Fermion Masses and
Unification
Steve King
University of Southampton
Lecture III
Family Symmetry and Unification I
1.
2.
3.
4.
5.
Doublet-triplet splitting
Introduction to family symmetry
Froggatt-Nielsen mechanism
Gauged U(1) family symmetry and unification
SO(3) or A4 family symmetry and unification
Doublet-triplet splitting or light triplets?
Two possible types of solutions:
a
Give large GUT scale masses to
!
b
D; D
Doublet-Triplet splitting
Allow TeV scale masses to D; D but suppress interactions
!
Yukawa suppression is required
(discussion session?)
a
‘Solves’ Proton Decay and Unification problems
b
‘Solves’ Proton Decay problem but leaves Unification problem
Doublet-Triplet Splitting Problem
Nontrivial to give huge masses to
D; D but not hu ; hd
e.g. most simple mass term would be
!
M GU T 55
in
SU(5)
M GU T hu hd + M GU T DD
Minimal superpotential contains:
)
2
D D (¹ + ¸ m) + hu hd (¹ ¡ ¸ m)
3
GUT
EW scale
Need to fine tune  =  m to within 1 part in 1014 to achieve » TeV light Higgs
Missing Partner Mechanism
Pair up H with a G representation (e.g. 50 of SU(5) ) that contains
(colour) triplets but not (weak) doublets
Suppose superpotential contains:
Under SU(5) !
SM : 50 contains (3,1) but not (1,2)
Then < 75 > in (1; 1) 0 direction gives mass couplings toD; D
)
Nothing for Higgs hu , hd
Problems:
to couple to
Large rank representations
 problem for Higgs mass…
Introduction to Family Symmetry
We would like to account for the hierarchies embodied in the textures
0   


Y u   3  2  2 
 3  2 1 


3
3
 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
SUSY GUTs can describe but not explain such hierarchies
To understand such hierarchies we shall introduce a family
symmetry that distinguishes the three families
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. Idea is to use  to explain the textures.
What is a suitable family symmetry?
In SM the largest family symmetry possible is the symmetry of the
kinetic terms
3
   D 
i 1

i
,   Q, L,U , D , E , N  U (3)
c
i
c
c
c
6
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) …
N.B. If family symmetries are gauged and broken at high energies then
no direct low energy signatures
Candidate Family Symmetries (incomplete)
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
U(1) Family Symmetry
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

M
  
2
  
3
  
4
  
6
H 2 3    H 2 2    H 1 3    H 1 2    H 1 1
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 


Froggatt-Nielsen Mechanism
What is the origin of the higher order operators?
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
0
H
M
M Q
Q2
 Q0
 Q0
 1
0
Uc
U c3
Q2
U1 U1
c
c
U c3
Ibanez, Ross; Kane, SFK, Peddie, Velasco-Sevilla
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 Yf is invariant under the transformations
This means that in practice it is trivial to satisfy
for any choices of charges
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. There is
also an independent motivation for non-Abelian symmetry from neutrino physics…
Lepton mixing is large
Andre de Gouvea
Valle et al
e.g. Tri-bimaximal
Harrison, Perkins, Scott
Large Lepton Mixing From the See-Saw
Heavy Majorana
M RR
X
  0
0

0
Y
0
0
0 
Z 
Dirac  A1
mLR   A2
A
 3
B1
B2
B3
  A12 B12 C12   A1 A2 B1B2 C1C2 


Light Majorana
    

X
Y
Z
X
Y
Z




2
2
2


A
B
C
1 T
2
2
3 
m LL  mLR M RR
mLR  
.


X Y

Z





.
.


C1 
C2 
C3 
 A1 A3 B1B3 C1C3  




Y
Z 
 X

 A2 A3 B2 B3 C2C3  




Y
Z 
 X

 A32 B32 C32  
    
Z  
X Y
Each element has three contributions, one from each RH neutrino.
If the right-handed neutrino of mass X dominates and A1=0 then we have
approximately only (2,3) elements with m1,2¿ m3 and tan 23¼ A2/A3
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
Large lepton mixing motivates non-Abelian family symmetry
Need
0

YLR
  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
SO(3) family symmetry
Suppose that left handed leptons are triplets under SO(3)
family symmetry and right handed leptons are singlets
 
L     3, eR , R  1
 e L
i
i
To break the family symmetry introduce three flavons 3, 23, 123
0
Real vacuum alignment
 



23
(a,b,c,e,f,h real)
e
f
 
a
 
 123   b 
c
 
0
 
 3   0 
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
L.23 R1 h
 123
 0
 i1
 ee
 fei1

a
 
  b 
c
 
aei 2
bei 2
cei 2
L.123 R2 h
0
 
 3   0 
h
 
0 
 1
0 M
hei3 
But this is not
sufficient to account
for tri-bimaximal
neutrino mixing
L.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 R2 h
F .3 R3 h
This requires a delicate vacuum alignment of flavon vevs
– see next lecture
Extra Slides
The  problem
•MSSM solves “technical hierarchy problem” (loops)
•But no reason why » msoft  the “ problem”.
•In the NMSSM =0 but S Hu Hd  <S> Hu Hd where <S>» 
•S3 term required to avoid a massless axion due to global U(1) PQ symmetry
•S3 breaks PQ to Z3 resulting in cosmo domain walls (or tadpoles if broken)
•One solution is to forbid S3 and gauge U(1) PQ symmetry so that the
dangerous axion is eaten to form a massive Z’ gauge boson  U(1)’ model
•Anomaly cancellation in low energy gauged U(1)’ models implies either extra
low energy exotic matter or family-nonuniversal U(1)’ charges
•For example can have an E6 model with three complete 27’s at the TeV scale
with a U(1)’ broken by singlets which solve the  problem
•This is an example of a model where Higgs triplets are not split from doublets
E6SSM= MSSM+3(5+5*)+Singlets
E 6  SO (10)  U (1)
Right handed neutrinos
are neutral under:
MString
MGUT
Right handed
neutrino masses
TeV
MW
SO (10)  SU (5)  U (1)
U (1) N 
E8 £ E8 ! E6
E6 ! SU(5)£U(1)N
Quarks,
leptons
15
4
U (1)  14 U (1) 
! SM £ U(1)N
27', 27 '
Triplets
Singlets
H’,H’-bar Incomplete
and Higgs and RH s
multiplets
(required for
unification)
U(1)N broken, Z’ and triplets get mass,  term generated
SU(2)L£ U(1)Y broken
SFK, Moretti, Nevzorov
Family Universal Anomaly Free Charges:
Most general E6 allowed couplings from 273:
 term
FCNC’s due to
extra Higgs
Allows p and
D,D* decay
Triplet mass terms
Rapid proton decay + FCNCs extra symmetry required:
•Introduce a Z2 under which third family Higgs and singlet are even all else
odd  forbids W1 and W2 and only allows Yukawa couplings involving third
family Higgs and singlet
•Forbids proton decay and FCNCs, but also forbids D,D* decay so Z2 must
be broken!
•Yukawa couplings g<10-8 will suppress p decay sufficiently
•Yukawa couplings g>10-12 will allow D,D* decay with lifetime <0.1 s
(nucleosynthesis)
This works because D decay amplitude involves single g while p decay
involves two g’s
Unification in the MSSM
2 loop, 3(MZ)=0.118
3
2
1
MSUSY=250 GeV
Blow-up of GUT region
Unification with MSSM+3(5+5*)
2 loop, 3(MZ)=0.118
3
2
1
250
GeV
1.5
TeV
Blow-up of GUT region
SUSY with 3x27’s at TeV scale
E 6  SO (10)  U (1)
SO (10)  SU (4)PS  SU (2)L  SU (2)R
MPlanck
MGUT
E6! SU(4)PS£ SU(2)L £ SU(2)R
£ U(1)
SU(4)PS£ SU(2)L £ SU(2)R £ U(1) ! SM £ U(1)X
(4,2,1)  (4,1,2)(6,1,1)  (1, 2, 2) (1,1,1)  27 x three families
Right handed
neutrino masses
TeV
MW
Quarks,
leptons
Triplets
Singlet
and Higgs
U(1)X broken, Z’ and triplets get mass,  term generated
SU(2)L£ U(1)Y broken
Howl, SFK
Planck Scale Unification with 3x27’s
MPlanck
Low energy (below MGUT)
three complete families of 27’s of E6
High energy (above MGUT» 1016 GeV) this is embedded into a left-right
symmetric Pati-Salam model and additional heavy Higgs are added.
MPlanck