Confinement of bosonic and spinning particles in braneworlds
F. E. A. Souza G. Alencar L. F. F. Freitas R. R. Landim


Airways: Optimization-Based Planning of Quadrotor Trajectories according to High-Level User Goals
    
	Christoph Gebhardt[2] , Benjamin Hepp[2] , Tobias Nägeli, Stefan Stevšić, Otmar Hilliges

	Departement of Computer Science, AIT Lab

	ETH Zürich


    Received: date / Revised version: date
========================================================================================================================================================





§ INTRODUCTION

Since the emergence of the Kaluza-Klein models <cit.>, higher dimension gravitational scenarios have attracted attention of physicists. In this context, L. Randall and R. Sundrum (RS) proposed the first braneworld models <cit.>. Even today, RS models play an important role in the extra dimension context. Mainly, because they allow us recover the well-known  gravitational theory, even with an infinitely large extra dimension. This is possible because gravity is confined on a 4D hypersurface (3-brane) embedded in a 5D spacetime. Inspired by RS models, many other braneworld models with confined gravity were proposed in different dimensional configuration <cit.>. For all these models, gravity is confined on a 3-brane, allowing us to recover the 4D gravitational theory. However, the same is not true for the matter fields, as it was largely verified for all the above models <cit.>. 


The confinement of the fields in these braneworld leads, evidently, to the localization of the particles related to these fields. However, this interpretation can be obtained only in quantum level. The discussion about dynamics and the confinement of a classical test particle was little discussed in the literature. In fact, dynamics of particles was largely discussed for 5D Kaluza-Klein models <cit.>. About confinement, this study was little addressed. For RS models, ref. <cit.> discussed the geodesic motion of `free' test particles in RS-II delta-like braneworld. From this analysis, the authors showed that the motion in the extra dimension y is decoupled of the other dimensions and it is given by 

    |y(t)|=1/2kln(1-v^2k^2t^2).

Where k^2>0 and is related to the cosmological constant. Therefore, ordinary matter (v^2<0) is inevitably expelled into the extra dimension and it cannot be confined. A similar discussion was also performed in ref. <cit.> for the thick brane model <cit.>. But for this, the authors do not found an analytical solution for y(t). They just found an effective `potential' provided by the curved spacetime and, from this, the confinement was discussed. Just like the delta-like model, the confinement of the `free' particle on this thick brane cannot be attained. Therefore, a localization mechanism to trap the test particle on the 3-brane is necessary. In Ref. <cit.>, the authors provided the confinement through the coupling of the particle with a scalar field ϕ. This interaction is made by modifying the `free' particle action through a redefinition of the mass in 5D given by M_0→√(M_0^2+h^2ϕ^2). This mechanism provides the confinement for some values of mass. 

In this context, we will discuss the confinement of a classical test particle in 5D braneworlds. In doing this, let us propose a new localization mechanism for massive test particles which will allow us the confinement for any value of mass. Beyond this, we will also discuss the confinement of a spinning particle (superparticle) proposed by L. Brink et.al. in Ref. <cit.>. This model presents some interesting features in 4D and it was not explored yet in a braneworld context. This paper is organized as follows. First, we discuss the bosonic test particle. In doing this, we will make some comments about the results found in the literature. Next, we present and develop the new localization mechanism. Finally, we conclude with the discussion about the confinement of the spinning particle.




§ CONFINEMENT OF BOSONIC TEST PARTICLE

In this section, we will discuss the confinement of a bosonic test particle in codimension one braneworlds. In order to get a more general analysis, let us consider a generic background metric given by

    ds^2=e^2A(y)η_μνdx^μdx^ν+e^2B(y)dy^2.

Warp factors A(y) and B(y) are functions only the extra coordinate y, which is considered infinitely large. In addition, η_μν is the Minkowski metric with signature (-,+,+,+). As mentioned above, this metric is considered just as a background, therefore, we will not consider possible backreaction effects of the particles on it. This generic shape of the metric will allow us to discuss a variety of braneworlds, among them, the delta-like RS-II model <cit.> and also some thick brane models <cit.>.

To start this discussion about the confinement of a bosonic test particle, let us use an alternative form for the action of a free relativistic particle given by

    S=1/2∫[E^-2g_PQẋ^Pẋ^Q-M^2]E dτ.

In this action, E is the vierbien, ẋ^P=dx^P/dτ and M is the particle mass in 5D. For the free case, the particle mass is a constant M_0. However, as we will also discuss the interacting case (localization mechanism), let us consider that this mass is a function of the extra coordinate M=M(y). Generally, the action used to discuss the dynamics of relativistic particles is that in Eq. (<ref>). However, action (<ref>) presents an important advantage to our discussion because it allows us to study the massless case. Just like Eq. (<ref>), action (<ref>) is invariant by the arbitrary parameter transformation
τ→τ' =f(τ),
since the vierbein E changes as
E→ E'=dτ/dτ'E.
This vierbein plays the role of a tetrad on the worldline of the test particle. However, as we will show below, E is not really a dynamic variable and it can be chosen arbitrarily. The choice of vierbein means setting the parameterization (`gauge' condition).


From the action (<ref>), we can obtain the equations of motion

    δ S/δ E=0 → g_PQẋ^Pẋ^Q+E^2M^2(y)=0,
    δ S/δ x^N=0 →D/Dτ[E^-1ẋ^N]+δ_y^NEMM'e^-2B=0.


Note that, by replacing the equation (<ref>) in the action (<ref>), we get

    S=-∫ M(y)√(-g_PQẋ^Pẋ^Q) dτ,

which gets the common action for a test particle when M(y)=M_0. This show that the actions (<ref>) and (<ref>) are equivalent. Below, let us discuss the general procedure for the confinement of the bosonic particle.



 §.§ The General Procedure


When discussing the localization of a test particle, we must analyze its motion in the extra dimension y. If we can associate a potential to the motion in y, the particle will be confined when this potential has a minimum located over the brane, namely, y=0. In general, the localization of the test particle cannot be achieved for the free case, i.e., M(y)=M_0 <cit.>. Therefore, we discuss the general case with arbitrary M(y) to include possible localization mechanisms. 

By using the metric (<ref>), the Christoffel symbols are given by

    Γ^ρ_MN   =   [δ_N^ρδ_M^y+δ_M^ρδ_N^y]A',
    Γ^y_MN   =   δ_M^yδ_N^yB'-δ_M^μδ_N^νg_μνe^-2BA'.

With this, our equations for x^P in Eq. (<ref>) becomes

    D/Dτ[E^-1ẋ^μ]=d/dτ[E^-1ẋ^μ]
    +2A'E^-1ẋ^μẏ=0,
          D/Dτ[E^-1ẏ]+EMM'e^-2B=d/dτ[E^-1ẏ]+E^-1B'ẏ^2
          -E^-1A'g_μνẋ^μẋ^νe^-2B+ EMM'e^-2B=0.

In the above equations, prime means derivative regarding extra coordinate y.  

The first step is to choose a `gauge' condition and fix E. It is convenient to define it such that the parameter τ can be identified with the proper-time on the brane. This can be attained by choose E=e^2A(y). By using this gauge and equation (<ref>) we can show that 

    g_μνD/Dτ[E^-1ẋ^ν]=d/dτ[e^-2Ag_μνẋ^ν]
    ẏe^2B[D/Dτ(e^-2Aẏ)+e^2AMM'e^-2B]=
     =1/2d/dτ[e^2(B-A)ẏ^2+M(y)^2e^2A].
 
Therefore, from Eqs. (<ref>) and (<ref>) we get 

    e^-2Ag_μνẋ^ν=p_μ=constant,
     
    e^2B(y)-2A(y)ẏ^2+M(y)^2e^2A=-C^2=constant.
 
Where in the first equation we used the definition of the four-momentum 

    p_μ=dL/dẋ^μ=e^-2Ag_μνẋ^ν.
 
In order to interpret the constant C^2, we can use Eq. (<ref>) and (<ref>), thus, such constant is given by C^2=p^2. Therefore, the equation (<ref>) can be interpreted as the total energy E=T+U and, from this, we can get an effective `potential'. In doing this, we get 
    U_eff=M(y)^2e^2A(y)-M(0)^2.
 
Therefore, the localization of massive particles can be taken out just from the analysis of this potential. Below, let us discuss the confinement and the effective mass observed over the brane.

First, the particle will be confined on a brane placed at y=0 only if the potential (<ref>) satisfies

    U'_eff(0)=2M(y)M'(y)e^2A(y)(0)=0 → M'(0)=0

where we have used the fact that A'(0)=0 in braneworld models<cit.>. Beyond this, by using the above condition it must also satisfy 

    U”_eff=2e^2A(y)[MM”+M^2A”]_y=0>0.

We should remember that the above conditions are necessary but not sufficient to confine a particle. If the effective potential has a volcano like shape, a particle with large momentum in extra dimension could escape. Therefore, given that U_eff has a maximum at y_max, the particle will be trapped if ẏ_max=0. Given this, an important question is: what is the observed effective mass of the test particle? This can be determined from Eq. (<ref>) and using that p^2=-m^2_eff. From the above discussion, we can replace y_max in Eq. (<ref>) with ẏ_max=0 and get

    m^2_eff=M^2(y_max)e^2A(y_max).

Thus, the above expression gives us the effective mass of the particle and it is closely related to the return point y_max. Evidently, if the return point y_max does not exist, the effective particle mass cannot be obtained from (<ref>). For this case, the particle is not confined and the concept of effective mass over the brane is meaningless. Now we can apply the above procedure. 




 §.§ The Free Particle

A simple application of the above conditions is the free particle, with M=M_0=constant. For this massive case, condition (<ref>) is satisfied and (<ref>) imposes the condition

    . A”(y)|_y=0>0.

However, to build the braneworld model, the Einstein's equations provide us with A”(y)<0 <cit.>. In this way, condition (<ref>) cannot be satisfied and the localization cannot be attained for any massive test particle. This result is not new and was already discussed in the literature for the delta-like Randall-Sundrum model <cit.> and also in Ref. <cit.> for the thick brane model <cit.>. 

However, the authors of Refs. <cit.> did not consider the massless case. This is because they started from the action (<ref>), which is ill-defined for M=0. With our approach, this extension can be obtained trivially. For the massless case, the potential is null and the massless particle can be not be confined.





§ LOCALIZATION MECHANISMS FOR MASSIVE PARTICLES

In this section we must  consider some models to confine test particles.   



 §.§ Coupling With Scalar Field

As we saw previously, the confinement of free massive particles cannot be attained for RS-like models. To solve this problem, in Ref. <cit.>, the authors provided the confinement of test particles by coupling it with a scalar field ϕ in thick brane scenarios. This interaction is introduced in the action (<ref>) by modifying the particle mass as 

    M(y)^2=M_0^2+h^2ϕ^2.

This mechanism is based on the Yukawa interaction used to provide the spinor mass in field theory. With this mechanism, our conditions (<ref>) and (<ref>) for confinement can be translated to 

    M_0<√(3)h/κ,

where, κ is the gravitational coupling constant in 5D. Thus, this mechanism provides us with an upper limit for the allowed values of confined mass. This upper limit, of course, can be as large as we wish because there is no constraint on the free parameter h. 



 §.§ Coupling With Dilaton Field

It is a know fact that the dilaton field can be used to localize fields in RS-like models. Based on this, in this section we investigate if this kind of coupling can localize a test particle. In Ref. <cit.>, the authors discuss the dynamics of particles in a context other than braneworld scenario. In such study, they provide the interaction of particles with the dilaton field by replacing M_0→ M_0 e^λπ.
Where π is the dilaton. Based on this, let us use this same proposal to discuss the confinement of the massive test particle. For the models where the dilaton is present<cit.>, this field is proportional to the warp factor A(y) and given by π(y)=-√(3M_p^3)A(y), where M_p is the 5D Planck scale. Therefore, we get the effective potential 

    U_eff=M_0^2(e^2(1-λ√(3M_p^3))A-1).

With this,  the conditions (<ref>) and (<ref>) will be translated to 

    A'(0)=0,    -[λ√(3M_p^3)-1]A”(0)>0.

Therefore, since A'(0)=0, the first condition is satisfied and as we already discussed for the free case, all the brane models with localized gravity obey A”(0)<0. Therefore, the above condition will be satisfied only when

    λ>1/√(3M_p^3).

Note that we did not need to know the exact form of the warp factor A(y), but only its behavior at y=0. This result is interesting because, unlike the result (<ref>), obtained by using the localization mechanism (<ref>), the confinement is obtained for any value of the particle's mass M^2_0>0. In view this, the number of free parameters is reduced and the confinement depends only on the gravitational constant. We also point that U_eff in Eq. (<ref>) is unbounded from above. Therefore, we get that the dilaton field provides the confining of particles with arbitrary mass. 




§ SPINNING PARTICLE

All the above discussion was performed for a bosonic test particle. Thus, it does not consider any information about the particle's spin. Commonly, `classical description' of spin is carried out in the framework of fields. In fact, there is no classical description of the spin. However, by starting from a suitable Lorentz group representation for classical fields, spin description can be attained after the appropriate quantization process. For the test particle case, spin degree of freedom can be included by using Grassmann variables. These models are called pseudomechanic or superparticle model. This was performed in 4D for flat <cit.> and also curved <cit.> spacetimes, however, this was not discussed yet in braneworlds context. Based on these references, we will discuss below the confinement of spinning particles in codimension one braneworld. In doing this, let us start from the action

    S=1/2∫[E^-1g_PQẋ^Pẋ^Q-EM^2_0-ig_PQψ^PDψ^Q/Dτ.
    .-iξξ̇-iχ(E^-1g_PQẋ^Pψ^Q+M_0ξ)]dτ.

Where,

    Dψ^Q/Dτ=ψ̇^Q+Γ^Q_SRψ^Sẋ^R

and this covariant derivative is necessary because, unlike x^N, variable ψ^N is a vector by general coordinate transformations. Here, evidently, this action deserves some explications. First, coordinates x^μ, y and E are commuting variables (c-numbers), and ψ^N, ξ and χ are anticommuting variables (a-numbers). In this configuration, variables x^μ and y describe the geodesic motion, E is the vierbein, quantities ψ^N and ξ will be related to the spin description and χ is necessary by symmetry requirement. Just like the action (<ref>), the action presented above is invariant by reparametrization τ→τ'=f(τ). Beyond this, action (<ref>) is also invariant by the local SUSY transformations,

    δ x^N=iαψ^N,    δ E=iαχ,    δχ=2α̇,
    δξ =M_0α+i/M_0Eαξ(ξ̇-1/2M_0χ),
    δψ^N=α E^-1(ẋ^N-i/2χψ^N).

Where α is a real anticommuting parameter. For a more detailed discussion about this pseudomechanic, we recommend <cit.>. Next, we discuss the equation of motion for the action (<ref>). In doing this, our objective is to verify if the new spin variables can modify the effective potential U_eff(y) in order to provide the confinement.


From action (<ref>), equations of motion can be obtained, and they are given by

    δ S/δ E   →    g_PQ[ẋ^Pẋ^Q-iχẋ^Pψ^Q]+E^2M^2_0=0,
    δ S/δχ   →    g_PQẋ^Pψ^Q+EM_0ξ=0,
    δ S/δξ   →   ξ̇-1/2M_0χ=0,
    δ S/δψ^P   →   Dψ^P/Dτ-1/2E^-1χẋ^P=0,
    δ S/δ x^N   →   D/Dτ[E^-1g_PNẋ^P]-i/2d/dτ[χ E^-1g_PNψ^P]
          +i/2R_NSQRψ^Qψ^Rẋ^S=0.


Before to discuss the confinement of the spinning particle, it is convenient to choose `gauge' conditions for the two symmetries mentioned above. Let us choose E=e^2A, as in the bosonic particle and χ=0 <cit.>. Therefore, the above equations can be simplified as

    g_PQẋ^Pẋ^Q+M_0^2 e^4A=0,
    
    g_PQẋ^Pψ^Q+M_0e^2Aξ=0,
    Dψ^P/Dτ=0,
    D/Dτ[e^-2Aẋ^N]+i/2R^N_ SQRψ^Qψ^Rẋ^S=0.

Where, variable ξ is a constant. Here, we can briefly discuss the `meaning' of the above equations. In fact, it is not possible to understand the variables ψ^Q or ξ classically. However, after quantization is performed, variables ψ^Q and ξ can be identified with Dirac's gamma matrix <cit.>. In this way, the word `meaning' here is related to this interpretation in quantum level after quantization. Equations (<ref>) and (<ref>) will describe the spin dynamics, with Eq. (<ref>) playing the role of the Dirac equation. Unlike the bosonic particle case discussed previously, now geodesic equation (<ref>) presents a coupling between the spin and the curvature Riemann tensor. In fact, Eqs. (<ref>)-(<ref>) would be analogous to the Mathisson-Papapetrou equations which describe a spinning body in a curved spacetime <cit.>. Below, let us discuss the confinement.



 §.§ Confinement of Spinning Particle

As we saw for the case of bosonic particles, we are interested in analyze equations for the component y. From the equation (<ref>), we get the equation for x^M given by

    D/Dτ[e^-2Aẏ]+i/2R^y_ μλρψ^λψ^ρẋ^μ
    +iR^y_ μ yρψ^yψ^ρẋ^μ=0,
    
    g_μνD/Dτ[e^-2Aẋ^ν]+iR_μ y yρψ^yψ^ρẏ
    +i/2R_μνραψ^ρψ^αẋ^ν=0.

By using the Christoffel symbols (<ref>) and (<ref>), the components of the Riemann tensor are given by

    R^ρ_ SQR=δ_S^y[δ_R^ρδ_Q^y-δ_Q^ρδ_R^y][A”+A'^2-A'B']
    +[δ_R^ρδ_Q^μ-δ_Q^ρδ_R^μ]δ_S^νg_μνe^-2BA'^2,
    
    R^y_ SQR=δ_S^ν[δ_R^yδ_Q^μ-δ_Q^yδ_R^μ][A”+A'^2-A'B']g_μνe^-2B.

And we get

    D/Dτ[e^-2Aẋ^ρ]+i[A”-A'B']ψ^yψ^ρẏ
    
    +iM_0ψ^ρξ e^2(A-B)A'^2=0,
    D/Dτ[e^-2Aẏ]=iM_0[A”+A'^2-A'B']e^2(A-B)ξψ^y.

Where, we already used the constraint (<ref>) to eliminate the dependence in ẋ^μ. To find the constants of motion, we need to use the equations for ψ^ρ. 

From Eq. (<ref>) and by using the Christoffel symbols (<ref>) and (<ref>), we get 

    ψ̇^ρ+[ψ^yẋ^ρ+ψ^ρẏ]A'=0,
    ψ̇^y+ψ^y(Ḃ+Ȧ)+ξ M_0e^2A-2BA'=0.

We can multiply Eq.(<ref>) by ξ and use the constraint (<ref>) to find the solution

    ψ^y(τ)ξ=ψ^y_0ξ e^-A-B.

Now, we can multiply Eqs. (<ref>)  and (<ref>) by ψ^y and ψ^ρ, respectively, to get
 
    d/dτ[ψ^yψ^ρ]+ψ^yψ^ρ(Ḃ+2Ȧ)+M_0ξ e^2A-2BA'ψ^ρ=0.

Finally, from the above equation, we can show that 

    d/dτ[ψ^yψ^ρe^2A+B]+M_0ξ e^4A-BA'ψ^ρ=0.



With this expression, we can discuss if the spinning superparticle can be confined. If we use our previous relations  (<ref>), (<ref>), (<ref>) and (<ref>) in equation (<ref>), we get

    d/dτ[ẋ^μ+iψ^yψ^μe^2AA']=dp^μ/dτ=0,
    d/dτ[e^2B-2Aẏ^2+M_0^2e^2A-2iM_0ξψ^y_0e^A-BA']=0.

Where, we define the conserved momentum

    p^μ=ẋ^μ+iψ^yψ^μe^2AA'

I order to interpret the second constant coming from the second equation in (<ref>), we must use our mass constraint and equation (<ref>) to get

    -p^2=e^2B-2Aẏ^2+M_0^2e^2A-2iM_0ξψ^y_0e^A-BA',

with p^2=η_μν p^μp^ν and we already used Eq. (<ref>). Thus, just as before, we can interpret the term on the right as a total energy and 

    U_eff=M_0^2[e^2A-2i/M_0ξψ^y_0 e^A-BA'].

First, when ψ^N is zero, the last term in right-hand of above equation is zero. Therefore, we get the results discussed previously for a bosonic test particle and given by Eq. (<ref>) with M(y)=M_0. Now, because the spin variable, we get a new peace in U_eff acting on the particle.  Thus, the spin modify the effective potential, due to the interaction of it with the curvature. With this we have the possibility of confining a spinning  particle. 



 §.§ Application

For the massless case, we see from Eq. (<ref>) that U_eff=0. Therefore, just as in the bosonic case, the spinning particle is not confined. However, for the massive case U_eff has a correction due to the Grassmann variables. Figure (<ref>) shows a plot of the effective potential (<ref>) for the thick brane presented in Ref. <cit.>, namely,

    A(y)=ln[^b(cy)],    B(y)=0.
 



As we can see, y=0 is not an extreme for this effective potential. But, there are two extreme points, and one of them is a minimum. Therefore, the spinning particle can be localized at the minimum of this potential. However, this localization does not happen at y=0, but at a point that is slightly displaced from the origin. Similar conclusions are also obtained for other models as, for example, the delta-like brane <cit.>, the thick brane presented in Ref. <cit.> or the thick brane with inner structure obtained in Refs. <cit.>. In fact, from Fig. (<ref>) we can conclude that any particle over the brane will escape to the extra dimension. Note that U_eff(y=0)>0 and for large y the potential is null. Therefore, any particle over the brane goes to infinity.
 
 
An interesting possibility is to look for models in which the minimum of U_eff coincides with the position of the brane. Since our U_eff is asymmetric we should look for brane models with the same property. They are called asymmetric braneworlds and were discussed by D. Bazeia et.al. in Refs. <cit.>. However, for these models we also discovered that the spinning particle is not localized. It seems that there is some general property of the potential (<ref>) that forbids the confining. Let us now show this. First note that  

    U'_eff=2M_0^2e^2AA'-2M_0^2δ e^A-B[A”+A'^2-A'B'].

Now we should remember that for any braneworld, we must have that at the position of the brane A'(y_B)=B'(y_B)=0,A”(y_B)>0. With this we get that

    U'_eff(y_B)=-2δ e^A-BA”≠ 0

and therefore at the position of the brane we never have a minimum of U_eff. However, we could have the possibility that the particle is trapped around the position of the brane and do not escape to infinity. In order to analyze this we use the general properties A(y_B)=0,e^A(∞)=0 to obtain  

    Δ U_eff≡ U_eff(∞)-U_eff(y_B)=-M_0^2.

Therefore the potential at infinity is always fewer than the potential over the brane and any spinning particle over the brane will escape to infinity. 



§ CONCLUSION

In this manuscript, we discussed the confinement of test particles in codimension one braneworlds. We discussed the bosonic test particle and carried out the discussion about confinement of the spinning particle (superparticle). Localization of bosonic test particles was already studied previously in the literature. For the delta-like RS-II model, Ref. <cit.> discussed the confinement of a free test particle. And also, for the thick brane presented in Ref. <cit.>, where it was studied the free case and also with a localization mechanism <cit.>.


In discussion of the bosonic test particle, we studied the massive and massless case. For the free massive case, we obtained the conserved quantities given in equations (<ref>) and (<ref>). From this, we showed that the motion in extra dimension can be decoupled of other coordinates and it was possible to identify the effective potential (<ref>), namely, 

    U_eff=M(y)^2e^2A(y)-M(0)^2.
 
With this potential, the confinement would be attained if the conditions U'_eff(0)=0 and U”_eff(0)>0 could be satisfied. For free massive particles these conditions imply that A”(y)>0. However, for all the braneworld model with localized gravity <cit.>, the Einstein's equations provide us with A”(y)<0. Therefore, the necessary condition to confine the massive particles, namely, A”(y)>0, cannot be satisfied. This result agrees with the one obtained in Refs. <cit.> by other means. However, the massless case has not been considered in Refs. <cit.>, since their actions is ill-defined for M=0. By using a quadratic action, we show that (<ref>) is valid for both cases. For M=0, we get that (<ref>) is zero and, thus, the above approach is not conclusive. However, as discussed from Eq. (<ref>), the massless particle cannot be localized over the brane.


Next, we proposed a new localization mechanism for the massive particle by replacing 

    M_0→ M(y)=M_0e^λπ(y),

 where, π(y) is a scalar field sometimes called dilaton. This new mechanism is an alternative to that presented in Ref. <cit.>, where it was provided a coupling with a scalar field given by M_0→√(M_0^2+h^2ϕ^2). With this mechanism, the confinement can be attained for particles with mass values that satisfy M_0<√(3)h/κ. As we showed, by using our localization mechanism, it was possible to obtain a modified effective potential, but now given by 

    U_eff=M_0^2(e^2(1-λ√(3M_p^3))A-1).
 
Once again, we apply the confinement conditions to obtain 

    λ>1/√(3M_p^3).

This result presents the interesting feature that it allows us to confine any value of mass (M_0^2>0). In this sense, the coupling with the dilaton field seems to provide a more efficient confinement of the massive particles.


To conclude the analysis, we discussed the spinning particle presented for 4D in <cit.>. This study was not yet performed in the context of braneworld. Just like the bosonic test particle, it was discussed the massive and the massless cases. We found the modified effective potential given by (<ref>), namely, 

    U_eff=M_0^2[e^2A-2i/M_0ξψ^y_0 e^A-BA'].

Modification in this potential is closely related the presence of the spin variables ψ^P and ξ, which introduces an interaction of it with the curvature Riemann tensor. The analysis of the confinement was performed graphically. From figures (<ref>), we concluded that the above potential allows us to localize the test particles with spin in a position close of the brane, but not over the brane. By analyzing the general features of the above potential we show that in fact any particle over the brane will escape to infinity. This is valid for a general warp factors A(y) and B(y). Beyond this, the above potential is also valid for the massless case, and we conclude that they are not localized. Therefore, neither the bosonic nor the spinning free particles can be localized over the brane. This is very different of the field description, where the free scalar field are localized, but spinor and gauge fields are not.


Finally, we should point that due to the presence of worldline supersymmetry the possibility of couplings is very constrained. For example, it is not possible to introduce a coupling with the scalar or dilaton fields above without breaking this symmetry. However, we can consider the case with more than one supersymmetry and even higher spins from the worldline viewpoint <cit.>. Another possibility is to consider spacetime supersymmetry from the wordline viewpoint, such as the Brink-Schwarz superparticle <cit.> or even the covariant pure spinor description <cit.>. Therefore, the introduction of Grassmann variables opens a new avenue of research about confining of test particles in braneworlds. 


The authors would like to thank Alexandra Elbakyan and sci-hub, for removing all barriers in the way of science. We acknowledge the financial support provided by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) through PRONEM PNE-0112-00085.01.00/16.

0
Kaluza
	Theodor K.
	Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.)961921966-972.

Klein01
	Oskar K.
	Zeitschrift für Physik A371926895-906.

Klein02
	Oskar K.
	Nature118192651.
 	
RS1
    Lisa R. Raman S.
	Phys. Rev. Lett.8319993370.
	       
RS2
   	Lisa R. Raman S.
	Phys. Rev. Lett.8319994690.
      
Gremm1
   	Martin G.
	Phys. Lett. B4782000434.
     
Gherghetta
   	Tony G. Mikhail S.
	Phys. Rev. Lett.852000240-243.
   
Arkani02
   	Nima A-H., Savas D., Gia D. Nemanja K.
	Phys. Rev. Lett.842000586-589.
     
Kehagias
   	Alex K. Kyriakos T.
	Phys. Lett. B504200138.
    
Carlos
   Beatriz de C. Jesus M. M.
   JHEP03112003040.

Bazeia:2002xg
	D. Bazeia, L. Losano J. M. C. Malbouisson
	Phys. Rev. D662002101701.
	
Bazeia1
   D. Bazeia A. R. Gomes
   JHEP04052004012.

Cendejas
  N. B. Cendejas and A. H. Aguilar
   Phys. Rev. D, 73 (2006). Erratum: Phys. Rev. D, 77 (2008).
    
Choudhury
   D. Choudhury and S. Sengupta
   Phys. Rev. D, 76, (2007) 064030.
        
Silva
   J. E. G. Silva, V. Santos and C. A. S. Almeida
   Class. Quant. Grav. 30 (2013) 025005.
   
Bajc:1999mh 
   B. Bajc and G. Gabadadze
   Phys. Lett. B, 474, (2000) 282.
   
Alencar:2014moa 
   G. Alencar, R. R. Landim, M. O. Tahim and R. N. Costa Filho
   Phys. Lett. B, 739, (2014) 125.

Zhao:2014iqa
   Z. H. Zhao, Q. Y. Xie and Y. Zhong
   Class. Quant. Grav. 32 (2015) 035020.

Alencar:2015rtc 
   G. Alencar, C. R. Muniz, R. R. Landim, I. C. Jardim and R. N. Costa Filho
   Phys. Lett. B, 759, (2016) 138.

Alencar:2017dqb 
   G. Alencar
   Phys.  Lett.  B, 773, (2017) 601.

Mendes:2017hmv 
   W. M. Mendes, G. Alencar and R. R. Landim
   JHEP, 1802, (2018) 018.

Freitas:2018iil 
   L. F. F. Freitas, G. Alencar and R. R. Landim
   JHEP, 1902, (2019) 035.

Fu:2015cfa 
   C. E. Fu, Y. X. Liu, H. Guo and S. L. Zhang
   Phys.  Rev.  D, 93, (2016) 064007.
   
Gegenberg 
   J. Gesenberg and G. Kunstatter
   Phys. Rev. A, 106, (1984).

Kovacs 
   D. Kovacs
   Gen. Rel. Grav. 16, (1984).

Cho:1991ab
    Y. M. Cho and D. H. Park
    Gen. Rel. Grav. 23, (1991).

Wesson:1999mc
    P. S. Wesson, B. Mashhoon, H. Liu and W. N. Sajko
    Phys. Lett. B, 456, (1999).

Ponce:2001wf
    J. Ponce de Leon
    Phys. Lett. B, 523, (2001) 311-316.
    
Ponce:2001ec
    J. Ponce de Leon
    Grav. Cosmol., 8, (2002) 272-284.

Ponce:2002rv
    J. Ponce de Leon
    Int. J. Mod. Phys. D, 12, (2003) 757-780.
    
Mueck:2000bb 
	W. Mueck, K. S. Viswanathan and I. V. Volovich
	Phys. Rev. D, 62, (2000) 105019.
	
Dahia:2007ep 
	F. Dahia and C. Romero
	Phys. Lett. B, 651, (2007) 232.
	    
Brink
    L. Brink, P. Di Vecchia and P. S. Howe
    Nucl. Phys. B, 118, (1977) 76-94.
	
Landim 
	R. R. Landim, G. Alencar, M. O. Tahim and R. N. Costa Filho
	JHEP, 1202, (2012) 073.
	
Kim:2003ch
    T. W. Kim and E. J. Son
    JHEP, 09, (2003) 040.
    
Berezin
	F.A. Berezin and M.S. Marinov
	Annals of Physics, 104, (1977) 336-362.

Ali
    M. H. Ali
    Inter. J. of Theor. Phys., 41, (2002) 2319-2338. 
    
Mathisson
    M. Mathisson
    Acta Phys. Polon., 6, (1937) 163-2900.
    
Papapetrou
    Papapetrou A.
    Proc. R. Soc. Lond. A, 209, (1951) 248–258.
    

Bazeia2003
    D. Bazeia, C. Furtado and A. R. Gomes
    JCAP, 02, (2004).
   
Bazeia2013
    D. Bazeia, R. Menezes and R. da Rocha
    Adv. High Energy Phys. 2014, (2014).

Bazeia2019
    D. Bazeia and D. A. Ferreira
    Annals Phys., 411, (2019).

Bazeia2020
      D. Bazeia, D. A. Ferreira and M. A. Marques
      Eur. Phys. J. Plus, 135, (2020).   


vanHolten:1995qt
J. W. van Holten,

doi:10.1142/9789812830425_0010
[arXiv:hep-th/9510021 [hep-th]].



Bastianelli:2008nm
F. Bastianelli, O. Corradini and E. Latini,

JHEP 11, 054 (2008)
doi:10.1088/1126-6708/2008/11/054
[arXiv:0810.0188 [hep-th]].



Brink:1981nb
L. Brink and J. H. Schwarz,

Phys. Lett. B 100, 310-312 (1981)
doi:10.1016/0370-2693(81)90093-9



Berkovits:2001rb
N. Berkovits,

JHEP 09, 016 (2001)
doi:10.1088/1126-6708/2001/09/016
[arXiv:hep-th/0105050 [hep-th]].

      


