Fermion Masses and Unification

Steve KingUniversity of Southampton

Lecture 2Grand Unified Theories

1. SU(5)2. Pati-Salam3. SO(10)4. Proton decay with triplets

Appendix on group theory

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

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

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

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)

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

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

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

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….

(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

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:

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

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

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

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

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

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

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

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

Appendix 1 Group Theory of SU(5)

Nevzorov

Gauge Sector of SU(5)

Summary of Matter and Gauge Sector of SU(5)