Gauge field emergence from Kalb-Ramond localization

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Gauge

field

emergence

from

Kalb–Ramond

localization

G. Alencar

a

,

∗

, R.R. Landim

a

,

M.O. Tahim

b

,

R.N. Costa Filho

a

a_{DepartamentodeFísica,UniversidadeFederaldoCeará,60451-970Fortaleza,Ceará,Brazil}

b_{UniversidadeEstadualdoCeará,FaculdadedeEducação,CiênciaseLetrasdoSertãoCentral, R.EpitácioPessoa,2554,63.900-000Quixadá,Ceará,Brazil}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received24October2014

Receivedinrevisedform22January2015 Accepted26January2015

Availableonline29January2015 Editor:M.Cvetiˇc

A new mechanism,valid for anysmooth version ofthe Randall–Sundrum model,ofgetting localized masslessvectorfieldonthebraneisdescribedhere.Thisisobtainedbydimensionalreductionofafive dimensionmassivetwoform,orKalb–Ramondfield,givingaKalb–Ramondandanemergentvectorfield infourdimensions.AgeometricalcouplingwiththeRicciscalarisproposedandthecouplingconstantis fixedsuchthatthecomponentsofthefieldsarelocalized.Thesolutionisobtainedbydecomposingthe fieldsintransversaland longitudinalpartsandshowingthatthisgives decoupledequations ofmotion for thetransversevectorand KRfieldsinfourdimensions.Wealsoprovesomeidentitiessatisfiedby thetransversecomponentsofthefields.Withthisispossibletofixthecouplingconstantinawaythat alocalized zero modefor bothcomponentson thebrane isobtained. Then, allthe aboveresults are generalizedtothemassivep-formfield.Itisalsoshownthatingeneralaneffectivep and(p−1)-forms cannot belocalizedonthebraneandwehavetosortoneofthemtolocalize.Therefore,wecannot have avectorandascalarfieldlocalizedbydimensionalreductionofthefivedimensionalvectorfield.Infact wefindtheexpressionp= (d−1)/2 whichdetermineswhatformswillgiverisetobothfieldslocalized. ForD=5,asexpected,thisisvalidonlyfortheKRfield.

©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

In Kaluza–Klein models with extra dimensions (string theory and others) the most basic tool is the decomposition of fields depending on the dimensions they are embedded andits tenso-rialcharacteristics.Forexample,workingin D

=

5 andtakingthe important field as gμν ,the dimensional reduction to D

=

4 will giveusagaina fourdimensionalgravitationalfield,a vectorfield, andascalarfield (the dilaton)asdynamicalactors. Enlargingthe numberofextra dimensionswe can addYang–Mills fieldsinthe procedureofdimensionalreductiontoD

=

4[1].Thesamecanbe madeto p-formfields.Forfermionfieldsthereisthespecific pro-cedure to obtain in lower dimensions severalkinds of fermionic fields(chiralornot,realornot). Wepresentinthiswork a simi-larprocedurethat canbe appliedto localize p-formfieldsinthe Randall–Sundrumscenarioofextradimensions

[2,3]

.Interestingly, the results are similar to the fermion case and by dimensional reduction we generallyhavethat some components ofthe lower

*

Correspondingauthor.

E-mailaddresses:geova@fisica.ufc.br(G. Alencar),rrlandim@gmail.com

(R.R. Landim),makarius.tahim@uece.br(M.O. Tahim),rai@fisica.ufc.br

(R.N. Costa Filho).

dimensional fields are not localized. It is important to mention thatthisprocedureactuallyprovidesanewmechanismtolocalize gaugevectorfields:fromaKalb–RamondfieldinD

=

5 wecan ob-tainthe4D Kalb–Ramondandanadditionallocalizedvector field. Wecanthinkthegaugefieldemergesinthismechanism.

The problem of gauge formfield localizationin several brane world scenarios has been studied along the last years. This is a necessary step to walk along since our four dimensional space– time presents us a propagating vector field, despite more possi-blesignalswhichcanbeinterpretedascomingfromother tensor gaugefields.Inthissense,itisalreadyunderstoodhowtolocalize thezeromodesofgravityandscalarfields

[3,4]

inapositive ten-sionbrane. However,theconformal invarianceofthebasicvector modelfallintoserious problemsforbuildingarealisticmodel be-cause the localization method gives no result. This problem has been approached in many ways. Some authors have introduced a dilaton couplinginorderto solveit

[5]

andother propose that astronglycoupledgaugetheoryinfivedimensionscangeneratea masslessphotoninthebrane

[6]

.Modificationsofthemodel con-sidering spherical branes, multiple branesor induced branes can befoundin

[7–16]

.

Beyond the gauge field (one form) other forms can be con-sidered. Infive dimensionswe can have yet the two,three, four and five forms. In D-dimensionswe can in fact think about the

http://dx.doi.org/10.1016/j.physletb.2015.01.041

0370-2693/©2015TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

existence of any p

≤

D. However, as we will see, they can be considered in a unified way.The analysis of localizability of the formfieldshasbeenconsideredin

[17]

whereithasbeenshown that in D-dimensionsonly the formswith p

< (D

−

3)/2 can be localized.However,itiswellknownthatintheabsenceofa topo-logical obstruction, the field strength ofa p-form is dual to the

(D

−

p

−

2)-form [18]. Usingthis theauthors in [19] found that alsofor p

> (D

−

1)/2, thefieldsare localized.Itis importantto point that in themodel proposed here the Hodge Duality is not validsinceweconsidermasstermsintheactionthatbreakthe du-ality.Beyondthezeromodelocalizationtheresonancesofp-forms hasalsobeenstudied

[20–24]

.

Anotherinteresting point of view is related to models where membranesaresmoothedout bytopologicaldefects [25–33].The advantageofthesemodelsisthatthe

δ-function

singularities gen-eratedby the brane in the RSscenario are eliminated. This kind ofgeneralizationalsoprovidesmethodsforfindinganalytical solu-tions

[34,35]

.Thisisanicecharacteristicifwewanttoputforward theideaofconsideringageometricalcouplingwiththeRicciscalar. The Ricci scalar can inform about possible space–time singulari-ties and, as we want avoid them, such a coupling is natural in thissense. We thereforeconsider this kindof coupling withthe gauge field, the Kalb–Ramond field and p-form fields in models withsmooth membranes.Thiskindofcoupling hasits originsin theDGPmodelanditsconsequences

[36]

.Oneofitsconsequences is a model of(quasi) localization of gauge fields [37] where the membraneis described by a delta function, i.e., a singular place thatcan be understoodusingthe Ricciscalar: infact wecan get that function as coming from a smooth model. The Ricci scalar, whenwemakethelimittotheRSmodel,giverisetoadelta func-tionandexplainthegeometricalcouplingwiththemembrane.

Other studiesusing a topologicalmass terminthe bulk were introduced,butwithoutgivingamasslessphotoninthebrane[38]. Most of these models introduce other fields or nonlinearities to the gauge field [39]. As a way to circumvent this, the authors in[40] introduced inthe action, beyondthe usual field strength (YM N

= ∂

[MXN]), a mass termin five dimensionsanda coupling withthebranegivenby

(M

2

+

cδ(z))GM NXMXN,whereXM isthe

vectorgauge field. Thisgivesa localized masslessphoton. Inthis modelthelocalizationisobtainedonlyforsomevaluesofthe pa-rameter c andfor a range in M. It is important to note that in thiscasethegauge symmetryislostduetheexistence ofamass termbutisrecoveredintheeffectiveactionofthezeromode.In thiscontext, a model has beenproposed in whichthe two cou-plings are replaced by a coupling with the Ricci scalar[41]. This is a very a natural way if we want to consider smooth version ofRSmodel.For obtaining their resultsthe authorsof [41] used the particular configurationof fields

∂

μ Aμ

=

A5

=

0. This is the

samegauge used inthe massless case. However, herewe havea massterm andthe gauge symmetry is lost.Therefore, the result obtainedbythemisnotgenerallyvalid.Asolutiontothisproblem wasfoundbythepresentauthorsin

[42]

.Weshowtherethatthe choice

∂

μ Aμ

=

A5

=

0,yetbeingvalidasa particularsolution,is

unnecessary.Weshowthatupondimensionalreductionofthefive dimensionalvector field( AM)wegetdecoupledequationsforthe

scalar( A5) andthe transverse vector ( Aμ)fields in four

dimen-sions.Forthisweprovesomeidentitiessatisfiedbythetransverse componentofthefield Aμ.Thenweobtainthatwejustcan local-izethezeromodeoftheAμ orofthescalarfield.

Inthepresentmanuscriptwe considerthesameprocedure to thetwoformfield,whichbydimensionalreductiongivesusatwo andan one form fields in four dimensions. In this case we ob-tainthatbothfieldsaresimultaneouslylocalizedonthefourbrane. Therefore,ascommentedbefore,wefindthatwecanhaveto dif-ferentsituations:upondimensionalreductionsomecomponentsof

thelowerdimensionalfieldsarenotlocalized.Aspecialcase hap-pensfortheKRfield in D

=

5.Tohaveabetter understandingof thiswe generalize ourresults tohigher dimensionsandconsider p-formsfieldson it.Wefindthat foreachspace–timedimension D we can have justone higher dimensional p-form which pro-videsbothcomponentsoflowerdimensionalformfieldslocalized. In fact we finda relation, givenby p

= (

D

−

1)/2, wherethis is valid.

The paper is organized as follows. In section two we review theresultsfortheoneformgaugefield.Insectionthreewestudy the generalization forthe Kalb–Ramond, or two formfield. After considering similardecompositionofthefield weshow thatthey aredecoupled.Bydimensionalreductionitisalsoshownthatwe can localize both,the gauge and theKalb–Ramond fields in four dimensions. In section four we generalize all the results to the p-formcase.

2. Theoneformcase

Herewemustreviewtheresultsfoundbytheauthorsina pre-viouswork

[42]

.Thegeometricalcouplingisproposedwithaction S1

= −



d5X

√

−

g1 4g M N_gP Q_Y M PYN Q

−

γ

1 2



d5x

√

−

g R gM NXMXN

,

(1)

whereds2

₌

_e2 A(z)

_(dxμdxμ

₊

_dz2

_).

_{Theequationsofmotionare}

∂

M

√

−

g gM OgN PYO P



= −

γ

1

√

−

g R gN PXP

,

(2)

andfromthe antisymmetryof Eq.(2)obtain thetransverse con-dition

∂

N

(

√

−

g R XN

)

=

0.Then split the field intwo parts Xμ

=

Xμ_L

+

Xμ_T, where L stands for longitudinal and T stands for transversalwithXμ_T

= (δ

νμ

−

∂ μ_∂ ν 2

)Xν and

XμL

=

∂μ_∂ ν 2 Xν .Withthis, Eq.(2)canbedividedintwo.ForN

=

5

∂

μYμ5

+

γ

1e2 AR

Φ

=

0 (3)

where

Φ

≡

X5andforN

=

ν

weget

eA

2

Xν_T

+



eA

∂

Xν_T





+

γ

1e3 AR XνT

+



eAY_L5μ





+

γ

1e3 AR XνL

=

0

,

(4) wheretheprimemeans a z derivative,andalllower dimensional indexwillbecontractedwith

η

μν .Yetformourtransversality con-ditionweget

e3 AR

∂

μXμ

= −



e3 AR

Φ



 (5) andusing theprevious definition andY5_Lμ

≡

X_Lμ

− ∂

μ

Φ

we can showthefollowingidentities

∂

μYμν

= 2

XνT

;

Y5μ

=

X μ T

+

Y 5μ L

;

Y μ5 L

=

∂

μ

2

∂

νYν 5

_.

(6) Usingnow

(3)

,

(5)

and

(6)

weget



eAY_Lμ5





= −

γ

1

∂

μ

2



e3 AR

Φ





= −

γ

1e3 AR XνL

,

andfinallyobtainfromEq.(4)theequationforthetransversepart ofthegaugefield

Finally separating the z dependence like Xμ_T

= ˜

Xμ_T

˜ψ(

z), using R

= −

4(2 A

+

3 A2

_)e

−2 A _{andperforming thetransformation}

˜ψ =

e−A2

_ψ

_{wegetthedesiredSchrödingerequationwithpotential}

U

=



1 4

+

12

γ

1



A2

+



1 2

+

8

γ

1



A (7)

whichislocalizedfor

γ

1

=

1/16 withsolutioneA.Herewecorrect

a misprint ofRef. [42] where we gavethe solution eA/2_.Forthe

scalarfieldwemustbecarefulsincewehave

2Φ −



∂

μAμ





₋

γ

1Re2 A

Φ

=

0

.

Performingtheseparation ofvariables

Φ

= Ψ (

z)φ (x),defining

Ψ

= (

e3 A_R)−1/2

_ψ

_{,usingEq.(5)andaftersomemanipulationswe}

getaSchrödingerequationforthemassivemodeofthescalarfield withpotentialgivenby

[42]

U

=

1 4



3 A

+ (

ln R

)





2

−

1 2



3 A

+ (

ln R

)





+

γ

1Re2 A

.

Withthispotential weseethat thezero modeofthe scalarfield solutionislocalizedfor

γ

1

=

9/16.Thisshowsusthatwe cannot

havebothfieldslocalized.

3. TheKalb–Ramondfieldcase

In this section we usethe same approach asbefore in order totry to localizethezero mode oftheKalb–Ramond field. Upon dimensional reduction of the KR field we are left with to kinds of terms,namely a Kalb–Ramond in four dimensions Bμν anda vector field Bμ5. We must remember that here we also do not

have gauge symmetry and we cannot choose B5μ

=

0. However, we canagainshow that thelongitudinal andtransversalpartsof thefield decouples and we getthe desiredresults.The action in thiscaseisgivenby

S2

=



d5x

√

−

g



−

1 24

(

YM1M2M3

)

2

₋

1 4

γ

2R

(

XM1M2

)

2



,

andtheequationsofmotionaregivenby 1

2

∂

M1

√

−

gYM1M2M3

−

_γ

2R

√

−

g XM2M3

=

0

.

(8)

IntheaboveequationalltheindexesareraisedwithgM N.Justlike inthecaseoftheoneformfield,theantisymmetryoftheequation gives usthe transverse condition

∂

M1

(R

√

g XM1M2

₎

=

_{0.Now we} proceedtofindthedecoupledequationsofmotion.Firstofallthe aboveequationmustbeexpanded.ForM2

=

μ

2 andM3

=

μ

3 we

obtain 1 2e −A

_∂

μ1Yμ1μ2μ3

+



e−AY5μ2μ3





−

_γ

2ReAXμ2μ3

=

0

;

(9)

andforM3

=

5 weget

1

2

∂

μ1Yμ1μ2

5

₋

_γ

2Re2 AXμ2

=

0

.

(10)

The transverse equation, differentlyfromthe vector case, will giverisetotwoequations.ForM4

=

5 weget

∂μ X

μ5

≡ ∂

μ Xμ

=

0, where we have used the previous definitions. Therefore, we see that thetransverse condition for ourvector field is naturally ob-tainedupondimensionalreduction.ForM4

=

μ

4 weget



ReAXμ4





+

_eA_R

_∂

μ1X

μ1μ4

=

₀

_.

₍₁₁₎

Justasinthecaseoftheone form,herewe haveeffective equa-tions that couple the Kalb–Ramond and the Vector field. Before proceeding to solve the equations we can further simplify them

if we take the longitudinal and transversal part of each field. As the vector field already satisfy the transverse condition we just need to perform this for the KR field by Xμ1μ2

=

_Xμ1μ2

L

+

Xμ1μ2 T ,definedas X μ1μ2 T

≡

Xμ1μ2

+

21

∂

[μ1

∂

ν1Xμ2]ν1 andX μ1μ2 L

≡

−

1 2

∂

[μ1

∂ν

1Xμ2] ν1_{.Observingthat}

∂

μ1Y μ1μ2μ3

=

₂



_Xμ2μ3 T

;

∂

μ1Y μ1μ2

=

₂



_Xμ2 T

,

where Yμν

= ∂

[μ Xν],we seethat the first termof Eq.(9), is al-ready decoupledfromthe longitudinalpart.However, thesecond term is not decoupled because Y5μν

₌

_Y5μν

L

+

2∂X μν T , then our equationsbecome e−A



Xμ2μ3 T

+ ∂



e−A

∂

Xμ2μ3 T



−

γ

2ReAXμ_T2μ3

+

1 2

∂



e−AY5μ2μ3 L



−

γ

2ReAXμ_L2μ3

=

0 (12) and 1 2

∂

μ1Y μ1μ2 L

−

γ

2Re2 AXμ2

=

0

.

(13)

ItisclearlyfromEq.(12)thatwehaveacouplingbetweenthe transversalpart ofthe field, thelongitudinal part,andthegauge field. FromEq.(13)we seethat thegaugefield iscoupledto the longitudinal partof the KR field.As in the caseofthe one form fieldweshouldexpectthatwehavetwouncoupledeffective mas-siveequationsforthegaugefieldsXμ1μ2

T andXμ sincebothsatisfy

thetransversecondition infourdimensions.Toprovethisweuse

∂μ X

μ

=

0 toshowthat Yμ1μ25 L

= −

1

2

∂

[μ1

∂

νYμ2]ν

=

2

γ

2Re2 A

∂

[μ1_Xμ2]

2

,

whereinlastequalitywehaveusedEq.(10).Nowwecanusethis andEq.(11)toshowthat



eAYμ1μ25 L





₌

₂

γ

2ReA

∂

[μ1

_∂

_ν 1Xμ2]ν1

2

= −

2

γ

2ReAXμL1μ2

andthistermcancelsthelongitudinalpartofthemassterm.Then wegetthefinalformoftheequationofmotion

e−A



Xμ1μ2 T

+



e−A

∂

Xμ1μ2 T





₋

γ

2ReAXμ_T1μ2

=

0

.

Imposing the separation of variables in the form Xμ1μ2 T

(z,

x)

=

f

(z) ˜

Xμ1μ2

T

(x)

weobtainthefollowingmassequation



e−Af

(

z

)





−

γ

2ReAf

(

z

)

=

2m2Xe−Af

(

z

),

usingthetransformation f

(z)

=

eA/2

_ψ(z)

_{wegetthestandard}

po-tential,plusthecorrection U

(

z

)

=



A2 4

−

A 2

+

γ

2Re 2 A



=



1 4

+

12

γ

2



A2

+



−

1 2

+

8

γ

2



A

.

The zero mode solution is of the form eb A _{which if plugged in}

the above equation givesus

γ

2

=

5/16 and we getthe integrand

e4 A rendering a localized zero mode. Now we must analyze the localizability of the vector field. In order to decouple the vector field andthelongitudinalpart ofKRfield wecan useEq.(11)in

(13)weget



Xμ2 T

+

R−1e−A



ReAXμ2







−

_γ

2Re2 AXμ2

=

0

.

(14)

Now separating the variables Xμ1

=

_{u(z) ˜Xμ}1

_(x)

_{we get the} massequationforthevectorfield



R−1e−A



ReAu

(

z

)









−

γ

2Re2 Au

(

z

)

=

2m21u

(

z

).

(15)

The above equation can be cast in a Schrödinger form by using thegeneraltransformationfoundin

[42]

,oru(z)

= (

ReA

)

1/2

ψ

.The finalpotentialisgivenby

U

=

1 4



A

+ (

ln R

)





2

−

1 2



A

+ (

ln R

)





+

γ

2Re2 A

.

Inthiswaywe seethat foranysmooth version of RSmodelthe abovepotential is identicaltothat ofthe Kalb–Ramondcaseand we have a localized solution. In this sense, we can say that the vector field emerges in D

=

4 from the localizationof the Kalb– Ramondfield.Inthenext sectionitwillbeclearwhyjustforthe KRfieldinfivedimensionswecanhavebothfieldslocalized.

4. Thep-formfieldcase

In this section we further develop the previous methods in order to generalize our results to the p-form field case in a

(D

−

1)-brane.Theactionisgivenby

Sp

= −

1 2p

!



dDx

√

−

g



₍

_Y M1...Mp+1

)

2

(

p

+

1

)

!

+

γ

pR

(

XM2...Mp+1

)

2



,

(16) where YM1...Mp+1

= ∂

[M1XM2...Mp+1]. The equations of motion are givenby 1 p

!

∂

M1

√

−

gYM1...Mp+1

−

_γ

pR

√

−

g XM2...Mp+1

=

0

.

(17)

Similarlytotheone andtwoformcase, fromtheaboveequation wegettheidentity

Re(D−p)A

∂

ν2_X ν2N3...Np+1

+

Re(D−p)AX5N3...Np+1



₌

0

.

(18) Now we can obtain the equations of motion by expanding Eq.(17).Wearriveatjusttwokindsofterms,wherenoneofthe indicesis5,giving 1 p

!

e αpA

_∂

μ1

Yμ1μ2...μp+1

+

1 p

!



eαpA_Y5μ2...μp+1





−

γ

pRe(αp+2)AXμ2...μp+1

=

0

,

(19)

with

α

p

=

D

−

2(p

+

1).Whenoneoftheindicesis5 weget

1

p

!

∂

μ1Yμ1μ2

...μp5

−

_γ

pRe2 AXμ2...μp5

=

0

.

(20)

JustlikeintheKalb–Ramondcase,thetransverseequation

(18)

giverise totwo equations.Forthe indexwithdirection5 weget

∂μ

1Xμ1

...μp−15

_{≡ ∂}

μ1Xμ1

...μp−1

₌

_{0, wherewe have usedour} pre-vious definitions. Therefore we see that the transverse condition forour

(p

−

1)-formfield isnaturally obtainedupon dimensional reduction.Foraindexnotequalto5 weget



Re(αp+2)A_Xμ1...μp−1





+

_Re(αp+2)A

_∂

μpX

μ1...μp

=

₀

_.

₍₂₁₎

First of all, we must split the field as done before by defin-ing Xμ1...μp T

≡

Xμ1...μp

+

(− 1)p 2

∂

[μ1

∂

ν1Xμ2 ...μp]ν1 _{and X}μ1...μp L

≡

(−1)p−1 2

∂

[μ1

∂ν

1Xμ2

...μp]ν1_{.Observingnowthat}

∂

μ1Y μ1μ2...μp+1

= 

_Xμ2...μp+1 T

;

∂

μ1Y μ1μ2...μp

= 

_Xμ2...μp T

,

(22)

weseethatthefirsttermofEq.(19),justlikeinthelastsection,is alreadydecoupledfromthelongitudinalpart.However,thesecond termisnotdecoupledandifusethefactthat

Y5μ1...μp

=

_Y5μ1...μp

L

+

p

!

X

μ1...μp

T (23)

wecanwriteEq.(19)as eαpA



_Xμ1...μp T

+



eαpA

_∂

_Xμ1...μp T



_

−

γ

pRe(αp+2)AXμ2 ...μp+1 T

+

1 p

!



eαpA_Y5μ1...μp L





−

γ

pRe(αp+2)AXμL1...μp

=

0

,

(24) and

(20)

as 1 p

!

∂

μ1Y μ1μ2...μp L

−

γ

pRe 2 A_Xμ2...μp

=

₀

_.

₍₂₅₎

Therefore,weseeclearlyfromEq.(24)thatwehaveacoupling betweenthetransversalpartofthe p-formfield,the longitudinal part and the

(p

−

1)-form field. From Eq. (25) we see that the

(p

−

1)-formiscoupledtothelongitudinalpartofthep-formfield. Asinthecaseoftheoneformfield,weshouldexpectthatwehave to uncouple the effective massive equations for the gauge fields Xμ1μ2...μp

T andXμ2...μp sincebothsatisfy thetransversecondition

infourdimensions.Letswalkalongandprovethisnow.Firstofall notethatusing

∂

μ2Xμ2

...μp

=

_{0 wecanshowthat}

Yμ1...μp

=

(

−

1

)

p−1

2

∂

[μ1

∂

νYμ2...μp]ν (26)

andwegetanidentitysimilartothatforthegaugefield Yμ1...μp5

L

=

p

!

γ

p

Re2 A

2

∂

[μ1Xμ2...μp]

,

(27)

whereinthelast equationwe haveusedEq.(20).Usingnow the transverseequation

(21)

weobtain



eαpA_Yμ1...μp5 L





₌

_p

_!

γ

pRe(αp+2)AXμ1 ...μp L (28)

and we get the equation of motion for the transversal part of p-form eαpA



_Xμ1...μp T

+



eαpA

_∂

_Xμ1...μp T





₋

γ

pRe(αp+2)AXμT1...μp

=

0

.

Imposing now the separation of variables in the form Xμ_T1...μp

(z,

x)

=

f

(z) ˜

Xμ_T1...μp

(x)

weobtainthemassequation



eαpA_f





−

_γ

pRe(αp+2)Af

=

m2Xp

!

eαpAf

,

(29)

wheretheprimesmeansderivativewithrespecttoz.Now,making f

(z)

=

e−αpA/2

_ψ

_andusinge2 A_R

_{= −(}

_D

₋

₁₎

_[

_{2 A}

₊₍

_D

₋

_2)A2

_]

_,we

canwritetheaboveequationinaSchrödingerformwithpotential givenby U

(

z

)

=



_α

2 p 4

+ (

D

−

1

)(

D

−

2

)

γ

p



A2

+



α

p 2

+

2

(

D

−

1

)

γ

p



A

.

(30) The localized zero mode solution is given by ep A with

γ

p

=

[(

D

−

2)

−

2

α

p

)/4(D

−

1). Forthe

(p

−

1)-formwe have,

impos-ing theseparationofvariables Xμ2...μp

(z,

x)

=

u(z) ˜Xμ2...μp

(x)

and

from

(20)

and

(21)

themassequation



Re−(αp+2)A



_Re(αp+2)A_u

₍

_z

₎









−

_γ

pRe2 Au

(

z

)

=

m2p−1u

(

z

).

(31)

Just as in the last two section we see that we just have to use u(z)

= (

Re(D−2p)A

)

1/2

ψ

in

(31)

togetaSchrödingerequationwith potential

U

=

1 4

(

2

α

p

+

1

)

A

+ (

ln R

)



2

−

1 2

(

2

α

p

+

1

)

A

+ (

ln R

)



+

γ

pRe2 A

.

(32)

From the above equation we see that we can recover all the previous cases.We alsoanalysethe localizabilityofthefield ina verysimpleway.ForanymetricwhichrecoverstheRSforlarge z wegettheasymptoticpotential

U

(

z

)

=

1 4

(

2

α

p

+

1

)

2 A2

−

1 2

(

2

α

p

+

1

)

A

+

γ

pRe2 A

.

(33)

The solution to the above equation is found by fixing

γ

p

=

(D

+

2

+

2

α

p

)/4(D

−

1).Thereforewe canseethat theonlycase

for localizing both fields happens for p

= (

D

−

1)/2. Now it is clearwhyfor D

=

5 we havethat KR field providesthe localiza-tionofbothfields.Thisisthe resultwe wanttostress here.This ispossibleduetothegeometricalcouplingandthefieldsplitting described.

5. Conclusionsandperspectives

In this paper we have developed the idea that a geometrical coupling with the Ricci scalar can solve the problem of gauge field localization.We firstshowedthat foranyformfieldwe can obtain decoupled equations of motion for the longitudinal and transverse components of the fields. We studied first the sim-plestcases,namelytheVectorandKalb–Ramondfields.Fromthese we can understand how a generalization to p-forms can be ob-tained.Some points are worthwhile noting. First, we have found that forsome specific value of couplingconstant we can getthe localizationofanyp-form.However,the

(p

−

1)-formobtainedby dimensionalreductioncannot besimultaneouslylocalized.Despite of this, something very interesting happensin the Kalb–Ramond casein D

=

5. Herewe getthatthrough adimensionalreduction we naturally havethe KRand thegauge field localized.This isa very important resultsince this gives a richer possibility of dy-namicscomingfromauniquefieldinfivedimensions.Infact,this canbeseenasanewmechanismtolocalizethegaugevectorfield. Asa byproductIt isalsointerestingto observethatfor p

=

0 we get

γ

0

= −(

D

−

2)/4(D

−

1) what is exactly the conformal

cou-plingtothescalarfield.Itremainstoanalyzeothercharacteristics likeresonantmodesinthissituation.Thequestionaboutfermions withsimilarcouplingscanbeinterestingtoanotherstudy.Wecan askhere,becauseofthefactofnon-localizationatthesametime of fields coming from the procedure explained, ifthere is some physical criteria to choose one field or another. These are good questionstothinkaboutandarelefttofutureworks.

Acknowledgements

The authors would like to thanks the referee for the care-ful reading of the manuscript, suggestingmodifications and cor-rections which improved it. We also acknowledge the financial support providedby Fundação Cearense de Apoio ao Desenvolvi-mento Científico e Tecnológico (FUNCAP), the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and FUN-CAP/CNPq/PRONEX.

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