Research ArticleA Note on Asymmetric Thick Branes

D Bazeia12 R Menezes23 and R da Rocha4

1 Departamento de Fısica Universidade Federal da Paraıba 58051-970 Joao Pessoa PB Brazil2 Departamento de Fısica Universidade Federal de Campina Grande 58109-970 Campina Grande PB Brazil3 Departamento de Ciencias Exatas Universidade Federal da Paraıba 58297-000 Rio Tinto PB Brazil4 CMCC Universidade Federal do ABC 09210-170 Santo Andre SP Brazil

Correspondence should be addressed to D Bazeia dbazeiagmailcom

Received 4 April 2014 Revised 20 May 2014 Accepted 22 May 2014 Published 11 June 2014

Academic Editor Rong-Gen Cai

Copyright copy 2014 D Bazeia et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study asymmetric thick braneworld scenarios generated after adding a constant to the superpotential associated with the scalarfield We study in particular models with odd and even polynomial superpotentials and we show that asymmetric brane can begenerated irrespective of the potential being symmetric or asymmetric We study in addition the nonpolynomial sine-Gordon likemodel also constructed with the inclusion of a constant in the standard superpotential and we investigate gravitational stabilityof the asymmetric brane The results suggest robustness of the new braneworld scenarios and add further possibilities of theconstruction of asymmetric branes

1 Introduction

The concept of braneworld with a single extra dimension ofinfinite extent [1] plays a fundamental role in high energyphysics and cosmology In such braneworld scenario theparticles of the standard model are confined to the branewhile the graviton can propagate in the whole space [1ndash6]see also [7ndash14] for further details

Most braneworld scenarios assume a Z2-symmetric

brane as motivated by the Horava-Witten model [15 16]Notwithstanding more general models that are not directlyderived from M-theory are obtained by relaxing the mirrorsymmetry across the brane [17ndash40] Asymmetric branes areusually considered in the literature as a framework to modelswhere the Z

2symmetry is not required The parameters

in the bulk such as the gravitational and cosmologicalconstants can differ on either side of the brane The termcoined ldquoasymmetricrdquo branes means also that the gravitationalparameters of the theory can differ on either side of thebrane In some cases the Planck scale on the brane differson either side of the domain wall [39ndash41] One of the promi-nent features regarding asymmetric braneworld models isthat they present some self-accelerating solutions without

the necessity to consider an induced gravity term in the braneaction [33]

Braneworld models without mirror symmetry have beeninvestigated in further aspects For instance different blackhole masses were obtained on the two sides on the brane [17]as well as different cosmological constants on the two sides[19ndash24] In addition one side of the brane can be unstableand the other one stable [35 36] If the Z

2symmetry is

taken apart it is also possible to get infrared modificationsof gravity comprehensively considered in the asymmetricmodels in [39 40] and in the hybrid asymmetric DGPmodelin [30] as well In both models the Planck mass and thecosmological constant are different at each side of the braneIn the asymmetric model there is a hierarchy between thePlanck masses and the AdS curvature scales on each side ofthe brane [34]

Asymmetric braneworlds can be further described bythick domain walls in a geometry that asymptotically is led todifferent cosmological constants being de Sitter (dS) in oneside and anti-de Sitter (AdS) in the other one along the per-pendicular direction to the wall The asymmetry regardingthe braneworld arises as the scalar field interpolates betweentwo nondegenerate minima of a scalar potential without Z

2

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 276729 8 pageshttpdxdoiorg1011552014276729

2 Advances in High Energy Physics

symmetry Asymmetric static double-braneworlds with twodifferentwallswere also considered in this context embeddedin an AdS

5bulk Furthermore an independent derivation of

the Friedmann equation was presented in the simplest formfor anAdS

5bulk with different cosmological constants on the

two sides of the brane [38] Finally a braneworld that acts asa domain wall between two different five-dimensional bulkswas considered as a solution to Gauss-Bonnet gravity [19 33]Models with moving branes have been further considered[42] in order to break the reflection symmetry of the branemodel [17 20 22 23 43ndash45] Other interesting models thatdeal with asymmetric branes have been done some of themcan be found in [46ndash51]

In the current work we will further study asymmetricbranes focusing on general features which we believe can beused to better qualify the braneworld scenario as symmetricor asymmetric The key issue is to add a constant to thesuperpotential that describes the scalar field which affectsall the braneworld results In particular one notes that itexplicitly alters the quantum potential that responds for sta-bility evincing and unraveling prominent physical featuresas the asymmetric localization of the graviton zero modeWe analyze all the abovementioned features of asymmetricbranes in the context of superpotentials containing odd andeven power in the scalar field up to the third- and fourth-order power respectively and the sine-Gordon model

The investigation starts with a brief revision of theequations that govern the braneworld scenario under inves-tigation getting to the first-order framework where first-order differential equations solve the Einstein and fieldequations if the potential has a very specific form Wedeal with asymmetric branes in the scenario where thebrane is generated from a kink of two distinct and well-knownmodels with the superpotentials being odd and evenrespectively Subsequently the sine-Gordon model is alsostudied and the associated linear stability is investigated Thequantum potential and the graviton asymmetric zero modeare explicitly constructed

2 The Framework

We start with a five-dimensional action in which gravity iscoupled to a scalar field in the form

S = int1198895119909radic10038161003816100381610038161198921003816100381610038161003816 (minus

1

4119877 +L) (1)

where

L =1

2nabla119886120601nabla119886120601 minus 119881 (120601) (2)

and 119886 = 0 1 4We are using 4120587119866 = 1 with field and spaceand time variables dimensionless for simplicity By denoting119881120601= 119889119881119889120601 the Einstein equations and the Euler-Lagrange

equation are 119866119886119887= 119879119886119887and nabla119886nabla

119886120601 + 119881120601= 0 respectively We

write the metric as

1198891199042= 1198902119860120578120583]119889119909120583119889119909

]minus 1198891199102 (3)

where 119860 = 119860(119910) describes the warp function and onlydepends on the extra dimension119910 Taking the scalar fieldwithsame dependence we obtain

11986010158401015840+2

312060110158402= 0 (4a)

11986010158402minus1

612060110158402+1

6119881 (120601) = 0 (4b)

12060110158401015840+ 412060110158401198601015840minus 119881120601= 0 (4c)

where prime stands for derivative with respect to the extradimension As one knows solutions to the first-order equa-tions

1206011015840=1

2119882120601 (5a)

1198601015840= minus

1

3119882(120601) (5b)

also solve (4a) (4b) and (4c) if the potential has thefollowing form

119881 =1

81198822

120601minus1

31198822 (6)

where119882 = 119882(120601) is a function of the scalar field 120601This resultshows that if one adds a constant to 119882 it will modify thepotential and consequently all the results that follow fromthe model

To better explore this idea we study three distinctmodelswhich generalize previous investigations

21 The Case of an Odd Superpotential Let us introduce thefunction

119882119888(120601) = minus

2

31206013+ 2120601 + 119888 (7)

where 119888 is a real constantThis119882 is an odd function for 119888 = 0The potential given by (6) has now the form

119881 (120601) =1

2(1 minus 120601

2)2

minus1

3(119888 + 2120601 minus

2

31206013)2

(8)

It has the Z2symmetry only when 119888 = 0 the symmetry is

now 120601 997891rarr minus120601 and 119888 997891rarr minus119888 There are minima at 120601plusmn= plusmn1 with

119882119888(120601plusmn) = plusmn

4

3+ 119888 (9)

The maxima depend on 119888 and are solutions of the algebraicequation 41206013max minus 39120601max minus 2119888 = 0 see Figure 1

We focus attention on the scalar field There is a topolog-ical sector connecting the minima In this sector there aretwo solutions (kink and antikink) which can be mapped toone another when 120601 997891rarr minus120601 or 119910 997891rarr minus119910 Thus we study solelythe kink like solution The first-order equation for the field

Advances in High Energy Physics 3

minus4

minus15 minus1 minus05

minus3

minus2

minus1

0

0 05 151 2

1

2

120601

Figure 1 The potential given by (8) for 119888 = 0 (black line) 119888 = 23(dashed curve) 119888 = 43 (dot-dashed line) and 119888 = 2 (dotted line)

does not depend on the parameter 119888 and is provided by 1206011015840 =1 minus 1206012 The solution for this equation is

120601 (119910) = tanh (119910) (10)

with 120601(plusmninfin) = 120601plusmn= plusmn1 It obeys 120601(119910) = minus120601(minus119910) and it

connects symmetric minimaThe warp function can be obtained by using (5b)

119860 (119910) = minus1

9tanh2 (119910) + 4

9ln (sech (119910)) minus 119888

3119910 (11)

with 119860(0) = 0 We note that the behavior of this functionoutside the brane can be written as

119860plusmninfin(119910) = minus

1

3119882(120601plusmn)10038161003816100381610038161199101003816100381610038161003816 = minus

1

3(4

3plusmn 119888)

10038161003816100381610038161199101003816100381610038161003816 (12a)

and asymptotically the five-dimensional cosmological con-stant is

Λ5plusmn

equiv 119881 (120601 (plusmninfin)) = minus1

3(4

3plusmn 119888)2

(12b)

If 119888 = 0 the brane is symmetric connecting two regionsin the AdS

5bulk with the same cosmological constant

Λ5plusmn

= minus1627 If |119888| gt 43 the warp factor 1198902119860 divergesIn all other cases the warp factor describes asymmetricbraneworlds If |119888| = 43 the brane connects an AdS

5bulk

and a five-dimensional Minkowski (M5) bulk with Λ

5+=

minus6427 [Λ5+

= 0] and Λ5minus

= 0 [Λ5minus

= minus6427] when119888 = 43 [119888 = minus43] If 0 lt |119888| lt 43 the brane connectstwo distinct AdS

5bulk spaces

In Figure 2 we plot the profile of the warp factor forsome values of 119888 The solid curve represents the symmetricprofile (Brane I)The dashed curve represents the asymmetricbrane that connects two AdS

5bulk spaces with 119888 = 23

Λ5+= minus43 and Λ

5minus= minus427 (Brane II) The dot-dashed

curve represents the case where the brane connects an AdS5

to aM5bulk (Brane III) Finally the dotted curve represents

the divergent warp factor with 119888 = 2

minus6 minus4 minus2 00

025

050

075

1

125

150

175

2 4 6 8

y

Figure 2 The warp factor 1198902119860 with 119860 given by (11) for 119888 = 0 (solidline) 119888 = 23 (dashed line) 119888 = 43 (dot-dashed line) and 119888 = 2(dotted line) We fix 119860(0) = 0

minus3 minus2 minus1 0 1 2 3 4

120601

minus02

00

02

04

06

08

10

Figure 3 The energy density given by (14) for 119888 = 0 (solid line)119888 = 23 (dashed line) and 119888 = 43 (dot-dashed line)

The energy density is given by

120588 (119910) = minus1198902119860L (13)

and it also depends on 119888

120588 (119910) = 1198902119860(1198882minus16

81+4

27sech4 (119910) + 31

81sech6 (119910)

minus4119888

27tanh (119910) (2 + sech2 (119910)))

(14)

It is symmetric only for 119888 = 0 When 119888 = 0 there existcontributions to the asymmetry from both the warp factor1198902119860 and the Lagrange density L In Figure 3 we depict theprofile for Branes I (solid curve) II (dashed curve) and III(dot-dashed curve) Therefore for 119888 = 0 the warp factor andthe energy density are asymmetric

This model shows that although the kink like solutionconnects symmetricminima the potential is asymmetric andgives rise to braneworld scenario which is also asymmetricunless 119888 = 0

4 Advances in High Energy Physics

minus1

minus0100

minus0075

minus0050

0050

minus0025

0025

0075

0100

minus05 05 150

0

1

120601

Figure 4 The potential given by (16) for 119888 = minus12 (solid line) 119888 =minus14 (long dashed line) 119888 = minus18 (dashed line) 119888 = 0 (dot-dashedline) and 119888 = 120 (dotted line)

22 The Case of an Even Superpotential Let us now investi-gate a different model which presents kink like solution thatconnects asymmetric minima It is defined by

119882119888(120601) = minus

1

21206014+ 1206012+ 119888 (15)

This119882 is an even function of 120601 The potential given by (6) isnow

119881 (120601) =1

21206012(1 minus 120601

2)2

minus1

3(119888 + 120601

2minus1

21206014)2

(16)

It is depicted in Figure 4 and it has the Z2symmetry for any

value of the parameter 119888 The minima are 120601minus= minus1 120601

0= 0

and 120601+= 1 with

119882119888(120601plusmn) =

1

2+ 119888 119882

119888(1206010) = 119888 (17)

There are two topological sectors The first connects theminima 120601

minusand 120601

0 while the second connects the minima

1206010and 120601

+ Note that these sectors can be mapped by taking

the transformation 120601 997891rarr minus120601 In each sector there are twosolutions (kink and antikink) they can be mapped into eachother with the coordinate transformation 119910 997891rarr minus119910 Thuswe only study one of these solutions Using the first-orderequation 1206011015840 = 120601(1 minus 1206012) we find the solution

120601 (119910) =radic2

2radic1 + tanh (119910) (18)

which connects asymmetric minimaHere the warp function is given by

119860 (119910) = minus1

24tanh (119910) + 1

12ln (sech (119910)) minus ( 1

12+119888

3) 119910

(19)

and presents the asymptotic values

119860plusmninfin

(119910) = minus1

3[1

4plusmn (

1

4+ 119888)]

10038161003816100381610038161199101003816100381610038161003816 (20a)

02

0

0

04

06

08

12

14

1

minus75 75 10minus5 5minus25 25

y

Figure 5 The warp factor 1198902119860 with 119860 given by (19) for 119888 = minus12

(solid line) 119888 = minus14 (long dashed line) 119888 = minus18 (dashed line)119888 = 0 (dot-dashed line) and 119888 = 120 (dotted line)

The bulk cosmological constant is provided by

Λ5plusmn

= minus1

3[1

4plusmn (

1

4+ 119888)]

2

(20b)

For 119888 lt minus12 and 119888 gt 0 the warp factor diverges We obtainthe asymmetric branes separating two bulk spaces M

5and

AdS5 for 119888 = minus12 and 119888 = 0 In the first case Λ

5+= 0 and

Λ5minus= minus112 while in the second case Λ

5+= minus112 and

Λ5minus= 0 The energy density is

120588 (119910) = 1198902119860[39

64minus119888

12minus1198882

9minus (

31

48minus119888

18) tanh (119910)

+ (119888

36minus35

288) tanh2 (119910) + 17

144tanh3 (119910)

minus1

576tanh4 (119910) ]

(21)

In Figures 5 and 6 we depict the warp factor and the energydensity for some values of 119888

This model is different from the previous one Thepotential is always symmetric and the kink like solutionconnects asymmetric minima The model gives rise to warpfactor that is always asymmetric although at 119888 = minus14 itconnects similar AdS

5bulk spaces

3 Stability

An issue of interest concerns stability of the gravity sector ofthe braneworldmodel For themodels studied in the previoussection such investigations can be implemented numerically[52] an issue to be described in the longer work underpreparation [53] Here however to gain confidence on thestability of the suggested braneworld scenarios we developthe analytical procedure we get inspiration from the firstwork in [7ndash14] where it is shown that the sine-Gordonmodel

Advances in High Energy Physics 5

0075

minus0050

0050

minus0025

0025

0125

0100

0150

0

0minus4 minus3 minus2 minus1 1 2 3 4

120601

Figure 6 The energy density given by (21) for 119888 = minus12 (solid line)119888 = minus14 (long dashed line) 119888 = minus18 (dashed line) and 119888 = 0

(dot-dashed line)

is good model for analytical investigation Thus we considerthe sine-Gordon-type model with

119882119888(120601) = 2radic

3

2sin(radic3

2120601) + 119888 (22)

with |119888| le radic6 We note that for 120601 being small themodel is similar to the case of an odd superpotential studiedpreviously thus the current investigation engenders resultsthat are also valid for the model investigated in Section 21The point here is that we want to study stability analyticallyso the sine-Gordon model is the appropriate model

We use (5a) and (5b) to obtain analytic solutions

120601 (119910) = radic3

2arcsin (tanh (119910)) (23a)

119860119888(119910) = 119860

0(119910) minus

1

3119888119910 (23b)

1198600(119910) = minus ln (sech (119910)) (23c)

where 120601(119910) and 1198600(119910) are the standard solutions of the sine-

Gordon model for 119888 = 0To study stability we follow [52] We work in the trans-

verse traceless gauge to decouple gravity from the scalar fieldAlso we have to change from 119910 to the conformal coordinate119911 = 119911(119910) to get to a Schroedinger like equation with aquantum mechanical potential This investigation cannot bedone analytically anymore unless we take 119888 = 0 Thuswe resort to the approximation taking 119888 very small andexpanding the results up to first-order in 119888 In this case wecan write the conformal coordinate as

119911 (119910) = 119891 (119910) +119888

3(119910119891 (119910) minus int119889119910119891 (119910)) (24a)

119891 (119910) = int119889119910119890minus119860(119910)

(24b)

We note that 119891(119910) is an even function while the 119888-term con-tribution is odd Therefore 119911(119910) is an asymmetric functionThe inverse is

119910 = 119891minus1(119911) minus

119888

3[

1

1198911015840 (119909)(119909119911 minus int119889119909119891 (119909))]

119909=119891minus1(119911)

(25)

After some algebraic calculations we could find the associ-ated potential

119880 (119911) =3

211986010158401015840+9

411986010158402+ 119888 119880119888(119911) (26)

where119880119888(119911) is the contribution up to first-order in 119888The sine-

Gordon-type model is nice and we could find the explicitresults the conformal coordinate 119911 = int 119890minus119860(119910) 119889119910 in (24a) iscomputed as

119911 (119910) = sinh (119910) + 1198883(119910 sinh (119910) minus cosh (119910)) (27)

Since 119888 is small the above expression can be inverted to give

119910 (119911) = arcsinh (119911) minus 1198883

[[

[

1

radic1 + (119911)2

times((119911) arcsinh (119911) minus radic1 + (119911)2)]]

]

(28)

Therefore

119860 (119911) = minus1

2ln (1 + (119911)2)

minus 119888(arcsinh (119911)1 + (119911)

2+

119911

radic1 + (119911)2

)

(29)

It follows from (26) that the quantum potential is writtenas

119880 (119911) =3

4

(51199112 minus 2)

(1 + 1199112)2+ 119888 119880119888(119911) (30a)

119880119888(119911) =

1

(1 + 1199112)3( (1 minus 6119911

2) arcsinh (119911)

+7119911radic1 + 1199112)

(30b)

It is depicted in Figure 7 The maxima of the potential 119880(119911)are provided by

119911plusmn= plusmn

3

radic5minusradic70119888

9450(27radic5arcsinh( 3

radic5)radic70 minus 1330)

asymp plusmn134164 minus 068398119888

(31)

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Superconductivity

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2 Advances in High Energy Physics

symmetry Asymmetric static double-braneworlds with twodifferentwallswere also considered in this context embeddedin an AdS

5bulk Furthermore an independent derivation of

the Friedmann equation was presented in the simplest formfor anAdS

5bulk with different cosmological constants on the

two sides of the brane [38] Finally a braneworld that acts asa domain wall between two different five-dimensional bulkswas considered as a solution to Gauss-Bonnet gravity [19 33]Models with moving branes have been further considered[42] in order to break the reflection symmetry of the branemodel [17 20 22 23 43ndash45] Other interesting models thatdeal with asymmetric branes have been done some of themcan be found in [46ndash51]

In the current work we will further study asymmetricbranes focusing on general features which we believe can beused to better qualify the braneworld scenario as symmetricor asymmetric The key issue is to add a constant to thesuperpotential that describes the scalar field which affectsall the braneworld results In particular one notes that itexplicitly alters the quantum potential that responds for sta-bility evincing and unraveling prominent physical featuresas the asymmetric localization of the graviton zero modeWe analyze all the abovementioned features of asymmetricbranes in the context of superpotentials containing odd andeven power in the scalar field up to the third- and fourth-order power respectively and the sine-Gordon model

The investigation starts with a brief revision of theequations that govern the braneworld scenario under inves-tigation getting to the first-order framework where first-order differential equations solve the Einstein and fieldequations if the potential has a very specific form Wedeal with asymmetric branes in the scenario where thebrane is generated from a kink of two distinct and well-knownmodels with the superpotentials being odd and evenrespectively Subsequently the sine-Gordon model is alsostudied and the associated linear stability is investigated Thequantum potential and the graviton asymmetric zero modeare explicitly constructed

2 The Framework

We start with a five-dimensional action in which gravity iscoupled to a scalar field in the form

S = int1198895119909radic10038161003816100381610038161198921003816100381610038161003816 (minus

1

4119877 +L) (1)

where

L =1

2nabla119886120601nabla119886120601 minus 119881 (120601) (2)

and 119886 = 0 1 4We are using 4120587119866 = 1 with field and spaceand time variables dimensionless for simplicity By denoting119881120601= 119889119881119889120601 the Einstein equations and the Euler-Lagrange

equation are 119866119886119887= 119879119886119887and nabla119886nabla

119886120601 + 119881120601= 0 respectively We

write the metric as

1198891199042= 1198902119860120578120583]119889119909120583119889119909

]minus 1198891199102 (3)

where 119860 = 119860(119910) describes the warp function and onlydepends on the extra dimension119910 Taking the scalar fieldwithsame dependence we obtain

11986010158401015840+2

312060110158402= 0 (4a)

11986010158402minus1

612060110158402+1

6119881 (120601) = 0 (4b)

12060110158401015840+ 412060110158401198601015840minus 119881120601= 0 (4c)

where prime stands for derivative with respect to the extradimension As one knows solutions to the first-order equa-tions

1206011015840=1

2119882120601 (5a)

1198601015840= minus

1

3119882(120601) (5b)

also solve (4a) (4b) and (4c) if the potential has thefollowing form

119881 =1

81198822

120601minus1

31198822 (6)

where119882 = 119882(120601) is a function of the scalar field 120601This resultshows that if one adds a constant to 119882 it will modify thepotential and consequently all the results that follow fromthe model

To better explore this idea we study three distinctmodelswhich generalize previous investigations

21 The Case of an Odd Superpotential Let us introduce thefunction

119882119888(120601) = minus

2

31206013+ 2120601 + 119888 (7)

where 119888 is a real constantThis119882 is an odd function for 119888 = 0The potential given by (6) has now the form

119881 (120601) =1

2(1 minus 120601

2)2

minus1

3(119888 + 2120601 minus

2

31206013)2

(8)

It has the Z2symmetry only when 119888 = 0 the symmetry is

now 120601 997891rarr minus120601 and 119888 997891rarr minus119888 There are minima at 120601plusmn= plusmn1 with

119882119888(120601plusmn) = plusmn

4

3+ 119888 (9)

The maxima depend on 119888 and are solutions of the algebraicequation 41206013max minus 39120601max minus 2119888 = 0 see Figure 1

We focus attention on the scalar field There is a topolog-ical sector connecting the minima In this sector there aretwo solutions (kink and antikink) which can be mapped toone another when 120601 997891rarr minus120601 or 119910 997891rarr minus119910 Thus we study solelythe kink like solution The first-order equation for the field

Advances in High Energy Physics 3

minus4

minus15 minus1 minus05

minus3

minus2

minus1

0

0 05 151 2

1

2

120601

Figure 1 The potential given by (8) for 119888 = 0 (black line) 119888 = 23(dashed curve) 119888 = 43 (dot-dashed line) and 119888 = 2 (dotted line)

does not depend on the parameter 119888 and is provided by 1206011015840 =1 minus 1206012 The solution for this equation is

120601 (119910) = tanh (119910) (10)

with 120601(plusmninfin) = 120601plusmn= plusmn1 It obeys 120601(119910) = minus120601(minus119910) and it

connects symmetric minimaThe warp function can be obtained by using (5b)

119860 (119910) = minus1

9tanh2 (119910) + 4

9ln (sech (119910)) minus 119888

3119910 (11)

with 119860(0) = 0 We note that the behavior of this functionoutside the brane can be written as

119860plusmninfin(119910) = minus

1

3119882(120601plusmn)10038161003816100381610038161199101003816100381610038161003816 = minus

1

3(4

3plusmn 119888)

10038161003816100381610038161199101003816100381610038161003816 (12a)

and asymptotically the five-dimensional cosmological con-stant is

Λ5plusmn

equiv 119881 (120601 (plusmninfin)) = minus1

3(4

3plusmn 119888)2

(12b)

If 119888 = 0 the brane is symmetric connecting two regionsin the AdS

5bulk with the same cosmological constant

Λ5plusmn

= minus1627 If |119888| gt 43 the warp factor 1198902119860 divergesIn all other cases the warp factor describes asymmetricbraneworlds If |119888| = 43 the brane connects an AdS

5bulk

and a five-dimensional Minkowski (M5) bulk with Λ

5+=

minus6427 [Λ5+

= 0] and Λ5minus

= 0 [Λ5minus

= minus6427] when119888 = 43 [119888 = minus43] If 0 lt |119888| lt 43 the brane connectstwo distinct AdS

5bulk spaces

In Figure 2 we plot the profile of the warp factor forsome values of 119888 The solid curve represents the symmetricprofile (Brane I)The dashed curve represents the asymmetricbrane that connects two AdS

5bulk spaces with 119888 = 23

Λ5+= minus43 and Λ

5minus= minus427 (Brane II) The dot-dashed

curve represents the case where the brane connects an AdS5

to aM5bulk (Brane III) Finally the dotted curve represents

the divergent warp factor with 119888 = 2

minus6 minus4 minus2 00

025

050

075

1

125

150

175

2 4 6 8

y

Figure 2 The warp factor 1198902119860 with 119860 given by (11) for 119888 = 0 (solidline) 119888 = 23 (dashed line) 119888 = 43 (dot-dashed line) and 119888 = 2(dotted line) We fix 119860(0) = 0

minus3 minus2 minus1 0 1 2 3 4

120601

minus02

00

02

04

06

08

10

Figure 3 The energy density given by (14) for 119888 = 0 (solid line)119888 = 23 (dashed line) and 119888 = 43 (dot-dashed line)

The energy density is given by

120588 (119910) = minus1198902119860L (13)

and it also depends on 119888

120588 (119910) = 1198902119860(1198882minus16

81+4

27sech4 (119910) + 31

81sech6 (119910)

minus4119888

27tanh (119910) (2 + sech2 (119910)))

(14)

It is symmetric only for 119888 = 0 When 119888 = 0 there existcontributions to the asymmetry from both the warp factor1198902119860 and the Lagrange density L In Figure 3 we depict theprofile for Branes I (solid curve) II (dashed curve) and III(dot-dashed curve) Therefore for 119888 = 0 the warp factor andthe energy density are asymmetric

This model shows that although the kink like solutionconnects symmetricminima the potential is asymmetric andgives rise to braneworld scenario which is also asymmetricunless 119888 = 0

4 Advances in High Energy Physics

minus1

minus0100

minus0075

minus0050

0050

minus0025

0025

0075

0100

minus05 05 150

0

1

120601

Figure 4 The potential given by (16) for 119888 = minus12 (solid line) 119888 =minus14 (long dashed line) 119888 = minus18 (dashed line) 119888 = 0 (dot-dashedline) and 119888 = 120 (dotted line)

22 The Case of an Even Superpotential Let us now investi-gate a different model which presents kink like solution thatconnects asymmetric minima It is defined by

119882119888(120601) = minus

1

21206014+ 1206012+ 119888 (15)

This119882 is an even function of 120601 The potential given by (6) isnow

119881 (120601) =1

21206012(1 minus 120601

2)2

minus1

3(119888 + 120601

2minus1

21206014)2

(16)

It is depicted in Figure 4 and it has the Z2symmetry for any

value of the parameter 119888 The minima are 120601minus= minus1 120601

0= 0

and 120601+= 1 with

119882119888(120601plusmn) =

1

2+ 119888 119882

119888(1206010) = 119888 (17)

There are two topological sectors The first connects theminima 120601

minusand 120601

0 while the second connects the minima

1206010and 120601

+ Note that these sectors can be mapped by taking

the transformation 120601 997891rarr minus120601 In each sector there are twosolutions (kink and antikink) they can be mapped into eachother with the coordinate transformation 119910 997891rarr minus119910 Thuswe only study one of these solutions Using the first-orderequation 1206011015840 = 120601(1 minus 1206012) we find the solution

120601 (119910) =radic2

2radic1 + tanh (119910) (18)

which connects asymmetric minimaHere the warp function is given by

119860 (119910) = minus1

24tanh (119910) + 1

12ln (sech (119910)) minus ( 1

12+119888

3) 119910

(19)

and presents the asymptotic values

119860plusmninfin

(119910) = minus1

3[1

4plusmn (

1

4+ 119888)]

10038161003816100381610038161199101003816100381610038161003816 (20a)

02

0

0

04

06

08

12

14

1

minus75 75 10minus5 5minus25 25

y

Figure 5 The warp factor 1198902119860 with 119860 given by (19) for 119888 = minus12

(solid line) 119888 = minus14 (long dashed line) 119888 = minus18 (dashed line)119888 = 0 (dot-dashed line) and 119888 = 120 (dotted line)

The bulk cosmological constant is provided by

Λ5plusmn

= minus1

3[1

4plusmn (

1

4+ 119888)]

2

(20b)

For 119888 lt minus12 and 119888 gt 0 the warp factor diverges We obtainthe asymmetric branes separating two bulk spaces M

5and

AdS5 for 119888 = minus12 and 119888 = 0 In the first case Λ

5+= 0 and

Λ5minus= minus112 while in the second case Λ

5+= minus112 and

Λ5minus= 0 The energy density is

120588 (119910) = 1198902119860[39

64minus119888

12minus1198882

9minus (

31

48minus119888

18) tanh (119910)

+ (119888

36minus35

288) tanh2 (119910) + 17

144tanh3 (119910)

minus1

576tanh4 (119910) ]

(21)

In Figures 5 and 6 we depict the warp factor and the energydensity for some values of 119888

This model is different from the previous one Thepotential is always symmetric and the kink like solutionconnects asymmetric minima The model gives rise to warpfactor that is always asymmetric although at 119888 = minus14 itconnects similar AdS

5bulk spaces

3 Stability

An issue of interest concerns stability of the gravity sector ofthe braneworldmodel For themodels studied in the previoussection such investigations can be implemented numerically[52] an issue to be described in the longer work underpreparation [53] Here however to gain confidence on thestability of the suggested braneworld scenarios we developthe analytical procedure we get inspiration from the firstwork in [7ndash14] where it is shown that the sine-Gordonmodel

Advances in High Energy Physics 5

0075

minus0050

0050

minus0025

0025

0125

0100

0150

0

0minus4 minus3 minus2 minus1 1 2 3 4

120601

Figure 6 The energy density given by (21) for 119888 = minus12 (solid line)119888 = minus14 (long dashed line) 119888 = minus18 (dashed line) and 119888 = 0

(dot-dashed line)

is good model for analytical investigation Thus we considerthe sine-Gordon-type model with

119882119888(120601) = 2radic

3

2sin(radic3

2120601) + 119888 (22)

with |119888| le radic6 We note that for 120601 being small themodel is similar to the case of an odd superpotential studiedpreviously thus the current investigation engenders resultsthat are also valid for the model investigated in Section 21The point here is that we want to study stability analyticallyso the sine-Gordon model is the appropriate model

We use (5a) and (5b) to obtain analytic solutions

120601 (119910) = radic3

2arcsin (tanh (119910)) (23a)

119860119888(119910) = 119860

0(119910) minus

1

3119888119910 (23b)

1198600(119910) = minus ln (sech (119910)) (23c)

where 120601(119910) and 1198600(119910) are the standard solutions of the sine-

Gordon model for 119888 = 0To study stability we follow [52] We work in the trans-

verse traceless gauge to decouple gravity from the scalar fieldAlso we have to change from 119910 to the conformal coordinate119911 = 119911(119910) to get to a Schroedinger like equation with aquantum mechanical potential This investigation cannot bedone analytically anymore unless we take 119888 = 0 Thuswe resort to the approximation taking 119888 very small andexpanding the results up to first-order in 119888 In this case wecan write the conformal coordinate as

119911 (119910) = 119891 (119910) +119888

3(119910119891 (119910) minus int119889119910119891 (119910)) (24a)

119891 (119910) = int119889119910119890minus119860(119910)

(24b)

We note that 119891(119910) is an even function while the 119888-term con-tribution is odd Therefore 119911(119910) is an asymmetric functionThe inverse is

119910 = 119891minus1(119911) minus

119888

3[

1

1198911015840 (119909)(119909119911 minus int119889119909119891 (119909))]

119909=119891minus1(119911)

(25)

After some algebraic calculations we could find the associ-ated potential

119880 (119911) =3

211986010158401015840+9

411986010158402+ 119888 119880119888(119911) (26)

where119880119888(119911) is the contribution up to first-order in 119888The sine-

Gordon-type model is nice and we could find the explicitresults the conformal coordinate 119911 = int 119890minus119860(119910) 119889119910 in (24a) iscomputed as

119911 (119910) = sinh (119910) + 1198883(119910 sinh (119910) minus cosh (119910)) (27)

Since 119888 is small the above expression can be inverted to give

119910 (119911) = arcsinh (119911) minus 1198883

[[

[

1

radic1 + (119911)2

times((119911) arcsinh (119911) minus radic1 + (119911)2)]]

]

(28)

Therefore

119860 (119911) = minus1

2ln (1 + (119911)2)

minus 119888(arcsinh (119911)1 + (119911)

2+

119911

radic1 + (119911)2

)

(29)

It follows from (26) that the quantum potential is writtenas

119880 (119911) =3

4

(51199112 minus 2)

(1 + 1199112)2+ 119888 119880119888(119911) (30a)

119880119888(119911) =

1

(1 + 1199112)3( (1 minus 6119911

2) arcsinh (119911)

+7119911radic1 + 1199112)

(30b)

It is depicted in Figure 7 The maxima of the potential 119880(119911)are provided by

119911plusmn= plusmn

3

radic5minusradic70119888

9450(27radic5arcsinh( 3

radic5)radic70 minus 1330)

asymp plusmn134164 minus 068398119888

(31)

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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PhotonicsJournal of

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Journal of

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ThermodynamicsJournal of

Advances in High Energy Physics 3

minus4

minus15 minus1 minus05

minus3

minus2

minus1

0

0 05 151 2

1

2

120601

Figure 1 The potential given by (8) for 119888 = 0 (black line) 119888 = 23(dashed curve) 119888 = 43 (dot-dashed line) and 119888 = 2 (dotted line)

does not depend on the parameter 119888 and is provided by 1206011015840 =1 minus 1206012 The solution for this equation is

120601 (119910) = tanh (119910) (10)

with 120601(plusmninfin) = 120601plusmn= plusmn1 It obeys 120601(119910) = minus120601(minus119910) and it

connects symmetric minimaThe warp function can be obtained by using (5b)

119860 (119910) = minus1

9tanh2 (119910) + 4

9ln (sech (119910)) minus 119888

3119910 (11)

with 119860(0) = 0 We note that the behavior of this functionoutside the brane can be written as

119860plusmninfin(119910) = minus

1

3119882(120601plusmn)10038161003816100381610038161199101003816100381610038161003816 = minus

1

3(4

3plusmn 119888)

10038161003816100381610038161199101003816100381610038161003816 (12a)

and asymptotically the five-dimensional cosmological con-stant is

Λ5plusmn

equiv 119881 (120601 (plusmninfin)) = minus1

3(4

3plusmn 119888)2

(12b)

If 119888 = 0 the brane is symmetric connecting two regionsin the AdS

5bulk with the same cosmological constant

Λ5plusmn

= minus1627 If |119888| gt 43 the warp factor 1198902119860 divergesIn all other cases the warp factor describes asymmetricbraneworlds If |119888| = 43 the brane connects an AdS

5bulk

and a five-dimensional Minkowski (M5) bulk with Λ

5+=

minus6427 [Λ5+

= 0] and Λ5minus

= 0 [Λ5minus

= minus6427] when119888 = 43 [119888 = minus43] If 0 lt |119888| lt 43 the brane connectstwo distinct AdS

5bulk spaces

In Figure 2 we plot the profile of the warp factor forsome values of 119888 The solid curve represents the symmetricprofile (Brane I)The dashed curve represents the asymmetricbrane that connects two AdS

5bulk spaces with 119888 = 23

Λ5+= minus43 and Λ

5minus= minus427 (Brane II) The dot-dashed

curve represents the case where the brane connects an AdS5

to aM5bulk (Brane III) Finally the dotted curve represents

the divergent warp factor with 119888 = 2

minus6 minus4 minus2 00

025

050

075

1

125

150

175

2 4 6 8

y

Figure 2 The warp factor 1198902119860 with 119860 given by (11) for 119888 = 0 (solidline) 119888 = 23 (dashed line) 119888 = 43 (dot-dashed line) and 119888 = 2(dotted line) We fix 119860(0) = 0

minus3 minus2 minus1 0 1 2 3 4

120601

minus02

00

02

04

06

08

10

Figure 3 The energy density given by (14) for 119888 = 0 (solid line)119888 = 23 (dashed line) and 119888 = 43 (dot-dashed line)

The energy density is given by

120588 (119910) = minus1198902119860L (13)

and it also depends on 119888

120588 (119910) = 1198902119860(1198882minus16

81+4

27sech4 (119910) + 31

81sech6 (119910)

minus4119888

27tanh (119910) (2 + sech2 (119910)))

(14)

It is symmetric only for 119888 = 0 When 119888 = 0 there existcontributions to the asymmetry from both the warp factor1198902119860 and the Lagrange density L In Figure 3 we depict theprofile for Branes I (solid curve) II (dashed curve) and III(dot-dashed curve) Therefore for 119888 = 0 the warp factor andthe energy density are asymmetric

This model shows that although the kink like solutionconnects symmetricminima the potential is asymmetric andgives rise to braneworld scenario which is also asymmetricunless 119888 = 0

4 Advances in High Energy Physics

minus1

minus0100

minus0075

minus0050

0050

minus0025

0025

0075

0100

minus05 05 150

0

1

120601

Figure 4 The potential given by (16) for 119888 = minus12 (solid line) 119888 =minus14 (long dashed line) 119888 = minus18 (dashed line) 119888 = 0 (dot-dashedline) and 119888 = 120 (dotted line)

22 The Case of an Even Superpotential Let us now investi-gate a different model which presents kink like solution thatconnects asymmetric minima It is defined by

119882119888(120601) = minus

1

21206014+ 1206012+ 119888 (15)

This119882 is an even function of 120601 The potential given by (6) isnow

119881 (120601) =1

21206012(1 minus 120601

2)2

minus1

3(119888 + 120601

2minus1

21206014)2

(16)

It is depicted in Figure 4 and it has the Z2symmetry for any

value of the parameter 119888 The minima are 120601minus= minus1 120601

0= 0

and 120601+= 1 with

119882119888(120601plusmn) =

1

2+ 119888 119882

119888(1206010) = 119888 (17)

There are two topological sectors The first connects theminima 120601

minusand 120601

0 while the second connects the minima

1206010and 120601

+ Note that these sectors can be mapped by taking

the transformation 120601 997891rarr minus120601 In each sector there are twosolutions (kink and antikink) they can be mapped into eachother with the coordinate transformation 119910 997891rarr minus119910 Thuswe only study one of these solutions Using the first-orderequation 1206011015840 = 120601(1 minus 1206012) we find the solution

120601 (119910) =radic2

2radic1 + tanh (119910) (18)

which connects asymmetric minimaHere the warp function is given by

119860 (119910) = minus1

24tanh (119910) + 1

12ln (sech (119910)) minus ( 1

12+119888

3) 119910

(19)

and presents the asymptotic values

119860plusmninfin

(119910) = minus1

3[1

4plusmn (

1

4+ 119888)]

10038161003816100381610038161199101003816100381610038161003816 (20a)

02

0

0

04

06

08

12

14

1

minus75 75 10minus5 5minus25 25

y

Figure 5 The warp factor 1198902119860 with 119860 given by (19) for 119888 = minus12

(solid line) 119888 = minus14 (long dashed line) 119888 = minus18 (dashed line)119888 = 0 (dot-dashed line) and 119888 = 120 (dotted line)

The bulk cosmological constant is provided by

Λ5plusmn

= minus1

3[1

4plusmn (

1

4+ 119888)]

2

(20b)

For 119888 lt minus12 and 119888 gt 0 the warp factor diverges We obtainthe asymmetric branes separating two bulk spaces M

5and

AdS5 for 119888 = minus12 and 119888 = 0 In the first case Λ

5+= 0 and

Λ5minus= minus112 while in the second case Λ

5+= minus112 and

Λ5minus= 0 The energy density is

120588 (119910) = 1198902119860[39

64minus119888

12minus1198882

9minus (

31

48minus119888

18) tanh (119910)

+ (119888

36minus35

288) tanh2 (119910) + 17

144tanh3 (119910)

minus1

576tanh4 (119910) ]

(21)

In Figures 5 and 6 we depict the warp factor and the energydensity for some values of 119888

This model is different from the previous one Thepotential is always symmetric and the kink like solutionconnects asymmetric minima The model gives rise to warpfactor that is always asymmetric although at 119888 = minus14 itconnects similar AdS

5bulk spaces

3 Stability

An issue of interest concerns stability of the gravity sector ofthe braneworldmodel For themodels studied in the previoussection such investigations can be implemented numerically[52] an issue to be described in the longer work underpreparation [53] Here however to gain confidence on thestability of the suggested braneworld scenarios we developthe analytical procedure we get inspiration from the firstwork in [7ndash14] where it is shown that the sine-Gordonmodel

Advances in High Energy Physics 5

0075

minus0050

0050

minus0025

0025

0125

0100

0150

0

0minus4 minus3 minus2 minus1 1 2 3 4

120601

Figure 6 The energy density given by (21) for 119888 = minus12 (solid line)119888 = minus14 (long dashed line) 119888 = minus18 (dashed line) and 119888 = 0

(dot-dashed line)

is good model for analytical investigation Thus we considerthe sine-Gordon-type model with

119882119888(120601) = 2radic

3

2sin(radic3

2120601) + 119888 (22)

with |119888| le radic6 We note that for 120601 being small themodel is similar to the case of an odd superpotential studiedpreviously thus the current investigation engenders resultsthat are also valid for the model investigated in Section 21The point here is that we want to study stability analyticallyso the sine-Gordon model is the appropriate model

We use (5a) and (5b) to obtain analytic solutions

120601 (119910) = radic3

2arcsin (tanh (119910)) (23a)

119860119888(119910) = 119860

0(119910) minus

1

3119888119910 (23b)

1198600(119910) = minus ln (sech (119910)) (23c)

where 120601(119910) and 1198600(119910) are the standard solutions of the sine-

Gordon model for 119888 = 0To study stability we follow [52] We work in the trans-

verse traceless gauge to decouple gravity from the scalar fieldAlso we have to change from 119910 to the conformal coordinate119911 = 119911(119910) to get to a Schroedinger like equation with aquantum mechanical potential This investigation cannot bedone analytically anymore unless we take 119888 = 0 Thuswe resort to the approximation taking 119888 very small andexpanding the results up to first-order in 119888 In this case wecan write the conformal coordinate as

119911 (119910) = 119891 (119910) +119888

3(119910119891 (119910) minus int119889119910119891 (119910)) (24a)

119891 (119910) = int119889119910119890minus119860(119910)

(24b)

We note that 119891(119910) is an even function while the 119888-term con-tribution is odd Therefore 119911(119910) is an asymmetric functionThe inverse is

119910 = 119891minus1(119911) minus

119888

3[

1

1198911015840 (119909)(119909119911 minus int119889119909119891 (119909))]

119909=119891minus1(119911)

(25)

After some algebraic calculations we could find the associ-ated potential

119880 (119911) =3

211986010158401015840+9

411986010158402+ 119888 119880119888(119911) (26)

where119880119888(119911) is the contribution up to first-order in 119888The sine-

Gordon-type model is nice and we could find the explicitresults the conformal coordinate 119911 = int 119890minus119860(119910) 119889119910 in (24a) iscomputed as

119911 (119910) = sinh (119910) + 1198883(119910 sinh (119910) minus cosh (119910)) (27)

Since 119888 is small the above expression can be inverted to give

119910 (119911) = arcsinh (119911) minus 1198883

[[

[

1

radic1 + (119911)2

times((119911) arcsinh (119911) minus radic1 + (119911)2)]]

]

(28)

Therefore

119860 (119911) = minus1

2ln (1 + (119911)2)

minus 119888(arcsinh (119911)1 + (119911)

2+

119911

radic1 + (119911)2

)

(29)

It follows from (26) that the quantum potential is writtenas

119880 (119911) =3

4

(51199112 minus 2)

(1 + 1199112)2+ 119888 119880119888(119911) (30a)

119880119888(119911) =

1

(1 + 1199112)3( (1 minus 6119911

2) arcsinh (119911)

+7119911radic1 + 1199112)

(30b)

It is depicted in Figure 7 The maxima of the potential 119880(119911)are provided by

119911plusmn= plusmn

3

radic5minusradic70119888

9450(27radic5arcsinh( 3

radic5)radic70 minus 1330)

asymp plusmn134164 minus 068398119888

(31)

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

minus1

minus0100

minus0075

minus0050

0050

minus0025

0025

0075

0100

minus05 05 150

0

1

120601

Figure 4 The potential given by (16) for 119888 = minus12 (solid line) 119888 =minus14 (long dashed line) 119888 = minus18 (dashed line) 119888 = 0 (dot-dashedline) and 119888 = 120 (dotted line)

22 The Case of an Even Superpotential Let us now investi-gate a different model which presents kink like solution thatconnects asymmetric minima It is defined by

119882119888(120601) = minus

1

21206014+ 1206012+ 119888 (15)

This119882 is an even function of 120601 The potential given by (6) isnow

119881 (120601) =1

21206012(1 minus 120601

2)2

minus1

3(119888 + 120601

2minus1

21206014)2

(16)

It is depicted in Figure 4 and it has the Z2symmetry for any

value of the parameter 119888 The minima are 120601minus= minus1 120601

0= 0

and 120601+= 1 with

119882119888(120601plusmn) =

1

2+ 119888 119882

119888(1206010) = 119888 (17)

There are two topological sectors The first connects theminima 120601

minusand 120601

0 while the second connects the minima

1206010and 120601

+ Note that these sectors can be mapped by taking

the transformation 120601 997891rarr minus120601 In each sector there are twosolutions (kink and antikink) they can be mapped into eachother with the coordinate transformation 119910 997891rarr minus119910 Thuswe only study one of these solutions Using the first-orderequation 1206011015840 = 120601(1 minus 1206012) we find the solution

120601 (119910) =radic2

2radic1 + tanh (119910) (18)

which connects asymmetric minimaHere the warp function is given by

119860 (119910) = minus1

24tanh (119910) + 1

12ln (sech (119910)) minus ( 1

12+119888

3) 119910

(19)

and presents the asymptotic values

119860plusmninfin

(119910) = minus1

3[1

4plusmn (

1

4+ 119888)]

10038161003816100381610038161199101003816100381610038161003816 (20a)

02

0

0

04

06

08

12

14

1

minus75 75 10minus5 5minus25 25

y

Figure 5 The warp factor 1198902119860 with 119860 given by (19) for 119888 = minus12

(solid line) 119888 = minus14 (long dashed line) 119888 = minus18 (dashed line)119888 = 0 (dot-dashed line) and 119888 = 120 (dotted line)

The bulk cosmological constant is provided by

Λ5plusmn

= minus1

3[1

4plusmn (

1

4+ 119888)]

2

(20b)

For 119888 lt minus12 and 119888 gt 0 the warp factor diverges We obtainthe asymmetric branes separating two bulk spaces M

5and

AdS5 for 119888 = minus12 and 119888 = 0 In the first case Λ

5+= 0 and

Λ5minus= minus112 while in the second case Λ

5+= minus112 and

Λ5minus= 0 The energy density is

120588 (119910) = 1198902119860[39

64minus119888

12minus1198882

9minus (

31

48minus119888

18) tanh (119910)

+ (119888

36minus35

288) tanh2 (119910) + 17

144tanh3 (119910)

minus1

576tanh4 (119910) ]

(21)

In Figures 5 and 6 we depict the warp factor and the energydensity for some values of 119888

This model is different from the previous one Thepotential is always symmetric and the kink like solutionconnects asymmetric minima The model gives rise to warpfactor that is always asymmetric although at 119888 = minus14 itconnects similar AdS

5bulk spaces

3 Stability

An issue of interest concerns stability of the gravity sector ofthe braneworldmodel For themodels studied in the previoussection such investigations can be implemented numerically[52] an issue to be described in the longer work underpreparation [53] Here however to gain confidence on thestability of the suggested braneworld scenarios we developthe analytical procedure we get inspiration from the firstwork in [7ndash14] where it is shown that the sine-Gordonmodel

Advances in High Energy Physics 5

0075

minus0050

0050

minus0025

0025

0125

0100

0150

0

0minus4 minus3 minus2 minus1 1 2 3 4

120601

Figure 6 The energy density given by (21) for 119888 = minus12 (solid line)119888 = minus14 (long dashed line) 119888 = minus18 (dashed line) and 119888 = 0

(dot-dashed line)

is good model for analytical investigation Thus we considerthe sine-Gordon-type model with

119882119888(120601) = 2radic

3

2sin(radic3

2120601) + 119888 (22)

with |119888| le radic6 We note that for 120601 being small themodel is similar to the case of an odd superpotential studiedpreviously thus the current investigation engenders resultsthat are also valid for the model investigated in Section 21The point here is that we want to study stability analyticallyso the sine-Gordon model is the appropriate model

We use (5a) and (5b) to obtain analytic solutions

120601 (119910) = radic3

2arcsin (tanh (119910)) (23a)

119860119888(119910) = 119860

0(119910) minus

1

3119888119910 (23b)

1198600(119910) = minus ln (sech (119910)) (23c)

where 120601(119910) and 1198600(119910) are the standard solutions of the sine-

Gordon model for 119888 = 0To study stability we follow [52] We work in the trans-

verse traceless gauge to decouple gravity from the scalar fieldAlso we have to change from 119910 to the conformal coordinate119911 = 119911(119910) to get to a Schroedinger like equation with aquantum mechanical potential This investigation cannot bedone analytically anymore unless we take 119888 = 0 Thuswe resort to the approximation taking 119888 very small andexpanding the results up to first-order in 119888 In this case wecan write the conformal coordinate as

119911 (119910) = 119891 (119910) +119888

3(119910119891 (119910) minus int119889119910119891 (119910)) (24a)

119891 (119910) = int119889119910119890minus119860(119910)

(24b)

We note that 119891(119910) is an even function while the 119888-term con-tribution is odd Therefore 119911(119910) is an asymmetric functionThe inverse is

119910 = 119891minus1(119911) minus

119888

3[

1

1198911015840 (119909)(119909119911 minus int119889119909119891 (119909))]

119909=119891minus1(119911)

(25)

After some algebraic calculations we could find the associ-ated potential

119880 (119911) =3

211986010158401015840+9

411986010158402+ 119888 119880119888(119911) (26)

where119880119888(119911) is the contribution up to first-order in 119888The sine-

Gordon-type model is nice and we could find the explicitresults the conformal coordinate 119911 = int 119890minus119860(119910) 119889119910 in (24a) iscomputed as

119911 (119910) = sinh (119910) + 1198883(119910 sinh (119910) minus cosh (119910)) (27)

Since 119888 is small the above expression can be inverted to give

119910 (119911) = arcsinh (119911) minus 1198883

[[

[

1

radic1 + (119911)2

times((119911) arcsinh (119911) minus radic1 + (119911)2)]]

]

(28)

Therefore

119860 (119911) = minus1

2ln (1 + (119911)2)

minus 119888(arcsinh (119911)1 + (119911)

2+

119911

radic1 + (119911)2

)

(29)

It follows from (26) that the quantum potential is writtenas

119880 (119911) =3

4

(51199112 minus 2)

(1 + 1199112)2+ 119888 119880119888(119911) (30a)

119880119888(119911) =

1

(1 + 1199112)3( (1 minus 6119911

2) arcsinh (119911)

+7119911radic1 + 1199112)

(30b)

It is depicted in Figure 7 The maxima of the potential 119880(119911)are provided by

119911plusmn= plusmn

3

radic5minusradic70119888

9450(27radic5arcsinh( 3

radic5)radic70 minus 1330)

asymp plusmn134164 minus 068398119888

(31)

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

0075

minus0050

0050

minus0025

0025

0125

0100

0150

0

0minus4 minus3 minus2 minus1 1 2 3 4

120601

Figure 6 The energy density given by (21) for 119888 = minus12 (solid line)119888 = minus14 (long dashed line) 119888 = minus18 (dashed line) and 119888 = 0

(dot-dashed line)

is good model for analytical investigation Thus we considerthe sine-Gordon-type model with

119882119888(120601) = 2radic

3

2sin(radic3

2120601) + 119888 (22)

with |119888| le radic6 We note that for 120601 being small themodel is similar to the case of an odd superpotential studiedpreviously thus the current investigation engenders resultsthat are also valid for the model investigated in Section 21The point here is that we want to study stability analyticallyso the sine-Gordon model is the appropriate model

We use (5a) and (5b) to obtain analytic solutions

120601 (119910) = radic3

2arcsin (tanh (119910)) (23a)

119860119888(119910) = 119860

0(119910) minus

1

3119888119910 (23b)

1198600(119910) = minus ln (sech (119910)) (23c)

where 120601(119910) and 1198600(119910) are the standard solutions of the sine-

Gordon model for 119888 = 0To study stability we follow [52] We work in the trans-

verse traceless gauge to decouple gravity from the scalar fieldAlso we have to change from 119910 to the conformal coordinate119911 = 119911(119910) to get to a Schroedinger like equation with aquantum mechanical potential This investigation cannot bedone analytically anymore unless we take 119888 = 0 Thuswe resort to the approximation taking 119888 very small andexpanding the results up to first-order in 119888 In this case wecan write the conformal coordinate as

119911 (119910) = 119891 (119910) +119888

3(119910119891 (119910) minus int119889119910119891 (119910)) (24a)

119891 (119910) = int119889119910119890minus119860(119910)

(24b)

We note that 119891(119910) is an even function while the 119888-term con-tribution is odd Therefore 119911(119910) is an asymmetric functionThe inverse is

119910 = 119891minus1(119911) minus

119888

3[

1

1198911015840 (119909)(119909119911 minus int119889119909119891 (119909))]

119909=119891minus1(119911)

(25)

After some algebraic calculations we could find the associ-ated potential

119880 (119911) =3

211986010158401015840+9

411986010158402+ 119888 119880119888(119911) (26)

where119880119888(119911) is the contribution up to first-order in 119888The sine-

Gordon-type model is nice and we could find the explicitresults the conformal coordinate 119911 = int 119890minus119860(119910) 119889119910 in (24a) iscomputed as

119911 (119910) = sinh (119910) + 1198883(119910 sinh (119910) minus cosh (119910)) (27)

Since 119888 is small the above expression can be inverted to give

119910 (119911) = arcsinh (119911) minus 1198883

[[

[

1

radic1 + (119911)2

times((119911) arcsinh (119911) minus radic1 + (119911)2)]]

]

(28)

Therefore

119860 (119911) = minus1

2ln (1 + (119911)2)

minus 119888(arcsinh (119911)1 + (119911)

2+

119911

radic1 + (119911)2

)

(29)

It follows from (26) that the quantum potential is writtenas

119880 (119911) =3

4

(51199112 minus 2)

(1 + 1199112)2+ 119888 119880119888(119911) (30a)

119880119888(119911) =

1

(1 + 1199112)3( (1 minus 6119911

2) arcsinh (119911)

+7119911radic1 + 1199112)

(30b)

It is depicted in Figure 7 The maxima of the potential 119880(119911)are provided by

119911plusmn= plusmn

3

radic5minusradic70119888

9450(27radic5arcsinh( 3

radic5)radic70 minus 1330)

asymp plusmn134164 minus 068398119888

(31)

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

minus10 minus75 minus5 minus25

minus15

minus05

minus1

1

0

0

25 5 75

05

10

z

Figure 7 Plot of the quantum mechanical potential in (30a) and(30b) for 119888 = 0 (solid line) and 119888 = 15 (long dashed line)

In the limit 119911 rarr infin the asymptotic value of the potentialreads

119880 (119911) asymp15

41199112minus1

1199114(9 + 119888 (6 ln (2119911) minus 7)) + O(

1

1199116) (32)

It explicitly shows how the asymmetric contribution entersthe game asymptotically

The quantummechanical analogous problem is describedby the equation

(minus1198892

1198891199112+ (119911))120595

119899= 1198962120595119899 (33)

This Hamiltonian can be factorized as up to first-order in 119888

minus1198892

1198891199092+ 119880 (119909) = (minus

119889

119889119909+3

21198601015840)(

119889

119889119909+3

21198601015840) (34)

where 119860 = 119860(119911) is given by (29) It is then nonnegative sothere are no negative bound states In fact the graviton zeromode is 120595

0(119911) which solves the above equation for 119896 = 0 it

is given explicitly by

1205950(119911) = exp(3

2119860 (119911))

= (1 + (119911)2)minus34

times (1 minus119888

3(arcsinh (119911)1 + 1199112

+119911

radic1 + 1199112))

(35)

and is depicted in Figure 8 showing its asymmetric behaviorIt has no node indicating that it is the lowest bound state asexpected

4 Concluding Remarks

In this work we investigated the presence of asymmetricbranes constructed from the addition of a constant 119888 to119882(120601)the function that defines the potential in the form

119881 (120601) =1

81198822

120601minus1

31198822 (36)

minus10 minus75 minus5 minus25 0 25 5 75 10

z

0

02

01

03

04

05

06

07

Figure 8 Plot of the normalized graviton zeromode1205950(119911) for 119888 = 0

(solid line) and for 119888 = 15 (long dashed line)

This expression controls the scalar field model that generatesthe thick brane scenario and it is important to reduce theequations to its first-order form

1206011015840=1

2119882120601 119860

1015840= minus

1

3119882 (37)

The main results of the current work show that wecan construct asymmetric brane irrespective of the scalarpotential being symmetric or asymmetric

We have investigated two distinct models of the 1206014 and1206016 type andwe have explicitly shown how the constant addedto119882(120601) works to induce asymmetry to the thick braneworldscenario Moreover we also studied the sine-Gordon-typemodel focusing on the stability of the gravitational scenarioThe nice properties of the sine-Gordon model very muchhelped us to implement analytical calculations showingstability of the graviton zero mode despite the presence ofan asymmetric volcano potential

Evidently the asymmetry induced in the thickbraneworld scenario may be phenomenologically relevantsince it has to be consistent with the AdS

5bulk curvature

and to the experimental and theoretical limits of the branethickness [54 55]

When finishing this study we become aware of the work[56] which investigates similar possibilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank CAPES and CNPq for partialfinancial support

References

[1] L Randall and R Sundrum ldquoAn alternative to compactifica-tionrdquo Physical Review Letters vol 83 no 23 pp 4690ndash46931999

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

[2] W D Goldberger and M B Wise ldquoModulus stabilization withbulk fieldsrdquoPhysical ReviewLetters vol 83 pp 4922ndash4925 1999

[3] O DeWolfe D Z Freedman S S Gubser and A KarchldquoModeling the fifth dimensionwith scalars and gravityrdquoPhysicalReview D Particles Fields Gravitation and Cosmology vol 62no 4 Article ID 046008 pp 1ndash16 2000

[4] C Csaki J Erlich T J Hollowood and Y Shirman ldquoUniversalaspects of gravity localized on thick branesrdquo Nuclear Physics Bvol 581 no 1-2 pp 309ndash338 2000

[5] C Csaki J Erlich C Grojean and T J Hollowood ldquoGeneralproperties of the self-tuning domain wall approach to thecosmological constant problemrdquoNuclear Physics B vol 584 no1-2 pp 359ndash386 2000

[6] C Csaki ldquoTASI lectures on extra dimensions and branesrdquohttparxivorgabshep-ph0404096

[7] M Gremm ldquoFour-dimensional gravity on a thick domain wallrdquoPhysics Letters B vol 478 no 4 pp 434ndash438 2000

[8] F A Brito M Cvetic and S C Yoon ldquoFrom a thick to a thinsupergravity domain wallrdquo Physical Review D vol 64 ArticleID 064021 2001

[9] M Cvetic and N D Lambert ldquoEffective supergravity forsupergravity domain wallsrdquo Physics Letters B vol 540 no 3-4pp 301ndash308 2002

[10] A Campos ldquoCritical phenomena of thick branes in warpedspacetimesrdquo Physical Review Letters vol 88 no 14 Article ID141602 4 pages 2002

[11] A Melfo N Pantoja and A Skirzewski ldquoThick domain wallspacetimes with and without reflection symmetryrdquo PhysicalReview D vol 67 no 10 Article ID 105003 6 pages 2003

[12] D Bazeia F A Brito and J R Nascimento ldquoSupergravity braneworlds and tachyon potentialsrdquo Physical Review D vol 68 no8 6 pages 2003

[13] D Bazeia C Furtado and A R Gomes ldquoBrane structure froma scalar field in warped spacetimerdquo Journal of Cosmology andAstroparticle Physics no 2 article 002 9 pages 2004

[14] D Bazeia and A R Gomes ldquoBloch branerdquo Journal of HighEnergy Physics vol 2004 no 5 article 012 2004

[15] P Horava and E Witten ldquoHeterotic and type I string dynamicsfrom eleven dimensionsrdquo Nuclear Physics B vol 460 no 3 pp506ndash524 1996

[16] P Horava and E Witten ldquoEleven-dimensional supergravity ona manifold with boundaryrdquo Nuclear Physics B vol 475 no 1-2pp 94ndash114 1996

[17] P Kraus ldquoDynamics of anti-de Sitter domain wallsrdquo Journal ofHigh Energy Physics vol 1999 no 9912 article 011 1999

[18] D Ida ldquoBrane-world cosmologyrdquo Journal of High EnergyPhysics vol 2000 no 0009 article 014 2000

[19] N Deruelle and T Dolezel ldquoBrane versus shell cosmologies inEinstein and Einstein-Gauss-Bonnet theoriesrdquo Physical ReviewD vol 62 no 10 Article ID 103502 8 pages 2000

[20] W B Perkins ldquoColliding bubble worldsrdquo Physics Letters B vol504 no 1-2 pp 28ndash32 2001

[21] P Bowcock C Charmousis and R Gregory ldquoGeneral branecosmologies and their global spacetime structurerdquoClassical andQuantum Gravity vol 17 no 22 pp 4745ndash4763 2000

[22] B Carter J-P Uzan R A Battye and A Mennim ldquoSimulatedgravity without true gravity in asymmetric brane-world scenar-iosrdquo Classical and Quantum Gravity vol 18 no 22 pp 4871ndash4895 2001

[23] H Stoica S-H H Tye and I Wasserman ldquoCosmology in theRandall-Sundrum brane world scenariordquo Physics Letters B vol482 no 1ndash3 pp 205ndash212 2000

[24] L A Gergely ldquoGeneralized Friedmann branesrdquo Physical ReviewD Third Series vol 68 no 12 Article ID 124011 13 pages 2003

[25] S A Appleby and R A Battye ldquoRegularized braneworlds ofarbitrary codimensionrdquo Physical Review D Particles FieldsGravitation and Cosmology vol 76 no 12 Article ID 12400916 pages 2007

[26] G Konas ldquoHow to go with an RG flowrdquo Journal of High EnergyPhysics vol 2001 no 08 article 041 2001

[27] H Collins and R Holdom ldquoBrane cosmologies without orb-ifoldsrdquo Physical Review D vol 62 Article ID 105009 2000

[28] Yu V Shtanov ldquoClosed system of equations on a branerdquo PhysicsLetters B vol 541 no 1-2 pp 177ndash182 2002

[29] Y Shtanov A Viznyuk and L N Granda ldquoAsymmetric embed-ding in brane cosmologyrdquo Modern Physics Letters A Particlesand Fields Gravitation Cosmology Nuclear Physics vol 23 no12 pp 869ndash878 2008

[30] C Charmousis R Gregory and A Padilla ldquoStealth accelerationand modified gravityrdquo Journal of Cosmology and AstroparticlePhysics vol 2007 no 10 article 006 2007

[31] K Koyama and K Koyama ldquoBrane-induced gravity fromasymmetric warped compactificationrdquo Physical Review D vol72 Article ID 043511 2005

[32] H Zhang Z-K Guo C Chen and X-Z Li ldquoOn asymmetricbrane creationrdquo Journal of High Energy Physics vol 2012 article19 2012

[33] K Nozari and T Azizi ldquoPhantom-like effects in an asymmetricbrane embedding with induced gravity and the Gauss-Bonnetterm in the bulkrdquoPhysica Scripta vol 83 Article ID015001 2011

[34] E OrsquoCallaghan R Gregory andA Pourtsidou ldquoThe cosmologyof asymmetric Brane modified gravityrdquo Journal of Cosmologyand Astroparticle Physics vol 2009 no 0909 article 020 2009

[35] Y Shtanov V Sahni A Shaeloo and A Toporensky ldquoInducedcosmological constant and other features of asymmetric braneembeddingrdquo Journal of Cosmology andAstroparticle Physics vol2009 no 0904 article 023 2009

[36] K Koyama A Padilla and F P Silva ldquoGhosts in asymmetricbrane gravity and the decoupled stealth limitrdquo Journal of HighEnergy Physics vol 2009 no 3 article 134 19 pages 2009

[37] R Guerrero R O Rodriguez and R S Torrealba ldquode Sitter anddouble asymmetric brane worldsrdquo Physical Review D vol 72Article ID 124012 2005

[38] L A Gergely and R Maartens ldquoAsymmetric brane-worlds withinduced gravityrdquo Physical Review D vol 71 no 2 Article ID024032 7 pages 2005

[39] A Padilla ldquoCosmic acceleration from asymmetric branesrdquoClassical andQuantumGravity vol 22 no 4 pp 681ndash694 2005

[40] A Padilla ldquoInfra-red modification of gravity from asymmetricbranesrdquo Classical and Quantum Gravity vol 22 pp 1087ndash11042005

[41] R Gregory and A Padilla ldquoNested braneworlds and strongbrane gravityrdquo Physical Review D vol 65 no 8 Article ID084013 4 pages 2002

[42] A Kehagias and E Kiritsis ldquoMirage cosmologyrdquo Journal of HighEnergy Physics vol 1999 no 11 article 022 1999

[43] J P Uzan ldquoAsymmetric brane-world scenarios and simulatedgravityrdquo International Journal of Theoretical Physics vol 41 no11 pp 2299ndash2309 2002

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

[44] R A Battye B Carter A Mennim and J-P Uzan ldquoEinsteinequations for an asymmetric brane-worldrdquo Physical Review DThird Series vol 64 no 12 Article ID 124007 19 pages 2001

[45] A-C Davis S Davis W B Perkins and I R Vernon ldquoCosmo-logical phase transitions in a brane worldrdquo Physical Review Dvol 63 Article ID 083518 2001

[46] R Guerrero R O Rodriguez R Torrealba and R Ortiz ldquoDeSitter and double irregular domainwallsrdquoGeneral Relativity andGravitation vol 38 no 5 pp 845ndash855 2006

[47] A de Souza Dutra A C Amaro de Faria Jr and M HottldquoDegenerate and critical Bloch branesrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 78 no 4Article ID 043526 9 pages 2008

[48] Z H Zhao Y X Liu and H T Li ldquoFermion localizationon asymmetric two-field thick branesrdquo Classical and QuantumGravity vol 27 no 18 Article ID 185001 2010

[49] Y X Liu C E Fu L Zhao and Y S Duan ldquoLocalization andmass spectra of fermions on symmetric and asymmetric thickbranesrdquo Physical Review D vol 80 Article ID 065020 2009

[50] Y-X Liu Z-H Zhao S-W Wei and Y-S Duan ldquoBulk matterson symmetric and asymmetric de Sitter thick branesrdquo Journalof Cosmology and Astroparticle Physics vol 2009 no 02 article003 2009

[51] Q-Y Xie J Yang and L Zhao ldquoResonance mass spectra ofgravity and fermion on Blochrdquo Physical Review D vol 88Article ID 105014 2013

[52] D Bazeia A R Gomes and L Losano ldquoGravity localizationon thick branes a numerical approachrdquo International Journalof Modern Physics A vol 24 no 6 pp 1135ndash1160 2009

[53] D Bazeia L Losano R Menezes and R da Rocha ldquoStudy ofmodels of the sine-Gordon type in flat and curved spacetimerdquoEuropean Physical Journal C vol 73 no 7 pp 1ndash8 2013

[54] D J Kapner T S Cook E G Adelberger et al ldquoTests of thegravitational inverse-square law below the dark-energy lengthscalerdquo Physical Review Letters vol 98 Article ID 021101 2007

[55] JMHoff da Silva andRD Rocha ldquoEffectivemonopoles withinthick branesrdquo EPL vol 100 no 1 Article ID 11001 2012

[56] A Ahmed L Dulny and B Grzadkowski ldquoGeneralizedRandall-Sundrum model with a single thick branerdquo EuropeanPhysical Journal C vol 74 Article ID 2862 18 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of