Unified flavor symmetry from warped dimensions

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Unified

flavor

symmetry

from

warped

dimensions

Mariana Frank

a

,

∗

,

Cherif Hamzaoui

b

,

Nima Pourtolami

a

,

Manuel Toharia

a

a_{DepartmentofPhysics,ConcordiaUniversity,7141SherbrookeSt.West,Montreal,Quebec,H4B1R6,Canada}

b_{GroupedePhysiqueThéoriquedesParticules,DépartementdesSciencesdelaTerreetdeL’Atmosphère,UniversitéduQuébecàMontréal,CasePostale8888,} Succ.Centre-Ville,Montréal,Québec,H3C3P8,Canada

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received31October2014

Receivedinrevisedform14January2015 Accepted21January2015

Availableonline23January2015 Editor: M.Cvetiˇc

Keywords:

Warpedextradimensions Neutrinos

Fermionmasses

Inamodelofwarpedextra-dimensionswithallmatterfieldsinthebulk,weproposeascenariowhich explains allthe massesand mixingsofthe SMfermions.Inthisscenario,the sameflavor symmetric structure is imposed on all the fermions of the Standard Model (SM), including neutrinos. Due to the exponentialsensitivityonbulkfermionmasses, asmallbreaking ofthissymmetrycan begreatly enhancedandproduceseeminglyun-symmetrichierarchicalmassesandsmallmixinganglesamongthe charged fermionzero-modes(SMquarks andcharged leptons),thuswashingoutvisibleeffects ofthe symmetry.IftheDiracneutrinosaresufficientlylocalizedtowardstheUVboundary,andtheHiggsfield leaking intothebulk,theneutrinomasshierarchyand flavorstructurewill stillbelargely dominated and reflectthefundamentalflavorstructure,whereaslocalizationofthequarksectorwouldreflectthe effectsoftheflavorsymmetrybreakingsector.Weexplorethesefeaturesinanexamplebasedon which afamilypermutationsymmetryisimposedinbothquarkandleptonsectors.

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

1. Introduction

The original motivation for warped extra-dimensions, or

Randall–Sundrummodels(RS),wastoaddressthehierarchy prob-lem. In RS the fundamental scale of gravity is exponentially re-duced fromthe Planck mass scale to a TeV size due to a Higgs sector localized nearthe boundaryof the extra dimension [1]. If SMfermionsare allowedtopropagateintheextradimension[2], andbecome localized towardseither boundary, the scenarioalso addresses the flavor problem of the SM and suppresses generic flavor-violating higher-order operators present in the original RS setup. However, KK-mediated processes still generate dangerous contributions to electroweak and flavor observables (including dangerousdeviations tothe Z bb coupling)

¯

[3–5],pushingthe KK scale to5–10 TeV [6]. Theusual mechanismsemployed to lower the KK scale involve using a custodial gauge SU

(2)

R symmetry

[7], which insures a small contribution to electroweak precision parameters, lowering the KK scale to around 3 TeV, and render-ing this scenario visible at the LHC. Alternatively, introducing a dilatonic scalar such that the warping of the 5-th dimension is

*

Correspondingauthor.

E-mailaddresses:mariana.frank@concordia.ca(M. Frank),

hamzaoui.cherif@uqam.ca(C. Hamzaoui),n_pour@live.concordia.ca(N. Pourtolami), mtoharia@physics.concordia.ca(M. Toharia).

strongly modified near theinfrared (IR) brane whilebehaving as an Anti-de-Sitterspaceneartheultravioletregion,(so-called soft-wallmetrics)isanothersolution[8].There,thehierarchyproblem imposes constrains on the Higgs profile which are stronger than in the original model, while electroweak constraints are milder, allowingaKKscaleaslowas1–3 TeV.

Onerealizationofthewarpedspacemodelisbasedonthe so-called flavor anarchy [5], in which one assumes that no special structuregovernstheflavorofYukawacouplingsandbulkfermion masses, as natural

O(

1) values for these 5D parameters already generate viable masses and mixings. However the neutrino sec-tormustbehavedifferently,firstduetothepossibilityofMajorana mass terms, andsecond because thissetup generates large mass hierarchiesandsmallmixingangles,atoddswithneutrino obser-vations.Aninterestingpropertyofflavoranarchywarpedscenarios wasinvestigatedin[9],forthecaseofabulkHiggswavefunction leaking into the extra dimension. There one would obtain small mixing angles and hierarchical masses for all charged fermions, and at the same time very small Dirac masses, with large mix-ing angles andnegligible mass hierarchy for neutrinos.Thus the flavor anarchy paradigm could still work in these scenarios. The caseofMajorananeutrinosinthissetupwasalsoaddressedin[9]

where smallandalmost degenerate massescan be generatedfor IR localized Majorana neutrinos, out of higher-order interactions in the Lagrangian. The problemis that the warpingnecessary to

http://dx.doi.org/10.1016/j.physletb.2015.01.025

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

obtain the correctneutrino masses wouldbe too small, and not enoughto solve the hierarchyproblem. Adding UV-localized Ma-jorana masses (via some SU

(2)

triplet representation in the UV region)canalsogeneratesmallandalmostdegeneratemasses[10]. Asour goalhereis totreat neutrinoson equalfooting asquarks andchargedleptons,weprefernottoaddneutrino-specific exten-sions andassume that Dirac neutrinomasses give the dominant contributiontoneutrinomasses.

Herewepresentanalternativescenariowhereinsteadof adopt-ing flavor anarchy, we propose that all fermions share the same flavorsymmetry.Weassumethattheflavorviolatingeffectsinthe 5D Lagrangian can be parametrizedby asmall coefficient whose sizeis controlled by a ratioof scales, φ_Λnn, with

φ

the vacuum

expectationvalue (VEV) ofsomeflavonfield,and

Λ

some cut-off massscale,ortheKKmassofsome otherflavonfields.Thissmall breakingoftheflavorsymmetry isenoughtoreproduce correctly theflavorstructureoftheSMinboththequarkandleptonsectors. Weproceedbyintroducingthemodel,followedbyan exampleof aflavorsymmetrytoshowcaseourresults.

2. Themodel

The(stable)staticspacetimebackgroundis:

ds2

=

e−2 A(y)

η

μνdxμdxν

−

dy2

,

(1) wheretheextracoordinate y rangesbetweenthetwo boundaries aty

=

0 and y

=

yTeV,andwhere A

(

y

)

isthewarpfactor respon-sibleforexponentially suppressing massscales at differentslices oftheextradimension.IntheoriginalRSscenario A

(

y

)

=

ky,with k thecurvaturescaleoftheAdS5interval,whileingeneralwarped scenarios A

(

y

)

isamoregeneral(growing)functionof y.The ap-pealofmorecomplicatedmetricsliesonthepossibilityofhaving light KK resonances matter fields (

∼

1 TeV), whilekeeping flavor andprecisionelectroweakboundsatbay [8,11].Forsimplicitywe willtake A

(

y

)

=

ky,butwiththeassumption thatthesame gen-eralfeatures ariseinmorecomplicatedmetricscenarios(thiswill beaddressedin alonger companionpaper). Assuminginvariance undertheusualSMgaugegroup,the5DquarkLagrangianis

L

q

=

L

kinetic

+

MqiQ

¯

iQi

+

MuiU

¯

iUi

+

MdiD

¯

iDi

+



Y_{i j}5D uHQ

¯

iUj

+

h.c.



+



Y_{i j}5D dHQ

¯

iDj

+

h.c.



(2)

where Qi, Ui and Di are 5D quarks (doublets and singlets un-der SU

(2)).

In the lepton sector, we assume that Majorana mass terms are forbidden, and so the Lagrangian can be trivially ob-tainedfromthe previous one by substituting Qi by Li, Ui by Ni and Di by Ei, where Li are lepton doublets, and Ni and Ei are neutrinoand lepton singlets,respectively. The Higgs field H is a bulkscalarthatcanacquireanontrivialvacuumexpectationvalue (VEV) v

(

y

)

=

v0eaky, and is exponentially localized towards the TeVboundary, withdelocalization controlled by theparameter a. SuchnontrivialexponentialVEVs appearnaturallyinwarped back-grounds with simple scalar potentials and appropriate boundary conditions[8,12,13].

Thisextra dimensionalscenariohastwosourcesofflavor. One arisesfrom the usual Yukawa couplings Yu

i j, Ydi j, Yi je and Y νi j (di-mensionlessparameters definedin unitsofthecurvatureout the dimension-full 5D Yukawa couplings as Y5D

i j

=

√

kYi j). The other flavorparameterscomefromthefermionbulkmassterms, diago-nalinflavorspace,takentobeconstantbulktermswritteninunits ofthecurvaturek,i.e.Mi

=

cik (Mi

=

Mqi

,

Mui

,

Mdi

,

MLi

,

Mνi

,

Mei).

It has been observed before in [9] that, whenever the bulk Higgslocalizationparametera issmallenoughincomparisonwith theci parameters (i.e.,for theHiggssufficiently delocalizedfrom

Fig. 1. Effective4DYukawacouplingsforfermionsasafunctionofthefermionbulk massparameterc.Forsimplicity,wehavetakenthec-parametersforthedoublet andthesinglettobeequal.

theTeVbrane),the4Deffectivemassesdependexponentiallyona ratherthan onthe ci parameters. Theeffective4D massesforall

theSMfermionsbecome

mt



vY

˜

33 cq3

,

cu3

<

1

/

2 (3)

(

mf

)

i j



v



(cLi− 1 2)

_

(cR j− 1 2)_Y

˜

i j a

>

cLi

+

cRj (4)

(

mν

)

i j



v



a−1Y

˜

i j a

<

cli

+

cνj

,

(5)

where mt is the top quark mass,

(

mf

)

i j represents mass ma-trices for light quarks and charged leptons, and

(

mν

)

i j is the Dirac neutrino mass matrix.Note that the couplings Y

˜

i j still re-tain a mild dependence on the ci-parameters as follows Y

˜

i j

=

Yi j



2(a−1)(1−2c_Li)(1−2c_{R j})

a−c_Li−c_{R j} , with Yi j being the original 5D Yukawa couplings. The parameters cLi

≡

cqi

,

cli correspond to the SU

(2)

doublets, and cRj

≡

cuj

,

cdj

,

cej

,

cνj are for the SU

(2)

singlets.

The warp factor



definedby the backgroundparameters as



=

e−kyTeV

∼

₁₀−15_{encapsulatesthehierarchybetweentheUV}

(grav-ity)brane andtheTeV(SM)brane. Thec-parameters dependence on massesisshowninFig. 1forthecaseofabranelocalizedHiggs VEV (a

=

30) and for a delocalized (bulk) Higgs VEV (a

=

1.9), where the appearance of a plateau in the neutrino mass region reflectstheinsensitivitytothevaluesoftheci’sinthatlimit.

Tensionarisessince,inordertogenerateviableneutrinomasses fromEq.(5),onerequiresthata

∼

1.80–1.95.Valuesofa

<

2 rein-troducesomeamountoftuninginthemodeland,forexample,for a

=

1.95,someostensiblyindependentparametersofthe5DHiggs potential mustbe fixedto beequal towithin about1%.However this same tuning will also be responsible for generating a light enoughHiggsmodecomparedtotheKKscale[8].Inmoregeneral warpedbackgroundsthistensionwouldeasilydisappearduetoan enlargedparameterspace,justifyingthechoicea

=

1.9 throughout therestofthepaper.

Weassume thatallYukawamatricesandfermionbulkmasses fromthe 5D Lagrangian sharethesame symmetry structure, fur-therbrokenbysmalltermsi.e.

cf

=

c0f

+ δ

cf (6)

forall fermions of the model with YY

=

Yu

,

Yd

,

Ye

,

Yν and with cf

=

cq

,

cu

,

cd

,

cl

,

ce

,

cν . The matrices YY0 andc0f are flavor sym-metric, whilethe small corrections

δc

f and

δ

YY quantifying the symmetrybreakingdonothaveapriori anyflavorstructure.

FromEqs.(3)–(5),thefermionmassesreceivedifferent correc-tionsduetoflavorviolatingterms:

mt

=

mt0

+ δ

mt cq3

,

cu3

<

1

/

2 (8)

(

mf

)

i j

= (m

f

)

0i j



(δc_Li+δc_{R j})

a

>

cLi

+

cRj (9)

(

mν

)

i j

= (

mν

)

0i j

+ δ(

mν

)

i j a

<

cli

+

cνj

.

(10)

The exponential sensitivity of the fermion masses on the

c-parameters is responsible for an exponential sensitivity of the symmetrybreakingterms.Since



∼

10−15_{,thecorrectionsto the} massmatricescausedby thesetermsareoforder

∼

10−15(δci+δcj)_,

whichmeansthattheycouldaccountfortheobservedhierarchies in thequark andchargedlepton masses, aslong asthe symme-trybreaking corrections

δ

ci remain between

−

0.1 and

+

0.1.The mixing angles are also exponentially sensitive to small symme-try breaking terms so that the mixing angles diagonalizing the massmatricesfromtheleftwillbe Vi j

∼



(δcLi−δcL j)for

(

i

<

j

)

for quarksandchargedleptons.Thuseffectsoftheoriginalsymmetry arewashedout inthequarkandchargedleptonsectors, whilein theDiracneutrinosectorthesensitivitytothesymmetrybreaking islinear(i.e.small).

3. Flavordemocracy

Toqualifyourassertions, wepresentanexampleofa symme-try with a democratic structure, broken by small perturbations. Weimposeasimplestructure forall theflavorparametersofthe model,namelyone whichremains invariantunderfamily permu-tations[14].Thisleadstoaflavorstructurewherethe5D Yukawa couplingsareinvariant underS3

×

S3,whilethe5Dfermionbulk massmatricesare invariantunder S3 only. Underthissymmetry, the5D Yukawacouplings andto 5D fermionbulk mass matrices willbedemocratic,andwillbeparametrizedas

Y_YD

∝



_{1 1 1} 1 1 1 1 1 1



and cD_f

=



_A f Bf Bf Bf Af Bf Bf Bf Af



.

(11)

Since the complete flavor structure is described by Eq. (11), we simultaneouslydiagonalizeallmatricesandobtain

Y0_Y

=

y0_Y



_{0 0 0} 0 0 0 0 0 1



and c0_f

=

⎛

⎝

c0_f 1 0 0 0 c0_f 1 0 0 0 c0_f 3

⎞

⎠ ,

(12)

where y0_Y

=

yu

,

yd

,

ye

,

yν are complex Yukawa couplings in the up,down,chargedleptonandneutrinoYukawasectors.Thematrix c0_f isdiagonal,withrealentriesc0_f

i.Democraticmassmatrices

pro-ducetwomassless fermionsandone massiveone.The0-thorder CKMandPMNSmatricescanbeparametrizedas

V_i0

=

⎛

⎝

cos

θ

0 i sin

θ

0 i 0

−

sin

θ

_i0 cos

θ

_i0 0 0 0 1

⎞

⎠ ,

(13)

wherei

=

CKM,PMNS.Bothmatricescontainanangle

θ

_i0whichis fixed not bythe S3

×

S3 symmetry, butby the symmetry break-ingterms,implementedinEqs.(6)and(7).Thesmallbreakingof thesymmetryisresponsibleforthe(small)liftofthezeromasses, yielding a viable neutrino spectrum, with one heavier and two lighter eigenstateswithsimilar masses,andwitha normal hierar-chy ordering: m3

∼

m0 ν



|



(2cl3−1)

−

₁

|

,

(14) m1

∼ δ

Yνm0ν m2

∼ δ

Yνm0ν

,

(15)

wherem0_ν

=

va−1_{.Wehavekepttheterminc}

l3 sinceitcanhave

an effectwhencl3



1

2.Thedataconstrainingneutrinomasses re-quirem0_ν

∼ (

0.05–0.1)eV,whichinturnfixesthesizeoftheHiggs localizationparametera.Theneutrinomassesdependon

δ

Yν but not on

δ

ci’s, whichremainbasically free (even the0-th orderc0νi

arerelativelyfree,aslongastheysatisfya

<

cν

+

cl).Forexample, one finds that, in orderto generate a viableneutrino mass hier-archy ratior

= (|

m2

|

2

− |

m1

|

2

)/(

|

m3

|

2

− |

m1

|

2

)

∼

0.03, one needs

δ

Yν



√

r

∼

0.17.

In the chargedlepton and in theup-and down-quarksectors, the massless statesare also liftedby the flavor symmetry break-ingleavingasuppressionproportionalto

δ

Y .Buthere,inaddition, theexponentialdependenceonthesymmetrybreakingparameters

δ

cfi createsahierarchyamongallthemasses.Thethirdgeneration

chargedfermionmassesare

mt

∼

m0t

,

(16)

mb

∼

m0b



δce3

,

(17)

mτ

∼

m0τ



(δcl3+δce3)

,

(18)

with the 0-th order masses1 m_t0

=

y0_uv, m0_b

=

y0_dvc0d3−1/2 and m0

τ

=

y0ev

(c0

l3+c0e3−1)_{.Thelighterfermionmassesbehaveas}

mf2

∼ δY

Ym 0 f1



(δc_L2+δc_R2)

_,

₍₁₉₎ mf1

∼ δ

YYm 0 f1



(δc_L1+δc_R1)

_,

₍₂₀₎

where mf2

≡

mc

,

ms

,

mμ, mf1

≡

mu

,

md

,

me and with m 0

f1

=

v(c0L1+c0R1−1)_{. Note that in the flavor symmetric limit, the}

elec-tron and muon, the down and strange quarks, and the up and

charmquarks,aremassless.Thesymmetrybreakingproduces non-zero masses proportional to the generic size of

δ

Y amongthese fermions,2 _{withanaddedsourceofhierarchyduetothe}

exponen-tialdependenceonthe

δ

ci.

Thus, it is straightforward in this scenario to obtain phe-nomenologically viablequark andlepton massesbyappropriately fixing the different

δ

ci within the constraint

|δ

ci

|



0.1. The hi-erarchies betweenfermion massesoccur naturally andare under control since they depend exponentially onsmall numbers (hier-archical butnot too hierarchical). Some massesand mixingsstill dependlinearlyon

δ

Y sothatthetypicalsizeoftheseterms can-not betoosmallsince,forinstance,

δ

Y



mt

/

mc inthe (extreme) limitwherethecharmquarkc-parametersaretop-like.

In this unified scenario, one can also generate the observed

mixing angles in the CKM andPMNS matrices. The CKM entries

become3 Vus

∼



(δcq1−δcq2)

,

(21) Vcb

∼ δ

Y



(δcq2−δcq3) (22) 1 _{Assumingthatc}0 q3,cu3< 1 2 andc 0 d3,c 0 e3,c 0 l3> 1 2.

2 _{ThevaluesofthelightquarkwavefunctionsontheTeVbraneareslightlyhigher} thaninusualRSscenariostoovercometheδY suppression.Thisinducesstronger couplingswithKKgluonsandthusgenericallyenhancepotentiallydangerousFCNC processes.Thesedangerouseffectscanbedealtwithbyinvokingadditionalflavor constrains,suchasenforcingexactlycd1=cd2[15],orgoingtomoregeneralwarped

backgroundswhereflavorboundscanbemuchmilder[8]. 3 _{Strictlyspeaking,throughoutthispaper,byV}

i jwedenotetheabsolutevalue

Table 1

Zerothorder5Dfermionc-parameters.Forsimplicity,wealsosetallthe0-thorder Yukawacoefficientstobeuniversal y0

u=y0d=y0ν=y0e=4.4 andtheHiggs

local-izationparametertoa=1.9. f q u d l ν e c0 f1(≡c 0 f 2) 0.55 0.60 0.60 0.55 5.00 0.60 c0 f3 0.40 0.40 0.50 0.40 2.00 0.60 Vub

∼ δ

Y



(δcq1−δcq3)

.

(23)

Withrespect tothe0-th orderCKMmatrixentriesfromEq.(13), Vusreceivesasuppressionexponentiallysensitivetothedifference betweentwosmalltermswithrespecttotheCKM,whichcan eas-ilyreproducetheCabibboangle.TheentriesVcb andVub,arelifted fromtheinitialzerovalue,andacquireadoublesuppression,one parametricsuppression, linearin thesmallYukawa perturbations

δ

Y

∼

0.1,caused by the broken S3 symmetry, in additionto the exponentialone. Assuming the usual ordering

δ

cq1

> δ

cq2

> δ

cq3,

theratio Vcb

/

Vuswill then beofthe correctorderofmagnitude forthetypicalsizefor

δ

Y and

δ

c.The expectedorderoftheratio Vub

/

Vcb

∼

Vus alsoremains realistic (uptoorder onefactors not takenintoaccountintheestimates).Thislastfeatureisgenericin theusualRSscenarios.

The parametric dependence of the PMNS entries is different fromthatoftheCKMmatrixelements:

Ve2

∼

sin

θ

ν0 (24) Ve3

∼ δ

Y13ν







(2cl3−1)

−

₁



₍₂₅₎ Vμ3

∼ δ

Y23ν







(2cl3−1)

−

₁



_.

₍₂₆₎

Contrarytothemixinganglesinthequarksector,thevalueofVe2 is not suppressed and remains generically of

O(

1), fixed by the structureoftheneutrinoYukawaflavorviolating matrix

δ

Y ν_{i j}.The entriesVe3andV μ3 areliftedfromzero,bothdependon

δ

Y and, notonlyaretheynotfurthersuppressedbyexponentialterms,but canactuallybeenhancedbyexponentialterms(aslongasthe ap-proximationremainsvalid).Inparticularifcl3



1/2,itispossible

toliftthe valuesofthemixingangles asshowninEqs. (25)and

(26).Thisfeatureisspecifictothecasea

<

cl3

+

cνj andcl3

<

1/2,

andis not a generic feature in usual RS scenarios. More precise (andlesscompact)formulaewillbepresentedelsewhere.

The observed mixing angles in the neutrino sector are most sensitive to the flavor structure of the neutrino Yukawa matrix

δ

Y ν , but do not depend much on the charged lepton Yukawa matrix

δ

Yl _{or on the}

_δ

_c

i. The bulk mass parameter of the third family lepton doublet should satisfy cl3

<

1/2 in order to easily

obtain larger mixing angles for small values of

δ

Y

∼

0.1 (given that V_μexp₃

∼

0.65 andthat V_e3exp

∼

0.15).Thisconditionisvery in-terestingasitis thesame inthequark sector, where we require cq3



1/2,to obtaina largetop quark mass.Thiscould be ahint

ofanadditionalfamilysymmetryamongtheSU

(2)

doubletsofthe thirdfamily.

ComparingtheexpressionsfortheVPMNSmixinganglesandthe neutrinomasses,theelement

δ

Y23 mustbelargerthattherestof

δ

Y sothat

δ

Y

∼ δ

Y13

∼

δY₄23.

Inthisscenario itis easy tofind aset of0-th order bulk pa-rameters that reproduce the SM andthat show the features de-scribedabove. For example,a representative point forwhich the SM masses and mixing matrices are only a small perturbation away (of order 10% around an S3 symmetric set of parameters), isshown inTable 1. Charged fermions results are not too sensi-tivetosmalldeviationsintheYukawacouplings(



0.1),andonce the

δ

ci’sarefixed,the

δ

Y ’scanevenbetakenrandomlyaslongas

they remainataround 10%.One canthen obtaingeneric charged fermionmassesandmixingsconsistentwiththeSM,andanylevel ofprecisionispossiblebytuningthesevalues.

Inthisparticularsetup,flavorviolationboundsarenotreduced, andtheoriginofCPphasesis leftunexplained sincethey appear from the flavor symmetry breaking terms. Nevertheless different flavorsymmetriescanalleviatethestringentboundscomingfrom FCNC’s(speciallyin K

− ¯

K mixing),ifforexampletheCPviolating phasesin the down sector are suppressed. A thorough search of symmetriesisbeyondthescopeofthispaper,butlookingforthis type ofeffect isdefinitely a goodguidingprinciple in thesearch offlavorsymmetriesinthiscontext,astheywouldallowforlower KKmasses.Also,asmentionedearlier,itispossibletoworkin sce-narioswhereflavorviolationandelectroweakprecisionconstraints aresatisfiedformuchlowerKKmassesthankstoamodificationof thebackgroundmetric.Suchastudyinthissamecontextofflavor symmetriesisunderway[16].

4. Conclusion

Inthisletterwehaveproposedageneralframeworkinwarped extra dimensions where the SM flavor structure is unified in all fermionsectors.Inthisscenario,allthefieldsarelocalizedinthe bulk andgoverned by the same flavor symmetry. Smallbreaking terms areintroduced forthe 5D bulk massand Yukawa parame-ters.Butwhilethequarkandchargedleptonsectorsaredominated by thesmallflavor breakinginthebulk c-parameters,the(Dirac) neutrinosectorisdominatedbyflavorsymmetrybreakingYukawa couplings,andthis resultisrobust. Neutrinoflavor structure and charged lepton/quark flavor structure do not impact negatively uponeachother.Apermutationsymmetrywasstudiedtoillustrate theidea, butother symmetriescan alsobeinvokedandexplored withinthisframework.Themaindifferencebetweentheneutrino versus thechargedleptonandquark sectorsstemsfromallowing the Higgsfield leak sufficientlyout ofthe TeVbrane sothat the neutrino sector looses sensitivity on the 5D bulk mass parame-ters. This scenario can explain both massesof charged fermions andneutrinos, aswell as reproducethe mixingmatrices inboth quark and lepton sectors. Within the flavor democracy example, weobtainedsimpleexpressionsforthePMNSmatrixelements, re-latingtheminasimplewaytothe5Dparameters,Eqs.(24)–(26). We also obtained simple estimates of the neutrino masses as a function oftheseparameters,Eqs. (14)–(15),andfoundthat they mustobeyanormal hierarchy.Wefoundalsothatthelocalization ofthe third generationoflepton doublets cl3

>

1/2 isquite

con-strainedandfinallythesetuppredictsthattheratiooftheVPMNS mixinganglesVe3

/

V μ3isnontriviallygivenbytheratioofthe fla-vorviolatingcorrectionstotheYukawacouplings

δ

Y13

/δ

Y23. Acknowledgements

WethankNSERCforpartialfinancialsupportundergrant num-berSAP105354.

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