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Fermion Masses and Unification
Lecture I Fermion Masses and MixingsLecture II UnificationLecture III Family Symmetry and UnificationLecture IV SU(3), GUTs and SUSY Flavour
Steve KingUniversity of Southampton
Lecture IFermion Masses and Mixings
1. The Flavour Problem and See-Saw
2. From low energy data to high energy data
3. Textures in a basis
Appendix 1 References
Appendix 2 Basis Changing
The Flavour Problem
1. Why are there three families of
quarks and leptons?
Generations of Generations of
matter matter
III III II II I I
tau
-neutrino
bbottom
ttop
muon
-neutrino
sstrange
ccharm
eelectron
ee-neutrino
Lep
tons
L
epto
ns
ddown
upu
Qua
rks
Qua
rks
Horizontal
Ver
tica
l
b
cs
ud
e
1
2
3
310110
210110
110
2101210
310
410
1210
1110
Family symmetry e.g. SU(3)
GUT symmetry e.g. SO(10)
1
t
The Flavour Problem2. Why are quark and charged
lepton masses so peculiar?
The Flavour Problem3. Why is lepton mixing so large?
c.f. small quark mixing
3
2
3 2
1
1
1ijV
Harrison, Perkins, Scott
e.g.Tri-bimaximal
Normal Inverted
The Flavour Problem5. Why are neutrino masses so small?
See-saw mechanism is most elegant solution
Type I see-saw mechanism Type II see-saw mechanism (SUSY)
R
LL
2II uLL
vm Y
M
The see-saw mechanism
L L
Heavy triplet
cRR R RM
1I TLL LR RR LRm m M m
Y
Type II Type I
P. Minkowski (1977), Gell-Mann, Glashow, Mohapatra, Ramond, Senjanovic, Slanski, Yanagida (1979/1980)
Lazarides, Magg, Mohapatra, Senjanovic, Shafi, Wetterich (1981)
See-Saw Standard Model (type I)Yukawa couplings to 2 Higgs doublets (or one with )
Insert the vevs
Rewrite in terms of L and R chiral fields, in matrix notation
12 . .E c
L LR R L LR R R RR Re m e m M H c
The See-Saw Matrix
1II TLL LL LR RR LRm m m M m
cLL L Lm
II c
c LL LR LL R T
LR RR R
m m
m M
Dirac matrix
Heavy Majorana matrix
Light Majorana matrix
Diagonalise to give effective mass
Type II contribution (ignored here)
Lepton mixing matrix VMNS
†
0 0
0 0
0 0
L R
eE EE
LR
m
V m V m
m
†L LEMNSV V V
23 13 12MNSV V V V
Neutrino mass matrix (Majorana)
1
2
3
0 0
0 0
0 0
L LTLL
m
V mV m
m
Atmospheric Reactor Solar
Oscillation phase
13 12
23 12
23 13
13 13 12 12
23 23 12 12
23 23 13 13
1 0 0 0 0
0 0 1 0 0
0 0 10 0
i i
i iMNS
i i
c s e c s e
V c s e s e c
s e c s e c
13 23 12
Defined as
Can be parametrised as
Quark mixing matrix VCKM
†
0 0
0 0
0 0
L R
uU UU
LR c
t
m
V m V m
m
†LLU DCKM VV V
23 13 12CKMV R V R
†
0 0
0 0
0 0
L R
dD DD
LR s
b
m
V m V m
m
13 13 12 12
23 23 12 12
23 23 13 13
1 0 0 0 0
0 0 1 0 0
0 0 0 0 1
i
CKMi
c s e c s
V c s s c
s c s e c
Defined as
Can be parametrised as
Phase convention independent
Neutrino Masses and Mixings
Normal Inverted
Andre de Gouvea
12
23
13
13 0.1
2.4 0.1
0.20 0.05
c.f. quark mixing angles
Renormalisation Group running
MEW
MSUSY M
1 M2 M
3M
U
1016 [GeV]102
RH neutrino masses
Parameter at MU
RG running Parameter at MEW
RGEs for gauge couplings (to
one loop accuracy)
194110 6( , , 7)ab
SM beta functions MSSM beta functions
Symmetric hierarchical matrices with 11 texture zero motivated by
| | dus
s
mV
m
12
12 22
0LR
mm
m m
This motivates the symmetric down texture at GUT scale of form
223( )| |
DLR
cbb
mV
m 313( )
| |DLR
ubb
mV
m
Hierarchical Symmetric Textures
Gatto et al
3 3
3 2 2
3 2
0
1
dLRY
¼ 0.2 is the Wolfenstein Parameter
3 3
3 2 2
3 2
0
1
0.15DLR
b
m
m
3 3
3 2 2
3 2
0
0.05
1
ULR
t
m
m
Up quarks are more hierarchical than down quarks
This suggests different expansion parameters for up and down
4 2: : : :1d s bm m m 4 2: : : :1u c tm m m
Detailed fits require numerical (order unity) coefficients
With SUSY thresholds Ross and Serna
1( ) 1, ( ) , ( ) 3
3b s d
GUT GUT GUTe
m m mM M M
m m m
Georgi-Jarlskog
Final remarks on choice of basis We have considered a particular choice of quark texture in a particular basis
But it is shown in the Appendix that all choices of quark mass matrices that lead to the same quark masses and mixing angles may be related to each other under a change of basis.
For example all quark mass matrices are equivalent to the choice
3 3
3 2 2
3 2
0
1
0.15DLR
b
m
m
3 3
3 2 2
3 2
0
0.05
1
ULR
t
m
m
0 0
0 0
0 0
dDLR CKM s
b
m
m V m
m
0 0
0 0
0 0
uULR c
t
m
m m
m
However this is only true in the Standard Model, and a given high energy theory of flavour will select a particular preferred basis. Also in the see-saw mechanism all choices of see-saw matrices are NOT equivalent.
Appendix 1 ReferencesW. De Boer hep-ph/9402266
S.Raby ICTP Lectures 1994
G.G.Ross ICTP Lectures 2001
J.C. Pati ICTP Lectures 2001
S. Barr ICTP Lectures 2003
S. Raby hep-ph/0401115
S.Raby PDB 2006
A. Ceccucci et al PDB 2006
G. Ross and M. Serna 0704.1248
D. Chung et al hep-ph/0312378
S.F. King hep-ph/0310204
Appendix 2 Basis Changing
2.1 Quark sector
2.2 Effective Majorana sector
2.3 See-saw sector
S.F. King hep-ph/0610239