Fermion Masses

G.G.Ross, Corfu, September 2005

0u uij ijM Y H

0d dij ijM Y H

, ,u i c j d i c jYukawa ij ijL Y Q u H Y Q d H

The Standard Model

DATA :

iu?

1GeV

i id

il

1210 910 610 310 310

Masses

Mixing

1 0.218 0.224 0.002 0.005

0.218 0.224 1 0.032 0.048

0.004 0.015 0.03 0.048 1CKMV

†u uL Diag RM V M V

†d dL Diag RM U M U

†CKM L LV V U

DATA :

iu?

1GeV

i id

il

1210 910 610 310 310

Masses

Mixing

1 0.218 0.224 0.002 0.005

0.218 0.224 1 0.032 0.048

0.004 0.015 0.03 0.048 1CKMV

0.79 0.88 0.48 0.61 0.2

0.27 0.49 0.45 0.71 0.52 0.82

0.28 0.5 0.51 0.65 0.57 0.81MNSV

2 13 3

1 1 16 3 2

1 1 16 3 2

0

10-1

10-2

10-3

10-4

eV

Bi-Tri Maximal Mixing …Non AbelianStructure?

Hierarchy Democracy

2 22

2 2

1 2 1/ 2

1 1 21

a aa

a a

Data Theory ( )ijY

†... - CKM L Lunderconstrained need mixing angl V V Ues but on mea dly sure

u dC

Perturbative analysis (hierarchical basis)

M MD M m #

(small) off diagonal

2 , ,( )u d u dL L DV MM V M V V M M

2( )( )D D L L DM M m M M m V V M

V ij Om V ii 1 Om2 #

22 | | ( ) 0xi i ij ij j i ji ji

j j

M M m m M M m V

2 2( )i ij ik k i ki kj ik jk ij jk k

M V m M M m V m m V M

V ij m ijMj m ji

Mi

Mi2 Mj

2

m ikMk mki MimkjMj m jk

Mk

Mi2 Mj

2Mk2 Mj

2

m ikm jk

Mi2 Mj

2 Om3

# V12CKM

m12d

Ms m 12

u

Mc m 23

dm 13u

MbM t m 21

d

Ms

Md

Ms . . .

V21CKM

m12d

Ms m 12

u

Mc m 13

dm 23u

MbM t . . . .

#

Diagonal

Off diagonal

Quark Textures

4 3 3

2 2?

1? ?

d

b

M

m

dm usV

cbV

sm

ubV

23

33 33

12

2

32

2

2 22

13 1

3

1

3

3 33

23

12

13

s

b

d

s

d

b

mm m

mm m

mm

mm

mm

m

m

mmm m

V

V

V

Hierarchical :11 12 13

22 2321

31 3 32 3

m m

m mm

m

m

M

mm

Expansion parameter 0.15

Quark Textures

4 3 3

2 23

11

d

b

M

m

dm usV

cbV

sm

ubV

23

33 33

12

2

32

2

2 22

13 1

3

1

3

3 33

23

12

13

s

b

d

s

d

b

mm m

mm m

mm

mm

mm

m

m

mmm m

V

V

V

Hierarchical :11 12 13

22 2321

31 3 32 3

m m

m mm

m

m

M

mm

Det[ ]M

Expansion parameter 0.15

d

b

3 3

3 2 2

20

320 2

Mm

0 1.5 0.4

1.5 1.3 0.15

0.4 1.3 1

i

i

e

e

Symmetric fit

Asymmetric fit

d

b

3

3 2Mm

0 1.7 0

1.7 0 5

0 0.3 1

u

t

3

3Mm

0 2 0

0 2

0 0.6 1

( )ij ijTheory YDat Ma

u

t

3 3

3 2 2

3 2

Mm

0

0.05

1

?

?

? ?

RobertsRomaninoGGRVelasco Sevilla

203 3

3 2

0

2

3 22

0 1.5 0.4

1.5 1.3

0.4 1.3 1

d

b

i

Mm

i

e

e

1id uus

s c

m mV e

m m

13 2 1

23

13 2

23

( )ub u

cb c

td d

ts s

V m s iV m s

V m s iV m s

e

e

Texture zero

id uus

s c

m mV e

m m

| (0.216 0.2140.217 0.2 ) (0.0722 0.076) |. . iec f 00.213 0.223, 90

Fritzsch,WeinbergRoberts, Romanino,GGR,Velasco

CP SM phase/

3

23

3

2

? ? 1

0d

b

M

m

5

GATTO, SARTORI, TONINsymmetric

d

b

3 3

3 2 2

20

320 2

Mm

0 1.5 0.4

1.5 1.3 0.15

0.4 1.3 1

i

i

e

e

Symmetric fit

Asymmetric fit

d

b

3

3 2Mm

0 1.7 0

1.7 0 5

0 0.3 1

u

t

3

3Mm

0 2 0

0 2

0 0.6 1

u

t

3 3

3 2 2

3 2

Mm

0

0.05

1

?

?

? ?

RobertsRomaninoGGRVelasco Sevilla

Origin of structure?quarkM

Origin of structure?quarkM

, (specific string calc ulations + IRFP OK ) t bh gThird generation heavy

(?) b th h h (GUT?)

Pendleton, RossHillKobayashi, Kubo, Mondragon, Zoupanos

32 2 23

ln( ) 98

2

thg

t

dg h

dt

2 23

2( )

9th g

Infra red fixed point

3

3( )3 (0)

( )

(1 ( ) )

t

i

i

QFP Y

tt B

Y

sin205 210tm GeV

spontaneously broken family symmetry - mass matrix elements ordered : by

n

L RH M

spatial separ n atio

Yijk hqu n H2

DM , a a,ijk

dn e

Aijk n

2 e 2 i ijkn. #

Froggatt-Nielsen

Origin of structurequarkM

, (specific string calc ulations + IRFP OK ) t bh gThird generation heavy

(?) b th h h (GUT?)

0.2

Hierarchy :

… dimension of operator

… order of radiative corrections2 2( /16 )nh

Textures and flavour models

?u GUTM M

?d lGUTM M

(2) (2) ?L RSU SU(2) (2) ?L RSU SU

Family?Family?

Symmetries Coherent picture of quark and lepton masses and mixing?

Symmetries and textures

Hierarchical structure strongly suggests a broken symmetry

0 0 0

0 0 0

0 0 1

2

0 0 0

0 0

0 0 1

a

0

2(1),OM

a

MESSENGER SECTOR?

i X

j

H

Xij

H

Mm

FAMILY SYMMETRY? Froggatt, Nielsen

6Abelian, Non-Abelian (U(3))

Abelian Family symmetry (1)U

X XM M

1 -1

3 4

3 2

4

0

1

R

R

R

d

s

b

L L Ld s bdR L

b

M

m

-3 2 1

-3 2 1

8( )O

Ibanez GGR

3(40.4 1.8)10 d

s

mcb mV O

CKM

2,HiggsQ

d

b

3 3

3 2 2

20

320 2

Mm

0 1.5 0.4

1.5 1.3 0.15

0.4 1.3 1

i

i

e

e

Symmetric fit

Asymmetric fit

d

b

3

3 2Mm

0 1.7 0

1.7 0 5

0 0.3 1

u

t

3

3Mm

0 2 0

0 2

0 0.6 1

( )ij ijTheory YDat Ma

u

t

3 3

3 2 2

3 2

Mm

0

0.05

1

?

?

? ?

Is one favoured?

4 3 3

2

23

2

2

1

d

b

M

m

( , 0 0)dm S Q

0 0

( )

S Q

A

( )

CP

A

?

SUGR

Gauge mediation

bad for unific

A

ation

Vives, GGR

( , )..

i j ci j aMP Q q H

Cannot align Yukawa and A terms

(3 ( , )) *c c

ij ij a ij ai j i jA Y Q q H i j P Y Q q H

Symmetric structure?

With similar structure for charged leptons Neutrino mixing not related to large RH mixing

Favours near symmetric structure

, ,u c t, ,u c t

, ,..g W

, ,..g W

d

d

s

s

Probes RH angles

e , ,e

W W

2 2

2 2

312 10 ,c u

c u

m m

m m

q

2 2

2 2

312 10e

e

m m

m

l

m

,

0.973 0.975 0.217 0.222 0.0023 0.004

| | 0.21 0.24 0.9 1.2 0.038 0.041

0.006 0.01 0.026 0.04 0.84 1.14

( ) (1.8 5.3) ( ) (0.9 2.9)

( ) (35 100) ( ) (0.53

( ) (2.8 3.0)

i ju d CKM

d Z u Z

s Z c Z

b Z

V

M M MeV M M MeV

M M MeV M M

M M GeV

2

/

1 ( / )

0.68)

( ) 168 180

22.7 0.08

/ 0.553 0.043

/ 8.23 1.5

s d

u d

t Z

m m

m m

u d

c s

GeV

M M GeV

Q

m m

m m

me 0.51099907 0.00000015MeV

m 105.658389 0.000034MeV

m 1777.05 0.27MeV

Quark masses and mixing angles

Lepton masses and mixing angles

0.79 0.88 0.48 0.61 0.2

0.27 0.49 0.45 0.71 0.52 0.82

0.28 0.5 0.51 0.65 0.57 0.81MNSV

?Q L

3 3

2 23

2 2

?

?

1

0

? ?

lM

m

( ) 1bX

mM

m

( ) ( ) |X

l dMDet M Det M

Extension to charged leptons – GUT?

?

3 3

2 23

2 2

0

?

3 ?

?

?

1

lM

m

( ) 1bX

mM

m

( ) ( ) |X

l dMDet M Det M

1( )

3s

X

mM

m

Georgi Jarlskog

( ) 3dX

e

mM

m

Extension to charged leptons – GUT?

Possible with “see-saw”

Extension to neutrinos???

???dDM M

b e a

i1R

j

1iDM

1D jM

1M

TD DM MM M

d

b

3 3

3 2 2

3

20

0 22

Mm

0 1.5 0.4

1.5 1.3

0.4 1.3 1

i

i

e

e

Majorana Mass1

2 1 2 3

3

M

M

M M M M

M

M

Dirac Mass

100

10-1

10

10-2

10-3

10-4

eV

King, GGR

CP violation SUSY CP problem - 1/ 2( ),A Arg Am -21/ 2( ) ? <10B Arg Bm

conserved by compactification

- spontaneously broken on family symmetry breaking

CP

( , )..

i j ci j aMP Q q H

Cannot align Yukawa and A terms

(3 ( , )) *c c

ij ij a ij ai j i jA Y Q q H i j P Y Q q H

CP violation from Yukawa sector4 3 3

2

23

2

2

1

d

b

M

m

( , 0 0)dm S Q

0 0

( )

S Q

A

( )

CP

A

OK with suitable family symmetry

In string theory CP is a discrete gauge symmetry : Dine, Leigh, MacIntyre

Vives, GGR

, close to experimental limitse EDM

Fermion Masses II

iu?

1GeV

i id

il

1210 910 610 310 310

Masses

Mixing

1 0.218 0.224 0.002 0.005

0.218 0.224 1 0.032 0.048

0.004 0.015 0.03 0.048 1CKMV

0.79 0.88 0.48 0.61 0.2

0.27 0.49 0.45 0.71 0.52 0.82

0.28 0.5 0.51 0.65 0.57 0.81MNSV

2 13 3

1 1 16 3 2

1 1 16 3 2

0

10-1

10-2

10-3

10-4

eV

Bi-Tri Maximal Mixing …Non AbelianStructure?

Textures and flavour models

?u GUTM M

?d lGUTM M

Symmetries Coherent picture of quark and lepton masses and mixing?

6Family ( (3))U

(5), (10)SU SO

Abelian Family symmetry (1)U

X XM M

1 -1

3 4

3 2

4

0

1

R

R

R

d

s

b

L L Ld s bdR L

b

M

m

-3 2 1

-3 2 1

8( )O

CKM

2,HiggsQ

D-term problems for FCNC

† † 22 † † 2 2 † † 2 †| | ( )i i i i i i i ii i i

g Q q q g Q q q m m Qq q

Family dependent soft mass

Murayama

, ,u c t, ,u c t

, ,..g W

, ,..g W

d

d

s

s

Probes RH angles

e , ,e

W W

2 2

2 2

312 10 ,c u

c u

m m

m m

q

2 2

2 2

312 10e

e

m m

m

l

m

Discrete family symmetry

Gauge mediated breaking22...( ) smallm m

Gravity mediation 22...( ) and common radiative correctionsm m

Abelian Family symmetry (1)U

X XM M

1 -1

3 4

3 2

4

0

1

R

R

R

d

s

b

L L Ld s bdR L

b

M

m

-3 2 1

-3 2 1

8( )O

Ibanez GGR

3(40.4 1.8)10 s

b

mcb mV O

CKM

2,HiggsQ

Abelian Family symmetry (1)U

X XM M

1 -1

12

12

3 4

3 2

4

0

1

2R

R

R

d

s

b

L L Ld s bdR L

b

M

m

-3 2 1

-3 2 1

8( )O

Ibanez GGR

3(40.4 1.8)10 s

b

mcb mV O

CKM

2,HiggsQ

Need special choice of O(1) coefficients

DATA :

iu?

1GeV

i id

il

1210 910 610 310 310

Masses

Mixing

1 0.218 0.224 0.002 0.005

0.218 0.224 1 0.032 0.048

0.004 0.015 0.03 0.048 1CKMV

0.79 0.88 0.48 0.61 0.2

0.27 0.49 0.45 0.71 0.52 0.82

0.28 0.5 0.51 0.65 0.57 0.81MNSV

2 13 3

1 1 16 3 2

1 1 16 3 2

0

10-1

10-2

10-3

10-4

eV

Bi-Tri Maximal Mixing …Non AbelianStructure?

?u GUTM M

?d lGUTM M

Symmetries Coherent picture of quark and lepton masses and mixing?

(10) (3) familySO SU G

6Family ( (3))U

(10)SO

Simple(?) flavourGroup restricting Yukawa couplings

Non-Abelian family symmetry

(10) (3) familySO SU G

3 23 123 45, , ,i i i H

, No mass while SU(3) unbrok3 e16, nci i

Spontaneous symmetry breaking

(1,3) (1,3) (1,3) 45,1

2

0 0 1

0 1 1

1 1 1

M M M

c.f. Georgi-Jarskog

(4) (2) (2)L RSU SU SU

3 3 23 23 45 23 123 123 232 3 2 2

1 1 1 1i j c i j c i j c i j cY i j i j i j i jP H HH H H

M M M M

only terms allowed by G

I de Medeiros, GGRKing, Vives, Velasco Sevilla…

(1) (1) 'G R U U

23 23 45 23 123 123 233 2 23 32

1 1 1 1i i j c i j c i j cY i j i j i j

j ci jP HH H H

M M MH

M

D

3

Mm

0 0 0

0 0 0

0 0 1

3

0

0

1

Dirac mass structure

23 23 453 3 23 123 123 232 2 23

1 1 11i j c i j c i j cY

i j cii j i j i jjP H H H

M MHH

MM

D

3

d

e2 2

2

Mm

a

a 3

0 0 0 1

0

a0 0

1

a a

a

45 3( ) 2 RH B L T

123

1

1

1

Dirac mass structure

. .c f Georgi Jarlskog

3 3 23 23 45 23 123 123 232 22 3

1 11 1 i j c i j ci j

i j c i j cY i j i j i jP H H H

M MHH

M M

D

3

3 3

3 2 3 2 3

3 2 3

Mm

0

1

a a

a

123

1

1

1

23

0

1

1

Dirac mass structure

i X

j

H

d l

u

0.15

0.05

0.02

Messenger masses break (10)SO

D

3

3 31

3 3 32 1 2 3

3 33

Mm

0

1M

M

M M M M

M

M

6 62 2 2

3 2 1

1( ) ( ) ( )eL

M M M

6

2

( )( )eM

Neutrino masses

3 3 23 23 45 23 123 123 232 3 2 2

1 1 1 1i j c i j c i j c i j cY i j i j i j i jP H HH H H

M M M M

1M

TD DM MM M

Structure determined from symmetries too

Actually cancelled by small off diagonalMajorana terms

b e a

Bi-Tri Maximal Mixing

Vacuum alignment 3 23 123, ,i i i

2

0 0 1

0 1 1

1 1 1

00

1 2

i

13

i

1 2

45 33

SU(3) SU(2) ..

SU(3) SU

(2)' ..

q , l

' '

i

???

Discrete (Non Abelian) family symmetry?

2. . ( ) :Ze g i e d d

M d a b

d b a

1

2

3

0 0

0 0

0 0

T

m

U M U m

m

cos sin 0

sin / 2 cos / 2 1/ 2

sin / 2 cos / 2 1/ 2

U

Bi-maximal

23 12 13/ 4, , 0

Vacuum alignment Altarelli,Feruglio

21 2 3 3 0 4 0' ' ' ' '

3 i i

gP g g

1 2 32 3i ii

M gP

1 2 3 2 1 3 3 1 21 2 3

0; 0; 0P P P

M g M g M g

(v,v,v), vM

g

4A

23 4

02 3 1 3 1 2

1 2 3

' ' 0; ' ' 0; ' ' 0; ''

'' ' i i

P P P Pg g gg g

3

4

2' (v',0,0), ( ' ' ) undetermined at tree levelgi igu

( , ), ( 3, )i i i i i iQ u d L l

1 1 1 2 2 2 3 3 3, , , , , ,1, 1', 1''c c c c c c c c cu d l u d l u d l

4 irreducible r 1,1',1eps '', 3Ma;Altarelli,Feruglio

4A

Grand Unified

Bi-Tri-maximal mixing

X

(10)SO G 3 3 2 (3)Z Z Z SU

1 2 3 Allows symmetric invariant - vacuum alignment

(10) (3) familySO SU G since only small high dimension corrections

Grand Unified

Bi-Tri-maximal mixing

King,GGR

Summary

Due to the see-saw mechanism, the quark, charged lepton and neutrino masses and mixing angles can be consistent with a similar structure for their Dirac mass matrices.

Hints at an underlying (spontaneously broken) family symmetry? SO(10)XSU(3)?

Extraction of Yukawa couplings crucial to understanding fermion structure

Bounded off diagonal terms (anti)symmetric, hermitian SUSY data

Texture and texture zero hints at underlying structure

Family symmetry ? GUT symmetry?

22, , , , ,R R R Rim q u d l e

232 2

2 2 22 2 2 2 2 2 21 13 23 ,, , , '

R RL

i ii i q l d uMM M

m m m m m

gives a new solution to the SUSY FCNC and CP problems(3) familySU

Summary

Due to the see-saw mechanism, the quark, charged lepton and neutrino masses and mixing angles can be consistent with a similar structure for their Dirac mass matrices.

Hints at an underlying (spontaneously broken) family symmetry? SO(10)XSU(3)?

) (3 familySU gives a new solution to the SUSY FCNC and CP problems

Extraction of Yukawa couplings crucial to understanding fermion structure

Bounded off diagonal terms (anti)symmetric, hermitian SUSY data

Texture and texture zero hints at underlying structure

Family symmetry ? GUT symmetry?

A discrete non-Abelian subgroup can give vacuum alignment neededFor bi-tri-maximal mixing

0c ci i i

Q u Hdm m m m m

3 0c ci i id HQ u

m m m m m

3 3(1 0.2)Q Qim m

3 family symmetry brea king

Ramage,GGR

Field SU3 f SU4PS SU2L SU2R R U1 U1

3 4 2 1 1 0 0 c 3 4 1 2 1 0 0 l 3 4 1 2 0 0 0 l 3 4 1 2 0 0 0 H 1 1 2 2 0 4 4 H45 1 15 1 3 0 2 2 3 3 1 1 1 0 2 3 3 3 1 1 3 1 0 2 2 2 3 1 1 1 0 1 1 2 3 1 1 1 0 1 1 23 3 1 1 1 0 4 3 23 3 1 1 1 0 1 1 123 3 1 1 1 0 0 1 123 3 1 1 1 0 3 3

(4) (2) (2) (3) (1) (1) 'L R FSU SU SU SU R U U

PM 1M

i ic j j

c #

1M5

23i i

c 23j j

c k 123 k l 3 l 1M5

i ic 23

j jc k 123 k 23

l 3 l #

1M5

123i i

c 123j j

c k 23 k l 3 l 1M5

i ic 123

j jc 123

k 23 k l 3 l 1M5

i ic 123

j jc k 23 k 123

l 3 l #

Majorana Mass Matrix

MNR M3

1 R d

4 d5 1

R d

4 d5 3

R d

4 d5

1 R d

4 d5 2

R d

4 d4 4

R d

4 d4

3 R d

4 d5 4

R d

4 d4 1

#

23, ,

, ,R

R

u du dM

0c ci i i

Q u Hdm m m m m

3 0c ci i id HQ u

m m m m m

3 3(1 0.2)Q Qim m

3 family symmetry brea king

Ramage,GGR

( , ), ( 3, )i i i i i iQ u d L l

1 1 1 2 2 2 3 3 3, , , , , ,1, 1', 1''c c c c c c c c cu d l u d l u d l

2 / 3 / 3 / 3

/ 3 2 / 3 / 3

/ 3 / 3 2 / 3

a d d d

M d d a d

d a d d

1

1

1

3 v 0 0

0 3 v 0

0 0 3 v

cLl

f

M f

f

4 irreducible r 1,1',1eps '', 3

3 3 1 1' 1'' 3 3

4. . ( )e g i Ai (Non-Abelian discrete subgroup (3) of (12) (3))familySO SU

1/ 2 1/ 2W WI Ii iH H

1 1 1 2 2 3 3 1

22 1 1 2 2 3 3 2

23 1 1 2 2 3 3 3

. .

cY

c

c

L f L L L l

f L L L l

f L L L l h c

1 1 2 2 3 3

2 3 3 2

' /

( ) / . .YL a L L L L L L

d L L L L h c

1WI

Charged leptons Neutrinos

(v,v,v)

3 3 1' 11 3'' 1' WI

i

' (v ',0,0)

Ma;Altarelli,Feruglio

3

( , ), ( 3, )i i i i i iQ u d L l

1 1 1 2 2 2 3 3 3, , , , , ,1, 1', 1''c c c c c c c c cu d l u d l u d l

1

1

1

3 v 0 0

0 3 v 0

0 0 3 v

cLl

f

M f

f

2 13 3

1 1 16 3 2

1 1 16 3 2

0

MNSU

1

2

3

0 0

0 0

0 0

m

M m

m

Bi-tri maximal

Harrison, Scott-Ma

4 irreducible r 1,1',1eps '', 3

3 3 1 1' 1'' 3 3

4. . ( )e g i Ai (Non-Abelian discrete subgroup (3) of (12) (3))familySO SU

1 1 2 2 3 3

2 3 3 2

' /

( ) / . .YL a L L L L L L

d L L L L h c

1WI

Charged leptons Neutrinos 3 3 1' 11 3'' 1WI

i

(v ',0,0)

Ma;Altarelli,Feruglio

1 1 1 2 2 3 3 1

22 1 1 2 2 3 3 2

23 1 1 2 2 3 3 3

. .

cY

c

c

L f L L L l

f L L L l

f L L L l h c

1/ 2 1/ 2W WI Ii iH H

(v,v,v) 3

Phenomenological implications :

13 40 if 0.m

2 23 @0.94 sin 2 1, 0.5 tan 0.520.16eU

?Q L , , ,1 1 2 1 3 1

, , ,1 2 2 2 3 2,

, , ,1 3 2 3 3 3

v v v

v v v

v v vc

u d u d u d

u d u d u d

Qu du d u d u d

f f f

M f f f

f f f

† 2 †; , ; ;

2

1 1 11

13

1L u d CKM L u L dU U U U I

100

10-1

10

10-2

10-3

10-4

eVDegenerate Normal Inverted

Degenerate, normal andinverted spectra possible

No connection between quark,charged leptonand neutrino masses