Localization and mass spectra of various matter fields on scalar-tensor brane

Recently, a new scalar-tensor braneworld model was presented in [Phys. Rev. \textbf{D 86} (2012) 127502]. It not only solves the gauge hierarchy problem but also reproduces a correct Friedmann-like equation on the brane. In this new model, there are two different brane solutions, for which the mass spectra of gravity on the brane are the same. In this paper, we investigate localization and mass spectra of various bulk matter fields (i.e., scalar, vector, Kalb-Ramond, and fermion fields) on the brane. It is shown that the zero modes of all the matter fields can be localized on the positive tension brane under some conditions, and the mass spectra of each kind of bulk matter field for the two brane solutions are different, which implies that the two brane solutions are not physically equivalent.


Review of the scalar-tension brane model
Let us consider a scalar-tensor brane generated by a real scalar field φ nonminimally coupled to gravity. The action for such a system is given by [30] where R is the five-dimensional scalar curvature and M * is the fundamental scale of gravity. The line-element for a five-dimensional space-time describing a Minkovski brane with an S 1 /Z 2 orbifold extra dimension is assumed as where the conformal coordinate z ∈ [−z b , z b ] is related to the physical extra dimension coordinate y by a coordinate transformation dy = a(z)dz. According to the symmetric of the space-time, the solution of the background scalar field φ depends on extra dimension only. The field equations in the bulk derived from the action (2.1) under these assumptions reduce to the following second-order differential equations: 2kφ ′′ + 2k 2 + 4k + 3 φ ′2 + 4k(ln a) ′ φ ′ + 6 a ′′ a = 0, (2.3) φ ′ − 2(ln a) ′ (4k + 3)φ ′ + 6(ln a) ′ = 0, (2.4) (4k + 3) 2φ ′′ + kφ ′2 + 6(ln a) ′ φ ′ − 4k 2 a ′′ a + (ln a) ′2 = 0, (2.5) where the prime denotes the derivative with respect to the coordinate z. From Eq. (2.4), one has φ ′ = 2(ln a) ′ , or (4k + 3)φ ′ = −6(ln a) ′ . (2.6) We do not consider the trivial case k = −3/4 for the second equation in (2.6) because the corresponding solution for a(z) is just a constant. The following two independent brane solutions were found in Ref. [30]. Solution I: where the parameters satisfy β > 0 and k < −3/2. Solution II: where β > 0 and −3/2 < k < −3/4. In the model, there are two thin branes: a positive tension brane located at the origin z = 0 and a negative one at another orbifold fixed poiont z = z b , which is the same as the case of the RS1 model. However, our world in this model is located at the positive tension brane rather than on the negative one in order to solve the gauge hierarchy problem. As a result, a correct Friedmann-like equation on the brane can be obtained (for detail see Ref. [30]).
Stability and the zero mode of the gravitational perturbation on the scalar-tensor brane have been analyzed in Ref. [30]. Here we give a brief review of the mass spectrum of the gravitational perturbation for the two solutions (solution I and solution II).
The analyzing of a full set of fluctuations of the metric around the background is very complex. However, the problem can be simplified when one only considers the transverse and traceless (TT) part of the metric fluctuation. So, we consider the following TT tensor perturbation of the metric (2.2): ds 2 = a 2 (z) η µν +h µν (x, z) dx µ dx ν + dz 2 , (2.9) where the tensor perturbationh µν satisfies the TT condition [45]:h µ µ = ∂ νh µν = 0. The equation forh µν is given by [30] h ′′ µν + 3 a ′ ah ′ µν + kφ ′h′ µν + (4)h′ µν = 0. (2.10) By performing the following decomposition h µν (x, z) = ε µν (x)J − 3 2 (z)h(z), (2.11) where the function J(z) is defined as J(z) = a(z)e kφ/3 , and its explicit expressions for solution I and solution II are same: J(z) = (1 + β|z|) 1 3 , (2.12) we get from Eq. (2.10) the Klein-Gordon equation (4) ε µν (x) = m 2 ε µν (x) for the fourdimensional gravity ε µν (x), and a Schrödinger-like equation for the KK mode h(z): Here m is the mass of the four-dimensional graviton (the gravitational KK excitation) and the effective potential is given by (2.14) The explicit expressions of V G (z) for both solutions are the same: where the second delta function in the right hand side of the above equation comes from the Z 2 symmetry with respect to the brane located at z = z b . Then, the values of the effective potential V G (z) at z = 0 and z = z b are from which we can deduce that the gravity zero mode may be localized on the negative tension brane located at z = z b . By setting m = 0 in Eq. (2.13), one can easily get the normalized zero mode It is clear that the zero mode is localized on the negative tension brane for the finite z b , but cannot be normalized anymore when the extra dimension is infinity, which is very different from the RS1 model. By solving the Schrödinger-like equation (2.13), the spectrum of gravity is [30]: where x n satisfies J 1 (x n ) = 0, and x 1 =3.83, x 2 =7.02, x 3 =10.17, · · · . Here, J 1 (x) is the Bessel function of the first kind. In order to solve the hierarchy problem, we need to set all the fundamental parameters including the five-dimensional scale of gravity M * , the parameter β, and the Higgs vacuum expectation value v 0 , to be about the TeV scale and βz b ≈ 10 16 [30]. So we have β ≈ 10 12 eV, z b ≈ 10 4 eV −1 , and m n (z) = x n × 10 −4 eV. Note that the mass spectra for both the two brane solutions are the same and the mass spacing of KK gravitons is much smaller than that of the RS1 model with the TeV scale spacing. So we cannot distinguish the two brane solutions from the gravitational mass spectrum. However, we will see in the following sections that the mass spectra of various matter fields are different for the two brane solutions, and so they are not physically equivalent.

Localization and Mass Spectra of various matters on the scalar-tensor brane
In this section, we would like to investigate localization and mass spectra of various bulk matter fiedls on the scalar-tensor brane by deriving the effective potentials of the KK modes of various bulk matter fiedls in the corresponding Schrödinger-like equations. The bulk matter fiedls include the spin-0 scalar, spin-1 vector, KR, and spin-1/2 fermion fields. For the spin-1/2 fermion field, we will introduce a new scalar-fermion coupling in order to localize the fermion on the brane. Here, we implicitly assume that the various bulk matter fields considered below are perturbations around the background space-time and they make little backreaction to the bulk energy so that the brane solutions given in section 2 remain valid.

Spin-0 scalar field
We first consider the localization of the massless real scalar field on the scalar-tensor brane reviewed in section 2. The five-dimensional action for a massless real scalar field coupled to the dilaton field φ is given by where λ is the coupling constant between the scalar and dilaton fields. By considering the action (3.1) and the metric (2.2), the equation of motion for the scalar field is read as In the following, we will discuss localization and mass spectrum of the scalar field for solution I and solution II. Solution I: With the KK decomposition of the scalar field where the function A(z) is defined as A(z) = ln a(z), the equation of motion for the extradimensional part χ n (z) of the scalar KK mode is recast into the following Schrödinger-like equation −∂ 2 z + V 0 (z) χ n (z) = m 2 χ n (z), (3.4) where m is the mass of the scalar KK mode and the effective potential V 0 (z) is given by By introducing the orthonormality condition we get the four-dimensional effective action of a massless and a series of massive scalar fields from the five-dimensional one: The explicit expression of the effective potential V 0 (z) for solution I is The values of V 0 (z) at z = 0 and |z| = z b read In order to localize the scalar zero mode on the positive tension brane located at z = 0, the effective potential V 0 (z) should be negative at z = 0. With k < − 3 2 and β > 0 for solution I, the condition is turned out to be By setting m = 0 in Eq. (3.4) and noting that the boundaries of the extra dimension are at z = 0 and z = z b , we get the normalized zero mode of the scalar field when λ = −3 − k: where the normalization coefficient is In order to localize the scalar zero mode on the positive tension brane, we need the following condition which is consistent with the condition (3.11). We note here that, when there is no coupling between the scalar and dilaton fields (λ = 0), the scalar zero mode is also localized on the positive tension brane. Besides, if the extra dimension is infinite, the normalization condition +∞ −∞ χ 2 0 (z)dz = 1 should be satisfied, which shows that the localization condition for the scalar zero mode is much stronger: λ > −3 − k.
When the coupling constant λ = −3 − k, the normalized scalar zero mode χ 0 (z) is For this special coupling, the zero mode can also be localized on the positive tension brane when z b is finite. But it cannot anymore when the extra dimension is infinite. In order to get the massive spectrum of the scalar KK modes, we assume that the extra dimension is finite. The Z 2 symmetry requires that the KK modes satisfy ∂ z (e − 2λ+3 2 A(z) χ n ) = 0 at the boundary z = 0, with which the general solution of Eq. (3.4) is turned out to be χ n (z) = N (1 + β|z|) 16) where N is the normalization coefficient, J P 1 s (z) and Y P 1 s (z) are respectively the Bessel functions of first and second kinds, and . (3.17) The Z 2 symmetry also requires that the KK modes satisfy ∂ z (e − 2λ+3 2 A(z) χ n (z)) = 0 at another boundary z = z b . When we consider the light modes in the long range case, i.e., m n /β ≪ 1 and 1 + βz b ≫ 1, the spectrum of the scalar KK modes can be determined by the following equation We can obtain the mass spectrum of the scalar field by numerical calculation. For example, when the parameters are set to β = 10 12 ev, z b = 10 4 ev −1 , λ = 1, k = −2, the mass spectrum is The explicit spectrum of the scalar field is shown in Fig. 1 for different λ and k, from which it can be seen that the mass gap between the massless mode and the first massive KK mode increases with the coupling constant λ and the non-minimal coupling parameter k. Besides, the mass spectrum is relatively sparse at the lower excited states but approaches equidistant for the higher excited states.

Solution II:
Similarly, for solution II, with the following decomposition it can also be shown that the extra-dimensional part χ n (z) of the scalar KK mode satisfies the Schrödinger-like equation (3.4) but with the following effective potential V 0 (z): By using the orthonormality condition (3.6), we get the same four-dimensional effective action given in (3.7).
Substituting the second solution of A(z) = ln a(z) in Eq. (2.8) into Eq. (3.21), we get the explicit expression of V 0 (z): The values of V 0 (z) at z = 0 and |z| = z b are In order to localize the scalar zero mode on the positive tension brane, the effective potential V 0 (z) should be negative at z = 0. The condition is turned out to be By setting m = 0 in Eq. (3.4) and assuming λ = 3 + 3k, we get the normalized scalar zero mode where the normalization coefficient is In order to localize the scalar zero mode on the positive tension brane, we need the following condition: which is same with the condition (3.25). So we reach the conclusion that when the extra dimension is finite, the effective potential can trap the massless scalar field on the positive tension brane providing the condition that λ > 3+4k 2 . It is interesting to note that when there is no coupling between the scalar and dilaton fields (λ = 0), the scalar zero mode can still be localized on the positive tension brane since 3 + 4k < 0 for solution II. When the negative tension brane is moved to infinite, the localization condition is much stronger: λ > 3 + 3k. In this case, the zero mode of a free scalar field (λ = 0) can be localized on the positive tension brane only if − 3 2 < k < −1.
When the coupling constant λ = 3 + 3k, the scalar zero mode χ 0 (z) is the same as that in Eq. (3.15) for solution I. For this special case and finite extra dimension, the scalar zero mode is also localized on the positive tension brane.
Here we also assume that the extra dimension is finite in order to get the explicit mass spectrum of the scalar KK modes. With the condition ∂ z e 6λ−3(3+4k) 2(3+4k) A(z) χ n z=0 = 0, which is required by the Z 2 symmetry, the general solution of Eq. (3.4) is where P 2 s ≡ P 1 s and .
With the second boundary condition ∂ z e 6λ−3(3+4k) 2(3+4k) A(z) χ n z=z b = 0, we consider the light modes in the long range case (m n /β ≪ 1 and 1 + βz b ≫ 1). The spectrum of the scalar field can be determined by the following equation The numerical scalar mass spectrum is shown in Fig. 2. One can see that the mass gap between the zero mode and the first massive KK mode is increased with the coupling constant λ but decreased with the parameter k, which is different from that in solution I.

Spin-1 vector field
Now we turn to the spin-1 vector field. Let's begin with the five-dimensional action of a vector field coupled to a dilaton field φ: is the field strength tensor and τ is the coupling constant between the vector and dilaton fields. Considering the explicit form of the metric (2.2), the equations of motion for the vector field are read as By using gauge freedom and the Z 2 symmetry of extra dimension, we can set the fourth component A 4 = 0. Next, we investigate localization of the zero mode and KK mass spectrum of the vector filed on the scalar-tensor brane for the two brane solutions. Solution I: By performing the following KK decomposition for the vector field one can show that the extra dimension part ρ n (z) of the vector KK mode satisfies the following Schrödinger-like equation where the effective potential V 1 (z) is given by With the orthonormality condition we can get the four-dimensional effective action for a series of vector fields: µ is the four-dimensional vector field strength tensor. Substituting solution I into Eq. (3.36), we get the explicit expression of the effective potential V 1 (z): The values of V 1 (z) at z = 0 and |z| = z b are In order to localize the vector zero mode on the positive tension brane, the effective potential V 1 (z) should be negative or have a well-like shape near z = 0. For k < − 3 2 and β > 0, the condition is turned out to be By setting m = 0 in Eq. (3.35), we get the normalized vector zero mode when τ = −2−k: where the normalization coefficient is given by Then the localization condition for the vector zero mode is turned out to be So, when the extra dimension is finite, the vector zero mode can be localized on the positive tension brane providing that the coupling constant between the vector and dilaton fields satisfies τ > − 1 2 . Clearly, the vector zero mode still can be localized on the brane even if there is no coupling between the vector and dilaton fields. When the extra dimension size is infinite, i.e., z b → ∞, the localization condition becomes τ > −2 − k.
When τ = −2 − k, the normalized vector zero mode ρ 0 (z) is It is localized on the positive tension brane only for the case of finite extra dimension. Next, we will investigate the massive KK modes of the vector field by assuming that the extra dimension is compact and finite. With the boundary condition ∂ z (e − 1+2τ 2 A(z) ρ n )| z=0 = 0, we get the general solution of Eq. (3.35): where .

(3.48)
With another boundary condition at z = z b : ∂ z (e − 1+2τ 2 A(z) ρ n )| z=z b = 0, the spectrum of the vector KK modes for light modes in long range case is determined by the following equation The mass spectrum is plotted in Fig. 3, which shows that the mass gap between the massless and first massive modes increases with the coupling constant τ and the parameter k. Besides, the spectrum interval approaches a constant for higher excited states.

Solution II:
Similarly, by making the KK decomposition of the vector field we can get the Schrödinger-like equation for the KK modes ρ n (z), which is also given by Eq. (3.35). Whereas the corresponding effective potential V 1 (z) is We also need the orthonormality condition in Eq.
Then the values of V 1 (z) at z = 0 and |z| = z b are where the normalization coefficient is given by (3.57) In order to localize the vector zero mode on the positive brane with finite extra dimension, the following condition should be satisfied: Clearly, the vector zero mode can still be localized on the positive tension brane even when there is no coupling between the vector and dilaton fields since −3/2 < k < −3/4. However, when the negative tension brane is moved to infinite, the localization condition is much stronger: τ > 6+5k 3 . For τ = 6+5k 3 , the zero mode ρ 0 (z) is the same as that in Eq. (3.46). The boundary conditions ∂ z (e − 3+4k−6τ 2(3+4k) A(z) ρ n )| z=0,z b = 0 will again decide the solution and mass spectrum of the vector KK modes. Considering the first boundary condition, the general solution of Eq. (3.35) for Soultion II read as where N is the normalization constant, and From the second boundary condition, when we consider the light modes in the long range case, the spectrum of the vector KK modes is determined by the following equation The spectrum is shown in Fig. 4, from which we can see that the mass of the first massive mode increases with the coupling constant τ but decreases with the parameter k. The mass gap between two adjoining KK modes, i.e., ∆m n = m n − m n−1 , becomes smaller and smaller with the increase of the level n and then trends to a constant.

Kalb-Ramond field
The action describing a five-dimensional KR field coupled with a dilaton field is given by  We choose the fourth component B α4 = 0 by using gauge freedom. Next, just like the vector field, we will discuss localization and mass spectrum of the KR KK modes for the two brane solutions considered in this paper.

Solution I:
With the KK decomposition of the KR field and providing the orthonormality condition for the KK modes Ω m and Ω n : dz Ω m (z)Ω n (z) = δ mn , (3.66) we get the following Schrödinger-like equation for the KR KK modes: Here the effective potential V KR (z) is Then the action of the KR field is reduced to withĥ (n) σαβ = ∂ σbαβ − ∂ βbασ the field strength of the four-dimensional KR field. The explicit expression of the effective potential V KR (z) for the KR KK modes is whose values at z = 0 and |z| = z b are respectively In order to get negative potential around z = 0, the coupling constant should satisfy the following constrain: By setting m = 0 in Eq. (3.67), we get the KR zero mode for the case of ζ = −1 − k where the normalization coefficient is In order to localize the KR zero mode (the four-dimensional massless KR field) on the brane located at z = 0, the following condition should be satisfied: Note that, different from the case of the vector field, the KR zero mode cannot be localized on the positive tension brane anymore if there is no coupling between the KR and dilaton fields (ζ = 0). When the size of extra dimension is infinite, the KR zero mode can be localized on the positive tension brane if ζ > −1 − k. When ζ = −1 − k, the zero mode Ω 0 (z) is It is clear that the KR zero mode for this case is also localized on the positive tension brane. The Z 2 symmetry implies that the KR KK modes should satisfy ∂ z (e − 7+2ζ 2 A(z) Ω n ) = 0 at z = 0, with which we get the general solution of Eq. (3.67): . (3.80) With another boundary condition ∂ z (e − 7+2ζ 2 A(z) Ω n ) = 0 at z = z b , we can obtain the mass spectrum from the following equation when we consider the light modes in the long range case: The mass spectrum for different values of the parameters is plotted in Fig. 5, from which we can see that m 1 increases with the coupling constant ζ and the parameter k, and the mass gap ∆m n approaches a constant for large level n.
The values of V KR (z) at z = 0 and |z| = z b are When − 3 2 < k < − 3 4 , in order to get negative potential around z = 0, the coupling constant should satisfy the following constrain: (3.87) By setting m = 0 in Eq. (3.67), when ζ = 3+k 3 , we get the solution of the KR zero mode with the coefficient N KR given by The condition for localizing the KR zero mode on the positive tension brane is turned out to be When the size of extra dimension is infinite, the KR zero mode can be localized on the positive tension brane only if ζ > 3+k 3 . When ζ = 3+k 3 , the zero mode Ω 0 (z) is the same as that in Eq. (3.77). . (3.93) The mass spectrum for the light modes in the long range case is determined by the following equation The spectrum is shown in Fig. 6. It can be seen that m 1 increases with the coupling constant ζ but decreases with the parameter k.

Spin-1/2 fermion field
Finally, we turn to investigate localization and mass spectrum of a spin-1/2 fermion field on the scalar-tensor brane. In order to localize a fermion on the thick brane generated by an odd scalar field φ(z), the Yukawa coupling should be introduced [13,14,23,25,[32][33][34][35][36][37][38][39][40][41][42]. But when the scalar field φ(z) is even, the Yukawa coupling (i.e. ηψF (φ)ψ) cannot preserve the Z 2 reflection symmetry of the fermion Lagrangian and hence cannot ensure localization of the fermion on the brane. Recently, the authors of Ref. [31] analyzed this problem and introduced a new localization mechanism to localize the fermion. Following this mechanism, we would like to analyze localization and spectrum of a fermion on the scalar-tensor brane by using the new scalar-fermion coupling form ηΨΓ M ∂ M F (φ)γ 5 Ψ.
In five dimensions, a fermion is a four-component spinor and its Then the five-dimensional Dirac equation is read as where γ µ ∂ µ is the Dirac operator on the brane. Now we study the above five-dimensional Dirac equation. Because of the Dirac structure of the fifth gamma matrix γ 5 , we expect the left-and right-handed projections of the fourdimensional part to behave differently. From the Dirac equation (3.97), we will search for solutions of the general chiral decomposition (3.98) wheref L,R (z) = a −2 (z)f L,R (z), ψ Ln (x) = −γ 5 ψ Ln (x) and ψ Rn (x) = γ 5 ψ Rn (x) are the lefthanded and right-handed components of a four-dimensional Dirac field, respectively, the sum over n can be both discrete and continuous. Here, we assume that ψ Ln (x) and ψ Rn (x) satisfy the four-dimensional massive Dirac equations γ µ ∂ µ ψ Ln (x) = m n ψ Rn (x) and γ µ ∂ µ ψ Rn (x) = m n ψ Ln (x). Then the left-and right-handed KK modes f Ln (z) and f Rn (z) satisfy the following coupled equations From the above coupled equations, we get the Schrödinger-like equations for the KK modes of the left-and right-handed fermions where the effective potentials are given by In order to obtain the standard four-dimensional action for the massless chiral fermion and massive fermions: we need the following orthonormality conditions for f Ln and f Rn : It can be seen from Eq. (3.102) that, in order to localize the left-or right-handed fermions, there must exist some kind of scalar-fermion coupling (η = 0), and the effective potential V L (z) or V R (z) should have a minimum at the location of the brane.
In this paper, we take F (φ(z)) = e vφ and investigate localization and spectrum of the fermion field for solution I and solution II.
Solution I: For the brane solution I, the explicit forms of the effective potentials (3.102) read as When v = (2k+3)/2, the left-and right-handed potentials in the bulk are the same constants: The values of V L,R (z) at z = 0 are The shapes of the potentials are shown in Fig. 7.
The solutions of the left-and right-handed fermion zero modes are It is clear that the left-and right-handed fermion zero modes cannot be localized on the positive tension brane at the same time. The left-handed fermion zero mode can be localized on the positive tension brane if ηv > 0 and the extra dimension is finite. In order to check whether the left-handed zero mode f L0 (z) can be localized when the extra dimension is infinite, we need to consider the normalization condition for the solution, which is given by Since 2k + 3 < 0 for solution I, the condition is turned out to be If the extra dimension is finite, then the right-handed fermion zero mode is localized on the negative tension brane under the condition ηv > 0. Next, we consider the massive fermion KK modes for the case of finite extra dimension. For simplicity, we only consider a free fermion, which means that the coupling constant is set to zero (η = 0). The boundary conditions are decided by the Z 2 symmetry: ∂ z (a −2 (z)f L,R (z))| z=0,z b = 0. The general solutions of the massive fermion KK modes are f Ln,Rn (z) = N cos(m n z) + 2β (3 + 2k)m n sin(m n z) . With the boundary condition at z = z b , the exact spectrum is determined by the following equation: (3.115) For those fermion KK modes satisfying m n ≪ 0.1TeV, the approximative mass spectrum is given by The mass spectrum numerically calculated from Eq. (3.115) and the approximative one given in Eq. (3.116) are plotted in Fig. 8. From Fig. 8(a), we reach the conclusion that the mass of the first massive fermion KK mode m 1 increases with the parameter k. From Fig. 8(b), we see that the approximative analytical spectrum is consistent with the exact numerical one. Solution II:  For the brane solution II, the explicit forms of the effective potentials (3.102) are The values of the effective potentials V L,R (z) at z = 0 are given by The left-and right-handed fermion zero modes can be solved When the extra dimension is finite, the left-handed fermion zero mode can be localized on the positive tension brane under the condition ηv > 0. When the extra dimension is infinite, we need the following normalization condition in order to localize the left-handed fermion zero mode: Since 2k + 3 > 0 for solution II, the condition is turned out to be For the massive KK modes, we consider the case of finite extra dimension. Similar to solution I, the KK modes of a free fermion are given by f Ln,Rn (z) = N cos(m n z) + 2(3 + 4k)β 3(3 + 2k)m n sin(m n z) . (3.125) The mass spectrum is determined by the following equation: tan(m n z b ) = 6(9 + 18 + 8k 2 )m n z b β 2 4(3 + 4k) 2 β 2 + 9(3 + 2k) 2 m 2 n z b β . (3.126) Note that m 1 decreases with k. The approximative spectrum is × 10 −4 eV, (n = 1, 2, · · · ) (3.127) which is the same as the case of solution I.

Discussions and conclusions
The scalar-tensor braneworld model presented in Ref. [30] not only solves the gauge hierarchy problem but also reproduces a correct Friedmann-like equation on the brane, and so overcomes the cosmological problem in the Randall-Sundrum model. In this model, there are two similar but different brane solutions. In each solution, there are two branes, one with positive tension and another with negative tension. Our world is confined on the positive tension brane. The tensor perturbation of the brane system is stable and the mass spectra of the gravitational KK modes for both brane solutions are the same. Therefore, one cannot distinguish the two solutions by the mass spectra of gravity.
In this paper, we investigated localization of the zero modes and mass spectra for various bulk matter fields (i.e., scalar, vector, KR, and fermion fields) on the scalar-tensor brane. For the scalar, vector, and KR fields, we considered their interaction with the background scalar field (the dilaton φ) that generates the brane. For the fermion, following Ref. [31], we introduced a new scalar-fermion couplingΨΓ M ∂ M e vφ γ 5 Ψ instead of the usual Yukawa coupling for the reason of the even parity of the dilaton φ. We found that the mass spectra of each bulk matter field for the two brane solutions are different, which implies that the two brane solutions are not physically equivalent.
It was found that the zero modes of various bulk matter fields can be localized on the positive tension brane for the two solutions under some conditions, which are collected in Table 1. It can be seen that the localization conditions for the case of infinite extra dimension are stronger than the case of finite extra dimension. When extra dimension is finite, the scalar and vector zero modes can be localized on the positive tension brane even if there is no interaction with the background scalar field, while the KR and fermion zero modes cannot be localized if there is no interaction.
The bound massive KK modes and discrete mass spectra were also obtained for the case of finite extra dimension. For the scalar, vector, and KR fields, we get the analytical solutions of their KK modes by solving the corresponding Schrödinger-like equations, and the numerical mass spectra by considering the boundary conditions. It was found that the mass of the first massive KK mode m 1 increases with the coupling constant and the parameter k for solution I, while it increases with the coupling constant but decreases with the parameter k Fermion L 1/2 =ΨΓ M D M + η∂ M e vφ γ 5 Ψ finite ηv > 0 ηv > 0 infinite η < 0, v < 0 η < 0, v < 0 for solution II. The mass gap between two adjoining KK modes becomes smaller and smaller with the increase of the level n and then trends to a constant.
For the fermion field, we did not find the analytic solution of the fermion KK modes when there exists coupling because the effective potential is too complex. Therefore, we only considered the free fermion and got the approximate analytical mass spectrum. It was found that the numerical m 1 slowly increases and decreases with the parameter k for solution I and solution II, respectively. The mass spectrum approaches equidistant for the higher excited states.