Fermion Masses and Unification

Steve King

University of Southampton

Lecture 2

Unification

G ! SU(3)c ­ SU(2)L ­ U(1)YSimple Group

Quarks and Leptons unified into representations of G

Most popular Groups are:

SU(5) Quarks, Leptons ! 10©5

SO(10) Quarks, Leptons, right-handed neutrino ºR ! 16

E6 Quarks, Leptons, exotics, SM singlet, ºR ! 27

G

G ! SU(3)c ­ SU(2)L ­ U(1)YSimple Group

Single coupling constant g

Applies to all three examples SU(5); SO(10); E6

spontaneously broken:G

TrfTa;Tbg=NG ±ab

) If no additional (non-SM singlet) fermions are added:

g1 ´

r53gY

Couplings assumed to ‘run’ to measured SM couplings

g3 =g2 =g1

g ! g3; g2; gY

At GUT energy.

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

Each­family­fits­nicely­into­the­SU(5)­multiplets­

N.B­in­minimal­SU(5)­neutrino­masses­are­zero.

Right-handed­neutrinos­may­be­added­to­give­neutrino­masses­but­they­are­not­predicted.­

SU(5)­GUT Georgi­and­Glashow

With­the­hypercharge­embedding­

Gauge Sector of SU(5)

Summary of Matter and Gauge 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

Higgs Sector of SU(5)

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-Salam­Partial­Unification(3) (2) (1)C L YSU SU U

(4) (2) (2)PS L RSU SU SU

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

Georgi-Jarlskog Textures

Gives good SUSY relations at MGUT: mb¼ m , ms ¼ m/3 , md ¼ 3me

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

0 0 0 0

0 , 3 0

0 0 1 0 0 1

d eY Y

(15,2,2)xa

15

1

1

1

3

V

Gives GJ factor of -3 for the lepton

(4,2,1)L

u

d

u

e

u

d d

(4,1,2)

R

u

d

u

e

u

d d

Summary­of­Pati-Salam

--­Predicts­RH­neutrinos­with­lepton­number­as­the­“fourth­colour”­

--­Allows­the­possibility­of­restoring­parity­if­LR­symmetry­is­imposed­

--­(Quark-lepton)­unification­of­16­family­into­two­LR­symmetric­reps

--­B-L­as­a­gauge­symmetry

--­Quantization­of­electric­charge­­Qe=­-Qp

--­Pati-Salam­can­be­unified­into­SO(10)(4,2,1) (4,1,2) 16

SO(10)­­GUT Georgi;­Fritzsch­and­Minkowski

The­16­of­SO(10)­contains­a­single­quark­and­lepton­family­and­also­predicts­a­single­right-handed­neutrino­per­family.­­

The­SU(5)­reps­are­unified­into­SO(10):

The­two­Higgs­doublets­are­contained­in­a­10­of­SO(10)

Neutrino­masses­in­SO(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

Dirac­mass

Heavy­Majorana­mass

SO(10)­contains­all­the­ingredients­for­the­see-saw­mechanism­and­tends­to­predict­a­hierarchical­pattern­of­neutrino­masses

0 335.0 10 ( )p e y SK

Like ‘matter’ particles,

G

Leads to new (triplet) particles D.

SU(5) SO(10) E6

All give new particles: D ´ (3;1)¡ 13

, (£3 in E6)

Problems: Spoil Unification of MSSM gauge couplings

Cause rapid proton decay

!Higgs must be embedded into representations of

e.g.

; D ´ (3;1) 13

1

2

5 uH

h

D

Say representation of

To produce SM Yukawa terms one generally uses termsFFH

Gives 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:

uchuQ; dchdQ; echdL

D;D

G

And quarks and leptons representation of

hu;hd;D;D ! H

! F G

DQQ; Ddcuc; ecDuc;QLD Proton decay

D D

1

DM

DQQ; Ddcuc; ecDuc;QLD

D-exchange generates superfield operators

In terms of scalar and fermion components some examples of dangerous operators are shown below

D

u u u u

p

K

p

K

D

1

DM

Minimal SU(5) is ruled out by proton decay -- but it gives unacceptable fermion masses anyway

2( )p K c loop RG matrix element

331.6 10 ( )p K y SK

Two possible types of solutions:

a Give large GUT scale masses to D;D

b Allow TeV scale masses to but suppress interactions D;D

! Doublet-Triplet splitting

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 but not D;D hu;hd

e.g. most simple mass term would be in SU(5)MGUT 55

! MGUT huhd + MGUTDD

Minimal superpotential contains:

Fine tuning to within 1 part in 1014

) DD(¹ +23¸m) + huhd(¹ ¡ ¸m)Superpotential:

SU(5) ! SM

GUT EW scale

Pair up H with a G representation that contains (colour) triplets

Take superpotential to contain:

e:g: minimal SU(5)

5 50< 75>+ 5 50< 75>

Under :SU(5) ! SM

) Nothing for Higgs hu , hd to couple to

Problems: Large rank representations

‘Missing – partner’ mechanism’

but not (weak) doublets (at least after G is broken).

Proton decay via triplet Higgsino from

And in direction gives mass couplings to <75> (1;1)0 D;D

effective term.

problem for Higgs mass

50 contains (3,1) but not (1,2)

•The­­problem­(light­Higgs­mass)­is­intimately­related­to­the­doublet-triplet­splitting­problem­(heavy­triplet­mass)

•One­approach­is­to­allow­both­light­Higgs­doublets­and­triplets

•Requirements:­generate­TeV­scale­mass­terms­for­the­light­Higgs­doublets­and­triplets,­suppress­proton­decay­due­to­triplet­exchange­while­allowing­triplets­to­decay­in­less­that­0.1­s­to­avoid­problems­with­nucleosynthesis

•The­Exceptional­Supersymmetric­Standard­Model­(ESSM)­is­an­example­of­a­model­with­extra­low­energy­exotic­matter­forming­complete­27’s­of­E6­plus­the­two­Higgs­doublets­of­the­MSSM:

[5*+10+­(5+5*)+1+1]xthree­families­+(H,H’)Quarks, leptons

Triplets,Higgs, singlets

27

Non-Higgs

15 14 4(1) (1) (1) NU U U

(10) (5) (1)SO SU U 6 (10) (1)E SO U

E6 ! SU(5)£U(1)N MGUT

TeV U(1)N broken, Z’ and triplets get mass, term generated

27',27'

Incomplete multiplets

(required for unification)

Right handed neutrino masses

MString E8 £ E8 ! E

6

Quarks, leptons

Triplets and Higgs

Singlets and RH s

H’,H’-bar

MW SU(2)L£ U(1)Y broken

Right handed neutrinos are neutral under:

ESSM= MSSM+3(5+5ESSM= MSSM+3(5+5**))+Singlets+Singlets

! SM £ U(1)N

Family Universal Anomaly Free Charges:

Most general E6 allowed couplings from 273:

Allows p and D,D* decay

FCNC’s due to extra Higgs

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 MSSMUnification in the MSSMBlow-up of GUT region

MSUSY=250 GeV

3

21

2 loop, 3(MZ)=0.118

Unification with Unification with MSSM+3(5+5MSSM+3(5+5**))

3

2

1

250 GeV

1.5 TeV

Blow-up of GUT region2 loop, 3(MZ)=0.118

MESSM= 3x27’s (no H,H’)MESSM= 3x27’s (no H,H’)

(10) (4) (2) (2)PS L RSO SU SU SU6 (10) (1)E SO U

MGUT

TeV U(1)X broken, Z’ and triplets get mass, term generated

Right handed neutrino masses

MPlanck

Quarks, leptons

Triplets and Higgs

Singlet

MW SU(2)L£ U(1)Y broken

E6! SU(4)PS£ SU(2)L £ SU(2)R

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

£­U(1)

Planck Scale Unification with Planck Scale Unification with 3x27’s3x27’s

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 MPlanck