Physics Letters B 680 (2009) 489–495

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Localization of matters on the bent brane in AdS5 bulk

Jun Liang ∗, Yi-Shi Duan

Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 July 2009Received in revised form 11 September2009Accepted 13 September 2009Available online 18 September 2009Editor: A. Ringwald

PACS:04.50.-h11.25.-w11.10.Kk

Keywords:Extra dimensionsAdS4 (dS4) braneLocalization of fieldsZero modes

We investigate the possibility of localizing various matter fields on a bent AdS4 (dS4) thick brane in AdS5.For spin 0 scalar field, we find a massless zero mode and an excited state which can be localized on thebent brane. For spin 1 vector field, there is only a massless zero mode on the bent brane. For spin 1/2fermion field, it is shown that, in the case of no Yukawa coupling of scalar-fermion, there is no existenceof localized massless zero mode for both left and right chiral fermions. In order to localize masslessfermions, some kind of Yukawa coupling must be included. We study two types of Yukawa couplings asexamples. Localization property of chiral fermions is related to the parameters of the brane model, theYukawa coupling constant and the cosmological constant of the 4-dimensional space–time.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In recent years, brane world received a considerable attentionfrom the physical community. In brane world scenario, our worldis a 4-dimensional submanifold (called “brane”) embedded in ahigher-dimensional space–time (called “bulk”), and is warped bya function of the extra dimension (call “warp factor”), as well asthe Standard Model particles are assumed to live on the brane,whereas the gravitational field is free to propagate in the bulk.Randall and Sundrum set forward a warped thin brane model [1,2], where the brane is mathematically represented by a Dirac deltafunction singularity in the higher-dimensional space–time. Subse-quently, various different thin and thick brane models [3–47] havebeen constructed using the 5- (or higher) dimensional Einstein-scalar set of equations. In most of the models, the brane is flatMinkowski brane, a few braneworld models considered the cur-vature of the embedded brane, namely, the brane is de Sitter oranti-de Sitter brane.

In the brane world scenario, an important issue is how grav-ity and different observable matter fields of the Standard Model

* Corresponding author.E-mail address: liangjunbeijing@yahoo.com.cn (J. Liang).

0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2009.09.023

of particle physics are localized on the brane. It has been shownthat, in the Randall–Sundrum model in 5-dimensional space–time,graviton and spin 0 field can be localized on a brane with posi-tive tension [48,2,49]. Spin 1 field cannot be localized either on abrane with positive tension or on a brane with negative tension(but spin 1 field can be localized on a string-like defect in high-dimensional space–time) [24]. And moreover spin 1/2 and 3/2can be localized on a negative-tension brane [49]; in order toachieve localization of fermions on a brane with positive tension,it seems that additional interactions except the gravitational inter-action must be including in the bulk.

Recently, we note that it has been pointed out that the mat-ter fields with spin 1 and 1/2 can be confined on an AdS4 branein AdS5 space–time [50]. The authors of Refs. [37,38] investigatedlocalization of 4-dimensional gravity on bent brane (for both dS4(Λ > 0) and AdS4 (Λ < 0) geometry) following a new route. Thepurpose of this Letter is to extend their analysis to the study of lo-calization of matters on the AdS4 (dS4) brane obtained in Ref. [37].The organization of the Letter is as follows: First, in Section 2 wegive a quick review of the AdS4 (dS4) thick brane solution in a 5-dimensional theory of gravity coupled with one scalar field. Thenin Section 3, we discuss localization of various different spin fieldson the AdS4 (dS4) brane. Finally, we present our summary in Sec-tion 4.

490 J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495

2. Review of AdS4 (dS4) thick brane solution

The action for 5-dimensional gravity coupled to a real scalarfield reads [37,38]

S5 =∫

d4x dy√−g

[−1

4R5 + 1

2gMN∂Mφ∂Nφ − V (φ)

], (1)

where g = det(gMN ), M, N = 0,1,2,3,4, det(gMN ) is the 5-dimensional metric with signature (+,−,−,−,−), R5 is its Ricciscalar, φ is a real scalar field depending on only the extra dimen-sion coordinate y, V (φ) is a self-interacting potential for the scalarfield, and 4πG = 1 is used. The element of the 5-dimensionalspace–time is assumed to take the form

ds25 = gMN dxM dxN = e2A(y) ds2

4 − dy2, (2)

where e2A(y) is the warp factor depending on only the extra di-mension, and ds2

4 represents the line element of the 4-dimensionalspace–time, which have the form

ds24 = gμν dxμ dxν = e−2

√Λx3

(dx2

0 − dx21 − dx2

2

) − dx23, (3)

or

ds24 = gμν dxμ dxν = dx2

0 − e−2√

Λx0(dx2

1 + dx22 + dx2

3

), (4)

for AdS4 or dS4 geometry, respectively, where Λ represents thecosmological constant of the 4-dimensional space–time.

We use Einstein’s equation for 4-dimensional bent space–timeto get [37]

A′′ + Λe−2A = −2

3φ′2, (5)

A′2 − Λe−2A = 1

6φ′2 − 1

3V (φ), (6)

for AdS4 geometry (Λ < 0) or dS4 geometry (Λ > 0).The presence of Λ makes the problem much harder. Following

Refs. [37,38], we introduce superpotential W = W (φ), and a newfunction of the scalar field Z = Z(φ) and assume that

A′ = −1

3W − 1

3ΛZ (7)

and

φ′ = 1

2Wφ + 1

2Λ(1 − s)Zφ, (8)

where s is a real parameter. In this case, the potential V (φ) isgiven by

V = 1

8

(Wφ + Λ(1 − s)Zφ

)(Wφ + Λ(1 + 3s)Zφ

)

− 1

3(W + ΛZ)2, (9)

and the following constraint

Wφφ Zφ + Wφ Zφφ + 2Λ(1 − s)Zφ Zφφ − 4

3W Zφ − 4

3ΛZ Zφ = 0

(10)

must be included. Following Refs. [37,38] let us consider a casegiven by Z = W and superpotential W = 3a sinh(bφ), where a and

b are real parameters. From (10) we have b = ±√

23c , where c =

[1 + (1 − s)Λ]/(1 + Λ). In this case we have

V = 3

4a2(1 + Λ)

[1 + (1 + 3s)Λ

]cosh2(bφ)

− 3a2(1 + Λ)2 sinh2(bφ), (11)

and the following solution can be obtained

A(y) = −1

2ln

{sa2(1 + Λ) sec2[a(1 + Λ)y

]}(12)

and

φ(y) = 1

barcsinh

{tan

[a(1 + Λ)y

]}, (13)

obviously, where we have to take s(1 + Λ) > 0.

3. Localization of various matters

In this section, we study whether various bulk fields with spinranging from 0 to 1 can be localized on the AdS4 (dS4) brane bymeans of only the gravitational interaction.

3.1. Spin 0 scalar field

In this subsection we study localization of a real scalar fieldon the AdS4 (dS4) brane described in previous section. Let us firstperform the coordinate transformation dz = e−A(y)dy to get a con-formally flat metric

ds25 = e2A(z)(gμν dxμ dxν − dz2). (14)

In the case we consider above section, in terms of z, (12) and (13)are expressed as

A(z) = −1

2ln

[sa2(1 + Λ) cosh2(qz)

](15)

and

φ(z) = q

bz, (16)

where q = √(1 + Λ)/s.

Let us consider the action of a massless real scalar coupled togravity

S0 = 1

2

∫d5x

√−g gMN∂MΦ∂NΦ, (17)

from which the equation of motion can be derived

1√−g∂M

(√−g gMN∂NΦ) = 0. (18)

By considering (14) the equation of motion (18) becomes

1√−g∂μ

(√−g gμν∂νΦ) − e−3A∂z

(e3A∂zΦ

) = 0. (19)

Let us decompose the field Φ as follows:

Φ(x, z) =∑

n

φn(x)χn(z), (20)

and demanding φn(x) satisfies

1√−g∂μ

(√−g gμν∂νφn(x)) = −m2

nφn(x), (21)

we obtain the equation for χn(z)

[∂2

z + 3(∂z A)∂z]χn(z) = −m2

nχn(z). (22)

Define χn(z) = e32 Aχn(z), we get the Schrödinger-like equation

[−∂2z + V (z)

]χn(z) = m2

nχn(z), (23)

J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495 491

Fig. 1. The zero mode χ00(z) (solid line) and the corresponding potential (24)(dashed line) with Λ = −0.2 and s = 0.8.

where mn is the mass of the Kaluza–Klein (KK) excitation and thepotential is given by

V (z) = 3

2∂2

z A + 9

4(∂z A)2. (24)

Substituting (15) into (24), we get

V (z) = 9

4q2 − 15

4q2 sech2(qz). (25)

This potential supports two bound state, one is the massless zeromode with m2

0 = 0, the other is an excited state with m21 = 2q2.

And the corresponding normalizable wave functions are

χ00(z) = N00

cosh3/2(qz), χ01(z) = N01

sinh(qz)

cosh3/2(qz), (26)

where N00 and N01 are normalization constants. As an example,we show the zero mode χ00(z) and the corresponding potential(24) in Fig. 1. In addition, the mass spectrum contains a continuumof states with m2 � 9

4 q2.

3.2. Spin 1 vector field

Let us start with the action of U(1) vector field

S1 = −1

4

∫d5x

√−g gMN g R S F M R F N S , (27)

where F MN = ∂M AN − ∂N AM as usual. From this action the equa-tion of motion is given by

1√−g∂M

(√−g gMN g R S F N S) = 0. (28)

In the background metric (14), this equation of motion (28) istransformed into

∂μ

(√−g gμν Fν4) = 0 (29)

and

1√−g∂μ

(√−g gμν gλρ Fνρ

) − e−A gλρ∂z(e A F4ρ

) = 0. (30)

Let us decompose the vector field as follows:

Aμ(x, z) = Σna(n)μ (x)ρn(z), (31)

using Fμν = ∂μ Aν − ∂ν Aμ , and choosing the gauge A4 = 0 [51],then (29) and (30) are changed into the following equations

1√ ˆ ∂μ

[√−g gμνa(n)ν (x)

]∂zρn(z) = 0 (32)

−g

Fig. 2. The zero mode (40) (solid line) and the corresponding potential (39) (dashedline) with Λ = −0.2 and s = 0.8.

and

1√−g∂μ

(√−g gμν gλρ f (n)νρ

)ρn(z)

− e−A gλρ∂z[e A∂zρn(z)

]a(n)ρ (x) = 0, (33)

where f (n)νρ = ∂νa(n)

ρ − ∂ρa(n)ν . We assume that a(n)

ρ and f (n)νρ satisfy

the following equations1

1√−g∂μ

(√−g gμνa(n)ν

) = 0 (34)

and

1√−g∂μ

(√−g gμν gλρ f (n)νρ

) = −m2n gλρa(n)

ρ . (35)

Obviously, (34) guarantees the validity of (32). Substituting (35)into (33), we get

[∂2

z + (∂z A)∂z]ρn(z) = −m2

nρn(z). (36)

Defining ρn = e A/2ρn , we get the Schrödinger-like equation for thevector field

[−∂2z + V (z)

]ρn(z) = m2

nρn(z), (37)

where the potential V (z) is given by

V (z) = 1

2∂2

z A + 1

4(∂z A)2. (38)

Substituting (15) into (38), we get

V (z) = 1

4q2 − 3

4q2 sech2(qz). (39)

In this case, we can get only one bound state, namely, the masslesszero mode

ρ0(z) = N1 cosh−1/2(qz), (40)

where N1 is a constant. And a continuum of modes with massesm2 � 1

4 q2. In Fig. 2 we show the zero mode (40) and the corre-sponding potential (39).

1 The (34) and (35) are the generalizations of Lorentz condition and Proca equa-tion, respectively.

492 J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495

3.3. Spin 1/2 spinor field

Now we are ready to consider spin 1/2 fermion. We intro-duce the vielbein E M

M (and its inverse E MM

) through the usual

definition gMN = E MM E N

NηMN , where M, N, . . . , denote the local

Lorentz indices. From the formula Γ M = E MM

Γ M where Γ μ =γ μ , μ = 0,1,2,3, Γ 4 = −iγ 5 and the usual Clifford algebra{Γ M ,Γ N } = 2gMN and {γ μ,γ ν} = 2ημν are obeyed, we haveΓ M = (e−A γ μ,−ie−Aγ 5), where γ μ = Eμ

ν γ ν , and the vielbein Eμν

and its inverse Eμν are introduced through the definition gμν =

Eμμ E ν

νημν . Our starting action is the Dirac action of a masslessspin 1/2 fermion coupled to gravity and scalar [24,8,11,52–57]

S1/2 =∫

d5x√−g

[Ψ iΓ M DMΨ − ηΨ F (φ)Ψ

]. (41)

From which the equation of motion is given by[iΓ M DM − ηF (φ)

]Ψ = 0, (42)

where DM = (∂M + ωM) = ∂M + 14 ωMN

M ΓMΓN is the covariant

derivative, where the spin connection ωM NM is defined as

ωM NM = 1

2E N M(

∂M E NN − ∂N E N

M

)

− 1

2E N N(

∂M E MN − ∂N E M

M

)

− 1

2E P M E Q N (∂P E Q R − ∂Q E P R)E R

M . (43)

The components of ωM are

ωμ = − i

2(∂z A)γμγ5 + ωμ, ωz = 0, (44)

where ωμ = 14 ω

μνμ ΓμΓν . And the Dirac equation (42) becomes

[iγ μ(∂μ + ωμ) + γ 5(∂z + 2∂z A) − ηe A F (φ)

]Ψ = 0. (45)

We assume that

iγ μ(∂μ + ωμ)ψLn(x) = mnψRn(x) (46)

and

iγ μ(∂μ + ωμ)ψRn(x) = mnψLn(x), (47)

where ψLn(x) and ψRn(x) are the left-handed and the right-handedcomponents of a 4-dimensional Dirac field, respectively. SinceEq. (45) involves the matrix γ5, it is convenient to split the full5-dimensional spinor in the general chiral decomposition as fol-lows:

Ψ (x, z) =∑

n

ψLn(x)αLn(z) +∑

n

ψRn(x)αRn(z), (48)

with γ 5ψLn(x) = −ψLn(x) and γ 5ψRn(x) = ψRn(x). We obtain thefollowing coupled eigenvalue equations[∂z + 2∂z A + ηe A F (φ)

]αLn = mnαRn (49)

and[∂z + 2∂z A − ηe A F (φ)

]αRn = −mnαLn. (50)

By defining αLn = e2AαLn , we get the Schrödinger-like equation forleft chiral fermions[−∂2

z + V L(z)]αLn = m2

nαLn, (51)

Fig. 3. The shape of potential (53) with Λ = −0.2, a = 1, b = √5/3, q = 1, s = 0.8

and η = 2.

where the potential V L(z) is given by

V L(z) = e2Aη2 F 2(φ) − e Aη∂z F (φ) − (∂z A)e AηF (φ). (52)

For right chiral fermions, the corresponding potential can be writ-ten out easily by replacing η → −η from the above potential (52).It can be seen clearly that, for localization of left (right) fermions,there must be some kind of Yukawa coupling since both the po-tentials for left and right chiral fermions are vanish in the case ofno coupling (η = 0). In this Letter we discuss two types of Yukawacouplings as examples.

(A) F (φ) = φ. This case is the usual Yukawa coupling, the po-tential for left chiral fermions is reduced to

V L(z) = η2q2z2

sa2b2(1 + Λ) cosh2(qz)− ηq

ab√

s(1 + Λ) cosh(qz)

+ ηq2z sinh(qz)

ab√

s(1 + Λ) cosh2(qz). (53)

For simplicity, we consider the case η > 0, ab > 0 and s > 0as examples. (For s > 0, we may have AdS4 or dS4 branes, be-cause the cosmological constant of the 4-dimensional space–timecan assume the values −1 < Λ < 0 or Λ > 0, respectively.) Thepotential (53) (the shape of this potential is plotted in Fig. 3) forleft chiral fermions has negative value at the location of the AdS4(AdS4) brane (at z = 0), which could trap the normalizable left chi-ral fermion zero mode solved in (49) by setting m0 = 0

αL0(z) = e2AαL0(z) ∝ exp

[−η

z∫dz′ e A(z′)φ(z′)

], (54)

we need to check whether the integral

+∞∫−∞

dz exp

[−2η

z∫dz′ e A(z′)φ(z′)

](55)

is convergent. This integral can be written as

+∞∫−∞

exp

{− 2η

ab(1 + Λ)arctan

[sinh(qz)

]}dz. (56)

Since limz→±∞ arctan[sinh(qz)] → π/2, we have

limz→±∞ exp

{− 2η

ab(1 + Λ)arctan

[sinh(qz)

]}

= exp

[− ηπ

]. (57)

ab(1 + Λ)

J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495 493

Fig. 4. The zero mode (60) (solid line) and the corresponding potential (58) (dashedline) with Λ = −0.2, a = 1, s = 0.8 and η = 1.

Obviously, as ηπ � ab(1 + Λ), then exp[− ηπab(1+Λ)

] ∼ 0, and thus,the integral (56) is convergent. That is to say, we can achieve thenormalizable left chiral fermion zero mode on the bent AdS4 (dS4)brane.

An analogous analysis can be done, it is found that we can alsoobtain the normalizable right chiral fermion zero mode on the bentbrane under certain condition.

(B) F (φ(z)) = sinh(bφ) = sinh(qz). For this case,2 for simplicity,we assume s > 0 (as we pointed out before, for s > 0, we mayhave AdS4 or dS4 branes. For s < 0 we only have AdS4 brane dueto the constraint of s(1 + Λ) > 0. The analysis for the case s < 0 issimilar), the potential (52) is reduced to

V L(z) = η2

sa2(1 + Λ)− η2 + a(1 + Λ)η

sa2(1 + Λ)sech2(qz). (58)

For right chiral fermions, the corresponding potential is given by

V R(z) = η2

sa2(1 + Λ)− η2 − a(1 + Λ)η

sa2(1 + Λ)sech2(qz). (59)

From (58) we get V L(0) = − ηq2

a(1+Λ)and V L(±∞) = η2

a2s(1+Λ)> 0.

Thus, for positive η, when a(1 + Λ) > 0, i.e., a > 0, the potential(58) for left chiral fermions has negative value at the location ofthe AdS4 (AdS4) brane (at z = 0), which could trap the normalizableleft chiral fermion zero mode

αL0(z) = NL0 cosh(qz)η

a(1+Λ) , (60)

where NL0 is a normalization constant, which represents the low-est energy eigenfunction of the Schrödinger-like equation (51)since it has no zeros. In Fig. 4 we show the zero mode (60) andthe corresponding potential (58).

In fact, in the case η > 0 and a(1 + Λ) > 0, the general boundstates for the modified Pöschl–Teller potential (58) can be ob-tained by using the traditional recipe of transforming the station-ary Schrödinger equation into a hypergeometric equation [58–60]

αLn(z) = [cosh(qz)

]1+ ηa(1+Λ)

{An 2 F1

[an,bn; 1

2;− sinh2(qz)

]

+ Bn sinh(qz) 2 F1

[an + 1

2,bn + 1

2; 3

2;− sinh2(qz)

]},

(61)

2 The Z2 symmetry of the bulk space–time with respect to the position of thebrane (at z = 0) puts a restriction on the choice of the form of the Yukawa coupling.If we demand such a symmetry for the potential V L(z), F (φ(z)) should necessarilybe an odd function of φ(z) [11].

where An and Bn are normalization constants, the parameters an

and bn are given by

an = 1

2(n + 1), (62)

bn = η

a(1 + Λ)+ 1

2(1 − n). (63)

The corresponding mass spectrum of the bound states is

m2n = q2n

[2η

a(1 + Λ)− n

](η > 0, n = 0,1,2, . . . < η/

[a(1 + Λ)

]). (64)

Obviously, from (64), if 0 < η � a(1 + Λ), there is just one boundstate, i.e., the zero mode. In order to get at least one boundexcited state for the left chiral fermion potential, the conditionη > a(1 + Λ) is needed. The total number of bound states is de-termined by the Yukawa coupling constant η, the parameter ofthe thick brane solution a and the cosmological constant of the4-dimensional space–time Λ, but is not related to the parameter s.We plot the potential (58) for different values of Λ, a, s and η inFig. 5. From this figure we note an interesting phenomenon, withdecreasing s, the potential well becomes deeper (with Λ, a and ηkept fixed), but the positions of bound states are also changed, asa consequence, the total number of bound states still remain un-changed (see Fig. 6). Besides, there is also a continuous spectrumof massive modes with m2 � η2/[a2s(1 + Λ)].

From (59) we get V R(0) = ηq2

a(1+Λ)and V R(±∞) = η2

a2s(1+Λ)> 0.

Thus, for positive η, when a(1+Λ) > 0, the potential (59) for rightchiral fermions is always positive at the location of the AdS4 (AdS4)brane (at z = 0), which cannot trap the normalizable left chiralfermion zero mode. We should point out that, although in the case0 < η < a(1 + Λ), we have V R(0) > V R(±∞) > 0 (see Fig. 7 (leftpanel)), thus it is impossible to get other bound states either; how-ever, when η > a(1 + Λ), we have 0 < V R(0) < V R(±∞), there isa potential barrier (see Fig. 7 (right panel)) which indicates thatthere may be some other bound states (nonzero mode). In fact,the corresponding mass spectrum is given by

m2n = q2(n + 1)

[2η

a(1 + Λ)− (n + 1)

](η > a(1 + Λ), n = 0,1,2, . . . < η/

[a(1 + Λ)

] − 1). (65)

From (64) the mass

m0 = q2[

2η

a(1 + Λ)− 1

]> q2 > 0, (66)

therefore, the ground state is not zero mode. As an example, weplot the potentials for left and right chiral fermions as well as cor-responding mass spectra for a = 1, s = 0.5, Λ = −0.2 and η = 4in Fig. 8. There are five and four bound states for left and rightchiral fermions, respectively. The massive left-handed and right-handed spectra are equal except that the potential for left-handedfermions confines an extra zero mode, which is accord with theexpectation of the supersymmetric quantum mechanics [61–63].

For positive η, when a(1 + Λ) < 0, i.e., a < 0, the potential forright chiral fermions has negative value at the location of the AdS4brane (z = 0), we can get the normalizable right chiral fermionzero mode

αR0(z) = NR0 cosh(qz)η

a(1+Λ) , (67)

where NR0 is a normalization constant.We only discuss the case for positive η above, for the case of

negative η, the discussion is similar.

494 J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495

Fig. 5. Plots of the potential (58). (a) The dashed line for η = 1, a = 1, s = 0.1 and Λ = −0.5, whereas the solid one for η = 1, a = 1, s = 0.1 and Λ = −0.1. (b) The dashedline for η = 1, a = 2, s = 0.1 and Λ = −0.1, whereas the solid one for η = 1, a = 1, s = 0.1 and Λ = −0.1. (c) The dashed line for η = 1, a = 1, s = 0.5 and Λ = −0.1,whereas the solid one for η = 1, a = 1, s = 0.1 and Λ = −0.1. (d) The dashed line for η = 0.5, a = 1, s = 0.1 and Λ = −0.1, whereas the solid one for η = 1, a = 1, s = 0.1and Λ = −0.1.

Fig. 6. Plots of the potential (58) and corresponding mass spectra for η = 4, a = 1, s = 0.5 and Λ = −0.2 (left panel), and η = 4, a = 1, s = 0.1 and Λ = −0.2 (right panel).

Fig. 7. Plots of the potential (59) for η = 0.5, a = 1, s = 0.1 and Λ = −0.2 (left panel), and η = 1, a = 1, s = 0.1 and Λ = −0.2 (right panel).

J. Liang, Y.-S. Duan / Physics Letters B 680 (2009) 489–495 495

Fig. 8. Plots of the potentials for left and right fermions as well as their corresponding mass spectra. The parameters and the coupling constant are chosen as a = 1, s = 0.5,Λ = −0.2 and η = 4.

4. Summary

In summary, we investigate the possibility of localizing vari-ous matter field on the AdS4 (dS4) thick brane in AdS5. For spin 0scalar field, we find a massless zero mode and an excited statewhich can be localized on the bent brane. For spin 1 vector field,there is only a massless zero mode on the bent brane. For spin 1/2fermion field, it is shown that, in the case of no Yukawa couplingof scalar-fermion, there is no existence of localized massless zeromode for both left and right chiral fermions. In order to localizemassless fermions, we introduce the Yukawa scalar-fermion cou-pling (ηΨ F (φ)Ψ ). We discuss two types of Yukawa couplings asexamples, and find localization property of chiral fermions is re-lated to the Yukawa coupling constant, the cosmological constantof the 4-dimensional space–time as well as the parameters of themodel.

Acknowledgements

Jun Liang wish to thank Prof. Yu-Xiao Liu for helpful discus-sions. Thanks also go to the referee for comments and suggestions.Jun Liang was supported by the Postdoctoral Education Fund ofLanzhou University, and Yi-Shi Duan was supported by the Na-tional Natural Science Foundation of China.

References

[1] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, hep-th/9905221.[2] L. Randall, R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, hep-th/9906064.[3] A. Karch, L. Randall, JHEP 0105 (2001) 008, hep-th/001156.[4] S. Kobayashi, K. Koyama, J. Soda, Phys. Rev. D 65 (2002) 064041, hep-th/

0107025.[5] D. Bazeia, F.A. Brito, A.R. Gomes, JHEP 0411 (2004) 070, hep-th/0411088.[6] A. Kehagias, K. Tamvakis, Phys. Lett. B 504 (2001) 38, hep-th/0010112.[7] E. Silverstein, Int. J. Mod. Phys. A 16 (2001) 641, hep-th/0109194.[8] C. Ringeval, P. Peter, J.P. Uzan, Phys. Rev. D 65 (2002) 044016, hep-th/0107025.[9] M. Grem, Phys. Lett. B 478 (2000) 434, hep-th/9912060.

[10] M. Grem, Phys. Rev. D 62 (2000) 044017, hep-th/0002040.[11] R. Koley, S. Kar, Class. Quantum Grav. 22 (2005) 753, hep-th/0407158.[12] M. Giovannini, Phys. Rev. D 64 (2001) 064023, hep-th/0106041.[13] M. Giovannini, Class. Quantum Grav. 20 (2003) 1063.[14] R. Koley, S. Kar, Mod. Phys. Lett. A 20 (2005) 363, hep-th/0407159.[15] M. Visser, Phys. Lett. B 159 (1985) 22.[16] M. Gogberashvili, P. Midodashvili, Phys. Lett. B 515 (2001) 447, hep-ph/

0005298.[17] M. Gogberashvili, D. Singleton, Phys. Rev. D 69 (2004) 026004, hep-th/0305241.[18] R. Koley, S. Kar, Phys. Lett. B 623 (2005) 244, hep-ph/0507277.[19] M. Gogberashvili, D. Singleton, Phys. Rev. D 64 (2001) 124004, hep-th/0107233.[20] C. Csáki, J. Erlich, T.J. Hollowood, Y. Shirman, Nucl. Phys. B 581 (2000) 309,

hep-ph/0001033.[21] C. Csáki, J. Erlich, C. Grojean, Nucl. Phys. B 604 (2001) 312, hep-ph/0012143.

[22] O. Castillo-Felisola, A. Melfo, N. Pantoja, A. Ramivez, Phys. Rev. D 70 (2001)104029, hep-th/0001033.

[23] S. Kar, gr-qc/0205020.[24] I. Oda, Phys. Lett. B 496 (2000) 113, hep-ph/0006203.[25] F. Leblond, R.C. Myers, D.J. Winters, JHEP 0107 (2001) 031, hep-th/0106040.[26] C.P. Burgess, J.M. Cline, N.R. Constable, H. Firouzjahi, JHEP 0201 (2002) 014,

hep-th/0112047.[27] M. Gogberashvili, P. Midodashvili, Europhys. Lett. 61 (2003) 308, hep-th/

0111132.[28] D.K. Park, A. Kim, Nucl. Phys. B 650 (2003) 114.[29] R. Koley, S. Kar, Class. Quantum Grav. 24 (2007) 79, hep-th/0611074.[30] O. Dewolfe, D.Z. Freedman, S.S. Gubser, A. Karch, Phys. Rev. D 62 (2000)

046008, hep-th/9909134.[31] D. Bazeia, F.A. Brito, J.R. Nascimento, Phys. Rev. D 68 (2003) 085007, hep-th/

0306284.[32] R. Koley, S. Kar, Phys. Lett. B 487 (2005) 1.[33] K. Farakos, G. Koutsoumbas, P. Pasipoularides, Phys. Rev. D 76 (2007) 064025,

arXiv:0705.2364 [hep-th].[34] M. Minamitsuji, W. Naylor, M. Sasaki, Phys. Lett. B 633 (2006) 607, hep-th/

0510117.[35] A. Campos, Phys. Rev. Lett. 88 (2002) 141602, hep-th/0111207.[36] S. Pal, S. Kar, Gen. Relativ. Gravit. 41 (2009) 1165, hep-th/0701266.[37] V.I. Afonso, D. Bazeia, L. Losano, Phys. Lett. B 634 (2006) 526, hep-th/0601069.[38] D. Bazeia, F.A. Brito, L. Losano, JHEP 0611 (2006) 064, hep-th/0610233.[39] A. Wang, Phys. Rev. D 66 (2002) 024024, hep-th/0201051.[40] N. Sasakura, JHEP 0202 (2002) 026, hep-th/0201130.[41] N. Sasakura, Phys. Rev. D 66 (2002) 065006, hep-th/0203032.[42] C. Csáki, J. Erlich, C. Grojean, T.J. Hollowood, Nucl. Phys. B 584 (2000) 359, hep-

ph/0004133.[43] D.Z. Freedman, C. Núñez, M. Schnabl, K. Skenderis, Phys. Rev. D 69 (2004)

104027, hep-th/0312055.[44] K. Skenderis, P.K. Townsend, Phys. Lett. B 468 (1999) 46, hep-th/9909070.[45] A. Melfo, N. Pantoja, A. Skirzewski, Phys. Rev. D 67 (2003) 105003, gr-qc/

0211081.[46] D. Bazeia, A.R. Gomes, JHEP 0405 (2004) 012, hep-th/0403141.[47] D. Bazeia, C. Furtado, A.R. Gomes, JCAP 0402 (2004) 002, hep-th/0308034.[48] S. Randjbar-Daemi, M. Shaposhnikov, Phys. Lett. B 492 (2000) 361, hep-th/

0008079.[49] B. Bajc, G. Gabadadze, Phys. Lett. B 474 (2000) 282, hep-th/9912232.[50] I. Oda, Phys. Lett. B 508 (2001) 96.[51] H. Davoudisal, J.L. Hewett, T.J. Rizzo, Phys. Lett. B 473 (2000) 43, hep-th/

9911262.[52] Y.X. Liu, X.H. Zhang, L.D. Zhang, Y.S. Duan, JHEP 0808 (2008) 041, arXiv:0803.

0098 [hep-th].[53] Y.X. Liu, X.H. Zhang, L.D. Zhang, Y.S. Duan, JHEP 0802 (2008) 067, arXiv:0708.

0065 [hep-th].[54] Y.X. Liu, X.H. Zhang, L.D. Zhang, Y.S. Duan, Phys. Rev. D 78 (2008) 065025,

arXiv:0804.4553 [hep-th].[55] Y.X. Liu, Z.H. Zhao, S.W. Wei, Y.S. Duan, JCAP 0902 (2009) 003, arXiv:0901.0782

[hep-th].[56] J. Liang, Y.S. Duan, Phys. Lett. B 678 (2009) 491.[57] J. Liang, Y.S. Duan, Europhys. Lett. 87 (2009) 40005.[58] J.I. Díaz, J. Negro, L.M. Nieto, O. Rosas-Oritz, J. Phys. A 32 (1999) 8447.[59] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, Pergamon Press, UK, 1965.[60] S. Flügge, Practical Quantum Mechanics, Springer-Verlag, New York, 1974.[61] M. Capdequi-Peyranere, Mod. Phys. Lett. A 14 (1999) 2657.[62] E. Witten, Nucl. Phys. B 185 (1981) 513.[63] E. Witten, Nucl. Phys. B 202 (1982) 253.