Transcript Document

Fermion Masses and
Unification
Steve King
University of Southampton
Lecture III
Family Symmetry and Unification
1.Introduction to family symmetry
2.Froggatt-Nielsen mechanism
3.Gauged U(1) family symmetry and its shortcomings
4.Gauged SO(3) family symmetry and vacuum alignment
5.A4 and vacuum alignment
6. A4 Pati-Salam Theory
Appendix A. A4
Appendix B. Finite Groups
1. Introduction to Family Symmetry
Recall Symmetric Yukawa textures
0 

Y u   3  2
 3  2

3
 

2
3
1 
 0  3  3


Y d   3  2  2 
 3  2 1 


 0 3

Y e    3 3 2
  3 3 2

3 

3 2 
1 
  0.05,   0.15
• Universal form for mass matrices, with Georgi-Jarlskog factors
• Texture zero in 11 position
ms
1
( M GUT )  ,
m
3
md
( M GUT )  3
me
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
3

c
c
c
c
6


D

,


Q
,
L
,
U
,
D
,
E
,
N

U
(3)
 i  i
i 1
In SO(10) ,  = 16, so the family largest symmetry is U(3)
Candidate continuous symmetries are U(1), SU(2), SU(3) or SO(3) etc
If these are gauged and broken at high energies then no direct low
energy signatures
GFamily
O(3) L  O(3) R
SO(3)
SU(3)
 27
SU(2)
S (3)L  S (3)R
A4  12
U(1)
S(3)
Nothing
2.Froggatt-Nielsen Mechanism
Simplest example is U(1) family symmetry spontaneously broken by a flavon vev
 0
For D-flatness we use a pair of flavons with opposite U(1) charges 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
Y   0 0 0 
0 0 1


The other Yukawa couplings are generated from higher order operators which
respect U(1) family symmetry due to flavon  insertions:
1  0  1  0  0

 
2
 
3
 
4
 
6
H 2 3    H 2 2    H1 3    H1 2    H11
M
M
M
M
M
When the flavon gets its VEV it generates
small effective Yukawa couplings in terms
of the expansion parameter


M
 6  4  3 


 Y   4  2  
 3  1 


What is the origin of the higher order operators?
To answer this Froggat and Nielsen took their inspiration from the
see-saw mechanism
H
H
2
M
M R
L
R
R L

H

M
M
2
H

 L L
M R


3

M
H 2 3
Where  are heavy fermion messengers
c.f. heavy RH neutrinos
There may be Higgs messengers or fermion messengers
H0

H1
H
1
2
 1
1
H0
M
MH
2
3
0
0
3
Fermion messengers may be SU(2)L doublets or singlets

1
H
M
M Q
Q2
Q0
Q0
 1
H0
0
U c3
Q2
Uc
U1 U1
c
c
U c3
3. Gauged U(1) Family Symmetry
Problem: anomaly cancellation of SU(3)C2U(1), SU(2)L2U(1) and U(1)Y2U(1)
anomalies implies that U(1) is linear combination of Y and B-L (only anomaly free
U(1)’s available) but these symmetries are family independent
Solution: use Green-Schwartz anomaly cancellation mechanism by which
anomalies cancel if they appear in the ratio:
Suppose we restrict the
sums of charges to satisfy
Then A1, A2, A3 anomalies are cancelled a’ la GS for any values of x,y,z,u,v
But we still need to satisfy the A1’=0 anomaly cancellation condition.
The simplest example is for u=0 and v=0 which is automatic in SU(5)GUT
since10=(Q,Uc,Ec) and 5*=(L,Dc)  qi=ui=ei and di=li so only two independent ei, li.
In this case it turns out that A1’=0 so all anomalies are cancelled.
Assuming
for a large top Yukawa we then have:
SO(10) further implies qi=ui=ei=di=li
F=(Q,L) and Fc=(Uc,Dc,Ec,Nc) 
In this case it turns out that A1’=0. PS implies x+u=y and x=x+2u=y+v.
So all anomalies are cancelled with u=v=0, x=y. Also h=(hu, hd) 
The only anomaly cancellation constraint on the charges is x=y which implies
Note that Y is invariant under the transformations
This means that in practice it is trivial to satisfy
Shortcomings of U(1) Family Symmetry
A Problem with U(1)
Models is that it is
impossible to obtain
 0 3 3


Y   3  2  2 
 3  2 1 


For example consider Pati-Salam
where there are effectively no
constraints on the charges from
anomaly cancellation
There is no choice of li and ei that can give the desired texture
e.g. previous example l1=e1=3, l2=e2=1, l3=e3=hf=0 gave:
 6  4  3 


Y   4  2  
 3  1 


The desired texture can be achieved with non-Abelian family symmetry. Another
motivation for non-Abelian family symmetry comes from neutrino physics.
Sequential dominance can account for large neutrino mixing
Diagonal RH nu basis
T
T
T
AA
BB
CC
mLL 


X
Y
Z
See-saw
Sequential
dominance
columns
Dominant Subdominant
m3
m2
V
Constrained SD
Decoupled
m1
L †
Tri-bimaximal
SFK
Large lepton mixing motivates non-Abelian family symmetry
Need

YLR
0
  A2
A
 3
B1
B2
B3

 
 
with
CSD
2$ 3 symmetry (from maximal atmospheric mixing)
1$ 2 $ 3 symmetry (from tri-maximal solar mixing)
Suitable non-Abelian family symmetries must span all three
families e.g.
SU (3)  SFK, Ross; Velasco-Sevilla; Varzelias
27
SO(3)
A4
SFK, Malinsky
4. Gauged SO(3) family symmetry
Left handed quarks and leptons are triplets under SO(3) family symmetry
Right handed quarks and leptons are singlets under SO(3) family symmetry
Antusch, SFK 04
To break the family symmetry introduce three flavons 3, 23, 123
Real vacuum alignment
(a,b,c,e,f,h real)
Barbieri, Hall, Kane, Ross
0
a
0
 
 
 23   e   123   b      0 
3
 
f
c
h
 
 
 
If each flavon is associated with a particular right-handed neutrino
1
1
1 i
i
1
i
2
23 HLi R 
123 HLi R 
3 HLi R3
M
M
M
then the following Yukawa matrix results
0
 
 23   e 
f
 

YLR
F .23 R1 h
 123
 0
 i1
 ee
 fei1

0
 
 3   0 
h
 
a
 
  b 
c
 
aei 2
bei 2
cei 2
But this is not
sufficient to account
for tri-bimaximal
neutrino mixing
0 

0 
hei3 
F .123 R2 h
F .3 R1 h
For tri-bimaximal neutrino mixing we need
0
 23   v 
 v 
 

LR
Y
F .23 R1 h
v
 123   v 
v
 
0
 3   0 
V 
 
 0 vei2 0 
 i1
 1
i 2
ve
0 
 ve
 vei1 vei2 Vei3  M


F .123 R2h
How do we achieve such
a vacuum alignment of
the flavon vevs?
F .3 R3h
The motivation for 123 is to give the second column
required by tri-bimaximal neutrino mixing
Vacuum Alignment in SO(3)
First set up an
orthonormal basis:
FA=0 flatness  <1>=1
FB=0 flatness  <2>=2
FC=0 flatness  <3>=3
1
1   0 
0
 
0
2   1 
0
 
SFK ‘05
1
3
2
FD=0 flatness 1 .2 =0
FE=0 flatness 1 .3 =0
FF=0 flatness 2 .3 =0
0
3   0 
1
 
1
3
2
Then align 23 and 123 relative to 1 , 2 , 3 using
additional terms:
FR=0  <23> gets vevs in the (2,3) directions
FT=0  <123> gets vevs in the (1,2,3) directions
(vevs of equal magnitude are required to minimize soft mass terms)
Finally 23 is orthogonal to 123 due to 123 . 23 =0
0
23   1 
 1 
 
123
 1
  1
 1
 
1
123
3
23
2
5. A4 and Vacuum Alignment
We can replace SO(3) by a discrete A4 subgroup:
A4  SO (3)
De M.Varzielas, SFK, Ross
A4 is similar to the semi-direct product
Same invariants as A4
2=12+22+32 ,
3 =1 2 3
The main advantage of using discrete family symmetry groups is that
vacuum alignment is simplified…
The Diamond (A4) Crystal Structure
0,1,1
1,1,1
Varzielas, SFK, Ross, Malinsky
Radiative Vacuum Alignment
A nice feature of MSSM is radiative EWSB Ibanez-Ross
(s)top loops drive
mH2 u   2
0
MZ
Q3L
MGUT
H
mH2 u negative
H
tR
Similar mechanism can be used to drive flavon vevs using D-terms
Leads to desired vacuum alignment with discrete family symmetry A4
negative

3
123
for negative 
for positive 
 23 T  v(0,1, 1) for positive 123
Ma; Altarelli, Feruglio; Varzeilas, Ross, SFK, Malinsky
Comparison of SO(3) and A4
Symmetry group of
the tetrahedron
A4
Discrete set of
possible vacua
SFK, Malinsky
6. A4 Pati-Salam Theory
A4  SU (4)PS  SU (2)L  SU (2) R
u
FL  (3, 4, 2,1)  
d
u
FRi  (1, 4,1, 2)i  
d
Dirac Operators:
u
d
u
d
u
d
u
d
 
i
e   L
 
i
e   R
Dirac Operators:
Further Dirac Operators required for quarks:
Dirac Neutrino matrix:
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
The Messenger Sector
Dirac:
Majorana:
Including details of the messenger sector:
Messenger masses:
Appendix A. A4
SFK, Malinsky hep-ph/0610250
Appendix B. Finite Groups
Ma 0705.0327