Fermion Masses and Unification
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Fermion Masses and Unification
Steve KingUniversity of Southampton
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Lecture 2Grand Unified Theories
1. SU(5)2. Pati-Salam3. SO(10)4. Proton decay with triplets
Appendix on group theory
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Grand Unified Theories (GUTs)Basic idea is to embed the SM gauge group into a simple gauge group G with a single coupling constant, broken at a high energy scale
Motivations1. Continuation of process of unification of physics starting with Maxwell
2. Remarkable fit of SM multiplets into Pati-Salam, SU(5), SO(10), E6…
3. Unification of gauge couplings at high energy scale MGUT
4. Charge quantization: equality of electron and proton charges
5. High energy fermion mass relations e.g. mb=m
R
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Candidate GUTs
E 6
(5) (1)SU U (3) (3) (3)C L RSU SU SU
(4) (2) (2)PS L RSU SU SU
(3) (2) (2) (1)C L R B LSU SU SU U
(3) (2) (1)C L YSU SU U
(5)SU
(10)SO
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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.
SU(5) GUT Georgi and Glashow
With the hypercharge embedding
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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
Higgs Sector of SU(5)
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The Yukawa superpotential for one family with Higgs H=5, H*=5*
good almost good
c.f. Georgi-Jarlskog relations at MGUT:
Fermion Masses in SU(5)
( )c c cu u d d dH Qu H Qd H Le
, , 33b s d e
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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 H=5H plus H*=5H*
With some tuning (see later) one can achieve light Higgs doublets which can develop weak scale vevs v5 ¿ v24
Breaking SU(5)
24H 5H
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0 335.0 10 ( )p e y SK
Proton Decay in Non-SUSY SU(5)
Decay modes
Gauge bosons in adjoint of SU(5) contain SM gauge bosons G,W,B plus new gauge bosons X,Y
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Proton Decay in SUSY SU(5)
There are also in addition dimension 5 proton decay operators arising from colour triplet exchange (see later)
However the main drawback of SU(5) is that it does not predict right-handed neutrinos….
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(4,2,1)L
ud
ue
ud d
(4,1,2)
R
ud
ue
ud d
-- Predicts RH neutrinos with lepton number as the “fourth colour”
-- 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
Pati-Salam Partial Unification
(4) (2) (2)PS L RSU SU SU
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Symmetry Breaking of Pati-SalamIn order to achieve the desired symmetry breaking
We introduce Higgs:
Electroweak symmetry breaking is then achieved via the Higgs bidoublet:
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The Yukawa superpotential for one family
a xx aW F F h
2 1 1 2c c c cQh u Qh d Lh e Lh
u d e at the GUT scale
Could work for the third family, but certainly not for all three families
u d eij ij ij ijY Y Y Y at the GUT scale is bad
d eij ijY Y at the GUT scale is almost good
Fermion Masses in Pati-Salam
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Georgi-Jarlskog Textures
Gives good SUSY relations at MGUT:
12 1 2 21 2 1
33 3 3 22 2 2
a x a xx a x a
a x a xx a x a
W F F h F F h
F F h F F
12 12
21 22 21 22
33 33
0 0 0 00 , 3 0
0 0 0 0
d eY Y
(15,2,2)xa 15
11
13
V
Gives GJ factor of -3 for the lepton
, , 33b s d e
Such a texture can be achieved from the superpotential
where
Consider the following texture
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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)
Fermion masses arise from the coupling
2 1 1 216.16.10 c c c cH Qh u Qh d Lh e Lh
c.f. Pati-Salam
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Neutrino masses in SO(10)
0
16.16.10H R LR L RL
He m
H
16.16.126 126H H R R
216.16.16 16 16H H HR RM M
Dirac mass
Heavy Majorana mass
SO(10) contains all the ingredients for the see-saw mechanism and tends to predict a hierarchical pattern of neutrino masses
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This Leads to new (colour triplet) particles D
SU(5) SO(10) E6
All give new colour triplet particles: D ´ (3;1)¡ 13
(£3 in E6)Problems: Spoil Unification of MSSM gauge couplings
Cause rapid proton decay
Low energy MSSM Higgs doublets must be embedded into representations of the GUT group
e.g.
; D ´ (3;1) 13
1
2
5 uH
HD
Troublesome Colour Triplet Higgs
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Say representation of
To produce SM Yukawa terms one generally uses termsFFHGives following SM interactions:
e:g: 10 f or SO(10)
e:g: 16 f or SO(10)
But also gives ‘dangerous’ terms involving with SM particles:D;D
G
And quarks and leptons representation of! F G
DQQ; Ddcuc; ecDuc;QLD Proton decay
Proton Decay with Triplet Higgs
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D D
1
DM
DQQ; Ddcuc; ecDuc;QLDD-exchange generates superfield operators
In terms of scalar and fermion components some examples of dangerous operators are shown below
D
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2( )p K c loop RG matrix element
331.6 10 ( )p K y SK
Minimal SU(5) turns out to be ruled out by proton decay -- but it gives unacceptable fermion masses anyway
1
DM
u u u u
p
K
p
K 1
GUTM
1
softM
Thus p » MGUT2 Msoft
2 instead of MGUT4
Proton Decay with Dim 5 Operators
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Appendix 1 Group Theory of SU(5)
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Nevzorov
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Gauge Sector of SU(5)
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Summary of Matter and Gauge Sector of SU(5)