A viable RS Model for Quarks and Leptons with

T´ Flavor SymmetryFelix Yu

University of California, IrvinePheno 2010

M-C. Chen, K. T. Mahanthappa, FY – Phys. Rev. D 81, 036004 (2010)[arXiv: 0907.3963 [hep-ph]]

Motivation• Fermion mass hierarchy unexplained• Gauge hierarchy problem motivates new

physics at about TeV• Randall-Sundrum (RS1) model with bulk

fermions provides a good framework– Can get fermion mass hierarchy with O(1)

coefficients– Need to suppress FCNCs

Randall-Sundrum• RS1 Model – warped geometry

– 5th dimension compactified via S1/2

– Higgs field confined to TeV brane (y = R), other fields propagate in bulk

– From compactification and boundary conditions, can find Fourier modes for bulk fields

– SM masses and mixings arise from zero modes• Integrate out y to find overlap between SM fields and

Higgs

Gherghetta, Pomarol (2000), Huber, Shafi (2000), Grossman, Neubert (1999)

222 dydxdxeds )y(

yk)y(

Randall, Sundrum (1999)

Flavor Changing Neutral Currents in RS• Change from gauge interaction basis to mass basis

• Generically get FCNCs if bulk masses are not equal• Solutions: (1) alignment, (2) degeneracy

233

222

211

4

00

00

00

)c,c(f

)c,c(f

)c,c(f

GgDiedy MM|y|k

mm V

)c,c(f

)c,c(f

)c,c(f

VGg

233

222

211

00

00

00†

The Finite Group T´• Double covering of A4

– A4 is the discrete invariant rotations of a tetrahedron

• Has two generators: S=(1234) (4321), T=(1234) (2314)– S2=R, T3=1, (ST)3=1, R2=1

• R=1: 1, 1´, 1´´, 3 (vector) [use for leptons]• R=-1: 2, 2´, 2´´ (spinorial) [use for quarks]

Frampton, Kephart (1995)

Assignment of T´ Representations• Motivated by neutrino mixing data: assign L ~ 3 (LH

lepton doublets), N ~ 3 (RH neutrinos) under T to obtain the tri-bimaximal mixing pattern– Introduce e ~ 1, ~ 1, ~ 1 for charged lepton masses– Tree-level lepton FCNCs are eliminated via degeneracy

(left-handed lepton doublets share a common bulk mass term) and alignment (right-handed lepton singlets can freely rotate)

• Motivated by quark masses, use 2 1 assignment– Tree-level quark FCNCs involving first and second

generations are eliminated via degeneracy (up- and down-type first two generations share a common bulk mass term)

• Require additional flavon fields to break T symmetry on the IR brane

.c.h

NLyHk

Ry

yyNNk

yL

Dc,SS,

SSD

b,Dc,SSD

a,SS,T

lepSS,,Yuk

5

55

1

1

Leptons in T´

.c.hLyLyeLyHk

)Ry(L DDDe

lepl,Yuk

5551)NcNcceceLcL(kL

LLLLL

NeLlepBulk

,Yukawal,YukawaBulkKineticleptonic

.c.hyyNLHk

)Ry(L DcDb,Dc,

DcDa,Dc,

lepDc,,Yuk

551

Purely Dirac neutrino masses

Seesaw type 1 neutrino masses

Quarks in T´: 2 1 Framework

.c.h)]TQyTQy

UQy)(UQy(H)[Ry(L

TUT

UUUUYukawa

331212

331212

)BcBDcD

TcTUcUQcQQcQ(kL

BD

TUQQBulk

331212 312

Down-type Yukawa Lagrangian is exactly analogous

YukawaBulkKineticquark LLLL

Parameter Counting• Input parameters (Naïve counting)

– Charged lepton: 8 (= 4 bulk + 3 Yukawa + 1 flavon)– Neutrino: [seesaw] 6 [7] (= 2 bulk + 2 [3] Yukawa + 2

flavon) – Quark: 24 = (6 bulk + 8 Yukawa + 10 flavon)

• Actual number of independent input combinations– 16 = Lepton matrix (3) + Neutrino matrix (2) + Quark

matrices (6 + 5)

• Contrast with anarchic case– 36 [30] for leptons, 36 for quarks

• Fit parameters– Lepton and quark masses (3 + 6 = 9)– CKM matrix (+ CP violating phase) (3 + 1 = 4)– Neutrino mixing angles (3)

16 Inputs, 16 Outputs

Results –Leptons

571260664960829250400000 .c,.c,.c,.c eL

Set all leptonic Yukawas to 1. Renormalization effects negligible.

Gives me=511 keV, m=105.7 MeV, m=1.777 GeV10

For normal hierarchy

For inverted hierarchy

Normal, Dc: msol2 = 7.6370 10-5 eV2, matm

2 = 2.4031 10-3 eV2

Inverted, SS: msol2 = 7.6560 10-5 eV2, matm

2 = –2.4009 10-3 eV2

Experimental: msol2 = 7.65 10-5 eV2, matm

2 = 2.40 10-3 eV2

Fusaoka, Koide (1998), Schwetz, Tortola, Valle (2008)

0944.0,1768.0,27000.1,40000.0 ,0,0 DcDcNL cc

Normal, SS: msol2 = 7.6520 10-5 eV2, matm

2 = 2.4001 10-3 eV2

06191.0,07427.0,40000.0,40000.0 ,0,0 SSSSNL cc

115241.0,02321.0,40000.0,40000.0 ,0,0 SSSSNL cc

For normal hierarchy

Results – Quarks

508.0350.0150.0

503.0512.0503.0

3

12

BTQ

DUQ

ccc

ccc

Bulk mass parameters

Flavons and Yukawas

060.0

00.1

540.0540.0

1135.0

00230.0

181.0

448.0

00200.0

00104.000143.0

3

3

0

0

00

0

0

B

T

D

D

D

U

UU

y

y

i

i

Prediction (3 TeV) Fit bounds

mu 1.49 MeV 0.75-1.5 MeV

md 2.92 MeV 2-4 MeV

mc 0.541 GeV 0.56 ± 0.04 GeV

ms 36.6 MeV 47 ± 12 MeV

mt 134.8 GeV 136.2 ± 3.1 GeV

mb 2.41 GeV 2.4 ± 0.04 GeV

Csaki, Falkowski, Weiler (2008)Other Yukawas set to 1

Results – CKM and Jarlskog

Fusaoka, Koide (1998), Charles, et al. (CKMfitter Group) (2009)

000078.0000047.0

0011.00020.0

00057.000064.0

0011.00019.0

00053.000052.0

0022.00022.0

00044.000032.0

0022.00022.0

00052.000052.0

999146.00404.000859.0

0412.097349.02250.0

00351.02251.097433.0

|| ExpCKMV

999176.00395649.000910164.0

040450.0973485.0225147.0

003464.0225305.0974282.0

|| TheoryCKMV Corrections to quark mixings from running are small.

Perform fit at mZ

510023 .Jth5450

250 10932 .

.ex .J

Leading FCNC Estimate• Leading contribution is from dim-6

operators arising from fermion zero-modes mixing with KK modes

• Scaled to Z-coupling, leading contribution is

• Using MKK ~ 3 TeV, kR ~ 11, v = 246 GeV:– coefficient is 2.96510-6 for u-c transition– coefficient is 4.15610-6 for d-s transition

2

42

222

)0()0(2)1(2

00 4 KK

kRkji

M

ve

Rk

fff

Conclusions• RS1 + T´ provides a framework for

realistic fermion masses and mixings– Motivated by neutrino mixings and quark

masses, we choose T´ representations• This choice eliminates tree-level lepton FCNCs

and first-second generation quark FCNCs

– Can fit for all SM fermion masses, CKM matrix, and Jarlskog invariant with 16 input parameter combinations

– Allows a low first KK mass scale, testable at colliders

Group Algebra of T´

ie

eiA

/i

/i

12

12

12

2

3

1

2 S=A1, T=A2,

2´ S=A1, T=2A2,

2´´ S=A1, T=A2

1 S=1, T=1,1´ S=1, T=,1´´ S=1, T=2

10

02A

122

212

221

3

1

2

2

2

S

200

00

001

T

Feruglio, Hagedorn, Lin, Merlo (2007)

3

Neutrino Constraints• Neutrino measurements (at 2)

• (at 1)• Well-fit by Tri-Bimaximal Mixing

(TBM) Harrison, Perkins, Scott (1999)

14012023

2 50 ...sin

0440032012

2 3040 ...sin

0160011013

2 010 ...sin

213161

213161

03132

///

///

//

UTBM

TBM can be easily obtained from A4 or T´ group symmetries

Schwetz, Tortola, Valle (2008)

Ma, Rajasekeran (2001)

Leptons in T´ 3

3

2

1

~

L

L

L

L

1

1

1

~

~

~e3

0

0

1

0 ~

1113333 AS

)c,c(fy

)c,c(fy

)c,c(fy

hM

LD

LD

eLD

e

e

5

5

5

0

00

00

00

2332111 1221331 1331221

T´ contraction:

Diagonal charged lepton mass matrixbecause of T´ assignments and flavon VEVs

.c.hLyLyeLyHk

)Ry(L DDDe

lepl,Yuk

5551

Quarks in T´: The 2 1 Framework

)c,c(fy)c,c(f

)c,c(fcos)c,c(f)c,c(fi

)c,c(fsin)c,c(fi

)c,c(fi

hM

TQT

TQU

UQUU

UQUQ

UQUU

UQUQ

U

312

31212

31212

30

0000

0000

02

12

1

2 1

Quarks in T´: The 2 1 Framework

)c,c(fy)c,c(f

)c,c(fcos)c,c(f)c,c(fi

)c,c(fsin)c,c(fi

)c,c(fi

hM

BQB

BQD

DQDD

DQDQ

DQDD

DQDQ

D

312

31212

31212

30

0000

0000

02

12

1

2 1

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