Fermion Masses and Unification

Steve King

University of Southampton

Lecture 4SU(3), GUTs and SUSY Flavour

1.SU(3) Family Symmetry

2.SU(3) £ SO(10) Model

3.Quark-lepton connections

4. SUSY Flavour Problem

5. SU(5) GUTs and Soft Masses

6. Family Symmetry and Soft Masses

7. Family Symmetry and SUSY CP

8. Do we need a family symmetry?

1. Gauged SU(3) family symmetry

Now suppose that the fermions are triplets of SU(3) i = 3

i.e. each SM multiplet transforms as a triplet under a gauged SU(3)

with the Higgs being singlets H» 1, , , , , 3c c c ci i i i i i iQ L U D E N

This “explains” why there are three families c.f. three quark colours in SU(3)c

0 0 0

0 0 0

0 0 1

2 2

2

0 0 0

0

0 1

23 0 0 0 0

0 0 0

0 0 0

3 0

The family symmetry is spontanously broken by antitriplet flavons

Unlike the U(1) case, the flavon VEVs can have non-trivial vacuum alignments.

We shall need flavons with vacuum alignments:

3>/ (0,0,1) and <23>/ (0,1,1) in family space (up to phases)

so that we generate the desired Yukawa textures from Froggatt-Nielsen:

3i

Frogatt-Nielsen in SU(3) family symmetry

(3)

0 0 0

0 0 0

0 0 0

SUtree levelY

In SU(3) with i=3 and H=1 all tree-level Yukawa couplings Hi j are forbidden.

2

1 i ji jH

M

In SU(3) with flavons the lowest order Yukawa operators allowed are:

3i

For example suppose we consider a flavon with VEV then this generates a (3,3) Yukawa coupling

23

3 32 2

0 0 01

0 0 0

0 0 1

i ji j

VH

M M

Note that we label the flavon with a subscript 3 which denotes the direction of its VEV in the i=3 direction.

3i

3 3(0,0,1)i V 3i

Next suppose we consider a flavon with VEV then this generates (2,3) block Yukawa couplings

223

23 232 2

0 0 01

0 1 1

0 1 1

i ji j

VH

M M

23 23(0,1,1)i V 23i

23

0 0 0

0 0 0

0 0

2 223 23

2 2 223 3 23

0 0 0

0

0

23 0 0 0 0

0 0 0

0 0 0

3 0

Writing and these flavons generate Yukawa couplings

22 33 2

V

M

22 2323 2

V

M

If we have 3 ¼ 1 and we write 23 = then this resembles the desired texture

3 3

3 2 2

3 2

0

1

Y

To complete the texture there are good motivations from neutrino physics for introducing another flavon <123>/ (1,1,1)

2. SU(3) £ SO(10) Model

Yukawa Operators Majorana Operators

Varzielas,SFK,Ross

Inserting flavon VEVs gives Yukawa couplings

After vacuum alignment the flavon VEVs are

Writing

Yukawa matrices become:

Assume messenger mass scales Mf satisfy

Then write

Yukawa matrices become, ignoring phases:

Where

..

.

..

From above we see that 12 3

e c

1212 3

de

3. Quark-Lepton Connections

Assume II: all 13 angles are very small

23

13 23

23

23

13

1

23

13 12

1312 12 2

23

1

23

( )13

( )

23 23

13 23 23

12

3 12

13 12 23 23112 122

E

E E

E E

E

E

i i

i i i

ii i

i

i

i

E

E E

s es e e

e e e

s e

c

e c s

s e c s c ce e

13

122

2

12

11

2

2cos

E

E

Assume I: charged lepton mixing angles are small

Charged Lepton Corrections and sum rule SFK,Antusch; Masina,….

23

13

1

1

12

23

3

2

2323

13

11 2

13

2

i

i

i

i

i

i

s e

e

s e

e

s e s e

†L LEMNS VV V

Note the sum rule212 131 cos

In a given model we can predict and .12

E 12

12 3 121 cos

The Neutrino Sum Rule

Measured by experiment – how well can this

combination be determined?

Predicted by theory e.g.1. bi-

maximal predicts 45o

e.g.2. tri-bimaximal

predicts 35.26o

Tri-bimaximal sum rule

.

.

12 1335.26 cos o

Antusch, Huber, SFK,

Schwetz

Bands show 3 error for a neutrino factory

determination of 13cos

1

13

2 2

12

3

33 ,

s

2 2

in 2 10

EC

12

23

13

33 5

45 10

13

Current 3

A Prediction

• In SUSY we want to understand not only the origin of Yukawa couplings

• But also the soft masses

4. The SUSY Flavour Problem

See-saw parts

The Super CKM Basis

Squark superfields

Quark mass eigenstates

Quark mass eigenvalues

Super CKM basis of the squarks(Rule: do unto squarks as we do unto quarks)

.

Squark mass matrices in the SCKM basis

Flavour changing is contained in off-diagonal elements of

Define parameters as ratios of off-diagonal elements to diagonal elements in the SCKM basis ij = m2

ij/m2diag

Down squark mass matrix in SCKM basis

Flavour changing observables in the down sector Ciuchini,Masiero,

Paradisi,Silvestrini, Vempati,Vives

e.g. slepton doublet mass matrix in charged lepton mass eigenstate basis

Off-diagonal slepton masses in this basis lead to LFV

R DH UH W W Le

e

12LL

2M2 2g v

2gh

11 31212 2

( ) 10 10LLLL

LL

BRm

e

Lepton Flavour Violation (LFV) results from off-diagonal soft masses in the basis where the charged Yukawas

are diagonal (leptonic analogue of SCKM basis)

Lepton Flavour ViolationCiuchini,Masiero,

Paradisi,Silvestrini,Vempati,Vives

Typical upper bounds on

Clearly off-diagonal elements 12 must be very small

Quarks

Leptons

5. SU(5) GUTs and Soft Masses

Ciuchini,Masiero, Paradisi,Silvestrini

,Vempati,Vives

LHC connection: need to measure squark and

slepton masses to relate quark and lepton

flavour violation

23dRR 23

dRRHadronic

constraints

12dRL

12dRL

Leptonic constraints

Leptonic constraints

Hadronic constraints

Ciuchini,Masiero, Paradisi,Silvestrini,

Vempati,Vives

Quark-lepton connection:

LFV processes can constrain Quark Flavour Violation via

GUTs

An old observation: SU(3) family symmetry predicts universal soft mass matrices in the symmetry limit

However Yukawa matrices and trilinear soft masses vanish in the SU(3) symmetry limit

So we must consider the real world where SU(3) is broken by flavons

6. Family Symmetry and Soft Masses

Soft scalar mass operators in SU(3)

Using flavon VEVs previously

Recall Yukawa matrices, ignoring phases:

Where

Under the same assumptions we predict:

In the SCKM basis we find:

Yielding small

parameters

The SUSY CP Problem

• Neutron EDM dn<4.3x10-27e cm

• Electron EDM de<6.3x10-26e cm

Abel, Khalil,Lebedev

Why are SUSY phases so

small?

g Rd

11

SCKMdm A

Ld

Ld

Rd

In the universal case 0d dij ijA AY

0

dSCKMd

ij s

b

y

A A y

y

11 511 10

SCKMd

LRd

m Am

m

0

210A

• Postulate CP conservation (e.g. real) with CP is spontaneously broken by flavon vevs

• This is natural since in the SU(3) limit the Yukawas and trilinears are zero in any case

• So to study CP violation we must consider SU(3) breaking effects in the trilinear soft masses as we did for the scalar soft masses

u dH H

Ross,Vives

7. Family Symmetry and SUSY CP

Soft trilinear operators in SU(3)

Using flavon VEVs previously

N.B parameters ci

f and i

f are real

0A

0A

Compare the trilinears to the Yukawas

They only differ in the O(1) real dimensionless coefficients

0A

Since we are interested in the (1,1) element we focus on the upper 2x2

blocks

The essential point is that , , , are real parameters and phases only appear in the (2,2,) element (due to SU(3) flavons)

Thus the imaginary part of Ad11 in the SCKM basis will be doubly

Cabibbo suppressed

0A

To go to SCKM we first diagonalise Yd

1†

2

00

0dd d

L Ris

yV V

ye

Then perform the same transformation on Ad

11ImSCKMdA

3

1 511 0 1 0 1 0 12

2

Im sin sin sind

SCKMddd d

A A A y A

c.f. universal case 11 0 1Im sinSCKMd

dA y A Extra suppression factor of 0.15

8. Do we need a family symmetry?

Ferretti, SFK, Romanino; Barr

One family of “messengers” dominates Three families of quarks and leptons

LM RM

Suppose Q D UM M M 0 0 0

0

0 1

ULR U U

QU

Y ab adM

cb

0

0

0 1

D DDLR D D

QD

ef ek

Y gf gkM

hf

then in a particular basis

, , , , , , , , , , 1Q QU D

U D

M Ma b c d e f g h k

M M

Not bad! But…

Accidental sym0

1

tan 50

u d

c s

t b

scb

b

m m

m m

m m

mV

m

tan 1C Need broken Pati-Salam…

Conclusion: partial success, but little predictive power esp. in neutrino sector