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Confinement and Effective Mass of Test Particlesin Thick Branes Coupled Non Minimally With theDilatonic Field
F.E.A. Souza,a,b G. Alencar,a R. R. Landima
aDepartamento de Física, Universidade Federal do Ceará, Caixa Postal 6030, Campus do Pici,60455-760, Fortaleza, Ceará, Brazil.
bLaboratório de Gravitação, Modelos de Campos e Cordas (LabGmc2), Universidade Federal doCeará, Caixa Postal 6030, Campus do Pici, 60455-760, Fortaleza, Ceará, Brazil.
E-mail: [email protected], [email protected],[email protected]
Abstract: In this manuscript we study the confinement of test particles in smooth Randall-Sundrum models. It is a known fact that free test particles can not be localized neitherin thin or thick branes. Recently a coupling of the particle to a scalar field has been usedto localize a limited range of masses. It is found that the threshold mass depends on thespecific model chosen. Up to now, no mechanism has been found that can trap test particlesof any mass to the brane. In order to solve this problem, we first find a general expressionfor the effective mass observed on the brane. Next we show that by coupling the particleto the dilaton we obtain the trapping of particles of any mass. We further show the factthat this result is valid for any smooth version which recover the RS model asymptotically.For this, we just need of general aspects of the field equations of motion. As an example,we apply the method to the specific cases of deformed and undeformed topological defects.
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Contents
1 Introduction 1
2 Review of Randall-Sundrum Model 42.1 Delta-like branes 42.2 Thick branes 42.3 Dilaton coupling 52.4 Deformed brane 6
3 Confinement of test particles in braneword 83.1 Analytical method 103.2 Effective potential method 113.3 Particle coupling with scalar field 13
4 General conditions to confinement and effective mass of test particles inbraneworld 174.1 General condition for confinement of test particles in braneworld 174.2 Effective mass 18
5 Confinement of test particles non minimally coupled with the dilaton 215.1 Dilaton coupling in deformed RS spacetime 21
6 Conclusion 23
1 Introduction
The main theory of extra dimensions is due Kaluza-Klein [1, 2], where they introducedan extra dimension in high energy Physics theory, in order to unify electromagnetism andgravitation. To recover the 4-dimensions Physics, it is imposed that this extra dimensionmust be compact with compactification radius in Planck length order [2].
In year 1999, Randall and Sundrum have proposed the possibility that our 4-dimensionaluniverse must be a hypersurface, by defining it as a brane of an Anti-de Sitter (AdS) 5-dimensional space-time with a non factorable metric [3]. They have found that Planck scaleis not fundamental, but a consequence of the larger size of the extra dimension [4]. Thisidea was first considered by ADD model [4]. In this scenario, matter and non gravitationalfields are confined in a 4-dimensional space-time by some mechanism in a way such thatthe extra dimension can not be measured by direct observations in a energy level bellowthe TeV scale, whereas the gravity permeates the whole space [4, 5]. Since this extra di-mensions can not be detected, how it can help us to solve some of the problems in fourdimensional physics? One of the most famous work in this direction, is due Randall and
– 1 –
Sundrum [5], pursuing to solve the hierarchy problem [6]. The Randall-Sundrum model isdivided in two parts, RS-I [5] and RS-II [3]. In RS-I, the extra dimension is compactified ina circle whose superior and inferior parts are identified. Formaly, this means that this extradimension lies in an orbifold S1/Z2, where S1 is an unidimensional sphere (e.x. a circle)and Z2 is the discrete group {−1, 1}; this construction implies two fixed points, one inorigin φ = 0 and other at the opposite extremity of the circle φ = π. In one of these points,the 4-dimensional hypersurfaces that we live, are contained. By analogy with membranesthat involve a volume, these (3 + 1)-dimensional world is also called of 3-branes. The RS-Imodel explain how an exponential hierarchy between mass scales can be generated. On theother hand, RS-II assumes that the extra dimension is large (called y) and there is just onedelta-like 3-brane localized at y = 0, where gravity is confined. It also explains how the 4Dgravity emerge in the Newtonian limit, becoming an alternative to compactification. Also,in RS-II, the brane metric is replaced by a scalar field φ, minimaly coupled with gravitation[7–11].
Since the extra dimension is not compact in RS-II, all of the matter fields (not only thegravity) must be confined on the brane in order to reproduce a realistic model. However,the RS-II model does not show any arguments that guarantee the localization of othersfields, such as gauge and fermionic. Seeking the solution of this problem, many works havebeen proposed in the literature [11–19].
For the case of test particles, there is a problem related to the confinement and stabilityon the brane, in both types of brane (delta [20] and thick [21]). This is because the minimumcoupling between the the test particle and gravity is not enough to confine it. Then, in theRS-II spacetime, it is necessary some confinement mechanism to trap test particles on thetwo kinds of branes. The simplest one, for both branes, is to invert the signal of the branetension. However, it causes the non localizability of gravity [12], making the mechanism notconsistent. For smooth brane spacetime, it is possible to create test particles confinementmechanisms as in [21], the particle confinement is possible if it interacts directly witha scalar field. This interaction is made by modifying the free particle action through aredefinition of the canonical moment in 5-dimension PAP
A = −(M20 + h2φ2), based in a
Yukawa interaction out of quantum level, allowing test particles to be confined on the branethat obeys a mass limit.
The motion of test particle in the extra dimension is decoupled from the motion in otherdimensions, and subjected to an effective potential [21] which arises from the interaction oftest particle with gravity (free test particle), or trough the association between the gravityand and other background field. The effective potential of free test particle is not sufficientto confine it. In the case of test particle coupled with the scalar field [21], the confinementis possible on a certain limit of mass of test particle in the background field. However, in[21], the authors do not provide any mechanism to evaluate the mass of test particle on a3-brane, they only provide the the mass confined in the bulk.
This paper is divided in two parts, first we developed a mechanism to obtain the massof a test particle on a 3-brane (we define it as effective mass), in the cases where the mass offree particle is coupled with gravity, and when the mass particle is associated with gravityand a background field. Then, we apply the mechanism to obtain the mass of test particles
– 2 –
in the following cases: free particle, test particle coupled with a scalar field and test particlecoupled with a dilaton field. In the second part, we propose the coupling between the testparticle and the dilaton field inspired in work [25].
This paper is organized as the following. In section (2) we make a review of the RS-IItype delta, thick branes, the model deformed by the dilaton and the model deformed by ancontrol parameter. In section (3) we review and analyze the confinement of test particlesin a RS II brane world with delta and thick branes, showing that the free test particle isnot confined. That way it is necessary a mechanism that makes the confinement possible.In section (4) we obtain the effective mass of the test particles observed on the brane tocases where the effective potential have a maximum point. In section section 5, we proposea mechanism where the test particles are coupled with the dilaton field in order to confinethem.
– 3 –
2 Review of Randall-Sundrum Model
In this section, a brief description of the braneworld models: delta-like, thick, deformedby scalar field (dilaton) and the brane as topological defect is going to be presented. Weshall use the Minkowsky metric ηµν = diag(−1, 1, 1, 1) on the brane.
2.1 Delta-like branes
Delta type branes are defined by the action [20]
S =
∫d4x
∫dy√−g(2M
(3)P R− Λ) + σ
∫y=0
d4x√−gb, (2.1)
where M (3)P is the Planck scale for theory, R is the Ricci scalar, Λ is the cosmological
constant, σ is the brane tension and gb is the induced metric on the brane, gb = g(x, y)|y=0.The metric of theory is of the form:
gAB = diag(e2a(y)ηµν , 1). (2.2)
The obtained Einstein’s equations are
Λ = −24M(3)P a′2, (2.3)
σ = 12M(3)P a′′ + 24M
(3)P a′2 + Λ. (2.4)
Solving the equation (2.3) we obtain the solution
a(y) = −k|y|, (2.5)
this is a warp factor, where k =
√−Λ
24M(3)P
. The metric (2.2) then takes the form
gAB = diag(e−2k|y|ηµν , 1) (2.6)
2.2 Thick branes
In the presence of a gravitacional field in a 5-dimensional spacetime, the action for ascalar field is [11]:
S =
∫dx5√−g(
2M(3)P R− 1
2∂Aφ∂
Aφ− U(φ)
). (2.7)
Where g is the metric determinant, R is the curvature scalar relative gAB, U(φ) correspondsto the potential of the scalar field and M (3)
P is the Planck constant for 5-dimensional space-time. We obtain from the action (2.7), that the energy moment tensor of the model isTAB = −2√
−gδ
δgAB(√−gLMatter)
TAB = ∇Aφ∇Bφ− gAB(
1
2∇Aφ∇Aφ+ U(φ)
), (2.8)
– 4 –
which allows us to obtain the Einstein equations GAB = 1
4M(3)P
TAB, while the dynamics of
the scalar field is governed by the equation
∇2φ− ∂U(φ)
∂φ= 0. (2.9)
In the absence of gravity for a scalar potential of the double well type U(φ) = λ4 (φ2−v2)2
the scalar field equation possesses bounce-like static solution depending only on the extradimension, which the simplest one is [11]
φB(y) = v tanh(cy), (2.10)
where a2 ≡ λv2/2. When introducing the metric (2.2), we obtain the Einstein equations
1
2(φ′)2 − U(φ) = 24M
(3)P (a′)2, (2.11)
1
2(φ′)2 + U(φ) = −12M
(3)P a′′ − 24M
(3)P (a′)2. (2.12)
When adding (2.11) and (2.12), we obtain the second order differential equation
(φ′)2 = −12M(3)P a′′. (2.13)
Applying (2.10) in (2.13), we get the solutions
a(y) = −β ln cosh2(cy)− β
2tanh2(cy), (2.14)
where β ≡ v2
36M(3)P
. Note that this represents a localized metric warp factor [11]:
e−2a(y) =e−β tanh2(cy)
(cosh2(cy))2β∝ e−4cβ|y| y →∞ (2.15)
2.3 Dilaton coupling
Let us consider the action [11]:
S =
∫dx5√−g{
2M(3)P R− 1
2(∂φ)2 − 1
2(∂π)2 − V (φ, π)
}, (2.16)
where φ is the scalar field minimally coupled with gravity to obtain the bounce-like config-uration and π (the other scalar field) is defined as dilaton.
The dilaton scalar field is obtained from the appropriated choice of the potential. Ingeneral, this potential is dependent of φ and π, so the resulting motion equations from(2.16) are
RAB −1
2gAB =
1
4M(3)P
{∂Aφ∂Bφ+ ∂Aπ∂Bπ − gAB
(1
2(∂φ)2 + ∂π)2 + V
)}, (2.17)
∇2φ− ∂V (φ)
∂φ= 0, (2.18)
∇2π − ∂V (π)
∂π= 0. (2.19)
– 5 –
The metric ansatz is given by
gAB = diag(e2a(y)ηµν , e2b(y)). (2.20)
The above metric was proposed in Reference [11] and can be lifted to six dimentions wherez parametrize an extra S1 direction. The solutions for the scalar fields depend only on theextra dimension y, and we obtain the set of Einstein equations
1
2(φ′)2 +
1
2(π′)2 − e2bV = 24M
(3)P (a′)2, (2.21)
1
2(φ′)2 +
1
2(π′)2 + e2bV = −12M
(3)P a′′ − 24M
(3)P (a′)2 + 12M
(3)P a′b′, (2.22)
φ′′ + (4a′ − b′)φ′ = ∂φV, (2.23)
π′′ + (4a′ − b′)π′ = ∂πV, (2.24)
of which three are independent. The solutions to the above system of equations can befound using the superpower method, for which φ′ = ∂W
∂φ . The specific solution we willconsider here corresponds to the potential
V = eπ/√
12M(3)P
{1
2
(∂W (φ)
∂φ
)2
− 5
32M(3)P
W (φ)2
}(2.25)
Noting that the solution is
π(y) = −√
3M(3)P a(y), (2.26)
b(y) =a(y)
4, (2.27)
a(y)′ = − 1
12M(3)P
W. (2.28)
and by choosing a specific W (φ), the solution is completely specified [11]. From the super-potential
W (φ) = vc
(1− φ2
3v2
)(2.29)
we obtain the solution
φB(y) = v tanh(cy),
a(y) = −β ln cosh2(cy)− β
2tanh2(cy),
π(y) =
√3M
(3)P β
(ln cosh2(cy) +
1
2tanh2(cy)
).
2.4 Deformed brane
The deformation method, studied in [27], is based in modifications of the potentials ofmodels containing solitons in order to produce new and unexpected solutions [28]. Let’suse the following metric ansatz
gAB = diag(e2as(y)ηµν , e2bs(y)), (2.30)
– 6 –
where s is a deformation parameter. The deformation parameter controls the deformedtopological defect that we want, in order to obtain different classes of membranes. To solvethis problem, we use the so-called superpotential function Ws(φ), defined by
φ′ =∂Ws
∂φ, (2.31)
using, then, the same approach of Kehagias and Tamvakis [11], the particular solutionfollows from choosing the potential
Vs = e
π√12M
(3)P
[1
2
(∂Ws
∂φ
)2
− 5
32M(3)P
Ws(φ)2
], (2.32)
and superpotential
Ws(φ) = cφ2
[2
2s− 1
(v
φ
) 1s
− 2
2s− 1
(φ
v
) 1s
], (2.33)
where c and v are parameters to adjust the dimensionality. As in [11], this potential givesthe desired soliton-like solution. In this way, it is easy to obtain first order differentialequations whose solutions comes from (2.31)-(2.33), namely
a′s =Ws
12M(3)P
, bs =as4, πs = −
√3M
(3)P as. (2.34)
The solution for these new set of equations are the following:
φ(y) = v tanh(cy), (2.35)
a1(y) = −β1 ln cosh2(cy)− β12
tanh2(cy), (2.36)
for s = 1, and
φ(y) = v tanhs(cys
)(2.37)
as(y) = −βs2
tanh2s(cys
)− 2sβs
2s− 1
{ln[cosh
(cys
)]−
s−1∑n=1
1
2ntanh2n
(cys
)}(2.38)
for s > 1, where β1 = v2
36M(3)P
and βs = v2
12M(3)P
s2s+1 [27]. Then, we obtain a set of second
order diferential equations similar to equation (2.13),
(φ′)2 = −12M(3)P a′′s . (2.39)
The values of s assumes only odd integer values positives, since the first derivative of thescalar function must behave like an step function [7]. And note that this warp factorrepresents a localized metric warp factor
e−2as(y) ∝ e−4s
2s−1βs|y| ≡ e−4csβs|y| y →∞, (2.40)
with cs = s2s−1 .
– 7 –
3 Confinement of test particles in braneword
In this section, we will give an overview on the confinement of test particles basedon the Randall-Sundrum (RS) model. In the remaining of this section, we use the termmassive test particles for particles with non-null bare mass and the term massless particlesfor particles with null bare mass.
For the RS model, consider the action for a test particle with mass M(y), where thebare mass is coupled with any field dependent of the extra dimension RS spacetime as thefollowing
S = −∫M(y)
√−gAB
dxA
dθ
dxB
dθdθ (3.1)
where θ is an arbitrary parameter along the wordline. The action (3.1) is invariant undertransformation in the form
θ → θ′ = f(θ) (3.2)
where f(θ) is an arbitrary function of θ.However, the action (3.1) is not well defined forM(y) = 0 i.e for massless test particles.
To solve this problem, the action of relativistic test particles is going to be written in thefollowing alternative form
S =1
2
∫E
(−E−2gAB
dxA
dθ
dxB
dθ+M2(y)
)dθ. (3.3)
In the above action E is a one-dimensional vierbein field and transforms into form E′(θ′) =dθ′
dθ E in order to maintain the invariance (3.2). The equation of motion for E is given by
gABdxA
dθ
dxB
dθ+ E2M2(y) = 0
which replaced in Eq. (3.3) gives us the original action (3.1). This new form has someadvantages over the original one (3.1), it allows to write the action for massless test particlecase and also it does not contains square root, which gives us simpler equations of motion[30].
The equations of motion of (3.3) in a 5-dimensional RS spacetime are
δS
δE= 0 → gAB
dxA
dθ
dxB
dθ+ E2M2(y) = 0, (3.4)
δS
δxµ= 0 → ηµν
d2xν
dθ2+ 2a′ηµν
dy
dθ
dxν
dθ− E−1dE
dθηµν
dxν
dθ= 0, (3.5)
δS
δy= 0 → g55
d2y
dθ2− a′gµν
dxµ
dθ
dxν
dθ
+1
2g′55
(dy
dθ
)2
− E−1dEdθg55
dy
dθ+ E2M(y)M ′(y) = 0, (3.6)
with a′ = da/dy and M ′(y) = dM(y)/dy.Since the equation of motion for E is purely algebraic, E does not represent a new
dynamical degree of freedom [30]. In general, for massive test particles we can fix the new
– 8 –
variable as E = 1M(y) and use the constraint gAB dxA
dθdxB
dθ +1 = 0. For massless test particles,
we can choose the new variable as E = 1 and the constraint gAB dxA
dθdxB
dθ = 0, for y = 0,M2(0) = M2
0 , whereM20 is the bare mass of the test particle. Then, if the bare mass is null,
the action (3.3) is not well defined for E = 1M(y) . Then, to avoid this problem, we choose
E = 1 and we use the following constraint
gABdxA
dθ
dxB
dθ+M2(y) = 0. (3.7)
To obtain the conserved quantities in directions xµ and y, we write equations (3.5) and(3.6) as total derivatives as the following.
Initially, to obtain the quantity conserved in xµ direction, we multiply the equation(3.5) by E−1e2a and write it as a total derivative,
d
dθ
[E−1e2a
dxµ
dθ
]→ E−1e2a
dxµ
dθ= pµ, (3.8)
which gives the conserved quantity in xµ direction
dxµ
dθ= e−2aηµνEpν . (3.9)
For extra dimension y, we replace (3.9) in the equation (3.6) and write it as the followingtotal derivative
d
dθ
[g55
(dy
dθ
)2
E−2 +M(y)2 + p2e−2a
]= 0, (3.10)
which gives us the conserved quantity
g55
(dy
dθ
)2
E−2 +M(y)2 + p2e−2a = C, (3.11)
where p2 = ηµνpµpν . Replacing p2e−2a = gµνdxµ
dθdxν
dθ E−2, see (3.8), in the equation (3.11)
we obtain the expression
g55
(dy
dθ
)2
E−2 +M(y)2 + gµνdxµ
dθ
dxν
dθE−2 = C
gABdxA
dθ
dxB
dθE−2 +M(y)2 = C, (3.12)
comparing (3.12) with the constrains equation (3.4) we conclude that C = 0 and thus (3.11)take the form
g55
(dy
dθ
)2
E−2 +M(y)2 + p2e−2a = 0. (3.13)
Multiplying (3.13) by a factor e2a, we obtain a conserved quantity in the extra dimensiondirection as following
E−2g55e2a
(dy
dθ
)2
+M2e2a = p2, (3.14)
– 9 –
this quantity can be interpreted as the total energy of the test particle in the extra dimensionand this shows that the motion of the particle along the extra dimension decouples fromthe motion in other directions.
Therefore, based in quantities (3.13) and (3.14) we can analyze the confinement andstability of the test particle analytically or by the effective potential method, respectively,as we will show below.
3.1 Analytical method
The confinement of free test particles, M(y)2 = M20 , in delta-like branes is described
analytically in reference [20]. The analytical description of the motion of test particles isbased on finding an expression that gives how test particles behaves in the neighborhoodof y = 0 (brane), allowing to evaluate whether or not the test particle can be confined onthe brane.
For this, they have applied the following non-affine parameter transformation t
dt
dθ= Ee−2a, (3.15)
which allowed the authors to obtain an analytical form of equations of motion and alsoto verify the behavior of the particle along the extra dimension. This transformation onlyapplies when y 6= 0, because in y = 0 the parametrization above is affine under certainconditions [24].
Following the work done in [20], and applying the chain rule in equation (3.9),
dxµ
dt
dt
dθ= e−2aηµνEpν ,
and replacing (3.15) in equation (3.9), it can be written as first order EDOs, as the following
xµ ≡ dxµ
dt= pµ → xµ = xµ0 + pµt, (3.16)
that is, the particle follows a flat geodesic in parameter t on the hypersurface y = y0.Now, to analyze the motion of free test particles in the extra dimension we can apply
the following chain rule to equation (3.13),
g55
(dy
dt
)2( dtdθ
)2
E−2 +M20 + e−2ap2 = 0,
using the non-affine parametrization (3.15) and choosing the parameter E as unity, weobtain (
dy
dt
)2
e−4a + e−2ap2 = −M20 . (3.17)
After some manipulations, we obtain the following integral form∫dy e−2a
(−M20 − e−2ap2)
12
=
∫dt, (3.18)
– 10 –
which has analytical solution for delta-like branes, while for thick branes the analytic solu-tion cannot exist. Taking a(y) as (2.5), the equation (3.17) can be written as
y2e4k|y| + e2k|y|p2 = −M20 , (3.19)
where y ≡ dydt and we can define y0 for a given initial time t0, so that −M2
0 = y20e4k|y0| +
e2k|y0|p2. Solving (3.18), we obtain an analytical form for equation of motion of free testparticles in the extra dimension as the following
e2k|y| = e2k|y0| + 2ky0e2k|y0|t− p2k2t2. (3.20)
The above equation is valid when y 6= 0. So, for RS model (k > 0), one ordinary particlep2 < 0, is repelled from the brane even if we assume |y0| → 0 at any time t, i.e. the particlewill not reach y = 0. In other words, the particle is repelled out of the brane, and in thelimit that |y0| → 0 and |y0| → 0 the equation (3.19) is valid if M2
0 > 0, that is, the particlein the bulk also is ordinary. For tachionics particles p2 > 0, they are attracted for the branein the limits |y0| → 0 and |y0| → 0; the equation (3.19) is valid if M2
0 < 0, that is, theparticle in the bulk also tachionic. While for massless particles p2 = 0, they will only beexpelled from the brane if y0 6= 0, since they are continuous, as we can infer from (3.6).One way to solve the problem of particle localization, is to invert the brane tension k → −k[20], but in that case then the gravity is no more localized [12].
3.2 Effective potential method
The confinement of test particles in delta-like branes is described in [21], where theyhave obtained an energy conservation equation of test particle in extra dimension trough thestandard action S = M0
∫(−gAB dxA
dθdxA
dθ )12dθ with the constraint gAB dxA
dθdxA
dθ = −ε, whereε assume the values {−1, 0, 1} representing tachions, light and causal particles, respectively.In this method, called effective potential method, the test particle appears under action bya effective potential that depends only on the extra dimension. However, this action is notwell defined for massless test particles; this because the standard action is not well defined.
To solve this issue relative to massless test particles, instead of applying the actionused in [21], we apply the action in form (3.3) and the constraint (3.7). Then, we applythe effective potential method using the new action (3.3), and we obtain the conservationequation (3.14). In order to evaluate the energy on the brane, we evaluate it at y = 0,
which give us the conservation equation equal to p2 =(dydθ
)2 ∣∣y=0
+M20 , where
(dydθ
)2 ∣∣y=0
can be interpreted as initial kinetic energy of the particle along the extra dimension, when
the particle was initially on the brane. Then, replacing p2 =(dydθ
)2 ∣∣y=0
+M20 in (3.14) and
using the metric (2.2), we obtain
e2a(dy
dθ
)2
=
(dy
dθ
)2 ∣∣y=0−M2
0 (e2a − 1), (3.21)
where equation (3.21) is the total energy of particle in the extra dimension, where the last
term is the effective potential. The particle can move in the region(dydθ
)2 ∣∣y=0−M2
0 (e2a −
– 11 –
1) ≥ 0, being(dydθ
)2 ∣∣y=0
the maximum value of the effective potential
Veff(ymax) =
(dy
dθ
)2 ∣∣y=0
. (3.22)
The equation (3.21), represents the motion of free particle under influence of an effectivepotential
Veff = M20 (e2a − 1), (3.23)
except for massless particles, that are not affected by the effective potential and stay onthe brane if y0 = 0, otherwise the particle is expelled from the brane, because the velocityof the massless test particle obey, from equation (3.21), the relation
e2a(dy
dθ
)2
=
(dy
dθ
)2 ∣∣y=0
, (3.24)
where in the limit y →∞,(dydθ
)2→∞, that is, the test particle does not return (see figure
1 for M20 = 0).
To investigate the motion of a test particle on the brane, we will analyze the firstand second derivatives of the effective potential around y = 0 and for M2
0 6= 0. The firstderivative is V ′eff = 2M2
0a′e2a, where the necessary condition for confinement is to have
y = 0 as a critical point, where to achieve this, it is required that a′(0) = 0. Thus, a(y)
must satisfy an initial condition a(0) = a′(0) = 0. Once the two conditions are satisfied, thesecond derivative of the effective potential, which is equal to V ′′eff(0) = 2M2
0a′′(0), provides
us information about stability of the critical point. Analysing the second derivative andaccording to equations (2.5) and (2.14), a′′(y) ≤ 0 for all y. Therefore, we can evaluate theparticle stability according to the value of M2
0 , being the point y = 0 an maximum pointfor causal particles (see figure 1 for M2
0 > 0) and minimum point for tachions (see figure 1for M2
0 < 0).For the thick brane warp factor defined in (2.14), we have
Veff = M20 (sech4β(cy)/eβ tanh2(cy) − 1).
The tachions are stably confined, while causal particles are not, that is, the gravity aloneis not sufficient to confine causal particles, as shown in the Fig. 1.
– 12 –
-4 -2 0 2 4
-3
-2
-1
0
1
2
3
Figure 1. Effective potential for a causal free particle test Veff for a particular choice of theconstants (β =
√22 , c = 1,M3
P =√
2,M20 = 3).
Then, it is necessary to introduce another mechanism for test particles to be stablyconfined on the brane. In [21], the author proposed a mechanism where the particle iscoupled non-minimally with the scalar field to solve the problem of the confinement.
3.3 Particle coupling with scalar field
The localization of general fields in the RS model, is only possible by implementationof specific mechanism which explain the field localization on the brane. As an example,in quantum regime, the Dirac spinor field can be trapped on the brane by a Yukawa-typeinteraction with the scalar field [31]. The action is expressed by
Sψ =
∫dyd4x(iψΓA∂Aψ − hφψψ). (3.25)
When the classical domain wall solution φB is considered, the field equation for ψ is
iΓA∂Aψ − hφBψ = 0, (3.26)
by analogy to the Dirac equation, we note that the scalar field generates mass for the five-dimensional fermion. Admitting that the five-dimensional fermion possesses an rest massM0, due to the Yukawa interaction the following relation is obtained
PAPA = −(M2
0 + h2φ2), (3.27)
where PA represents the 5D-momentum of the fermion. So, assuming that this mechanismcan emerge in a classical picture of test particles, the confinement is made possible by
– 13 –
allowing an direct interaction between the test particle and scalar field [21]. The interactionis proposed by redefinition of the action (3.3) modifying the mass through the Yukawainteraction
M0 → (M20 + h2φ2)
12 , (3.28)
then, the new action for the test particle assumes the form
S =1
2
∫ (−gAB
dxA
dθ
dxB
dθ+M2
0 + h2φ2)dθ, (3.29)
where we choose E = 1. For the action (3.29), we obtain the equations of motion,
d2y
dθ2+ a′
(M2
0 + h2φ2 +
(dy
dθ
)2)
+ h2φφ′ = 0, (3.30)
where the constraint gAB dxA
dθdxB
dθ = −(M20 + h2φ2, was used and the equation of motion
(3.30) for massless test particles becomes well defined.Multiplying (3.30) by the factor e2a dydθ , the first integral can be obtained, then we obtain
a conserved quantity [(dy
dθ
)2
+M20 + h2φ2
]e2a = σ. (3.31)
Evaluating in y = 0 we can rearrange it and obtain an energy equation in the direction ofthe extra dimension as below
e2a(dy
dθ
)2
=
(dy
dθ
)2 ∣∣∣y=0−[M2
0
(e2a − 1
)+ e2ah2φ2
], (3.32)
the last term is the effective potential
Veff(y) =[e2a(M2
0 + h2φ2)−M20
], (3.33)
which reduces to the free case (3.23), to M20 6= 0, when h = 0. We can obtain the extreme
points of the effective potential by evaluating its derivatives, as in the previous section.Taking the first derivative of the effective potential we obtain
V ′eff(y) = 2e2a[a′(M2
0 + h2φ2)
+ h2φφ′]. (3.34)
The real values for y that must satisfy V ′eff (y) = 0 are
y = 0; (3.35)
|ymax| =1
carctanh
[(3
2− M2
0
2h2v2+
1
2β
+
√−4h2v2β(h2v2 − 3M2
0β) + (−h2v2 +M20β − 3h2v2β)2
2h2v2β
) 12
(3.36)
, (3.37)
– 14 –
-6 -4 -2 0 2 4 6-5
0
5
10
15
Figure 2. Confining behavior of the effective potential Veff when the interaction is taken intoaccount (thin line) for a particular choice of the constants (c = 1, v = 6,M
(3)P =
√2 and M2
0 = 3),compared to the case when there is no interaction (tick line), i.e. for h = 0.
that represents the extreme points of the function. To evaluate the conditions for stableconfinement of the test particle, we must evaluate the extreme points in V ′′eff (y),
V ′′eff(y) = 2e2a{φ′2[h2 − 1
12M3P
(M2
0 + h2φ2)]
+ 2h2φφ′ + h2φφ′′}
+2a′V ′eff (y), (3.38)
where the relationship (2.13) has been used. We obtain one condition for the mass particle,so that y = 0 is a minimum point [21].
M20 < 12M
(3)P h2. (3.39)
The point y = 0 will represent a stable equilibrium point when the mass in the bulksatisfies inequality (3.39). However, the author does not evaluate the mass of the testparticle on the brane.
– 15 –
If the particle acquires velocity related to the extra dimension, the kinetic energy ofthe particle along the extra dimension should imply an increase in the effective mass, so themotion of the particle in the extra dimension is characterized by the increasing of the mass.We can thus analyze the effective mass of the particle in RS-II models, and then evaluatethe observable values. In the next section we will develop an mechanism showing we candetermine either the particle can be confined just by knowing the function describing theparticle mass in the bulk.
– 16 –
4 General conditions to confinement and effective mass of test particlesin braneworld
Following the discussion in Section 3, we need an expression that gives condition toconfine the test particle and the effective mass observed on the brane. In this direction,the authors in [24] define the effective mass through the constraint gAB dxA
dθdxB
dθ + 1 = 0,but they do not provide a solution. In this section, we obtain a general expression for theeffective mass of test particles observed on the brane and the confinement conditions.
Different from what occur with fields, where the field localization is based in the Kaluza-Klein modes [3], the effective mass of test particles is found by a affine parameterization inthe motion parameter θ to other parameter λ. Such motion is observed on the hypersurfaceat y = 0, called brane, and this model represents our world [24]. Then, to write the equationof motion (3.5) as
d2xµ
dλ2= 0, (4.1)
we need to find an affine transformation.Therefore, on the brane (y = 0), we must have the Minkowsky metric as the following
ηµνdxµ
dλ
dxν
dλ+m2
eff = 0, (4.2)
where m2eff is the effective mass of test particle on the brane. Then, by parametrization
(3.7) we obtain the equation
ηµνdxµ
dλ
dxν
dλ= −e−2a
(dθ
dλ
)2[M2(y) + g55
(dy
dθ
)2], (4.3)
and the action (3.3) is invariant under reparametrization E(λ) = dθdλE(θ) for the new
variable, and by constraints (3.7) and (4.2) we choose E(θ) = E(λ) = 1, therefore therelation under the parameters is given by
(dθdλ
)2= 1. Using the parametrization (4.2), the
effective mass of the test particle on the brane is
m2eff = M2(0) +
(dy
dθ
)2 ∣∣∣y=0
. (4.4)
From Eq. (4.4), the effective mass is dependent on the initial velocity of test particle in theextra dimension direction, and can be limited, since the maximum initial velocity dependson the effective potential (see section 3.2).
4.1 General condition for confinement of test particles in braneworld
In order to analyze the confinement of the most general mass of test particles, weassume a test particle with mass M(y) in action (3.3), where the test particle bare mass iscoupled with any field dependent on the extra dimension. Thus, the conjugated momentumin 5-D dimensional spacetime is PAPA = gAB
dxA
dθdxB
dθ = −M(y)2. Along this paper, theequation for conservation of energy along the extra dimension have the following form
e2a(dy
dθ
)2
=
(dy
dθ
)2 ∣∣∣y=0− Veff(y), (4.5)
– 17 –
which is equivalent from the one used [21], with the advantage of the potential in (4.5)obtained by the action (3.3) with E(θ) = 1 does not have M2
0 = 0 as singularity.Then, for a mass of the test particle in the bulk, a conserved quantity is given by
e2a(dy
dθ
)2
=
(dy
dθ
)2 ∣∣∣y=0−[e2aM2(y)−M2(0)
], (4.6)
whereVeff(y) = e2aM2(y)−M2(0), (4.7)
is the effective potential. The confinement of the test particle is granted on the brane if thefollowing conditions are satisfied:
(i) V ′eff(y)|y=0 = 0.
Taking the first derivative of effective potential
V ′eff(y) = 2(a′M2 +MM ′)e2a, (4.8)
and evaluating in y = 0, the condition (i) is satisfied ifM ′(0) = 0, once that a′(0) = 0.
(ii) V ′′eff(y)|y=0 > 0 .
Taking the second derivative of the effective potential
V ′′eff (y) = 2(a′′M2 + 2a′M ′M +M ′2 +MM ′′)e2a + 2a′V ′eff (y), (4.9)
and evaluating in y = 0, the condition (ii) is satisfied if the following inequality issatisfied
M(0)M ′′(0) +M2(0)a′′(0) > 0. (4.10)
In summary, the confinement of test particle on the brane dependent only on its massin the bulk. In the next section, we show how to obtain the effective mass of the testparticle on the brane.
4.2 Effective mass
According to equation (4.5), the particle can move in the region where(dydθ
)2 ∣∣∣y=0−
Veff(y) ≥ 0, therefore the maximum distance the geodesic can reach (i.e. y = ymax) is given
by the condition Veff(ymax) =(dydθ
)2 ∣∣∣y=0
. Then, replacing Veff(ymax) =(dydθ
)2 ∣∣∣y=0
in (4.7),
the maximum initial velocity of the particle is given by(dy
dθ
)2 ∣∣∣y=0
= e2a(ymax)M2(ymax)−M2(0). (4.11)
Then, assuming that the condition (4.10) is satisfied, we can replace (4.11) in theequation (4.4) to obtain the maximum effective mass of test particle observed on the brane,which is given by
m2eff = e2a(ymax)M2(ymax). (4.12)
– 18 –
This result agrees with [5], where any mass parameter m20 on the visible 3-brane, in the
fundamental higher-dimensional theory will correspond to a physical mass equals to
M2 = e2am20. (4.13)
The results found here, for effective mass of test particles (Eq.(4.12), has the followingconnections with existing effective mass of test particles in the literature.
• Free test particle: The case where the free particle with bare mass is given byM(y) =
M0, we obtain from (4.10), the following results
M20a′′(0) > 0, (4.14)
but a′′(y) < 0 for all y (2.13). Then, the free massive test particle is not confined, asview in Section 3.2, because the free test particle is subject to an effective potentialVeff = M2
0 (e2a − 1). So, y = 0 is an unstable equilibrium point (see Fig. 1). In thecase of a bare massless test particle, Eq. (4.10) is identically null, implying y = 0,which has also a unstable equilibrium point.
Therefore, and in agree with the conclusion of [20, 21], free test particles cannot beconfined on the brane. In order to confine test particles, it is required to couple thetest particle with some field which depends of the extra dimension while satisfyingequation (4.10).
• Test particle coupled with scalar field: For the test particle coupled with scalar field(3.27), its mass is given by M2(y) = M2
0 + h2φ2 [21]. Replacing it in the inequality(4.10), we obtain
h2 − M20
12M3P
> 0, (4.15)
and we obtain the same condition (3.39): M20 < 12M
(3)P h2, for non-null bare mass,
M20 6= 0 and
h2 > 0 (4.16)
for null bare mass, M20 = 0.
Then, a test particle in the bulk, with non-null bare mass, M0 6= 0, is observed on thebrane, where the effective mass (4.12) given by
m2eff < [12M
(3)P + φ2(ymax)]h2e2a(ymax), (4.17)
where ymax is given by (3.37), and represents the maximum point of the effective potential(3.33), which is the point where the equation (3.22) is satisfied (see Fig. 2, for h =
√3).
Then, if the test particle has a mass greater than that of inequality (4.17), the test particlecannot be stably confined.
In the case of test particles with null bare mass, M0 = 0, the effective mass on thebrane is obtained by taking the limit when M2
0 → 0 in the equation (4.17) and condition(4.16), so we have
m2eff < h2φ2(ymax); (4.18)
– 19 –
but φ(0) = 0, therefore, on the brane, the test particle is observed as a massless test particle
m2eff = 0. (4.19)
Besides that, the canonical momentum in the bulk (3.27), for M20 = 0 given by PAPA =
−h2φ2 is null on the brane, which agrees with the results obtained in [21], and is non nullout of the brane.
– 20 –
5 Confinement of test particles non minimally coupled with the dilaton
In Section (2.3), we have seen that the dilaton field acts in a way that it modifiesthe space-time metric, allowing us to choose an appropriate field of the form π(y) =
−√
3M3Pa(y) and b(y) = 1
4a(y) such that the equation (2.13) remains unchanged; we em-phasize that the free particle will also not be confined in this new scenario .
Here, we propose a new mechanism for the confinement of the test particles inspiredin [25], where they have used a conformal transformation of the Jordan to Einstein framegenerating a new field, called dilaton. In this scenario, all matter fields are coupled withthis new field. In the simplest case, where matter is phenomenologically represented by aset of mass point particles m, the matter action becomes
Sm = −∑∫
meλπds. (5.1)
This can yet be reformulated as a spacetime dependent mass in the conformal Einsteinframe, m = meλπ, where m is the constant mass in conformal Jordan frame [25, 26].
So, we propose the test particle non-minimally coupled with the dilation field in thefollowing form
M20 →M2
0 e2λπ, (5.2)
whose conjugated momentum is PAPA = −M20 e
2λπ(y), which makes possible to confine theparticle on the brane.
5.1 Dilaton coupling in deformed RS spacetime
The action for the test particle (3.3), for M2(y) = e2λπM20 , can be written as:
S =1
2
∫ (−gAB
dxA
dθ
dxB
dθ+M2
0 e2λπ
)dθ, (5.3)
where we choose the new variable E(θ) = 1. Using the condition for confinement of thetest particle (4.10) and replacing the mass of this test particle given by (5.2), we obtain theinequality
M20 (−λ
√3M
(3)p + 1)a′′(0) > 0. (5.4)
However, as a′′ < 0 for all y, the inequality above is satisfied only if
λ >1√
3M3P
, (5.5)
for a non-null bare mass of the test particle in the bulk (M20 6= 0). For the case where the
bare mass of test particle is null, M20 = 0, the inequality (5.4) is not satisfied, therefore the
massless test particle is not confined.Then, a test particle in the bulk with non-null bare mass, M0 6= 0, is observed on the
brane with maximum effective mass given by
m2eff = M2
0 e2a(ymax)(1−λ
√3M
(3)P ), (5.6)
– 21 –
-6 -4 -2 0 2 4 6
0
5
10
15
Figure 3. Confining behavior of the deformed effective potential Veff(s)when the interaction is
take into account (thin line) for a particular choice of the constants (M20 = 3, b = 1
2 , c = 1, M3P =
0.3, λ = 1.5), compared to the case when there is no interaction (tick line), i.e. for λ = 0.
where ymax can be obtained analyzing the effective potential, Eq. (4.7), that is given by
Veff(y) = M20 (e2a(1−λ
√3M
(3)P ) − 1), (5.7)
where the case for a free test particle is obtained when we take λ = 0. By analysing thesecond derivative of the effective potential (5.4)
V ′′eff(y) = 2(−√
3M3Pλ+ 1)a′′e2(λπ+a) + 4(λπ′ + a′)2e2(λπ+a),
we note that the first term is positive only if the relation (5.5) is satisfied, once that(φ′)2 = −12M
(3)P a′′ for all values of y and the second term is always positive, which means
that the second derivative of effective potential (5.4) is positive for all values of y, implyingin a maximum potential with ymax →∞ and Veff(ymax)→∞. Therefore, the effective massof the free particle coupled with the dilaton field on the brane can assume any value,
m2eff <∞. (5.8)
In conclusion, there is no restrictions on the mass of the test particle in the bulk, andit is valid for deformed brane in any value of s, and we were able to solve the problem ofthe limited mass, but we restricted the solutions in a way that light type test particles arenot stably confined on the brane.
– 22 –
6 Conclusion
In summary, in this paper, we developed a general condition for the confinement oftest particles on the brane, where it has been show that the confinement depends only theform of the coupling between the test particle and the field in the background. We alsoobtain a expression that provides the mass observed on the 3-brane where we propose anon minimal coupling between the test particle and the dilaton field to solve the problemof the effective mass limited for massive test particles on the brane.
Thus, in this paper the motion of test particles on the brane world RS-II type delta,thick and deformed models was reviewed. It is well known that, in these models, theminimal couple between test particle and gravity is not enough to trap a free particleif it is transversely disturbed [20]. Similarly as with some fields, (eg gauge field) wheresome mechanism is needed for it to be localised on the brane, with test particles it is alsonecessary to create mechanisms that allow its localization, since the minimum couplingbetween particles and gravity is not able to confine it.
For many cases, the most simple mechanism is to reverse the brane tension, includingthe case of test particle. The crucial problem in reversing the brane tension is the fact thatgravity is not localized [12], which is not physically acceptable. Therefore, the confinementof test particles in models with type delta branes has not been solved. In the case of testparticles in smooth models, a solution was proposed based in the Yukawa interaction, amechanism that solves the localization problem of Dirac spinor field on the brane throughan interaction between the spinor and the scalar field [21]. In this interaction the mass of thespinor becomes a function of the scalar field. Based on this idea and assuming this picturein classical level, in this paper, a new Lagrangian of test particle is defined, assuming thatmass depends of the scalar field in the form M0 → (M2
0 + h2φ2)12 , the mechanism allows
the confinement of test particles with mass that obey the relation M20 < 12M
(3)P h2.
We further develop the result obtained in work [21] where we obtain the effective massof the particle observed on the brane and we use a polynomial action for test particle (3.3),allowing to evaluate as massless test particles are observed on the brane. By moving in theextra direction, part of the total kinetic energy that was initially all concentrated on thebrane, is now converted into the energy of the extra dimension, being characterized by thegain of effective mass of the particle with maximum value given by m2
eff = M20 e
2a(ymax).
The maximum values to the mass in the bulk M02 < 12M
(3)P h2 that obeys the condition
M(0)M ′′(0) + M(0)2a′′(0) > 0 and guarantees the confinement of the test particle. Thus,the effective mass observed on the brane is m2
eff < [12M(3)P + φ2(ymax)]h2e2a(ymax) for a
non-null bare mass and m2eff < h2φ2(ymax) for a null bare mass of the test particle.
Finally, we show how to solve all problems relative to the confinement of the massivetest particle on the brane. We propose the coupling between the dilaton field and the massof the test particle in the bulk as given by M0 → eλπM0 making possible the confinement,because the new mass obeys the relationship M(0)M ′′(0) +M(0)2a′′(0) > 0 if λ > 1√
3M3P
.
The condition to λ is also obtained evaluating the effective potential that acts on the testparticle Veff = e2(λπ+a) − 1 such that its second derivative is positive at y = 0 ensuring a
– 23 –
minimum point at this point. Taking the second derivative of the effective potential
V ′′eff (y) = 2(−√
3M3Pλ+ 1)a′′e2(λπ+a) + 4(λπ′ + a′)2e2(λπ+a),
we note that the second term is always positive and if the condition for λ is satisfied andknowing that, by equation (φ′)2 = −12M
(3)P a′′, a′′ < 0 for every y then the maximum
point for the effective potential is given in ymax → ∞ (Fig. 3). The effective mass ofthe test particle observed on the brane is given, as we have seen, by m2
eff = M(0)2[1 +
Veff (ymax)]. And at the limit where ymax →∞ we have that the effective potential diverges,Veff (ymax)→∞. Therefore the effective mass of the particle observed on the brane is
meff <∞.
The problem is the confinement of massless test particles on the brane, whose confinementis not achieved. Without loss of generality, for the case of the topological defect we verifiedthat the test particles coupled with dilaton are also localized for any mass.
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