = 0 . (3.44)It can be seen that, for the left (right) chiral fermion localization, there must be somekind of scalar-fermion coupling. This situation can be compared with the one in the RSframework [20], where additional localization method [40] was introduced for spin 1/2– 13 –elds. Furthermore, F ( φ ( z )) must be an odd function of φ ( z ) when we demand that V L ( z )or V R ( z ) is Z -even with respect to the extra dimension z . In this paper, we will considertwo cases F ( φ ) = φ and F ( φ ) = sin( φφ ) cos − δ ( φφ ) as examples. For F ( φ ) = φ , we get acontinuous spectrum of KK modes with µ ≥
0. However, it is shown that even the masslessleft and right chiral modes can not be localized on the brane. For F ( φ ) = sin( φφ ) cos − δ ( φφ ),there exists a mass gap, and we get some discrete bound modes and a continuous spectrumof KK modes. For the first case F ( φ ) = φ , the explicit forms of the potentials (3.40) are V SL ( z ) = η φ cosh − δ (cid:18) βzδ (cid:19) arctan sinh (cid:18) βzδ (cid:19) + ηβφ δ cosh − − δ (cid:18) βzδ (cid:19) (cid:20) δ sinh (cid:18) βzδ (cid:19) arctan sinh (cid:18) βzδ (cid:19) − (cid:21) , (3.45) V SR ( z ) = V SL ( z ) | η →− η , (3.46)and V AL ( z ) = ηβ φ cosh − / (2 βz ) (cid:0) β + a arctan tanh( βz ) (cid:1) (cid:20) a arctan sinh(2 βz )+ ηφ p cosh(2 βz ) arctan sinh(2 βz ) ( δ = 12 ) (3.47)+ (cid:0) β + a arctan tanh( βz ) (cid:1)(cid:0) sinh(2 βz ) arctan sinh(2 βz ) − (cid:1)(cid:21) ,V AR ( z ) = V AL ( z ) | η →− η , (3.48)for the symmetric and asymmetric dS brane world solutions, respectively. -1.5 -1 -0.5 0 0.5 1 1.5-15-10-5051015 zV L ,V R -1.5 -1 -0.5 0 0.5 1 1.5-100102030 zV L ,V R Figure 8:
The shapes of the potentials V L (thick lines), V R (thin lines) for left and right chiralfermions for the case F ( φ ) = φ . The parameters are set to δ = 1 / , η = 3 , β = 3, a = 0 (left) and a = 3 (right). All potentials have the asymptotic behavior: V L,R ( z → ±∞ ) →
0. But for a givencoupling constant η and the parameter β , the values of the potentials for the left and right– 14 –hiral fermions at z = 0 are opposite. The shapes of the potentials are shown in Fig. 8 forgiven values of positive η and β . It can be seen that V L ( z ) is indeed a modified volcanotype potential. Hence, the potential provides no mass gap to separate the fermion zeromode from the excited KK modes, and there exists a continuous gapless spectrum of theKK modes for both the left chiral and right chiral fermions.For positive β and η , only the potential for left chiral fermions has a negative value atthe location of the brane, which could trap the left chiral fermion zero mode solved from(3.36a) by setting µ = 0: α L ( z ) ∝ exp (cid:18) − η Z z dz ′ e A ( z ′ ) φ ( z ′ ) (cid:19) . (3.49)In order to check the normalization condition (3.42) for the zero mode (3.49), we need tocheck whether the inequality Z dz exp (cid:18) − η Z z dz ′ e A ( z ′ ) φ ( z ′ ) (cid:19) < ∞ (3.50)is satisfied. For the integral R dz e A φ , we only need to consider the asymptotic characteristicof the function e A φ for z → ∞ . For symmetric dS brane case, noting that arctan(sinh z ) → π/ z → ∞ , we havee A φ = φ cosh − δ (cid:18) βzδ (cid:19) arctan (cid:18) sinh βzδ (cid:19) → π φ δ e − βz , (3.51)exp (cid:18) − η Z dz e A φ (cid:19) → exp (cid:16) δ πηφ e − βz /β (cid:17) → , (3.52)which indicates that the normalization condition (3.50) is not satisfied and the zero modeof the left chiral fermions can not be localized on the brane. For asymmetric case, wecan also get the same conclusion. This is different from the conclusion obtained in Refs.[30, 41], where the zero mode of the left chiral fermions can be localized on the Branes inthe Background of Sine-Gordon Kinks. For the case F ( φ ) = sin( φφ ) cos − δ ( φφ ), the potentials (3.40) are V SL ( z ) = η (cid:18) η − β + δηδ sech ( βz/δ ) (cid:19) , (3.53) V SR ( z ) = V SL ( z ) | η →− η , (3.54)and V AL ( z ) = ηβ sech(2 βz ) tanh(2 βz ) (cid:0) a + η sinh(2 βz ) (cid:1) ( β + a arctan tanh( βz )) − ηβ sech (2 βz ) β + a arctan tanh( βz ) , (3.55) V AR ( z ) = V AL ( z ) | η →− η , (3.56)for the symmetric and asymmetric dS brane world solutions, respectively.– 15 – . Symmetric dS brane We first investigate the case of symmetric dS brane. The values of the correspondingpotentials (3.53) and (3.54) at y = 0 and y = ±∞ are given by V SL (0) = − V SR (0) = − βηδ , (3.57) V SL ( ±∞ ) = V SR ( ±∞ ) = η , (3.58)i.e., both potentials have same asymptotic behavior when y → ±∞ , but opposite behaviorat the origin z = 0. The shapes of the two potentials are shown in Figs. 9 and 10 fordifferent values of β and η , respectively. -1.5 -1 -0.5 0 0.5 1 1.5-60-40-200204060 zV LS -1.5 -1 -0.5 0 0.5 1 1.5-60-40-200204060 zV RS Figure 9:
The shapes of the potentials V SL (left) and V SR (right) of left and right chiral fermionsfor the symmetric dS brane with different β . The parameters are set to δ = 1 / η = 6, and β = 1for thick lines, β = 3 for dashed lines and β = 5 for thin lines. -1 -0.5 0 0.5 1-40-20020406080 zV LS -1 -0.5 0 0.5 1-40-20020406080 zV RS Figure 10:
The shapes of the potentials V SL (left) and V SR (right) of left and right chiral fermionsfor the symmetric dS brane with different η . The parameters are set to δ = 1 / β = 3, and η = 9for thick lines, η = 6 for dashed lines and η = 3 for thin lines. Note that, for a positive coupling constant η , the potential for left chiral fermions hasa negative value at the location of the brane and a positive value far away from the brane– 16 –long the extra dimension, which can always trap the left chiral fermion zero mode: α SL = " β Γ( β + δη β ) δ √ π Γ( δη β ) cosh − δη/β ( βz/δ ) . ( η >
0) (3.59)The zero mode represents the lowest energy eigenfunction of the Schr¨odinger equation(3.38) since it has no nodes. The right chiral fermion zero mode for η > V SR in Figs. (9) and (10).It is clear that for µ Ln > η , we obtain the asymptotic plane waves. The general boundstates for the potential V SL (3.53) for left chiral fermions are found to be α SLn ∝ cosh δηβ (cid:18) βzδ (cid:19) F (cid:18) a n , b n ; 12 ; − sinh ( βz/δ ) (cid:19) , (3.60)for even n and α SLn ∝ cosh δηβ (cid:18) βzδ (cid:19) sinh (cid:18) βzδ (cid:19) F (cid:18) a n + 12 , b n + 12 ; 32 ; − sinh ( βz/δ ) (cid:19) , (3.61)for odd n , where F is the hypergeometric function, the parameters a n and b n are givenby a n = 12 ( n + 1) , b n = δηβ −
12 ( n − . (3.62)The corresponding mass spectrum of the bound states is µ Ln = β (2 δη − βn ) nδ , ( η > , n = 0 , , , ... < δηβ ) . (3.63)It shows that the ground state always belongs to the spectrum of V SL ( z ) for positive η ,which is just the zero mode (3.59) with µ L = 0. Since the ground state has the lowestmass square µ L = 0, there is no tachyonic left chiral fermion modes. Here, we supposethe number of bound states for left chiral fermions is N L . If 0 < η ≤ β/δ , there is onlyone bound state ( N L = 1), i.e., the zero mode (3.59). In order to get bound exited states( N L ≥ η > β/δ .In the case η >
0, the potential V SR ( z ) = η (cid:0) η + β − δηδ sech ( βz/δ ) (cid:1) for right chiralfermions is always positive near the location of the brane, which shows that it can not trapthe right chiral zero mode. For the case 0 < η < β/δ , we have V SR (0) ≥ V SR ( ±∞ ) > η = β/δ , the potential V SR is a positive constant: V SR ( z ) = η = β /δ ,and there is still no any bound state. However, provided η > β/δ , we will get a potentialwell since V SR (0) < V SR ( ±∞ ) (see Figs. 9 and 10), which indicates that there may be somebound states, but none of them is zero mode. The general bound states for the potential V SR are α SRn ∝ cosh δηβ (cid:18) βzδ (cid:19) F (cid:18) n , δηβ − n − sinh ( βz/δ ) (cid:19) (3.64)– 17 –or even n and α SRn ∝ cosh δηβ (cid:18) βzδ (cid:19) sinh (cid:18) βzδ (cid:19) F (cid:18) n , δηβ − n − sinh ( βz/δ ) (cid:19) (3.65)for odd n . The corresponding mass spectrum is µ Rn = ( n + 1) β (2 δη − ( n + 1) β ) δ , ( η > βδ , n = 0 , , , ... < δηβ − . (3.66)By comparing with the mass spectrum of left chiral fermions (3.66), we come to the con-clusion that the number of bound states of right chiral fermions N R is one less than thatof left ones, i.e., N R = N L −
1. If 0 < η ≤ β/δ , there is only one left chiral fermion boundstate (the zero mode). If η > β/δ , there are N L ( N L ≥
2) left chiral fermion bound statesand N L − α SR = " β Γ( δη β ) δ √ π Γ( δη β − ) cosh − δηβ (cid:18) βzδ (cid:19) , (cid:18) η > βδ (cid:19) (3.67)which is not zero mode any more because the mass is determined by µ R = β (2 δη − β ) /δ >β /δ >
0. In Figs. 13(a) and 13(b) we plot the potentials, the mass spectra and somebound states of left and right chiral fermions. For the case δ = 1 / , β = 1 , η = 11, thereare 6 and 5 bound states for the left and the right chiral fermions respectively and themass spectra are µ Ln = { , , , , , } ∪ [121 , ∞ ) , (3.68) µ Rn = { , , , , } ∪ [121 , ∞ ) . (3.69)
2. Asymmetric dS brane Now we turn to the case of asymmetric dS brane, for which the corresponding potentials(3.55) and (3.56) are obviously asymmetric and the solution of the bound states and massspectrum is very complex. The values of the potentials at z = 0 , ±∞ are given by V AL (0) = − V AR (0) = − βη,V AL (+ ∞ ) = V AR (+ ∞ ) = 16 β η ( aπ + 4 β ) < η , (3.70) V AL ( −∞ ) = V AR ( −∞ ) = 16 β η ( aπ − β ) > η . Both potentials have also same asymptotic behavior when z → ±∞ . However, comparedwith the symmetric potentials (3.53) and (3.54), V AL,R ( −∞ ) increase and V AL,R (+ ∞ ) de-crease for positive asymmetric factor a , which may reduce the number of the bound states.The shapes of the two potentials are shown in Figs. 11 and 12 for different values of β and η , respectively. Different from the symmetric potentials V SL,R ( z ), the asymmetric ones V AL,R ( z ) at z = ±∞ dependent on the parameter β unless the asymmetric factor a = 0.Hence, even at same η and a , the limits of V AL,R ( z ) at z → + ∞ and z → −∞ are different– 18 –or different β (see Fig. 11). For positive η , the right chiral fermion zero mode does notexist, but the left one is always exist and can be solved as α AL ( z ) ∝ exp (cid:18) − ηβ Z z dz ′ tanh(2 βz ′ ) β + a arctan tanh( βz ′ ) (cid:19) . (3.71) -1.5 -1 -0.5 0 0.5 1 1.5-50-250255075 zV LA -1.5 -1 -0.5 0 0.5 1 1.5-50-250255075 zV RA Figure 11:
The shapes of the potentials V AL (left) and V AR (right) of left and right chiral fermionsfor the asymmetric dS brane with different β . The parameters are set to δ = 1 / η = 6, a = 0 . β = 1 for thick lines, β = 3 for dashed lines and β = 5 for thin lines. -1 -0.5 0 0.5 1-50-250255075100 zV LA -1 -0.5 0 0.5 1-50-250255075100 zV RA Figure 12:
The shapes of the potentials V AL (left) and V AR (right) of left and right chiral fermionsfor the asymmetric dS brane with different η . The parameters are set to δ = 1 / β = 3, a = 0 . η = 9 for thick lines, η = 6 for dashed lines and η = 3 for thin lines. For µ n > β η / ( aπ + 4 β ) , we obtain the continuum of asymptotic plane waves.In order to obtain acceptable normalizable modes, µ n should be limited in the interval[0 , β η / ( aπ + 4 β ) ). Although the analytic massive modes can not be solved because ofthe complexity of the potentials, we can get the numerical solutions. The mass spectra arelisted in Tab. 1 for left chiral fermions and Tab. 2 for right ones for some given parameters.We also plot the potentials, mass spectra and part of the eigenfunctions in Fig. 13. Fromthese tables and Eq. (3.70), we can draw a conclusion: the number of the bound statesincreases with the coupling constant η but decreases with the asymmetric factor a .– 19 – N L V AL (+ ∞ ) V AL ( −∞ ) Mass spectrum µ Ln of bound states0.00 6 121.0 121.0 {
0, 40.00, 72.00, 96.00, 112.00, 120.00 } {
0, 40.00, 71.99, 95.94, 111.74 } {
0, 39.95, 71.67, 94.57 } {
0, 39.70, 69.95 } {
0, 38.79 } { } Table 1:
Mass spectrum of bound states for asymmetric potentials V AL ( z ). The parameters are setto δ = 1 / β = 1, η = 11. N L presents the number of bound states for left chiral fermions. a N R V AR (+ ∞ ) V AR ( −∞ ) Mass spectrum µ Rn of bound states0.00 5 121.0 121.0 { } { } { } { } { } { } Table 2:
Mass spectrum of bound states for asymmetric potentials V AR ( z ). The parameters are setto δ = 1 / β = 1, η = 11. N R presents the number of bound states for right chiral fermions. To close this section, we make some comments on the issue of the localization offermions. Localizing the fermions on branes or defects requires us to introduce other inter-actions besides gravity. More recently, Volkas et al had extensively analyzed localizationmechanisms on a domain wall. In particular, in Ref. [24], they proposed a well-definedmodel to localize the SM, or something close to it, on a domain wall brane. There are someother backgrounds, for example, gauge field [43], supergravity [44, 45] and vortex back-ground [46, 47, 48, 49], could be considered. The topological vortex coupled to fermionsmay result in chiral fermion zero modes [50].
4. Conclusion and discussion
In this paper, by presenting the shapes of the mass-independent potentials of KK modes inthe corresponding Schr¨odinger equations, we have investigated the localization and massspectra of various matter fields with spin 0, 1 and 1/2 on symmetric and asymmetric dS thick branes, where the asymmetric dS thick brane is constructed from the symmetric oneby using a same scalar (kink) with different potentials.For spin 0 scalars and spin 1 vectors, the potentials of KK modes in the correspondingSchr¨odinger equations are the modified P¨oschl-Teller potentials. They have a finite negativewell at the location of the brane and a finite positive barrier at each side which doesn’tvanishes. Such potentials suggest that there exist a mass gap and a series of continuous– 20 – = = (a) Left chiral fermions, a = 0 -3 -2 -1 0 1 2 3-100-50050100 n = = (b) Right chiral fermions, a = 0 -3 -2 -1 0 1 2 3-100-50050100150 n = = (c) Left chiral fermions, a = 0 . -3 -2 -1 0 1 2 3-100-50050100150 n = = (d) Right chiral fermions, a = 0 . -3 -2 -1 0 1 2 3-200-1000100200300 n = = (e) Left chiral fermions, a = 0 . -3 -2 -1 0 1 2 3-200-1000100200300 n = (f) Right chiral fermions, a = 0 . Figure 13:
The potentials V AL,R ( z ) (black thick lines), the mass spectrum µ L,R (thick gray lines)and some eigenfunctions (black thin lines) for asymmetric dS brane with β = 1, δ = 1 / η = 11and different a . spectrum starting at positive µ . It can be shown that the existence of such a mass gap isuniversal for all such dS branes.For the symmetric dS brane, the spectrum of scalar KK modes consists of a zero mode– 21 –nd a set of continuous modes, i.e., there is only one bound mode (the zero mode). Themassless mode is separated by a mass gap from the continuous modes. For the asymmetric dS brane with a small asymmetric factor, the spectrum is same as the symmetric case.However, for a large enough asymmetric factor, the spectrum of scalar KK modes containsa bound massive KK mode besides a zero mode and a set of continuous modes, namely,there are two bound modes. For spin 1 vectors, the spectra of KK modes on both dS branes are made up of a bound zero mode and a set of continuous ones. The asymmetricfactor does not change the number of the vector bound modes.It is shown that, without scalar-fermion coupling, there is no bound state for boththe left and right chiral fermions. Hence, in order to localize the massless and massiveleft or right chiral fermions on the branes, some kind of kink-fermion coupling should beintroduced. As examples, two types of kink-fermion couplings are investigated in detail.These situations can be compared with the case of the domain wall in the RS framework[20], where for localization of spin 1/2 field additional localization method by Jackiw andRebbi [40] was introduced.For the usual Yukawa coupling η ¯Ψ φ Ψ, the potential for only one of the left and rightchiral fermions has a finite well at the location of the brane and a finite barrier at each sidewhich vanishes asymptotically. It is shown that there is only one single bound state (zeromode) which is just the lowest energy eigenfunction of the Schr¨odinger equation for thecorresponding chiral fermions. Since the potentials for both left and right chiral fermionsvanish asymptotically when far away from the brane, all values of µ > µ >
0. Themassive KK modes asymptotically turn into continuous plane waves when far away fromthe brane [5, 10], which represent delocalized massive KK fermions.For the scalar-fermion coupling η ¯Ψ sin( φφ ) cos − δ ( φφ )Ψ with positive η , the potentialfor the left chiral fermions has a finite well at the location of the brane, and a finite positivebarrier at each side, which does not vanishes when far away from the brane. The potentialis the modified P¨oschl-Teller potential and suggest that there exist some discrete KK modesand a series of continuous ones. The discrete modes are bound states while the continuousones are not. The total number of bound states is determined by four parameters: δ , β , a and η . The number of bound states of right chiral fermions is one less than that of leftones. The number of the bound states increases with the coupling constant η . For the caseof the symmetric dS brane, if 0 < η < β/δ , there is only one left chiral fermion boundstate which is just the left chiral fermion zero mode; if η > β/δ , there are N L ( N L ≥ N L − dS brane scenario, the asymmetric factor a reduces the number of the bound fermion KKmodes. For large enough a , there would not be any right chiral fermion bound mode, butat least one left chiral fermion bound mode, i.e., the zero mode.For fermions, localization property is decided by the coupling of fermion and scalar.For the first type of Yukawa coupling, F ( φ ( z )) ∼ arctan(sinh z ) is a usual kink which isalmost a constant at large z . For the second type of coupling, F ( φ ( z )) is another kink likessinh z , which increases quickly with z . In short, at large z , the first coupling is invariant,– 22 –ut the second one becomes stronger. Hence, the two different types of Yukawa couplingsgive different localization properties for fermions.Finally, we give some brief discussion about graviton and gravitino localization on thestudied branes. The Schr¨odinger potentials of graviton and gravitino KK modes are thesame as that of scalar and fermion, respectively. Thus, for the symmetric dS brane and theasymmetric dS brane with small asymmetric factor, the spectrum of graviton KK modesconsists of a discrete zero mode and a set of continuous modes. While for a large enoughasymmetric factor, the spectrum of graviton KK modes contains a bound massive KKmode besides a zero mode and a set of continuous modes. The spectrum of gravitino issimilar to that of fermion. Acknowledgement
The authors are really grateful to the referee for his/her constructive comments and sug-gestions which considerably improved the paper. This work was supported by the NationalNatural Science Foundation of China (No. 10705013), the Doctor Education Fund of Ed-ucational Department of China (No. 20070730055) and the Fundamental Research Fundfor Physics and Mathematics of Lanzhou University (No. Lzu07002).
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