Standing Wave 6D

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A 6D standing-wave Braneworld.

L. J. S. Sousa a,b , J. E. G. Silva a , C. A. S. Almeida a

aDepartamento de Fsica, Universidade Federal do CearaCaixa Postal 6030, CEP 60455-760, Fortaleza, Ceara, Brazil

bInstituto Federal de Educacao, Ciencia e Tecnologia do Ceara (IFCE)Campus Caninde

Abstract

We constructed a six-dimensional version of the standing-wave model withanisotropic 4- brane generated by a phantom-like scalar field. The model rep-resents a braneworld where the compact (on-brane) dimension is assumed tobe sufficiently small in order to describe our universe (hybrid compactfication).The proposed geometry of the brane and its transverse manifold is non-static,unlike the majority of braneworld models presented in the literature. Further-more, we have shown that the zero-mode scalar field is localized around thebrane. While in the string-like defect the scalar field is localized on a branewith decreasing warp factor, here it was possible to perform the localizationwith an increasing warp factor.

1. Introduction

The idea that our world is a brane embedded in a higher-dimensional space-time has attracted the attention of the physics community in the last years,basically because the braneworld idea has brought solution for some intrigu-ing problems in the Standard Model (SM), as the hierarchy problem. Themainly theories that carrier the braneworld basic idea are the one first proposedby Arkani-Hamed, Dimopoulos and Dvali [1, 2, 3] and the so-called, Randall-Sundrum (RS) model [5, 4].

One important feature of these models is the assumption that all the matterfields are constrained to propagate only on the brane, whereas gravity is freeto propagate in the extra dimensions. However the presupposition that theStandard Model fields are initially trapped on the brane is not so obvious in

this framework. In this way, it is interesting to seek alternatives field theoreticlocalization mechanisms in braneworld scenarios [12, 13]. Therefore, the knackof a model to localize fields has been used as a parameter to evaluate it as apotential candidate of our universe.

URL: [email protected] (L. J. S. Sousa a,b ), [email protected] (J. E. G.Silva a ), [email protected] (C. A. S. Almeida a )

Preprint submitted to Elsevier September 5, 2012


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Particularly the RS model, which is a 5D theory, was quickly generalized to

six dimensions [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,24, 32, 33, 34, 35]. The scenarios where the brane has cylindrical symmetry arethe so-called string-like braneworlds. There is a broad variety of these modelsas the global string [12, 6, 9], the local string [10, 11, 14], the thick string[20, 21, 22, 19, 17, 16] and the supersymmetric cigar-universe [23].

There exist a vast literature concerned to the localization of fields both in5D [45, 46, 47, 48, 28, 29] as in 6D braneworld [10, 12, 13, 14, 33, 34, 35]. How-ever, at least we know, there is not yet a purely analytical geometry that trapsall the SM fields through only the gravitational field interaction. Hence, thequest for a model which is both analytical and that localizes all the SM fieldsby means of only the gravitational interaction is, in our point of view, a reason-able justification to continue studding field localization in different braneworldmodels.

In this spirit, it has been proposed some braneworld scenarios with non-standard transverse manifold. Randjbar-Daemi and Shaposhnikov has assumedit as a Ricci-flat or an homogeneous space and they obtained trapped masslessgravitational modes and chiral fermions as well [36]. Kehagias proposed a conicaltear-drop whose conical singularity drains the vacuum energy explaining thesmall value of the cosmological constant [37]. Gogberashvili et alhas achievedthree-generation for fermions on a 3-brane whose transverse space has an appleshape [25]. Other examples are the torus [39], a football-shape space [38] andsmoothed versions of the conifold, the resolved [40] and deformed [43, 41, 42].

In this paper we considered a bulk phantom-like scalar field in a 6D braneworldas a source of the brane. The phantom is a scalar field whose sign of the kineticterm in its Lagrangian is negative. This model exhibits instability issues, as un-

bounded negative energy, but phenomenologically it has been useful in differentscenarios [32]. In cosmology, the phantom scalar is a candidate to explain darkenergy and the accelerated expansion of the universe [44].

The use of bulk scalar fields to generate branes was introduced by [50, 51],and it has been largely studied in the literature [52, 53, 54, 55, 56, 57]. Thephantom field has been implemented in the six-dimensional context by Koleyand Kar [32], where it has shown the localization of gauge fields.

In the present model the field is not a phantom in the sense described above.Despite it has the wrong-sign characteristic of the phantom its energy is notever negative and then, there are no energy problems. Thus, it is more conve-nient call this source as a phantom-like scalar.

This exotic source was first proposed in five dimensions, the so-called standing-wave braneworld. This is a completely anisotropic one brane model whose

phantom-scalar is equivalent to a Weyl scalar [28, 29, 30]. The Weyl scalar,in its turn, is used in an extension of the RS model, the so-called pure-gravitybraneworld. In the standing-wave approach it was possible to localize variousfields, albeit it was not possible trap right-handed fermions [28, 29]. Besides,the study of massive modes was not addressed in this model.

The model built in this work consists of a 6D braneworld with an anisotropic4-brane, where the compact (on-brane) dimension is assumed to be sufficiently

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small in order to accomplish our universe (hybrid compactification). Anisotropic

six-dimensional braneworlds have already been addressed before in the thickstring-like models [20, 21, 22, 19, 16], where there are different warp factor fordifferent directions. Nonetheless, it is assumed the brane has a cosmologicalconstant what turned it a locally isotropic manifold. However, the standing-wave braneworld is a completely anisotropic manifold, unless for some pointscalled theAdSislands [28, 29].

Another appreciable feature of our model is its dynamics, in the sense thatboth metric and the bulk phantom-like scalar field are time dependent. Indeed,as in RS model, the most string-like models are static [18, 8, 23, 10, 20, 21, 12,22, 19, 11, 34, 9, 15, 17]. On the other hand, Gregory has proposed a time-dependent string-like brane yielding to gravity localization [6]. The differencehere is that both the 3-brane and the transverse manifold are time-dependent.This allows the exterior geometry of the brane to reflect the changes of the

brane, a key property of the string-like defects [7, 9, 11, 14].In spite of the complexity due the time-dependence and anisotropy, we have

obtained analytical solution for the warp factor and the phantom field. Thebulk is everywhere smooth converging asymptotically to an AdS6 manifold.From these solutions we have analysed some properties of the source. Theenergy-momentum components have compact support around the origin whatmeans that this geometry is created by a local source. Nevertheless, they do notsatisfy the energy conditions, what turns this an exotic scenario, as expected.However, even the thin string-like braneworlds does not obey the dominantenergy-condition [10, 15]. Besides, in the phantom 6D braneworld the energymomentum tensor violates all of energy conditions [32]. Although the bulkspacetime obtained in that setup is not dynamical stable, the authors argued

that it also occurs in other models [32].Once studied the geometry in details we have concerned with the behavior ofa minimally coupled scalar field in this model. As in the string-like model andthe 5D standing-wave this field has its massless mode trapped to the brane. Thedifference between the string-like and the 6D standing-wave approach is that inthe former the scalar field is trapped for a decreasing warp factor whereas inthe last there is a localized mode for a increasing warp factor.

This work is organized as follows: in section 2 we described the model andsolved the Einstein and scalar field equations obtaining the general static ex-pression for the phantom like scalar. In section 3 we have found the non-staticstanding waves and we have discussed its mainly characteristics. The localiza-tion of the zero mode scalar field have been done in the section 4. In the section5, some final remarks and conclusions were outlined.

2. The model

We begin by the general action in 6-dimensional space-time composed by thestandard Einstein-Hilbert action and a bulk massless scalar field action whichis time dependent and minimally coupled to gravity

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S=

1

226

d6

x (6)g(R 2) + gABAB V() (1)

where6 is the 6-dimensional gravitational constant, is the bulk cosmologicalconstant. The wrong signal in front of the kinetic term of the scalar fieldaction characterizes it as a phantom-like field.

The variation of this action with respect to the metric and the field giveus the following equations of motion

RMN 12

gMNR= gMN+ 26TMN, (2)

1g Mg gMNN= V

, (3)

whereM,N, ...denote D-dimensional space-time indices and TMNis the energy-momentum tensor.

We will begin by considering the general metric ansatz

ds2 =eA

dt2 eudx2 eudy2 e3udz2 dr2 R20eB+ud2, (4)where A(r) and B(r) are functions of r , only, and u is function of rand t. This ansatz generalizes the global string like defect consideredby [12, 6], for instance.

For a scalar field the energy-momentum tensor may be written as

TMN =MN gMN

1

2CC + V()

. (5)

from which the Einsteins equation may be rewritten as

RMN =26MN +

1

2gAB(

26V() + 6) (6)

From equations (2) and (6) the non zero components of the Riccitensor, forV() = 0, are given as

Rxx = Ryy =

1

4eA+u

4A2 + 2A

+ A

B 2eAu + (4A + B)u + 2u

= 1

2eA+u6 (7)

Rzz =

1

4eA3u

4A2 2A AB 6eAu + 3(4A + B)u + 6u

= 1

2eA3u6 (8)

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Rtt =

1

4 eA

4A

2

+ 2A

+ A

B

12eA

u2

= 2

62

+

1

2 eA

6 (9)

Rrt = 1

4u(A

B 12u) =26

(10)

Rrr =

1

4eA3u

4A2 B2 8A 2B + 2(A B)u 12u2

=26

2 12

6 (11)

R =

1

4R20e

B+u

B2 + 2B

+ 4A

B 2eAu + (4A + B )u + 2u

= 1

2R20e

B+u6 (12)

which may be simplified to

eAu +12

(4A

+ B

)u

+ u

= 0 (13)

Rrt = 14u(A

B

12u

) =26

(14)

and

1

4

6A + 5AB+

+

1

4

(6A

B)u + 2u 2eAu 12(eAu2 + u2)

=26(eA2 +

2) +1

26 (15)

For A = B equation (14) implies that has to be a phantom-likescalar (minus sign in front of the kinetic term in Lagrangian )which

relates to the function u as 326 u. In this case the equations forthe function u and the warp factor A are given, respectively, as

eAu +52

A

u

+ u

= 0, (16)

A

(r) 56

A2 +

63

= 0 (17)

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The solution to the last equation is than, unless integration constants,

A(r) = 65

log

cosh

5618

r

(18)

This warp factor represents a thick brane and the model proposedhere generalizes the 5D standing wave braneworld not only becausewe are working in six dimensions but in the sense that here we havea thick brane instead of a thin brane.

It is interesting to note that the field, in general, do not necessityto be a phantom one. This solution is a special case when we considerA= B, as we saw above. For A =B , in general, it is difficult to solvethe system of equations,(712). However one could find the functionu and the warp factors A, B and the components 2,

2 of the bulk

scalar and analyses the energy conditions. This job will be leave for another work because here we are interested to study field localizationin this scenario.

Hereinafter, we shall consider A(r)B(r) = 2ar and use the metric ansatz

ds2 =e2ar

dt2 eudx2 eudy2 e3udz2 dr2 R20e2ar+ud2, (19)wherea Rwith dimension [a] =M, 0 r < , 0


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From metric (19) and choosing V() = 0 the scalar field equation (3) reads

1g MggMNN= e2ar 5a = 0, (21)

where g is the bulk metric determinant. The scalar phantom field energy-momentum tensor is given by

TMN = MN +12

gMNCC. (22)

For further reference it is important to write here the bulk curvature scalar

R= 30a2 3e2aru2 + u2 (23)

From (2) the non-zero components of the Ricci tensor are

Rxx =1

2e2ar+u

10a2 + e2aru 5au u

= Ryy (24)

Rzz = 3

2e2ar3u

10

3 a2 e2aru + 5au + u

(25)

Rtt= 1

4

20a2e2ar 12u2 (26)

Rrt = 3uu

(27)

Rrr =

1

420a2 12u

2 (28)

R = R20

4 e2ar+u

20a2 10au 2u + 2e2aru

(29)

From the energy-momentum tensor (22) we may rewrite the Einstein equa-tion (6) in the following simpler form

RMN = MN+ 12

gAB6 (30)

where =

.Comparing (27) and the Ricci components coming from (30) we need to

impose that =

3uto obtain a solution. This imply that the ufield, according

to = , needs to satisfy the relation (21). This is possible if, by comparing(24) with Rxx in (30) we do

6= 10a2 (31)

Therefore, the differential equation for the u,which is a simplification of(16) is given by

e2aru(r, t) 5au(r, t) u(r, t) = 0, (32)

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where prime and dots mean differentiation with respect to r and t, respectively.

In order to solve equation (32) we proceed by separating the variable as follows

u(r, t) =g(t)(r), (33)

where we require that g(t) satisfies the following basic requisites: g g; gg. In other words, g(t) may be an exponential, sine, cosine , hyperbolic sine,hyperbolic cosine... function of time. Then, assuming thatg(t) satisfies someof this requirements, the equation to the new variable(r) reads

(r) + 5a

(r) + 2e2ar(r) = 0 (34)

where is a constant which follows from the g time derivative.In order to solve equation (34) we perform the following change of variables

z = a ear; = F h (35)

whereF =a

5/2e

5

2ar. This give us the well-known Bessel equation of order

52 , for the new variable z

2h(z) +1

zh(z) +

1 25

4

1

z2

h(z) = 0 (36)

where= /z. The general solution to this equation is given by

h(z) =AJ52

(z) + BY52

(z) (37)

where A and B are integration constants and J, Y are the Bessel function offirst and second kind, respectively. In term of the r variable the solution reads

(r) =C1e

5

2arJ5

2

(

aear) + C2e

5

2arY5

2

(

aear) (38)

where C1, C2 are the new constants. In order to obtain the u function 33 weneed to specifies the g function. In the next section we analyse the case whereg(t) is a sine function of time, which implies in a standing waves solution [27].

3. Standing waves solution

In order to obtain a standing wave solution we choose u(r, t) =sin(t)(r)where is given by (38) and is a constant. In this case = the solution is

(r) =C1e

5

2arJ5

2

(

aear) + C2e

5

2arY5

2

(

aear) (39)

Some properties of the functionu(r, t) may be found from the solution above.The first is the fact that the Y function, because its exponential dependence ofr, do not have divergence on the origin, as one usually find. For large r anda > 0 the term e

5

2ar in front ofYdominates and we do not have divergence

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forr , as ones expect. For a 0the function 39, for C1 = 0 or C2 = 0 will converge rapidly depending on thevalue of the ratio /a. The quantity of zeros will depend on the value ofa andprincipally on the value of this ratio. In this case we have a finite number ofzero. Fora


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this case the energy density assumes negative values which shows the exotic

nature of the source which is not a phantom field in the sense that is studied inthe literature [58, 28, 29, 31]. The dominant energy condition is not satisfied,which is common in phantom field theory, because, even in the case that theenergy density is positive it would be necessary it to be greater than the pressurecomponents. However, all of this quantities are finite which is less problematicin comparison with infinity unbounded energy momentum components. It isapplicable to say that the presence of models which do not satisfy the dominantenergy condition is frequent in the literature. We may cite the 5D version ofthis present model [27] as a case where the energy conditions is not satisfied. In6D braneworld there exist models that present this unorthodox feature, as theproposed by Koley-Kar [32] and the thin string-like brane [10, 15].

In the figure (3) we plot the bulk curvature scalar for three different valuesof : sectioned line for = 5.76; dotted line for = 9.09 and filed line for

= 12.3. We see that in all this cases R is finite and positive, which reveals adS scenario. Beside the fact that the bulk is a de Sitter space time, the brane isan AdS universe as can be seen in figure 4. In this graphic the values assumedby are: = 5.76; dotted line for = 9.09 and filed line for = 12.3.

From the figures (14) the brane behaves like a thick brane which is locatedin the origin and extends yourself to the position r = 1.5, approximately. Thisprofile and the behaviour of the energy components is consequence of the exoticsource that generate it. It is worthwhile to stress that this figures representtime-average of the quantity in question. Hence, it is possible, in some timeinterval, have the energy conditions satisfied.

0.2 0.4 0.6 0.8 1.0 1.2

6

4

2

Tr

Figure 1: T profile. = 5.76

0.2 0.4 0.6

12

10

8

6

4

2

Tr

Figure 2: T profile. = 9.09

In brane world scenarios there is an interest in the potential of the model tolocalizes the Standard Model (SM) fields. However, find a model where all of theSM fields are localized only by means of gravitational interaction (without the

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0.0 0.2 0.4 0.6 0.8

10

20

30

40

Figure 3: R profile. = 5.76, 9.09, 12.3

. . .

5

4

3

2

1

Figure 4: R(4) profile. = 5.76, 9.09, 12.3

necessity of additional fields, as the dilaton field) is not an easy task. The simpleextension of the RS model from 5D to 6D sometimes is sufficient to eliminatethe necessity of additional interaction in some cases, as can be seen in thisworks: in the two first we have 5D models where the localization of the gaugeand Kalb-Ramond fields, respectively, are possible only with the introduction ofthe dilaton field [48, 45] but in 6D the same fields are localized only by meansof gravity interaction [12, 13, 35]. Here, we have a generalization of the thinbrane world previously studied in six dimension[10, 14, 11, 15, 12, 13, 35, 6]

and the so-called standing waves brane world 5D model [27, 28, 29] whereit is possible to localize gravity, scalar, vector and tensor fields but it was notpossible to trap the right fermions [31]. We argue that this synthesis enabletrap SM fields without the necessity of any other interaction than gravity. Inthis work, we shall give the first step in this analysis beginning by the scalarfield whose localization will be studied in the next section.

4. Scalar field localization

In order to study the localization of the bulk scalar field consider the action

S=

1

2 d6xgg

MNMN (42)

from which we derive the equation of motion

1g MggMNN= 0 (43)

From the metric ansatz (19) we have thatg = e5ar. Thus, equation (43)

may be rewritten as

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2t e

u 2x+

2y e3u2z e

u

R20 2

= e3ar

e5ar

(44)

Consider a solution of the form

(xM) = (r, t)(x, y)(z)eil (45)

Substituting (45) in (44) yields the following system of equations

2x+

2y

+

p2x+ p

2y

= 0 (46)

2z +p2z= 0 (47)

2t + e

up2x+ p

2y

+ e3up2z+ l

2

eu

R20

= e3ar

e5ar

(48)

Now we turn back to (48) and we separate the variables by making (r, t) =eiEt (r), which will give us

e5ar(r)

e3arG(r)(r) = 0, (49)

where

G(r) =p2x+p

2y

eu 1+p2z e3u 1 + l2R20 eu (50)

For further convenience we write (49) in an analogue non-relativistic quan-tum mechanic problem by making the variable change (r) = e5/2ar(r).Therefore, we obtain

(r) V(r)(r) = 0, (51)where

V(r) = 25

4a2 + e2arG(r). (52)

In order to analyse the localization of the scalar field we need to obtain thert-dependent function in (48). We shall do it by means of equation 51, butthis will be done only for the scalar zero mode and s-wave, so we will assume

(l = 0) and E = p2x+ p2y + p2z. Furthermore, we shall consider the case where >> Ewhich permit us to perform the time-averaging ofV(r) reducing theproblem to just one variable, r . Then, using the following expressions

ebu =+n=0

(bu)n

n! (53)

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/2 2/0

[sin(t)]m = 22n(2n)!(n!)

2; (m= 2n) (54)

we find

ebu

= 1++n=1

(b)2n

22n(n!)2[C1e

5

2arJ5

2

(

aear)+C2e

5

2arY5

2

(

aear)]2n =I0(b(r))

(55)whereI0 is the modified Bessel function of the first kind. Even in this case, ascan be seen from the expression (55), find analytical solution for (51) is not soeasy. We shall accomplish it considering asymptotic approximations.

For C2 = 0 in (39), the u(r, t) will depends on first kind Bessel function

J52 . The expansion 55 will be given byebu

= 1 ++n=1

(bC1)2ne5anr

22n(n!)2 [J5

2

(

aear)]2n (56)

Let us study the behavior of equation 51 in two distinct regions: far from andnear to the brane. Forr + the arguments inJ5

2

goes to zero ((/a)ear 0) so the expression (56) will be approximated as

ebu 1. Consequently

equation (51) will assume the simple form

(r) 254

a2(r) = 0 (57)

whose solution ise5

2ar. We choose =e

5

2ar anda >0 which is naturally con-

vergent. This solution is the same find in 5D standing-wave, for the localizationof scalar field, in the same asymptotic limit considered here [29].

Forr 0 the equation 51 may be approximated by

(r)

65

2ca2r2 12car+ c

(r) = 0 (58)

This is a generalization of the same equation found for localization of scalarfield in 5D standing waves braneworld [29]. The constants c and c

are given,respectively, by

c= C1

(

7

2)2

2a5

(p2x+p2y+ 9p

2z) (59)

c

= 25

4a2 + c (60)

Equation (58) is nothing but the parabolic cylinder equation whose generalsolution is

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(r) = E1D

12(2c)

1/4a

653+

a

130c

r

+ E2D

i 12(2c)

1/4a

653+ i

a

130c

r

(61)

whereD is the parabolic cylinder function andE1, E2are integration constants.We see that E2 must be zero in order to have a real solution. The , indexesare given respectively by

= 64a

130c 144c + 130c130a

130c

(62)

=4225ac 72130c + 65130c

8450a

c (63)

For E2 = 0 and /a= 5, 76 it is possible to choose the overall constants inorder to expand our solution 58 as

(r) 2.83 18.7r+ 5.58r2 9.78r3 + 10.7r4 0[r]5 (64)In this case the function (58) assumes the form D0.53(1.5 + 2.85r). Theapproximation (64) and, consequently, the extra part of the scalar zero-modewave function (r) has a maximum at r = 0 and it falls of from the brane,assuming the asymptotic form e5ar which is in accordance with [29].

This result shows the localization of the scalar filed zero-mode in this model.

It is interesting to point out that the localization is obtained for an increasingwarp factor whereas in the thin string-like brane the localization of the zero-mode scalar field is obtained for a decreasing warp factor [12, 13].

5. Remarks and conclusions

We have constructed a six-dimensional standing-wave model where the braneis generated from a phantom-like scalar field. We have supposed the compact di-mension staying on the brane and small enough to accomplish an realistic model.The metric ansatz, unlike the most braneworld models present in the literature,is anisotropic and non-static and the compact dimension is time dependent.The bulk geometry tends asymptotically to spacetime with positive cosmologi-

cal constant. From the action composed by the gravity and a phantom-like fieldwe derived the equation of motion and we have evaluated the phantom scalarexpression.

The exotic source which generated the brane is well characterized by itsenergy density. As can be seen in the figures 1 and 2, this quantity has not afixed value varying in intensity and signal. While the traditional phantom fieldhas negative energy density, in the 6D standing-wave model this quantity may be

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positive and is finite. This property prevent the model from unbounded infinite

energy density. Far from the brane the bulk presents the features expected fora space with non-null cosmological constant, as expected. Once we know theimportance of phantom fields in cosmology, but in the same time, the difficultythat the phantom physic suffer, like the negative unbounded energy density, ourmodel may presents an alternative phantom model for phenomenology.

Moreover, in the context of braneworld it is suitable to investigate whether amodel is able to localize fields. In order to analyze if our solution yields the fieldlocalizations we studied the zero mode scalar field localization. The results haveshown that there is a zero mode localization for the scalar field. The solutionfound here is in accordance with the one first encountered in five dimensions[29].

In future works we intend to investigate the second part of the general solu-tion to our phantom-like scalar field. Further, we shall verify the possibility of a

brane with a decreasing warp factor. Moreover, we shall study the localizationof the overall SM fields, particularly the right-handed fermions which are notlocalized in the 5D version of this model,as was previously mentioned.

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