arX

iv:0

710.

2078

v1 [

hep-

ph]

10

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B N L-H ET -07/16

SLA C -PU B -12855

O n D irect Veri�cation ofW arped H ierarchy-and-Flavor M odels

Hoom an Davoudiasl,1,� Thom as G.Rizzoy,2,z and Am arjitSoni1,x

1Departm entofPhysics,Brookhaven NationalLaboratory,Upton,NY 11973-5000,USA

2Stanford Linear Accelerator Center,

2575 Sand HillRd., M enlo Park, CA 94025,USA

Abstract

W e considerdirectexperim entalveri�cation ofwarped m odels,based on the Randall-Sundrum

(RS)scenario,thatexplain gaugeand avorhierarchies,assum ingthatthegauge�eldsand ferm ions

oftheStandard M odel(SM )propagate in the 5D bulk.M oststudieshave focused on thebosonic

K aluza K lein (K K ) signatures and indicate that discovering gauge K K m odes is likely possible,

yet challenging,while graviton K K m odesare unlikely to be accessible at the LHC,even with a

lum inosity upgrade.W e show thatdirectevidence forbulk SM ferm ions,i.e. theirK K m odes,is

likely also beyond thereach ofa lum inosity-upgraded LHC.Thus,neitherthespin-2 K K graviton,

the m ostdistinctRS signal,northe K K SM ferm ions,directevidence forbulk avor,seem to be

within the reach ofthe LHC.W e then considerhadron colliders withps = 21,28,and 60 TeV.

W e �nd that discovering the �rstK K m odesofSM ferm ions and the graviton typically requires

the NextHadron Collider (NHC) withps � 60 TeV and O (1) ab� 1 ofintegrated lum inosity. If

the LHC yields hints ofthese warped m odels,establishing that Nature is described by them ,or

their4D CFT duals,requiresan NHC-classm achine in the post-LHC experim entalprogram .

y W ork supported in partby the Departm entofEnergy,ContractDE-AC02-76SF00515.�Electronicaddress:hoom an@ bnl.gov

zElectronicaddress:rizzo@ slac.stanford.edu

xElectronicaddress:soni@ bnl.gov

1

I. IN T R O D U C T IO N

The Randall-Sundrum (RS)m odel[1]was originally proposed to resolve the hierarchy

between the scales ofweak and gravitationalinteractions, m W � 102 GeV and �M P �

1018 GeV,respectively. The RS m odelisbased on a truncated AdS5 spacetim e,bounded

by two 4D M inkowskiwalls,often called UV (Planck)and IR (TeV)branes.Thecurvature

in 5D induces a warp factor in the m etric which redshifts scales oforder �M P at the UV

branetoscalesoforderm W attheIR brane.Sincethem etricdependsexponentially on the

5th coordinate,explaining m W =�M P � 10� 16 does not require hierarchic param eters. The

requisitebrane-separation wasshown to beeasily accom m odated early on [2].

Initially,itwasassum ed thatallStandard M odel(SM )�eldsresideattheIR brane.The

m oststriking and distinctsignatureofthism odelwould then beweak scalespin-2 Kaluza-

Klein (KK) excitations ofthe graviton,appearing as resonances in high energy collisions

[3].Itwassoon realized thatresolving thehierarchy only required theHiggsto belocalized

neartheIR brane[4]and SM gauge[5,6]and ferm ion [7]�eldscould propagatein the5D

bulk.Itwasshown thatplacing theferm ionsin thebulk providesa naturalm echanism for

generation ofSM ferm ion m assesand also suppression ofunwanted 4-ferm ion operators[8].

Thisisachieved by a m ild m odulation ofbulk ferm ion m assesthatcontroltheexponential

localization offerm ion zerom odes.AstheHiggsiskeptneartheIR brane,sm all4D Yukawa

couplingsare naturally obtained,iflight avorzero m odesare UV brane localized. Given

thecorrespondencebetween location in thebulk and scalein warped backgrounds,operators

containing light avorsaresuppressed by scalesm uch largerthan m W .

Theabovesetup,an RS-typegeom etry with a avored bulk,o�ersan attractivesim ulta-

neousresolution ofhierarchy and avorpuzzles.However,theexperim entalsignalsofthese

warped m odelsarenow m uch m oreelusive.Thisisbecausespreading thegauge�eldsover

the bulk and localizing the lightferm ionsnearthe UV brane suppressestheircouplingsto

IR-brane-localized KK m odes,them ain signaturesofwarped m odels.Therefore,theirpro-

duction via and decay into lightSM ferm ionsand gauge�eldsaresuppressed.Thisfeature

isgenericto warped m odelsofhierarchy and avor,largely independently oftheirdetails.

Recently,various studies have been perform ed to assess the prospects for discovering

thenew warped scenarios,given thattheold setofsignaturesarenow m ostly inaccessible.

Precision data require the new KK states to be heavier than roughly 2-3 TeV [9],even

2

assum ing new custodialsym m etries [16]. Generally speaking,ithasbeen shown thatthe

m ostlikely new statein thesem odelsto bediscovered attheLHC isthe�rstKK gluon [17,

18].TheanalysisofRefs.[17,18]suggeststhatKK gluonsasheavy as4 TeV willbewithin

the reach ofthe LHC.However,forthe KK m odes ofthe weak sector,the corresponding

reach isin the 2-3 TeV range[19].ForgaugeKK m assesin theabove ranges,thegraviton

KK m odesare m ostlikely notaccessible,even with an upgraded LHC lum inosity [20,21].

Thus,typically,thegaugeKK m odesm ay bediscovered,whiletheKK gravitonswhich are

them ostdistinctRS-typesignaturewillbeoutofreach,attheLHC.

In thiswork,we exam ine the discovery prospectsforthe KK m odesofthe SM ferm ion

sector. Observation ofthese states willprovide direct evidence for the presence ofSM

ferm ionsin the5D bulk,a necessary ingredientofthewarped avorscenarios.Brie y put,

we �nd that with gauge KK m asses set at 3 TeV,a currently acceptable value,and for

generic zero-m ode pro�les thatyield a realistic avor hierarchy,the SM KK ferm ions are

not accessible at the LHC,even after a lum inosity upgrade. Hence,we are faced with a

situation in which theKK m odesofthegraviton and SM ferm ionsarenotdiscovered during

the LHC program .However,itseem sreasonable to require directobservation ofthese KK

statesin orderto establish an RS-typewarped m odelasa theory ofhierarchy and avor.

Given ourconclusion thatthe LHC isunlikely to establish realistic warped m odels,we

set out to determ ine the m inim um requirem ents that a future m achine needs to m eet in

orderto m akethistask possible.W etake3 TeV to bea referencem assforthelowestgauge

KK m ode.Thissetsthem assscalesofallotherKK states,in thesim plestgeneric warped

m odels. W hatwe �nd isthat,typically,in orderto have �rm evidence forthe lowestKK

stateswith spins1/2,1,and 2,weneed centerofm assenergiesps� 60TeV and integrated

lum inositiesL � 1 ab� 1! Thissuggeststhatfuture lum inosity and energy upgradesofthe

LHC willm ost likely be insu�cient to verify allthe essentialfeatures ofrealistic warped

m odels. Therefore,ifthese m odels do describe Nature,the LHC willlikely �nd evidence

forthem . However,the NextHadron Collider(NHC),de�ned to haveps � 60 TeV and

L � 1 ab� 1,m ust be part ofthe post-LHC experim entalprogram aim ed at establishing

the underlying theory. W e also show thatifthese statesare notobserved atthe LHC the

eventualreach ofNHC forthegluon KK statesistypically in excessof10 TeV.

In thenextsection wereview key aspectsoftypicalW arped Hierarchy-and-FlavorM odels

(W HFM ).In section 3,westudy theprospectsforveri�cation ofrealisticW HFM atcolliders

3

and estim atetherequired param etersoftheNHC.Ourconclusionsarepresented in section

4.

II. W A R P ED H IER A R C H Y -A N D -FLAV O R M O D ELS

Here,wewillbrie y describethegenericpropertiesofW HFM .M uch ofwhatwillfollow

iswell-known from previousworksand ism ainly included to provide som e background for

ourfurtherdiscussions.

A . G eneralFeatures

TheRS m etricisgiven by [1]

ds2 = e

� 2����dx

�dx

�� r

2

cd�2; (1)

where � = krcj�j,k isthe 5D curvature scale,rc isthe radiusofcom pacti�cation,�� �

� � �,and a Z2 orbifolding ofthe5th dim ension isassum ed.

Tosolvethehierarchy problem ,theHiggsisassum ed tobelocalized neartheTeV-brane,

wherethereduced m etric\warps" hH i5 � �M P down to theweak scale:hH i4 = e� krc�hH i5.

For krc � 11:3,we then get hH iSM � hH i4 � 100 GeV.Originally,it was assum ed that

allSM contentresidesatthe IR-brane [1]. However,asthe cuto� scale in the 4D e�ective

theory is also red-shifted to near the weak-scale,this would lead to unsuppressed higher

dim ensionaloperators that result in large violations ofexperim entalbounds on various

e�ects,such asthose on avor-changing neutralcurrents. This problem can be solved by

realizing that points along the warped 5th dim ension correspond to di�erent e�ective 4D

scales. In particular,by localizing �rstand second generation ferm ionsaway from the IR-

brane,thee�ectivescalethatsuppresseshigherdim ensionaloperatorsm adeupofthese�elds

ispushed to m uch higherscales[8].In theprocessofsuppressing thedangerousoperators,

thissetup also leadsto a naturalm echanism forobtaining sm allferm ion m asses.

The above localization isachieved by introducing a 5D m assterm in the bulk foreach

ferm ion �eld [7].Letc� m =k,wherem isthe5D m assoftheferm ion.Each 5D ferm ion

hasleft-and right-handed com ponents L;R which can beexpanded in KK m odes

L;R(x;�)=

1X

n= 0

(n)

L;R(x)e2�

prcf(n)

L;R (�): (2)

4

TheKK wavefunctionsf(n)

L;Rareorthonorm alized

Z

d� e�f(m )

L;Rf(n)

L;R= �m n: (3)

Onecan then show thatthen 6= 0 m odesaregiven by [7]

f(n)

L;R=

e�=2

NL;Rn

Z 1

2� c(zn); (4)

where the norm alization N L;Rn is �xed by Eq.(3);throughout our work,Z a = Ja + bnYa

denotesa linearcom bination ofBesselfunctionsofordera. In Eq.(4),zn � (m n=k)e� and

m n istheKK m ass.Thezero-m odewavefunction isgiven by

f(0)

L;R =e� c�

NL;R

0

; (5)

with thenorm alization

NL;R

0 =

�ekrc�(1� 2c)� 1

krc(1=2� c)

�1=2

: (6)

Notethatin ourconvention,thesinglet(right-handed)and doublet(left-handed)zerom ode

wavefunctionsarede�ned with opposite signsform assparam eterscS and cD ,respectively.

Hence,forexam ple,UV-localization forthe singletand doubletzero m odescorrespond to

cS < �1=2 and cD > 1=2,respectively.

In theSM ,allSU(2)doubletsareleft-handed,while thesingletsareright-handed.One

can im pose a Z2 parity on bulk ferm ion �elds so that only the doublets have left-handed

zerom odesand only thesingletshaveright-handed zerom odes.However,both thedoublets

and singletshaveleft-and right-handed higherKK m odes.To projecta particular4D zero

m odechirality,Neum ann-likeboundary conditionsarechosen forthecorresponding �eld at

� = 0;�;the otherchirality willthen obey Dirichletboundary conditions. Forexam ple,if

D representsa weak SM doublet,werequire

(@� + rcm )f(n)

L= 0 ; f

(n)

R= 0 at � = 0;�: (7)

Theabovechoiceresultsin a left-handed zero m ode,given by Eq.(5).W ealso �nd

bL;Rn = �

J� (c� 1=2)(zn)

Y� (c� 1=2)(zn)at � = 0;�: (8)

Theaboveequations�x thewavefunctionsand them asseigenvalues,oncecisspeci�ed for

any SU(2) doublet D ;(L;R) equations result in the sam e KK m ass spectrum . Sim ilar

equationscan also bederived forsinglets S.

5

The above ferm ion pro�leslead to a naturalschem e forSM ferm ion m asses[7,8]. W e

willassum e thatthe Higgsison the IR-brane;thisisa very good approxim ation since the

Higgsm ustbehighly IR-localized.Then,a typicalYukawa term in the5D action willtake

theform

S5

Y=

Z

d4x d�

p�g

�5

kH (x) D

L SR �(� � �); (9)

where�5 � 1 isa dim ensionless5D Yukawa coupling and D ;S aredoubletleft-and singlet

right-handed 5D ferm ions,respectively. After the rescaling H ! ekrc�H ,the 4D action

resulting from Eq.(9)is

S4

Y=

Z

d4xp�g�4H (x)

(D ;0)

L (S;0)

R+ :::; (10)

wherethe4D Yukawa coupling forthecorresponding zero-m odeSM ferm ion isgiven by [8]

�4 =�5

krc

"

e(1� cD + cS )krc�

ND ;L

0 NS;R

0

#

; (11)

Thus,in thequark sector,thereare,in general,9di�erentvaluesforcD ;S:3forthedoublets

and 6forthesinglets.Onecan seethattheexponentialform ofthee�ectiveYukawacoupling

�4 can accom m odatealargehierarchyofvalueswithouttheneed forintroducingunnaturally

sm all5D param eters.

W ith theferm ionsin thebulk,thegauge�eldsm ustalsofollow.A 5D gauge�eld A M has

scalarand vectorprojectionsin 4D.Thescalarzero m odecorresponding to A � isprojected

out using Z2 parity or a Dirichlet boundary condition. As is well-known,the rem aining

projectionsA � can beexpanded in KK m odesas

A � =

1X

n= 0

A(n)� (x)

�(n)(�)prc

: (12)

Thegauge�eld KK wavefunctionsaregiven by [5,6]

�(n)

A=

e�

N An

Z1(zn) (13)

subjectto theorthonorm ality condition

Z �

� �

d� �(m )

A �(n)

A = �m n: (14)

Theaboveequation �xesthenorm alization N An and Neum ann boundary conditionsat� =

0;� �x thewavefunctionsand yield theKK m asses.

6

The5D action forthecoupling ofthebulk ferm ion to thegauge�eld A M isgiven by

S A = g5

Z

d4x d�V

�VMl� l

A M �; (15)

whereg5 isthe5D gaugecoupling,V isthedeterm inantofthef�unfbeinV Ml ,withl= 0;:::;3,

V M� = e��M� ,and V 4

4 = �1; l = ( �;i 5). Dim ensionalreduction ofthe action (15)then

yields the couplings ofthe ferm ion and gauge KK towers. W ith ourconventions,the 4D

gaugecoupling isgiven by g4 = g5=p2�rc,which �xesthecouplingsofalltheotherm odes.

Forcom pleteness,wealso writedown thewavefunction ofthegraviton KK m odes[3]

�(n)

G=

e2�

N Gn

Z2(zn) (16)

which obey Neum ann boundary conditionsand areorthonorm alized according to

Z �

0

d� e� 2�

�(m )

G �(n)

G = �m n: (17)

The graviton wavefunctions are �xed and the corresponding KK m asses are obtained,via

theboundary conditions,asbefore.

W HFM are subject to variousexperim entalconstraints,including those from precision

electroweak [9]and avor data [10,11]. A num ber ofm odels with a custodialSU(2)L �

SU(2)R bulk sym m etry havebeen proposed to addresstheseconstraints[12,13,14].In the

following,we willlim itthe scope ofourstudy to bulk SM withoutspecifying a particular

fram ework forsuch constraints.W ewillnotdiscussthephenom enology oftheextra exotica

in these m odels,as they do not change the qualitative picture for the SM KK partners

thatwe presenthere.Thiswillsu�ce to dem onstrate ourkey observations.Fora study of

possiblelightexoticquarksin som ewarped scenariosseeRef.[15].

B . R eference Param eters

To com ply with precision constraints,we willchoose the m assofthe �rstKK m ode of

gauge�eldstobe3TeV [9].Thisim pliesthatwehaveassum ed extra custodialsym m etries,

asexplained in the above. W e ignore brane localized kinetic term sforvariousbulk �elds,

asthey are m ostnaturally loop suppressed. Then,the RS geom etry �xesthe ratiosofall

KK m asses.In whatfollows,wewillalso ignorepotentialm ixing am ong variousKK m odes

thatcan induce sm allshiftsin theirm asses. These considerationsdo notchange ourm ain

7

conclusions regarding discovery reach forW HFM ,atcolliders. The m asses ofvariousKK

m odesaregiven by

m n = xn ke� krc�; (18)

whereforgauge�eldsxn = 2:45;5:56;8:70;:::and forthegraviton xGn = 3:83;7:02;10:17;:::.

ThevaluesofKK m assesforSM ferm ionsdependson thebulk m assparam eterc.In the

ferm ion sector,we willonly discussthereach forSM KK quarks,since we willconcentrate

on m ulti-TeV hadron colliders. Here,we choose cD � �cS � 0:6 for lightquarks. This

choice resultsin m assesoforder10-100 M eV forO (1)bulk Yukawa couplings. Thisrange

roughly coversthequark avors(u;d;s)which constitutethedom inantquark initialstates

forcolliderproduction ofnew physics.ItturnsoutthattheKK m odesofallquarks,except

forthesinglettop quark,areroughly degeneratein m asswith KK m odesofthegauge�elds.

To geta reasonabletop quark m ass,wechoosecDt � cSt � 0:4,giving a singlettop �rstKK

m assroughly 1.5 tim esthe�rstgaugeKK m ass.

To assess the relative signi�cance ofvarious production channels, we m ust know the

relevantcouplingsthatenterthecalculations(an earlierqualitativediscussion in asom ewhat

di�erentcontextcan befound in Ref.[23]).Again,weonly discusstypicalvaluestokeep our

discussion m ore generaland lessparam eter-speci�c. W e adoptthe notation glm n to denote

thecoupling ofthelth gaugeKK m odeto ferm ionsofthem th and nth KK levels;g000 � gSM .

W ewillfocuson thegluon and quark sectors,so gSM = gs,thestrong coupling constant.

First,wewillconsidersingleproduction ofKK quarks.Thiscannotproceed via fusion of

quark and gluon zero m odes,sincethisvertex iszero by orthogonality offerm ion KK wave-

functions.Next,isthepossibility ofproduction in association with a zero m odequark.The

gluon-m ediated diagram again giveszero,by orthogonality. However,KK-gluon-m ediation

givesa non-zero result. Given thatKK gluonsare IR-localized,the only feasible channels

involve the third generation zero m odes. W e �nd that the coupling g100 � gs=5,for light

quarks. Note that the initialstates cannot be gluons,by orthogonality ofthe gluon KK

m odes. For (t;b)L and tR ,we get g101 �

p2�gs. Hence,production in these channels is

roughly proportionalto (p2�g2s=5)

2 � g4s=4. Given thatthe KK m odesofthe singlettop

areabout1.5 heavierthan thedoublets,them ostprom ising channelissingleproduction of

a third generation doubletKK m odein association with a tora b.

Next,letusexam inepairproduction ofKK quarks.Gluon m ediated production can com e

from zerom odequark and gluon initialstates;each oftheseam plitudesisproportionaltog2s.

8

Thereisalso a KK gluon m ediated channel.Here,theam plitudeisroughly proportionalto

g2s;thisisduetoan approxim atecancellation ofvolum esuppression and enhancem entatthe

two verticesforthisprocess. Therefore,we see thatpair-production hasa num berofSM -

strength channelsavailableto it,unlikethesingleproduction.Thiscan o�setthekinem atic

suppression from producing two heavy states.In factwewilllaterseethatpair-production

willdom inatesingle-production with ourtypicalchoicesofparam eters.

Finally,sincewewillalso discussdiscovery reach forKK gravitons,wewillbrie y review

theirrelevantcouplings.Asiswell-known,lightquark zerom odeshaveanegligiblecoupling

to KK gravitons. This is because ofthe extrem e IR-localization ofthese KK gravitons,

com pared with KK gauge �elds. Thus,in generic W HFM ,only the coupling C A A G00n oftwo

gluonsto the KK graviton isim portantforitscolliderproduction [20,21]. In unitsofthe

graviton zero m odecoupling,1= �M P ,wehave[22]

CA A G00n = e

krc�2�1� J0

�xGn

��

krc� (xGn )

2jJ2(x

Gn )j

; (19)

wherethe�rstfew xGn weregiven above.

III. EX P ER IM EN TA L P R O SP EC T S FO R V ER IFIC AT IO N O F W H FM

Giventhediscussion intheprevioussection,the�rstthingweneedtodoistodem onstrate

thatsingle (associated)production ofKK ferm ionstogetherwith theircorresponding zero

m odesisrelatively suppressed in com parison totheproduction ofKK ferm ion pairs.W ewill

concentrate on the quark and gluon sectors. Thisreaction isdom inated by the subprocess

q�q ! g(1) ! q(1)�q(0)+ h:c:,where the �rstgluon KK m ode g(1) issom ewhato�-shell. (W e

assum ethewidthtom assratioofthisgluonKK tobe1/6inouranalysis,followingRef.[17].)

Tobeconcrete,wefocusourattention on theexcitationsoftheleft-handed third generation

quark doubletQ 3 = (t;b)TL sinceithasa largecoupling to g(1),asdiscussed above.

FortheLHC,theresultsofthesecalculationsforsingleproduction can beseen in Fig.1

while generalization to higher energy colliders can be seen in Fig.2. Note that no cuts

orbranching fractionshave been included in these calculations,in orderto avoid a m odel-

speci�c analysis. These rates willgo down ifwe assum e a m ore m assive gluon KK state.

In thiswork,wewillgenerally assum ethatO (100)eventswillbesu�cientto establish the

discovery ofaKK m ode.Theresultsin Figs.(1)and (2)representthecasewith theproduct

9

ofthe entrance and exitchannelcouplingsequalto g2s,the SM value. However,from our

discussion in theprevioussection,weexpectthata m orerealisticvalue,fortheseprocesses,

is g2s=2,suppressing the production by factorofabout4. The rates,ascan be seen from

the �gures,are notvery im pressive,even with thisoptim istic assum ption. Itisclearthat

thisprocesswillbeunobservableattheLHC even with alum inosity upgrade.Further,note

thatthecrosssection only increasesby afactoroforder� 20in thepeak region when going

fromps=14 to 60 TeV.

FIG .1:Ratesforthe associated production of�rsttL and bL K K excitationstogetherwith their

corresponding zero m odes at the LHC;the results for t(b)L correspond to to the higher (lower)

m em berofeach histogram pair,respectively. Here the �rstgluon K K m assis�xed at3 TeV and

we show the resultsasa function ofthe ferm ion pairm ass. The three setsofhistogram sassum e

thatthe bL K K m assis100 (200;300)G eV heavierthan the gauge K K whereasthe threshold for

tL issom ewhathigherdueto the largerzero m odetop m ass.

W enow turn to thepossibility offerm ion KK pairproduction which can arisefrom both

q�q and gg initialstatesasdiscussed above and ism ediated by the entire gluon KK tower,

including thezero m ode.(Notethatin ourcalculationsweincludeonly the�rstthreegluon

KK excitations as wellas the SM gluon.) The results ofthese calculations are shown in

Fig.3 and 4 with no cutsorbranching fractionsapplied. Again we see thatforthe LHC

the ratesare fartoo sm allto be usefulbutthey grow quite rapidly asthe colliderenergy

increases; without m uch e�ort we see thatps = 28 GeV is perhaps the bare m inim um

10

FIG .2: Sam e asthe last�gure butnow fordi�erentvaluesofps and taking the �rstgluon K K

and ferm ion K K m assesto bedegenerateat3 TeV.From bottom to top thehistogram scorrespond

tops= 14,21,28 and 60 TeV,respectively.

requirem entto observetheseKK statesand an even highervalueislikely to benecessary if

e�cienciesand branching fractionsaresuitably accounted for.

W ewillnow discussthelikely dom inantdecay channelsforKK quarks,withoutentering

intoadetailed analysis.W ewillconsiderthegaugeeigen-basispicture,forsim plicity.Given

thattheKK quarksaresom ewhatheavierthan thegluon KK m odes,qK K ! qgK K ,with q

a zero-m odequark,isa possible decay channel;letusdenote thispossibility aschannelA.

Anotherpotentially im portantdecay m odeisthrough thebrane-localized Yukawa coupling

to the Higgs: qiK K ! H qj

K Kwhere the �nalquark KK m ode is lighter than the one in

the initialsate;we willreferto thisaschannelB.ChannelC in the following willreferto

qiK K ! H qj with qj denotinganotherzero-m odequark.(Theweak-sectoranalogofchannel

A willbe suppressed by the weak coupling constantso we willignore itin the discussion

which follows.) Note thathere ifqi isa weak doubletthen qj isa weak singlet,and vice

versa. The subsequent decay ofthe gluon KK state in channelA producesthe �nalstate

qt�t.In channelB,dependingon whetherqj

K K decaysthrough channelsA orC,wegeteither

H qt�torH H q,respectively asa �nalstate.

Thecouplingsin channelsA and C arecontrolled by theoverlap ofthezero m odequark

with the IR brane states,gK K orH . ForchannelB this coupling is O (1),when allowed,

11

FIG .3:Ferm ion K K pairproduction rate atvariouscolliderenergiesasa function ofthe ferm ion

pair invariant m ass assum ing that the �rst gluon and ferm ion K K states are degenerate with a

m ass of3 TeV.From bottom to top the histogram s correspond tops = 14,21,28 and 60 TeV,

respectively.No cutshave been applied.The dotted histogram showsthe resultofonly including

zero m odegluon exchange.

FIG .4:Sam easthe previous�gurebutnow assum ing thatthe K K ferm ion is10% m orem assive

than the �rstgauge K K state.

12

sinceitinvolvesthreeIR-branestates(CFT com posites,in thedualpicture).Forexam ple,

ifa singlet quark is m uch m ore UV-localized than its doublet counterpart, the doublet

KK m ode willdecay through channelA,whereas the singlet KK m ode willdecay m ostly

through channelB (sinceitism orem assivethan thedoubletKK m ode).W ith ourreference

values,thesingletand doubletKK m odesoflight avorsaretaken to bedegenerate,which

essentially elim inateschannelB.However,thisisnottypically expected to bethecasein a

realisticsetup,wheredoubletand singletquarkshavedi�ering valuesofpro�leparam eterc.

W ehavealsoignored thee�ectofm ixingin them asseigen-basis,whereoneexpectsthatthe

degenerate KK m odesaresplitby o�-diagonalYukawa couplings[11].Exactly whatdecay

m odeswilldom inateforeach quark KK statedependson thecvaluesand possiblem ixing

angles. Nonetheless,given the above three channels,the typical�nalstatescorresponding

to the decay ofthe pair-produced quark KK m odesare expected to be given by 2� [qt�t],

2� [H qt�t],or2� [H H q]. Needlessto say,these are com plicated �nalstatesand require

furtherstudy regarding theirreconstruction and possiblebackgrounds.

A com pleteveri�cation ofW HFM ,based ontheRS picture,would requiretheobservation

ofthe�rstgraviton KK state.Thisisknown tobedi�cultatLHC energiesifthegaugeKK

m assis3 TeV;the graviton KK m assin thiscase is’ 4:7 TeV.A prom ising search m ode

isto look forthe process gg ! G (1) ! ZLZL [21]which isratherclean;here ZL denotes

a longitudinalZ which is IR-localized. The SM background arises from the conventional

tree-levelprocess q�q ! ZZ via t� and u� channeldiagram s which is highly peaked in

theforward and backward directionsand isreducible by strong rapidity cuts.Notethatas

theps ofthe colliderincreasesthe average collision energy also increases. Thisleadsto a

strongerpeaking ofthe SM ZZ background in both the forward and backward directions.

Sincethedecay productsoftheZ’sessentially follow thoseoftheiroriginalparentparticle

a tightening oftherapidity cutswillbenecessary asps increasesto m aintain a reasonable

signalto background ratio. In perform ing these calculations,given our assum ptions,the

only freeparam eteristheratio k= �M P .

Figs.5,6and 7show theresultsofthesecalculationsforthreedi�erentvaluesoftheratio

k= �M P = 0:5;1:0 and 0.1,respectively;asexpected,in allcasestheeventrateisfartoo low

attheLHC tobeobservable.Fork= �M P = 0:5,thegraviton KK isa reasonably well-de�ned

resonance structure which growsquite wide (narrow)when thissam e ratio equals1 (0.1).

Fora �xed rapidity cut,thesignaloverthebackground isisseen to dim inish asthecenter

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FIG .5:Production rateforthe�rstgraviton K K excitation decaying into two Z bosonsassum ing

a rapidity cutjyj< 2(1)on theZ’scorrespondingto thedotted(solid)histogram s.Thehistogram s

correspond,from bottom to top,to colliderenergiesofps = 14,21,28 and 60 TeV,respectively,

Z branching fractionsare notincluded and k= �M P = 0:5 hasbeen assum ed.

ofm assenergy ofthe collidergrowslarger. Ifwe identify the graviton through the decay

ZZ ! jj‘+ ‘� ,which hasa branching fraction ofa few percent,itisclearthataps � 60

TeV colliderwillbenecessary to observethisstate.Here,weignoredetection issuesrelated

to having collim ated jets from highly boosted Z’s. A realistic jet resolution could m ake

these conclusionslessoptim istic. However,we refrain from m aking assum ptionsaboutthe

detector capabilities and data analysis m ethods offuture NHC experim ents. Note thatif

k= �M P issu�ciently sm alleven thislarge colliderenergy willbe insu�cient asto discover

thegraviton KK resonanceasitbecom esquitenarrow.

It is also ofsom e interest to exam ine the possibility ofsearching for gluon KK states

via pairproduction now thatweareconsidering higherenergy colliders.Theproduction of

resonantsingle gluon KK statesism ade di�cultsince itcan only occurthrough the light

initialstate partonswhich have suppressed couplingsto the KK gluons. The possibility of

pairproduction avoidstheseissuesasitarisesfrom both gg aswellasq�q initialstatesand

occurs through the exchange ofthe com plete gluon KK tower;we include the �rst three

gluon KK states in the calculations below in addition to the SM gluon zero m ode. Once

produced,gluon KK statesalm ostuniquely decay to top quark pairs,in typicalscenarios

14

FIG .6:Sam easthe previous�gurebutnow with k= �M P = 1:0.

FIG .7:Sam easthe previous�gurebutnow with k= �M P = 0:1.

[17,18].Since2KK gluon statesarem ade,the�nalstateconsistsoftwopairsoftop quarks.

Fig.8 showstheresultsofthesecalculationsforthesam ecolliderenergiesasabove,i.e.,ps= 14,21,28 and 60 TeV.Ascan beseen here,thesesignalratescan bequitesigni�cant

once we go to energies substantially above that ofthe LHC.In order to determ ine how

signi�cant the resulting signalfrom these ratesiswe need to have an estim ate ofthe SM

background. W e willnot consider this background here and only provide the production

15

rate.

FIG .8:Sam easFig.3 exceptnow forthe pairproduction ofthelightestgluon K K state.

The q�q ! g(1) ! t�tgluon KK channelessentially sets the discovery reach for the RS

scenario at hadron colliders [17,18]. At the LHC,a 3 TeV KK gluon should be visible

abovetheusualtop quark pairSM background,with suitablerapidity cuts,afterbranching

fractionsand tagging e�ciencies are accounted for. Unfortunately,asin the graviton KK

ZZ m odeabove,asthecenterofm assenergy ofthecolliderincreasestheforward/backward

peaked SM gg;q�q ! t�t process grows quite rapidly relative to the KK gluon signalfor

�xed rapidity cuts.(Recallthatatthese energiesthedecay productsofthe top quark will

essentially follow the originaltop ightdirection.) Thisresultcan be seen quite explicitly

in Fig.9. Here we see thatthe obvious gluon KK peak structure slowly disappears with

increasing colliderenergy.

Ifwe want to ensure a signi�cant signalto background ratio for gluon KK states at

higherenergy collidersweneed totighten ourrapidity cutsfrom theusual‘centraldetector’

requirem ents,jyj< 2:5. In ouranalysisbelow we willassum e thatjyj< 1 to increase the

S/B ratio.Furtherm orewewillde�nethesignalregion to bein thet�tinvariantm assrange

within ��K K ,thegluon KK width,ofthegluon KK m asswith �K K =M K K = 1=6 assum ed

in ouranalysisbelow. Fig.10 showsthe resulting signalratesfollowing thisprocedure for

a range ofKK gluon m asses asa function ofthe collider energy;branching fractionsand

e�ciencieshavebeen ignored in obtaining theseresults.

16

FIG .9: The resonant production rate for the �rst gluon K K state in the t�t channelassum ing

jyj< 2:5 for the dotted case and jyj< 1 for the solid one. From bottom to top the histogram s

correspond tops= 14,21,28 and 60 TeV respectively. No e�cienciesorbranching fractionsare

included.

FIG .10:Signalrateforapossiblegluon K K resonanceasafunction ofthecolliderenergyem ploying

thecutsdescribed in thetext.Branching fractionsand e�ciencieshavebeen neglected.From top

to bottom the resultsare shown forgluon K K m assesin the range from 3 to 12 TeV in stepsof1

TeV.

17

FIG .11: Sam e as the previous�gure butnow showing the signalsigni�cance. Again,branching

fractionsand e�ciencieshave been neglected.

Fig.11 showsthesignalsigni�canceforgluon KK production fora rangeofm assesasa

function ofthehadron collidercenterofm assenergy.In a m ore realisticcalculation which

includestop quark branching fractionsf and b-tagging e�ciencies�,theresultsshown here

m ust be scaled bypf� � 0:15 [17]. Taking this factor into account we see that,e.g.,at

aps = 21(28;60)TeV colliderKK gluon m assesaslarge as5.5 (7,12)TeV m ay becom e

accessible.Onecould say thatthiscoverstheentire‘natural’param eterspaceforW HFM .

IV . C O N C LU SIO N S

Am ongstm odelsofphysicsbeyond theStandard M odel,W HFM areratheruniquetothe

extentthatthey arecapableofprovidingasim ultaneousnaturalresolution oftwoim portant

puzzles,nam elythehierarchyand the avorproblem s.In particular,thisisin sharp contrast

to supersym m etry,which isotherwisean extrem ely interesting theoreticalconstruct.

Thus,itisclearlyim portanttoestablish therequirem entsfordirectexperim entalveri�ca-

tion ofW HFM .Thesem orerecentdevelopm entsoftheoriginalRS m odel[1],which explain

gauge and avor hierarchies in one fram ework,have KK particles in the few TeV range.

Low energy precision testssuggestthatgaugeKK m assesin W HFM arelikely to beheavier

than about3TeV.Perhapsthem ostcom pelling and uniquesignatureofthesem odelsisthe

18

spin-2KK graviton.Unfortunately,itseem sattheLHC,even with an upgraded lum inosity,

KK gravitons,expected heretolieabove4.7TeV,arevery challenging to observeand likely

inaccessible.ProspectsforKK gluonsup tom assesaround 4TeV seem brighterattheLHC.

Thusitm ay wellbethatsom eearly indication oftheunderlying RS idea m ay �nd support

atthe LHC.However,though considerable work existsin the literature on warped bosonic

KK m odes,not m uch has been done for the SM ferm ion counterparts. The observation

ofthese ferm ionic KK m odescan in essence be taken asdirectexperim entalevidence for

warped bulk avor.

W ith thatperspective in m ind,in thiswork,wesetoutto providean exploratory study

oftheparam etersneeded forthenextgeneration ofm achinesthatcould providesigni�cant

experim entalsupportforthe W HFM ,and especially forgeneration of avorthrough bulk

localization. W e concentrate on hadron colliders only,as they are expected to yield the

largestkinem aticreach.Forde�niteness,throughoutournum ericalstudy here,wetakethe

lightest gauge KK m ass to be 3 TeV;SM ferm ionic KK m odes are always as m assive or

heavier.

First,we studied single KK ferm ion production. The m ostprom ising candidate in this

regard isthe KK m ode ofthe third generation doubletproduced in association with a tor

a b quark. W e found thatatthe LHC the prospectsfor�nding thisKK ferm ion through

single production are rathergrim . In fact,we showed thateven a 28 TeV m achine isonly

likely to seeatm osta handfulofcandidateeventsofthiscategory.Theprospectsim prove

signi�cantly fora 60 TeV m achinewherein a few tensofeventsarepossible.

Pairproduction ofKK ferm ionstypically seem stohaveoveran orderofm agnitudelarger

crosssection com pared tothesingle-production,forlargeps.Forpairproduction,a28TeV

m achine can give severaltens ofevents and 60 TeV produceshundreds ofsuch candidate

events.Onem ay expectthata sam pleofthislattersizeisneeded fora reliableveri�cation

ofthe bulk avor scenarios,after cuts and e�ciencies are taken into account;we do not

delveinto theseissuesin thisexploratory work.

W e have revisited the earlier study ofthe KK graviton through the \gold plated" ZZ

m ode[21].W e�nd thatforthisuniquechannel,only a60TeV m achinecan provideO (100)

eventsoveraplausiblerangeofk= �M P values.Thus,itseem sthatveri�cation oftheW HFM ,

in thesense ofdirectly m easuring the propertiesofsignature KK m odes,requireswhatwe

referto astheNextHadron Collider(NHC)withps� 60TeV and O (1)ab� 1 ofintegrated

19

lum inosity. W e have also extended earlierstudies ofthe resonance production ofthe �rst

KK gluon stateand itsdetection through a t�tpair.W e�nd thatcolliderswith energies14,

21,28 and 60 TeV can allow detection ofthe�rstKK gluon up to m asses4,5.5,7,and 12

TeV,respectively.

W ehenceconcludethatan NHC-classm achinem ustbeanintegralpartofthehighenergy

experim entalprogram ifhintsofW HFM are discovered atthe LHC.The sam e conclusion

holds for 4D m odels that explain hierarchy and avor in a sim ilar fashion and constitute

dualdynam icalscenarios,according to AdS/CFT correspondence [24,25].

A cknow ledgm ents

H.D.and A.S.aresupported in partby theDOE grantDE-AC02-98CH10886 (BNL).

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