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Family Symmetry Solution to the SUSY Flavour and CP
Problems
Plan of talk:Plan of talk:
I.I. Family SymmetryFamily Symmetry
II.II. Solving SUSY Flavour and CP Solving SUSY Flavour and CP ProblemsProblems
Work with and Michal Malinsky
3 3
3 2 2
3 2
0
1
uY
• Universal form for mass matrices, with Georgi-Jarlskog factors
• Texture zero in 11 position
Fermion mass spectrum well described by Symmetric Yukawa textures
3 3
3 2 2
3 2
0
1
dY
3 3
3 2 2
3 2
0
3 3
3 1
eY
0.05, 0.15
1( ) , ( ) 3
3s d
GUT GUTe
m mM M
m m
Introduction to Family Symmetry
G.Ross et al
•To account for the fermion mass hierarchies we introduce a spontaneously broken family symmetry
•It must be spontaneously broken since we do not observe massless gauge bosons which mediate family transitions
•The Higgs which break family symmetry are called flavons
•The flavon VEVs introduce an expansion parameter = < >/M where M is a high energy mass scale
•The idea is to use the expansion parameter to derive fermion textures by the Froggatt-Nielsen mechanism (see later)
In SM the largest family symmetry possible is the symmetry of the kinetic terms
36
1
, , , , , , (3)c c c ci i
i
D Q L U D E N U
In SO(10) , = 16, so the family largest symmetry is U(3)
Candidate continuous symmetries are U(1), SU(2), SU(3) etc
If these are gauged and broken at high energies then no direct low energy signatures
U(1)
SU(2)
SU(3) SO(3)
S(3)Nothing
(3) (3)L RO O
(3) (3)L RS S27
4 12A
Candidate Family Symmetries
Simplest example is U(1) family symmetry spontaneously broken by a flavon vev
For D-flatness we use a pair of flavons with opposite U(1) charges
0 ( ) ( )Q Q
Example: U(1) charges as Q (3 )=0, Q (2 )=1, Q (1 )=3, Q(H)=0, Q( )=-1,Q()=1
Then at tree level the only allowed Yukawa coupling is H 3 3 !
0 0 0
0 0 0
0 0 1
Y
The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon insertions:
2 3 4 6
2 3 2 2 1 3 1 2 1 1H H H H HM M M M M
M
When the flavon gets its VEV it generates small effective Yukawa couplings in terms
of the expansion parameter
6 4 3
4 2
3 1
Y
1 0 1 0 0
Froggatt-Nielsen Mechanism
What is the origin of the higher order operators?
To answer this Froggat and Nielsen took their inspiration from the see-saw mechanism
2
R
L L
H
M
2 3HM
Where are heavy fermion messengers c.f. heavy RH neutrinos
L LR R
M
H H
RM
2M
H
3
M
There may be Higgs messengers or fermion messengers
2
M
0H
30
1
0
2 3
1 0H
1H1H HM
Fermion messengers may be SU(2)L doublets or singlets
2QQ
M
0H
3cU0
Q
1
0Q 2Q
cU
M
0H
3cU1
cU
1
1cU
Gauged SU(3) family symmetry
Now suppose that the fermions are triplets of SU(3) i = 3
i.e. each SM multiplet transforms as a triplet under a gauged SU(3)
with the Higgs being singlets H» 1, , , , , 3c c c ci i i i i i iQ L U D E N
This “explains” why there are three families c.f. three quark colours in SU(3)c
0 0 0
0 0 0
0 0 1
2 2
2
0 0 0
0
0 1
23 0 0 0 0
0 0 0
0 0 0
3 0
The family symmetry is spontanously broken by antitriplet flavons
Unlike the U(1) case, the flavon VEVs can have non-trivial vacuum alignments.
We shall need flavons with vacuum alignments:
3>/ (0,0,1) and <23>/ (0,1,1) in family space (up to phases)
so that we generate the desired Yukawa textures from Froggatt-Nielsen:
3i
Frogatt-Nielsen in SU(3) family symmetry
(3)
0 0 0
0 0 0
0 0 0
SUtree levelY
In SU(3) with i=3 and H=1 all tree-level Yukawa couplings Hi j are forbidden.
2
1 i ji jH
M
In SU(3) with flavons the lowest order Yukawa operators allowed are:
3i
For example suppose we consider a flavon with VEV then this generates a (3,3) Yukawa coupling
23
3 32 2
0 0 01
0 0 0
0 0 1
i ji j
VH
M M
Note that we label the flavon with a subscript 3 which denotes the direction of its VEV in the i=3 direction.
3i
3 3(0,0,1)i V 3i
Next suppose we consider a flavon with VEV then this generates (2,3) block Yukawa couplings
223
23 232 2
0 0 01
0 1 1
0 1 1
i ji j
VH
M M
23 23(0,1,1)i V 23i
23
0 0 0
0 0 0
0 0
2 223 23
2 2 223 3 23
0 0 0
0
0
23 0 0 0 0
0 0 0
0 0 0
3 0
Writing and these flavons generate Yukawa couplings
22 33 2
V
M
22 2323 2
V
M
If we have 3 ¼ 1 and we write 23 = then this resembles the desired texture
3 3
3 2 2
3 2
0
1
Y
To complete the texture there are good motivations from neutrino physics for introducing another flavon <123>/ (1,1,1)
The motivation for 123 from tri-bimaximal neutrino mixing
3
3 3
3 3
0 0
0
1LRY
3
0
0
1
3
23
0
1
1
3123
1
1
1
For tri-bimaximal neutrino mixing we
need
A Realistic SU(3)£ SO(10) Model
Yukawa Operators Majorana Operators
Varzielas,SFK,Ross
Inserting flavon VEVs gives Yukawa couplings
After vacuum alignment the flavon VEVs are
Writing
Yukawa matrices become:
Assume messenger mass scales Mf satisfy
Then write
Yukawa matrices become, ignoring phases:
Where
• In SUSY we want to understand not only the origin of Yukawa couplings
• But also the soft masses
The SUSY Flavour Problem
See-saw parts
The Super CKM Basis
Squark superfields
Quark mass eigenstates
Quark mass eigenvalues
Super CKM basis of the squarks(Rule: do unto squarks as we do unto quarks)
.
Squark mass matrices in the SCKM basis
Flavour changing is contained in off-diagonal elements of
Define parameters as ratios of off-diagonal elements to diagonal elements in the SCKM basis ij = m2
ij/m2diag
Typical upper bounds on
Clearly off-diagonal elements 12 must be very small
Quarks
Leptons
An old observation: SU(3) family symmetry predicts universal soft mass matrices in the symmetry limit
However Yukawa matrices and trilinear soft masses vanish in the SU(3) symmetry limit
So we must consider the real world where SU(3) is broken by flavons
Solving the SUSY Flavour Problem with SU(3) Family Symmetry
Soft scalar mass operators in SU(3)
Using flavon VEVs previously
Recall Yukawa matrices, ignoring phases:
Where
Under the same assumptions we predict:
In the SCKM basis we find:
Yielding small
parameters
The SUSY CP Problem
• Neutron EDM dn<4.3x10-27e cm
• Electron EDM de<6.3x10-26e cm
Abel, Khalil,Lebedev
Why are SUSY phases so
small?
g Rd
11
SCKMdm A
Ld
Ld
Rd
In the universal case 0d dij ijA AY
0
dSCKMd
ij s
b
y
A A y
y
11 511 10
SCKMd
LRd
m Am
m
0
210A
• Postulate CP conservation (e.g. real) with CP is spontaneously broken by flavon vevs
• This is natural since in the SU(3) limit the Yukawas and trilinears are zero in any case
• So to study CP violation we must consider SU(3) breaking effects in the trilinear soft masses as we did for the scalar soft masses
u dH H
Ross,Vives
Solving the SUSY CP Problem with SU(3) Family Symmetry
Soft trilinear operators in SU(3)
Using flavon VEVs previously
N.B parameters ci
f and i
f are real
0A
0A
Compare the trilinears to the Yukawas
They only differ in the O(1) real dimensionless coefficients
0A
Since we are interested in the (1,1) element we focus on the upper 2x2
blocks
The essential point is that , , , are real parameters and phases only appear in the (2,2,) element (due to SU(3) flavons)
Thus the imaginary part of Ad11 in the SCKM basis will be doubly
Cabibbo suppressed
0A
To go to SCKM we first diagonalise Yd
1†
2
00
0dd d
L Ris
yV V
ye
Then perform the same transformation on Ad
11ImSCKMdA
31 511 0 1 0 1 0 12
2
Im sin sin sind
SCKMddd d
A A A y A
c.f. universal case 11 0 1Im sinSCKMd
dA y A Extra suppression factor of 0.15
Conclusions• SU(3) gauged family symmetry, when spontaneously
broken by particular flavon vevs, provides an explanation of tri-bimaximal neutrino mixing
• When combined with SUSY it gives approx. universal squark and slepton masses, suppressing SUSY FCNCs
• It also suppresses SUSY contributions to EDMs by an extra order of magnitude compared to mSUGRA or CMSSM remaining phase must be <0.1
• Maybe SUGRA can help with this remaining 10% tuning
problem – work in progress