Bounds on topological abelian string-vortex and string-cigar from information-entropic measure

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Bounds

on

topological

Abelian

string-vortex

and

string-cigar

from

information-entropic

measure

R.A.C. Correa

a

,

D.M. Dantas

b

,

C.A.S. Almeida

c

,

Roldão da Rocha

d

,

∗

a_{CCNH,UniversidadeFederaldoABC(UFABC),09210-580,SantoAndré,SP,Brazil} b_{UniversidadeFederaldoCeará(UFC),60455-760,Fortaleza,CE,Brazil}

c_{UniversidadeFederaldoCeará(UFC),DepartamentodeFísica,60455-760,Fortaleza,CE,Brazil}

d_{CentrodeMatemática,ComputaçãoeCognição,UniversidadeFederaldoABC(UFABC),09210-580,SantoAndré,SP,Brazil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received30December2015

Receivedinrevisedform17February2016 Accepted17February2016

Availableonline23February2016 Editor:N.Lambert

Keywords:

TopologicalAbelianstring-vortex Six-dimensionalbraneworldmodels Configurationalentropy

In thiswork we obtain bounds on the topological Abelian string-vortex and onthe string-cigar, by usinganewmeasureofconfigurational complexity,knownasconfigurationalentropy.Inthisway,the information-theoreticalmeasureofsix-dimensionalbraneworldsscenariosis capabletoprobesituations where theparameters responsible for the brane thicknessare arbitrary.The so-called configurational entropy(CE)selectsthebestvalueoftheparameterinthemodel.Thisisaccomplishedbyminimizing theCE,namely,byselectingthemostappropriateparametersinthemodelthatcorrespondtothemost organizedsystem,basedupontheShannoninformationtheory.Thisinformation-theoreticalmeasureof complexity providesacomplementaryperspectivetosituationswherestrictlyenergy-basedarguments are inconclusive. We show that the higher the energythe higher the CE, what shows an important correlationbetweentheenergyofthealocalizedfieldconfigurationanditsassociatedentropicmeasure.

©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

In1948, inaseminal work,Shannon [1]introducedthe infor-mationtheory,whosemaingoalwastointroducetheconceptsof entropyandmutualinformation,usingthecommunicationtheory. Therein,theentropywasdefinedtobeameasureof“randomness” ofarandomphenomenon.Thus,ifalittledealofinformation con-cerning a random variable isreceived, the uncertaintydecreases, whichmakes it possibletomeasure thedecrement inthe uncer-tainty,relatedtothequantityoftransmittedinformation.Inspired byShannon, GleiserandStamatopoulos(GS)latterlyintroduceda measureofcomplexityofalocalizedmathematicalfunction[2].GS proposed that the Fourier modes of square-integrable, bounded, mathematical functionscan be used to constructa measure, the so-calledconfigurationalentropy (CE).A single mode system has zeroCE,whereas thatonewhereallmodescontributewithequal weight hasmaximal CE. Inorder toapply such ideas to physical models,GS usedthe energydensityofa givenspatially-localized

*

Correspondingauthor.

E-mailaddresses:rafael.couceiro@ufabc.edu.br(R.A.C. Correa),davi@fisica.ufc.br (D.M. Dantas),carlos@fisica.ufc.br(C.A.S. Almeida),roldao.rocha@ufabc.edu.br (R. da Rocha).

field configuration,asa solutionoftherelatedpartial differential equation (PDE).Hence theCEcanbe usedtochoose thebest fit-tingtrialfunctionwithenergydegeneracy.

The CE has been already employed to acquire the stability bound forcompactobjects[3],toinvestigatethenon-equilibrium dynamics of spontaneous symmetry breaking [4], to study the emergence oflocalized objects during inflationary preheating[5]

andtodiscern configurationswithdegenerate-energyspatial pro-files [6].Moreover, solitons were studied in a Lorentz symmetry violating (LV) framework withthe aid of CE [7–10]. Inthis con-text,theCEassociatedtotravellingsolitonsinLVframeworksplays a prominentrole inprobing systemswhereinthe parameters are somehowarbitrary.Furthermore,theCEidentifiescriticalpointsin continuous phase transitions [11]. Moreover, the CE can be used to measuretheinformationalorganization inthestructure ofthe systemconfiguration forfive-dimensional(5D) thick scenarios. In particular, theCEplays an importantroleto decidethe most ap-propriate intrinsic parameters ofsine-Gordon braneworld models

[12],beingfurtherstudiedbothin f

(

R

)

[13]and f

(

R

,

T

)

[14] the-ories ofgravity.In whatfollows,we presentabriefdiscussion of 5DbraneworldmodelstotreattheCEinsix-dimensional(6D) sce-narios.

Randall–Sundrum (RS) models [15,16] proposed a warped braneworld scenario, wherein the gauge hierarchy problem is http://dx.doi.org/10.1016/j.physletb.2016.02.038

0370-2693/©2016TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

explained and the gravity zero mode is localized, reproducing four-dimensional (4D) gravity on the brane. The 5D bulk gravi-tonsprovideasmallcorrectionintheNewton law[16].However, thisthin model presents singularities and drawbacks concerning thenon-localizationofspingaugeandfermionfields[17].Tosolve theseproblems,somethickmodelswereproposed[18].

Soon after the worksof RS, an axially symmetric warped6D modelwasproposedbyGergheta–Shaposhnikov[19],called string-likedefect (SD).Thisscenariofurtherprovidedtheresolutionofthe masshierarchyanda smallercorrectionto theNewtonian poten-tial[19],besidesthenon-requirementoffinetuningbetweenthe bulkcosmologicalconstantandthebrane tension,forthe cancel-lationofthe4D cosmologicalconstant[19].Besides,the localiza-tionof gauge zero modesis spontaneouseven in thethin brane case[20,21].Fermionsfieldsare trapped througha minimal cou-plingwithanU

(

1

)

gaugebackgroundfield[22,23].Later,other 6D, sphericallysymmetric,modelswereemployedtoexplainthe gen-erations of fundamental fermions [24,25] and the resolution of themasshierarchyofneutrinosaswell [26].Nevertheless,theSD modelisathinmodelanditleadstosomeirregularities[27].Due toit,some6Dthickmodelswereproposedtosolvethese remain-ingissues[28–42].InRef. [28],atopologicalabelianHiggsvortex was used to constructa regular scenario in which thedominant energyconditions hold, howeversolely numerical solutions have beenfound.Similarly,Refs.[31,32],lookingforanexactvortex so-lution,show thattheenergydensityandtheangularpressureare similar.ThisconditionislikewiseverifiedfortheResolvedConifold scenario [37–39]. Finally, for the String-Cigar [33–36], the trans-versespaceisrepresentedbyacigarsoliton,whichisastationary solution forthe Ricci flow [43–45]. The dominant energy condi-tionsarealsosatisfiedinthismodel.

Therefore, in this paper we investigate the entropic measure both in theTorrealba topological Abelian string(TA) [31,32] and String-Cigar(HC)[33–36]in6Dscenariosdue to itsanalytic prop-erties.Themainaimsofourworkare tofindboundsfor6Dstring defectsbasedupontheCEconceptandtoestablishavalueforthe thicknessoftheconfigurationresponsibleforextremizingtheCE.

Thispaperis organized asfollows: in Sect.2 a briefly review of string-likedefectsispresent,whereasinSect.3theCEbounds theparametersofTAandHCscenarios.Weexposetheconclusions andperspectivesaccordinglyinSect.4.

2. String-likedefectinwarpedsixdimensions

Ametricansatz for6Dstring-likemodelsreads[19,20] ds2₆

=

σ

(

r

)η

μνdxμdxν

−

dr2

−

γ

(

r

)

d

θ

2 (1) where

η

μν

=

diag

(

+

1

,

−

1

,

−

1

,

−

1

)

.The radial coordinate is lim-itedto r

∈

[0

,

∞)

,whereasthe angularcoordinateisrestrictedto

θ

∈

[0

,

2

π

)

.The

σ

(

r

)

representsthedimensionlesswarpfactorand

γ

(

r

)

haslengthsquareddimension.

The4D Planckmass (MP) andthebulk Planckmass(M6) are

relatedthroughthevolumeofthetransverseofspaceas[19,33,35, 36]: M2_P

=

2

π

M4₆ ∞



0

σ

(

r

)



γ

(

r

)

dr

.

(2)

Inaddition, the energy-momentumtensor T_MN

=

diag

(

t0

,

t0

,

t0

,

t0

,

tr,tθ

)

componentsaregivenby[19,33]

t0

(

r

)

= −

1

κ



3

σ

 2

σ

+

3

σ



γ

 4

σ γ

+

γ

 2

γ

−

γ

2 4

γ

2



− ,

(3a) tr

(

r

)

= −

1

κ



3

σ

2 2

σ

2

+

σ



γ



σ γ



− ,

(3b) tθ

(

r

)

= −

1

κ



2

σ



σ

+

σ

2 2

σ

2



− ,

(3c) where the

κ

=

8π

M4₆ is the 6D gravitational constant,

is the 6D

(negative)cosmologicalconstantandtheprimedenotesthe deriva-tivewithrespecttotheradialcoordinater.

Toobtainaregulargeometry,theconditions[19,28,33,42]

σ

(

r

)





r=0

=

const

.,

σ



₍

_r

₎



_

r=0

=

0

,

γ

(

r

)





r=0

=

0

,



γ

(

r

)



_

_



r=0

=

1

,

(4) musthold.

For the vacuum solution, the warp factor for the Gergheta– ShaposhnikovStringLikeDefect (SD)modelisproposedas[19–23]:

σ

SD

(

r

)

=

e−cr

,

γ

SD

(

r

)

=

R20

σ

SD

(

r

)

(5)

wheretheparametersc isaconstant,whichconnectsthe6D New-tonian constant andthe6D cosmologicalconstant,and R0 is the

radiusofcompactificationoftransversespace.Seethat,inthelimit wherer

→

0,onlythefirstconditionofEq.(4)holds.

Following the perspective pointed by Ref. [19], Giovannini in adopted a 6D action [28], wherein the matter Lagrangian is an Abelian–Higgs model and the transverse space obeys the Abrikosov–Nielsen–Olesenansatz[28,31,32]:

φ (

r

, θ )

=

v f

(

r

)

e−ilθ l

∈ Z ,

A

θ

(

r

)

=

1

q[l

−

P

(

r

)

]

,

A

μ

=

Ar

=

0

,

where

φ

and

AM

arescalarandgaugefields,respectively.The con-dition v

=

1 is a length dimension L−2 constant. The functions f

(

r

)

and P

(

r

)

aresuchthat f

(

r

→

0

)

=

0, f

(

r

→ ∞)

=

1,whereas P

(

r

→

0

)

=

l andP

(

r

→ ∞)

=

0.

From constraints by thisansatz and the regular conditions in theEq.(4),thesolutionsoffieldsandwarpfactorsarenumerically obtainedin Ref.[28].On theother hand,by imposing conditions onthefunction P

(

r

)

≡

0,Torrealba[31,32]obtainedananalytical solution,namedTopologicalAbelianHiggsstring (TA):

σ

TA

(

r

)

=

cosh−2δ



β

r

δ



,

γ

TA

(

r

)

=

R20

σ

TA

(

r

) ,

(6)

where the parameter

β

is similar to the parameter c in the SD model,and

δ

isathicknessparameterwhich,forsmallvalues, re-producesthethinGergheta–ShaposhnikovmodelinEq.(5). More-over, Ref. [31] concludes that, forthe localizationof gauge fields zeromode,thethicknessofthemodelcannot exceedthevalue

δ <

5

β

4

π

q

2_v2

_.

₍₇₎

Now,intheTA(6)string,twooftheconditions(4)areverified. Inanotherapproach,thetransversespacecanalsobebuiltfor acigarsolitonsolutionofRicciflow[33–36]

∂

∂λ

gMN

(λ)

= −

2RMN

(λ) ,

with

λ

beingametricparameterRefs.[33–36]constructedthe ge-ometrynamed HamiltonStringCigar (HC),wherethewarp factors read

σ

HC

(

r

)

=

e−cr+tanh(cr)

,

γ

HC

(

r

)

=

tanh2cr

c2

σ

HC

(

r

).

(8)

Fig. 1.σ(r)warp-factorwith c=2β= δ =0.5.IntheTA(dashedlines)and HC model(thicklines)theregularityconditions(4)aresatisfiedforthisfactor.

Fig. 2.γ(r)angularfactorswithc=2β= δ =0.5 andR0=1.OnlyintheHCmodel (thicklines)theregularityconditions(4)holdstill.

Toobserve thecorrespondence betweenthe regular condition inEq.(4)andtheenergymomentumtensorweplotthe

σ

(

r

)

the warpfactors(5),(6)and(8)inFig. 1and

γ

(

r

)

inFig. 2,whereas the energy momentum tensor in Fig. 3for TA and HC in Fig. 4. ConcerningtheHCscenario,whereinallmetricconditions(4)hold, thedominantenergyconditiont0

≥ |

ti

|

,

(

i

=

r

,

θ )

[29,30,40]is

sat-isfied.

Inthenextsection, weshallanalyzethesestringmodels from theCEpointofview.

3. Configurationalentropyinthevortex-stringscenario

The configurational entropy (CE) [2] represents an original quantity,employedtoquantifytheexistenceofnon-trivialspatially localized solutions in field configuration space. The CE is useful tobound thestability ofvariousself-gravitatingastrophysical ob-jects[47],boundstatesinLVscenarios[7],incompactobjectslike Q-balls[3],andinmodifiedtheoriesofgravityaswell[13].TheCE islinkedtotheenergyofalocalizedfieldconfiguration,wherelow energysystemsarecorrelatedwithsmallentropicmeasures[2].

TheCEcanbeobtained[2]bytheFouriertransformofthe en-ergydensityt0

(

r

)

[12,13],yielding

F(

ω

)

= −

√1_2π

_∞

0 t0

(

r

)

eiωrdr.

It is worth to remarkthat we will consider structureswith spa-tially localized, square-integrable, bounded energy density func-tions t0

(

r

)

. The modal fraction reads [2–4,6] f

(

ω

)

= |

F(

ω

)

|

2

/

_∞

0 d

ω

|

F(

ω

)

|

2. Next, the normalized modal fraction is defined

asthe ratioof the normalized Fourier transformed function and its maximum value fmax, namely,

˜

f

(

ω

)

=

f

(

ω

)/

fmax. A localized

andcontinuous function

˜

f

(

ω

)

yields the following definition for the CE: S

( ˜

f

)

= −

∞



0 d

ω

˜

f

(ω)

ln

˜

f

(ω)



.

(9)

Fig. 3. tM(r)energy-momentumtensorinTAmodelwithβ=0.25,κ=R0=1 and δ=0.5.Heret0=tθ.

Fig. 4. tM(r)inHCmodelwithc=0.5 andκ=1.Thedominantenergycondition

issatisfied.

Therefore,weusethisconcepttoobtaintheCEintheAbelian string-vortex and the string-cigar contexts. By substituting the warpfactor(6)intheenergydensitygivenbyEq.(3a),ityields

t0

(

r

)

=

1

κ



5 2

+

1

β

 

2

β

sech



β

r

δ



2

.

(10)

It represents a localized density ofenergy, as can be verified in

Fig. 3.Now,theFouriertransformof(10)reads

F

(ω)

=

√

2

πδω(

5

δ

+

2

)

csch



π

δω

2

β



,

(11)

which is a localizedfunction having the normalizedmodal frac-tion:

˜

f

(ω)

=



π

δω

2

β

csch



π

δω

2

β



2

.

(12)

ForthenumericalevaluationofEq.(9),withtheinputofEq.(12), itisnecessarytoexplicitheretheexpressionoftheparameter

β

, asdefinedinRefs.[31,32]:

β

=



(

− )

κ

10

,

with 0

< β

≤

1 2

,

(13)

Let us remember that

κ

is the 6D gravitational constant, being

<

0 the6Dcosmologicalconstant.Besides,thisimposition over therangeoftheparameter

β

isnecessarytopreventvalueslarger thanPlanckmass[34].

HencetheprofileofCEinEq.(9)forthefunctioninEq.(12)is presentedintheFig. 5,forS

(δ)

,andintheFig. 6,forS

(β)

.

It is verified in Fig. 5 that the maximum of CE occurs for

Fig. 5. S(δ)configurationalentropyasafunctionofthethicknessparameterδ,for differentvaluesoftheparameterβ.

Fig. 6. S(β)configurationalentropyasafunctionoftheparameterβ,fordifferent valuesoftheparameterδ.

regions:thefirstonefor

δ

→

0,whichendorsesthethinGergheta– Shaposhnikovmodel of Eq.(5), andthe second one for

δ > δcrit

. However, an upper bound thickness limit is provided in Eq.(7). Thus, for

δ

=

0, the constraint on the TA model thickness with q

=

1 intheEq.(7)yields

0

.

09

β < δ <

0

.

40

β.

(14)

Furthermore, another important physical information is pre-sentedinFig. 6,wheretheminimalCEoccurswhentheparameter

β

tendstozero.PerceivethatthemasshierarchyofEq.(2)forthe TAmodelisexposedinEq.(6)as

M2_P

=

2

π

R0 3

√

π

β





3δ 2

+

1







3δ 2

+

1 2



M₆4

.

(15)

In thecase where MP

M6,the parameter

β

to tends to zero,

once

δ

isboundedbyEq.(7).Thus,theCEexhibitsthisstable be-haviour in Fig. 6 and to small values of

β

there corresponds to smallvaluesofCE.

Forthe HC model,where we haveonly the c parameter, the energydensityofEq.(3a)yields

t0

(

r

)

=

c2

κ

sech 2

₍

_cr

₎



7

+

13 2 tanh

(

cr

)

−

5 2sech 2

₍

_cr

₎

.

(16)

Again,theenergydensityislocalizedascan beverifiedinthe

Fig. 4.TheFouriertransformsofaboveequationreads

F

(ω)

=



π

2

ω

12c2



64c2

+

39ic

ω

−

5

ω

2



csch



_{π ω}

2c



,

(17)

Fig. 7. S(δ)configurationalentropyoftheHCstringmodel,asafunctionofthe parameterc.

anditsnormalizedmodalfractionyields

˜

f

(ω)

=

π

2

_ω

2



_4096c4

₊

_881c2

_ω

2

₊

₂₅

_ω

4



16 384c6 csch 2



π ω

2c



.

(18) Settingtheexpressionandtherangeofthec parameterasdefined inRefs.[19,34,36]

c

=



2

κ

5

(

− ),

with 0

<

c

≤

1 (19)

it is possible to plot the S

(

c

)

by integrating Eq. (18), using, (9).

Fig. 7 representstheresult. Byconsidering themass hierarchyof Eq.(2)inthemodelofEq.(8)providedby

M2_P

≈

4

π

R0 3 1 cM 4 6

,

(20)

theresultof MP

M6 isverifiedwhenc tends tozero.Thisalso

agreeswiththeprofileexhibitedforCEinFig. 7.

The intrinsicbraneworld model parametershave been further constrainedbyanalyzingtheexperimental,phenomenologicaland observational aspects in, e.g., [12,46].In particular, Ref. [12] pro-videsarefinedanalysiswhereintheCEfurtherrestrictstherange parameters ofa5D sine-Gordonthick braneworld model,namely, the AdS bulk curvature andthe braneworld thickness. Here this procedurewasappliedto6Dthickbraneworldmodelsandwe ver-ifiedfortheTAmodeltheconstraintsonthethicknessparameter

δ

intheEq.(14).Besides,inbothTAandHCmodels,theminimal CE reflectsthe expectedresultobtained fromthemass hierarchy inthesemodels.

4. Discussionandconclusions

In this work we have investigated the CE in the context of thetopologicalabelianstring-vortexandstring-cigarscenarios.We have shown that the information-theoretical measure of 6D di-mensionalbraneworldmodelsopensnewpossibilitiestophysically constrain, for example, parameters that are related to the brane thickness. TheCEprovides themostappropriate value ofthis pa-rameter that is consistent withthe best organizationalstructure. The informationmeasureofthe systemorganizationis relatedto modesinthebraneworld model.Hence theconstraintsofthe pa-rameters thatwe obtained, forthe TAandthe HCstring models, providetherangeofthe parametersassociatedtothe most orga-nizedbraneworldmodels,withrespecttotheinformationcontent ofthesemodels.TheCEdemonstratestheexpectedlimitof param-eters that agrees withthemass hierarchyof these6D models.It providesfurtherphysicalaspectstomodels,wherestrictly energy-basedargumentsdonotprovidefurtherconclusionsofthephysical parameters.

Acknowledgements

TheauthorsthanktheCoordenaçãodeAperfeiçoamentode Pes-soalde Nível Superior(CAPES), theConselho Nacional de Desen-volvimentoCientíficoeTecnológico(CNPq),andFundaçãoCearense de apoio ao Desenvolvimento Científico e Tecnológico (FUNCAP) for financial support. DMD thanks to Projeto CNPq UFC-UFABC 304721/2014-0.RdRisgratefultoCNPqgrantsNo.473326/2013-2, No. 303293/2015-2andNo. 303027/2012-6,andto FAPESP Grant No. 2015/10270-0. RACC also acknowledges Universidade Federal do Ceará (UFC) for the hospitality. We would like to thank the anonymousrefereeforusefulsuggestionsinthisletter.

References

[1]C.E.Shannon,BellSyst.Tech.J.27(1948)379.

[2]M.Gleiser,N.Stamatopoulos,Phys.Lett.B713(2012)304. [3]M.Gleiser,D.Sowinski,Phys.Lett.B727(2013)272. [4]M.Gleiser,N.Stamatopoulos,Phys.Rev.D86(2012)045004. [5]M.Gleiser,N.Graham,Phys.Rev.D89(2014)083502.

[6]R.A.C.Correa,A.deSouzaDutra,M.Gleiser,Phys.Lett.B737(2014)388. [7]R.A.C.Correa,R.daRocha,A.deSouzaDutra,Ann.Phys.359(2015)198. [8]A.deSouzaDutra,R.A.C.Correa,Phys.Rev.D83(2011)105007.

[9]R.A.C.Correa,A.deSouzaDutra,Adv.HighEnergyPhys.2015(2015)673716. [10]R.A.C.Correa,R.daRocha,A.deSouzaDutra,Phys.Rev.D91(2015)125021. [11]N.Stamatopoulos,M.Gleiser,Phys.Lett.B747(2015)125.

[12]R.A.C.Correa,R.daRocha,Eur.Phys.J.C75(2015)522.

[13]R.A.C.Correa,P.H.R.S.Moraes,A.deSouzaDutra,R.daRocha,Phys.Rev.D92 (2015)126005.

[14]R.A.C.Correa,P.H.R.S.Moraes,Configurationalentropyinf(R, T)branemodels, arXiv:1509.00732.

[15]L.Randall,R.Sundrum,Phys.Rev.Lett.83(1999)3370. [16]L.Randall,R.Sundrum,Phys.Rev.Lett.83(1999)4690. [17]B.Bajc,G.Gabadadze,Phys.Lett.B474(2000)282.

[18]V.Dzhunushaliev,V.Folomeev, M.Minamitsuji,Rept.Prog. Phys.73(2010) 066901.

[19]T.Gherghetta,M.E.Shaposhnikov,Phys.Rev.Lett.85(2000)240. [20]I.Oda,Phys.Lett.B496(2000)113.

[21]I.Oda,Phys.Rev.D62(2000)126009.

[22]Y.-X.Liu,L.Zhao,Y.-S.Duan,J.HighEnergyPhys.0704(2007)097. [23]Y.X.Liu,L.Zhao,X.H.Zhang,Y.S.Duan,Nucl.Phys.B785(2007)234. [24]S.Aguilar,D.Singleton,Phys.Rev.D73(2006)085007.

[25]M. Gogberashvili, P.Midodashvili, D. Singleton, J. High Energy Phys. 0708 (2007)033.

[26]J.M.Frère,M.Libanov,S.Mollet,S.Troitsky,J.HighEnergyPhys.1308(2013) 078.

[27]P.Tinyakov,K.Zuleta,Phys.Rev.D64(2001)025022.

[28]M.Giovannini,H.Meyer,M.E.Shaposhnikov,Nucl.Phys.B619(2001)615. [29]R.Koley,S.Kar,Class.QuantumGravity24(2007)79.

[30]L.J.S.Sousa,W.T.Cruz,C.A.S.Almeida,Phys.Rev.D89(2014)064006. [31]R.S.Torrealba,Phys.Rev.D82(2010)024034.

[32]R.S.Torrealba,Gen.Relativ.Gravit.42(2010)1831.

[33]J.E.G.Silva,V.Santos,C.A.S.Almeida,Class.QuantumGravity30(2013)025005. [34]F.W.V.Costa,J.E.G.Silva,D.F.S.Veras,C.A.S.Almeida,Phys.Lett.B747(2015)

517.

[35]D.F.S.Veras, J.E.G.Silva, W.T.Cruz, C.A.S.Almeida, Phys. Rev. D91 (2015) 065031.

[36]D.M.Dantas,D.F.S.Veras,J.E.G.Silva,C.A.S.Almeida,Phys.Rev.D92(2015) 104007;

D.M.Dantas,R.daRocha,C.A.S.Almeida,arXiv:1512.07888[hep-th]. [37]J.E.G.Silva,C.A.S.Almeida,Phys.Rev.D84(2011)085027.

[38]F.W.V.Costa,J.E.G.Silva,C.A.S.Almeida,Phys.Rev.D87(2013)125010. [39]D.M.Dantas,J.E.G.Silva,C.A.S.Almeida,Phys.Lett.B725(2013)425. [40]J.C.B.Araújo,J.E.G.Silva,D.F.S.Veras,C.A.S.Almeida,Eur.Phys.J.C75(2015)

127.

[41]L.J.S.Sousa,C.A.S.Silva,D.M.Dantas,C.A.S.Almeida,Phys.Lett.B731(2014) 64.

[42]I.Navarro,J.Santiago,J.HighEnergyPhys.0502(2005)007.

[43]P.P. Orth, P.Chandra, P.Coleman, J.Schmalian, Phys. Rev.Lett. 109(2012) 237205.

[44]N.Lashkari,A.Maloney,Class.QuantumGravity28(2011)105007. [45]M.Headrick,T.Wiseman,Class.QuantumGravity23(2006)6683. [46]J.M.HoffdaSilva,R.daRocha,Europhys.Lett.100(2012)11001. [47]M.Gleiser,N.Jiang,Phys.Rev.D92(2015)044046.