Transcript Document

Fermion Masses and
Unification
Steve King
University of Southampton
Lecture 4
Family Symmetry and Unification II
1.
2.
3.
4.
5.
SO(3) or A4 family symmetry and unification
SU(3) or 27 family Symmetry and unification
Quark-lepton connections
SUSY flavour and SU(3)
Do we need a family symmetry?
Appendix Finite Groups
Realistic SO(3)£ Pati-Salam Model
SFK, Malinsky
Model also works with SO(3) replaced by its A4 subgroup
u
FL  (3, 4, 2,1)  
d
u
d
u
d
u
i
i
FR  (1, 4,1, 2)  
d
u
d
u
d
 
i
e   L
 
i
e   R
Flavon vacuum alignment
0
3   0 
1
 
0
23   1 
 1 
 
 1
123   1
 1
 
Ma; Altarelli, Feruglio; Varzeilas, Ross, SFK, Malinsky
Comparison of SO(3) and A4
Symmetry group of
the tetrahedron
A4
Discrete set of
possible vacua
Dirac Operators:
Majorana Operators
Majorana Neutrino matrix:
M RR
 2
  23


 0
 0

0
 2
123

0
0
  H 2
0
M
1 
Dirac Neutrino matrix:
.
.
.
.
•CSD in neutrino sector due to vacuum alignment of flavons
• m3 » m2 » 1/ and m1» 1 is much smaller since  ¿ 1
•See-saw mechanism naturally gives m2» m3 since the  cancel
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)
 i  Qi , Li ,Ui c , Di c , Ei c , Ni c
3 with the Higgs being singlets H» 1
This “explains” why there are three families c.f. three quark colours in SU(3)c
The family symmetry is spontanously broken by antitriplet flavons
i  3
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:
 0 0 0  3  0
0 0 0


0 0 0


0 0 0
0 0 0


0 0 1


23  0  0
0
0  2

0  2

0
 2 
1 
Frogatt-Nielsen in SU(3) family symmetry
In SU(3) with i=3 and H=1 all tree-level Yukawa couplings Hi j are forbidden.
SU (3)
Ytree
 level
In SU(3) with flavons
 0 0 0
  0 0 0 
 0 0 0


 i  3 the lowest order Yukawa operators allowed are:
1 i j
  H i j
2
M
For example suppose we consider a flavon
generates a (3,3) Yukawa coupling
3i with VEV 3i
 (0, 0,1)V3 then this
 0 0 0 2
1 i j
V3


  H i j   0 0 0  2
2 3 3
M
0 0 1 M


Note that we label the flavon 3 with a subscript 3 which denotes the
direction of its VEV in the i=3 direction.
i
Next suppose we consider a flavon 23 with VEV 23i  (0,1,1)V23 then this
generates (2,3) block Yukawa couplings
i
 0 0 0 2
1
 0 1 1  V23
i
j


H



23 23
i j

 M2
M2
0 1 1


Writing
V32
  2
M
2
3
and
 0 0 0  3  0
0 0 0


0 0 0


If we have 3 ¼ 1
and we write 23 = 
then this resembles
the desired texture
V232
  2 these flavons generate Yukawa couplings
M
2
23
 0 0 0  23  0
0 0 0 


0 0  2 
3 

 0 3 3


Y   3  2  2 
 3  2 1 


0 0
0  2
23

0  2
23


 232 
 32   232 
0
To complete the texture there are
good motivations from neutrino
physics for introducing another
flavon <123>/ (1,1,1)
Varzielas,SFK,Ross
Realistic SU(3)£ SO(10) Model
Model also
works with
SU(3) replaced
by its 27
subgroup

Yukawa Operators

Majorana Operators
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
 3u   3d   3e  1
Quark-Lepton Connections
.
.
.
12e 
From above we see that
.
e
12d .
12 
3
c
3
Charged Lepton Corrections and  sum rule
SFK,Antusch;
Masina,….
Assume I: charged lepton mixing angles are small
 i 23
s23e
 i13
 13e
 i12

13e
s12
12e
 s23e

 s12
12e

 i 23

 i13
13

 i12
12
 23E c23e
VMNS  V V
EL
E
 i 23
E
E   i13
13 23
 c e
E

 23
)
E   i (12
12 23
 s e

E
13
)
E   i (  23
13 12 23
 c s e
E    i1E2
12 23 12
 c c e
Assume II: all 13 angles are very small
In a given model we can
12E

 13 
E
predict 12 and 12.
2
 12  12 
1E2
2
cos 
Note the sum rule
12  12  13 cos 
L †
The Neutrino Sum Rule
12  13 cos   12

Measured by experiment –
how well can this
combination be
determined?
Predicted by theory
e.g.1. bi-maximal
predicts 45o
e.g.2. tri-bimaximal
predicts 35.26o
Tri-bimaximal sum rule
12  35.26  13 cos
o
Bands show 3  error for
a neutrino factory
determination of 13cos 
Current 3
.
.
12  33  5
 23  45  10
13  13
A Prediction
Antusch,
Huber,
SFK,
Schwetz
13 
12E

C
 3 ,
2 3 2
 sin 2 213  102
Solving the SUSY Flavour Problem with
SU(3) Family Symmetry
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
This was discussed in SUGRA by Ross, Velasco-Sevilla and
Vives – here we re-examine this from a bottom-up point of
view using only symmetry properties of SU(3) or 27
Soft scalar mass operators in SU(3)
Using flavon
VEVs previously
Recall Yukawa matrices, ignoring phases:
Where
 3u   3d   3e  1
This predicts almost universal squark and slepton masses:
Do we need a family symmetry?
One family of “messengers” dominates
Three families of quarks and leptons
MR
ML
Suppose
U
YLR
MQ
0
0
  0 abU
 0 cb
U

U 
MQ
MU
, D 
MD
0 
 
ad U 
M
1  Q
MQ
MD
Ferretti, SFK, Romanino;
Barr
MU
YLRD
then in a particular basis
 0 ef  D
  0 gf  D
 0 hf 
D

, a, b, c, d , e, f , g , h, k 1
ek D 
 
gk D 
M
1  Q
mu  md  0
mc
mt
ms
mb
tan  50
ms
 Vcb
mb
Accidental sym
1
Not bad! But…
tan C
1
Need broken
Pati-Salam…
Conclusion: partial success, but little predictive power esp. in neutrino sector
Appendix Finite Groups
Ma 0705.0327