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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