Fermion Masses and
Unification
Steve King
University of Southampton
Lecture 2
Unification
Simple Group
G ! SU(3) c - SU(2) L - U(1) Y
Quarks and Leptons unified into representations of
Most popular Groups
G
G
are:
SU(5)
Quarks, Leptons
SO(10)
Quarks, Leptons, right-handed neutrino
E6
!
10 © 5
Quarks, Leptons, exotics, SM singlet,
º R ! 16
ºR !
27
G ! SU(3) c - SU(2) L - U(1) Y
Single coupling constant g
Simple Group
G
spontaneously broken:
g ! g3 ; g2 ; gY
Couplings assumed to ‘run’ to measured SM couplings
Tr f T a ; T bg = N G ±ab
)
If no additional (non-SM singlet) fermions are added:
r
g3 = g2 = g1
Applies to all three examples
At GUT energy.
g1 ´
SU(5); SO(10); E6
5
gY
3
GUTs
E6
SU (5)  U (1)
SO (10)
SU (4) PS  SU (2) L  SU (2) R
SU (3)C  SU (3) L  SU (3) R
SU (5)
SU (3)C  SU (2) L  SU (2) R U (1) B L
SU (3)C  SU (2) L U (1)Y
SU(5) GUT
Georgi and Glashow
With the hypercharge embedding
Each family fits nicely into the SU(5) multiplets
N.B in minimal SU(5) neutrino masses are zero.
Right-handed neutrinos may be added to give
neutrino masses but they are not predicted.
Gauge Sector of SU(5)
Summary of Matter and Gauge Sector of SU(5)
Higgs Sector of SU(5)
Candidate Higgs reps of SU(5) are contained in
matter bilinears constructed from 5* and 10
Minimal suitable Higgs reps for fermion
masses consist of 5H + 5*H
The smallest Higgs rep which contains a singlet under the SM
subgroup is the 24 Higgs rep and is a candidate to break SU(5)
The Higgs superpotential involving the minimal Higgs sector of SU(5)
consisting of the 24H plus 5H plus 5H*
With some tuning (see later) one can achieve light Higgs doublets which can
develop weak scale vevs
The Yukawa superpotential for one family with Higgs H=5, H*=5*
good
bad
c.f. good SUSY relations at MGUT: mb¼ m , ms ¼ m/3 , md ¼ 3me
Pati-Salam Partial Unification
SU (3)C  SU (2) L  U (1)Y
SU (4) PS  SU (2) L  SU (2) R
The Yukawa superpotential for one family
a
W  F F h
x
x a
 Qh2u c  Qh1d c  Lh1ec  Lh2 c 
u  d  e  
at the GUT scale
Could work for the third family, but certainly not for all three families

Y  Y  Y  Yij
u
ij
d
ij
e
ij
Y Y
d
ij
e
ij
at the GUT scale is bad
at the GUT scale is almost good
Georgi-Jarlskog Textures
 0 12
Y d   21 22
 0
0

0
12
 0
0  , Y e   21 322
 0
1 
0

0
0 
1 
Gives good SUSY relations at MGUT: mb¼ m , ms ¼ m/3 , md ¼ 3me
W  12 F1 a F2 x hax  21F2 a F1 x hax
a
a
33 F3 F h  22 F2 F2  x 

x
a
x
3 x a
(15, 2, 2)
x
a
1

 1


  V15 


1



3


Gives GJ
factor of -3 for
the lepton
Summary of Pati-Salam
-- Predicts RH neutrinos with lepton number as the “fourth colour”
u
(4, 2,1)  
d
u
u
d
d
 

e  L
u
(4,1, 2)  
d
u
u
d
d
 

e  R
-- Allows the possibility of restoring parity if LR symmetry is imposed
-- (Quark-lepton) unification of 16 family into two LR symmetric reps
-- B-L as a gauge symmetry
-- Quantization of electric charge  Qe= -Qp
-- Pati-Salam can be unified into SO(10)
(4, 2,1)  (4,1, 2)  16
SO(10) GUT
Georgi; Fritzsch and Minkowski
The 16 of SO(10) contains a single quark and lepton family and
also predicts a single right-handed neutrino per family.
The SU(5) reps are unified into SO(10):
The two Higgs doublets are contained in a 10 of SO(10)
Neutrino masses in SO(10)
16.16.10H  
 H0 
e  L    R  mLR  L R
H 
Dirac mass
16.16.126 H  126 H   R R
Heavy Majorana mass
16.16. 16H 16H
 16H 2

 R R
M
M
SO(10) contains all the ingredients for the see-saw mechanism and
tends to predict a hierarchical pattern of neutrino masses
  p   0e    5.0  1033 y ( SK )
Like ‘matter’ particles,
Higgs must be embedded into representations of
!
Leads to new (triplet) particles D. e.g.
SU(5)
SO(10)
All give new particles: D ´ (3; 1)
, ¡
Problems:
1
3
G
 hu 
5H   
D
E6
; D ´ (3; 1) 31
(£ 3 i n E 6)
1
Spoil Unification of MSSM gauge couplings
2
Cause rapid proton decay
Say hu ; hd ; D; D !
H representation of
G
e:g: 10 f or SO(10)
And quarks and leptons
! F
representation of
G
e:g: 16 f or SO(10)
To produce SM Yukawa terms one generally uses
Gives following SM interactions:
FFH
terms
uc hu Q; dc hd Q; ec hd L
But also gives ‘dangerous’ terms involving D; D with SM particles:
DQQ; Ddc uc; ec Duc; QLD
Proton decay
c c
c
c
D
DQQ; Dd u ; e Du ; QLD
D-exchange generates superfield operators
In terms of scalar and fermion
components some examples of
dangerous operators are
shown below
D
1
MD
D
u
u
u
u
K
p
D
K
p
 ( p  K  ) c2  loop  RG  matrix element
  p  K    1.6  1033 y ( SK )
1
MD
Minimal SU(5) is ruled out
by proton decay -- but it
gives unacceptable fermion
masses anyway
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
a
‘Solves’ Proton Decay and Unification problems
b
‘Solves’ Proton Decay problem but leaves Unification 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:
SU(5) ! SM
)
2
Superpotential: D D (¹ + ¸ m) + hu hd (¹ ¡ ¸ m)
3
GUT
Fine tuning to within 1 part in 1014
EW scale
‘Missing – partner’ mechanism’
e:g: mi ni mal SU(5)
Pair up H with a G representation that contains (colour) triplets
but not (weak) doublets (at least after G is broken).
Take superpotential to contain:
Under SU(5) !
5 50 < 75 > + 5 50 < 75 >
SM : 50 contains (3,1) but not (1,2)
And < 75 > in (1; 1) 0 direction gives mass couplings to
)
Nothing for Higgs hu , hd
Problems:
D; D
to couple to
Large rank representations
 problem for Higgs mass
Proton decay via triplet Higgsino from
effective term.
•The  problem (light Higgs mass) is intimately related to the
doublet-triplet splitting problem (heavy triplet mass)
•One approach is to allow both light Higgs doublets and triplets
•Requirements: generate TeV scale mass terms for the light Higgs
doublets and triplets, suppress proton decay due to triplet
exchange while allowing triplets to decay in less that 0.1 s to avoid
problems with nucleosynthesis
•The Exceptional Supersymmetric Standard Model (ESSM) is an
example of a model with extra low energy exotic matter forming
complete 27’s of E6 plus the two Higgs doublets of the MSSM:
[5*+10+ (5+5*)+1+1]xthree families +(H,H’)
Quarks, Triplets,Higgs,
leptons singlets
27
Non-Higgs
ESSM= 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
Family Universal Anomaly Free Charges:
Most general E6 allowed couplings from 273:
FCNC’s due to
extra Higgs
Allows p and
D,D* decay
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
MESSM= 3x27’s (no H,H’)
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
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