Fermion Masses and Unification

Steve King

University of Southampton

Lecture III Family Symmetry and Unification I

1. Doublet-triplet splitting

2. Introduction to family symmetry3. Froggatt-Nielsen mechanism4. Gauged U(1) family symmetry and unification 5. SO(3) or A4 family symmetry and unification

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 (discussion session?)

a ‘Solves’ Proton Decay and Unification problems

b ‘Solves’ Proton Decay problem but leaves Unification problem

!

Doublet-triplet splitting or light triplets?

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 + MGUT DD

Minimal superpotential contains:

Need to fine tune = m to within 1 part in 1014 to achieve » TeV light Higgs

) DD(¹ +23¸m) + huhd(¹ ¡ ¸m)

GUT EW scale

Doublet-Triplet Splitting Problem

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

) Nothing for Higgs hu , hd to couple to

Problems: Large rank representations

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

problem for Higgs mass…

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

Missing Partner Mechanism

3 3

3 2 2

3 2

0

1

uY

We would like to account for the hierarchies embodied in the textures

3 3

3 2 2

3 2

0

1

dY

3 3

3 2 2

3 2

0

3 3

3 1

eY

0.05,

0.15

Introduction to Family Symmetry

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.

In SM the largest family symmetry possible is the symmetry of the kinetic terms

36

1

, , , , , , (3)c c c ci i

i

D Q L U D E N U

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

What is a suitable family symmetry?

U(1)

SU(2)

SU(3) SO(3)

S(3)Nothing

(3) (3)L RO O

(3) (3)L RS S27

4 12A

Candidate Family Symmetries (incomplete)

Simplest example is U(1) family symmetry spontaneously broken by a flavon vev

For D-flatness we use a pair of flavons with opposite U(1) charges

0 ( ) ( )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

0 0 0

0 0 1

Y

The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon insertions:

2 3 4 6

2 3 2 2 1 3 1 2 1 1H H H H HM M M M M

M

When the flavon gets its VEV it generates small effective Yukawa couplings in terms

of the expansion parameter

6 4 3

4 2

3 1

Y

1 0 1 0 0

U(1) Family Symmetry

What is the origin of the higher order operators?

Froggat and Nielsen took their inspiration from the see-saw mechanism

2

R

L L

H

M

2 3HM

Where are heavy fermion messengers c.f. heavy RH neutrinos

L LR R

M

H H

RM

2M

H

3

M

Froggatt-Nielsen Mechanism

There may be Higgs messengers or fermion messengers

2

M

0H

30

1

0

2 3

1

0H

1H1H HM

Fermion messengers may be SU(2)L doublets or singlets

2QQ

M

0H

3cU0

Q

1

0Q 2Q

cU

M

0H

3cU1

cU

1

1cU

Gauged U(1) Family SymmetryProblem: anomaly cancellation of SU(3)C

2U(1), SU(2)L2U(1) and U(1)Y

2U(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:

Ibanez, Ross; Kane, SFK, Peddie, Velasco-Sevilla

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

A Problem with U(1) Models is that it is impossible to obtain

3 3

3 2 2

3 2

0

1

Y

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 texture6 4 3

4 2

3 1

Y

e.g. previous example l1=e1=3, l2=e2=1, l3=e3=hf=0 gave:

Shortcomings of U(1) Family Symmetry

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

Harrison, Perkins, Scott

e.g. Tri-bimaximal

Valle et al

Andre de Gouvea

0 0

0 0

0 0RR

X

M Y

Z

1 1 1

2 2 2

3 3 3

LR

A B C

m A B C

A B C

2 2 21 1 1 1 2 1 2 1 2 1 3 1 3 1 3

2 2 21 2 2 3 2 3 2 3 2 3

2 2 23 3 3

.

. .

TLL LR RR LR

A B C A A B B C C A A B B C C

X Y Z X Y Z X Y Z

A B C A A B B C Cm m M m

X Y Z X Y Z

A B C

X Y Z

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

Heavy Majorana Dirac

Light Majorana

Large Lepton Mixing From the See-Saw

columns

Sequential dominance can account for large neutrino mixing

T T T

LL

AA BB CCm

X Y Z See-saw

Sequential dominance Dominant

m3

Subdominant m2

Decoupled m1

Diagonal RH nu basis

Tri-bimaximal

†LV

Constrained SD

Large lepton mixing motivates non-Abelian family symmetry

1

2 2

3 3

0

LR

B

Y A B

A B

Need

Suitable non-Abelian family symmetries must span all three families e.g.

2$ 3 symmetry (from maximal atmospheric mixing)

1$ 2 $ 3 symmetry (from tri-maximal solar mixing)

SFK, Ross; Velasco-Sevilla; Varzelias

SFK, Malinsky

with CSD

27

4

(3)

(3)

SU

SO A

SO(3) family symmetrySuppose that left handed leptons are triplets under SO(3) family symmetry and right handed leptons are singlets

123

a

b

c

23

0

e

f

3

0

0

h

Real vacuum alignment (a,b,c,e,f,h real)

To break the family symmetry introduce three flavons 3, 23, 123

3, , 1i

iR R

L

L ee

But this is not sufficient to account for tri-bimaximal neutrino mixing

If each flavon is associated with a particular right-handed neutrino

then the following Yukawa matrix results

1 2 323 123 3

1 1 1i i ii R i R i RHL HL HL

M M M

2

2

2 3

1

1

0

0

0 i

iLR

i

i

i i

ee

fe

Y

he

ae

be

ce

123

a

b

c

23

0

e

f

3

0

0

h

123. RL h 2

123. RL h 13. RL h

1

M

2

2

1

3

2

1

0

0

0 i

Li

iR

i

i i

ve

v

ve

ve

ve

Y

e Ve

123

v

v

v

23

0

v

v

3

0

0

V

123. RF h 2

123. RF h 33. RF h

1

M

For tri-bimaximal neutrino mixing we need

This requires a delicate vacuum alignment of flavon vevs – see next lecture

Extra Slides

•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)

The The problemproblem

•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

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:

EE66SSM= MSSM+3(5+5SSM= 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

SFK, Moretti, Nevzorov

term 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 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

SUSY with 3x27’s at TeV SUSY with 3x27’s at TeV scalescale

(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

Howl, SFK