Family Symmetry Solution to the SUSY Flavour and CP Problems

Family Symmetry Solution to the SUSY Flavour and CP

Problems

Plan of talk:Plan of talk:

I.I. Family SymmetryFamily Symmetry

II.II. Solving SUSY Flavour and CP Solving SUSY Flavour and CP ProblemsProblems

Work with and Michal Malinsky

3 3

3 2 2

3 2

0

1

uY

• Universal form for mass matrices, with Georgi-Jarlskog factors

• Texture zero in 11 position

Fermion mass spectrum well described by Symmetric Yukawa 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

1( ) , ( ) 3

3s d

GUT GUTe

m mM M

m m

Introduction to Family Symmetry

G.Ross et al

•To account for the fermion mass hierarchies we introduce a spontaneously broken family symmetry

•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

•The idea is to use the expansion parameter to derive fermion textures by the Froggatt-Nielsen mechanism (see later)

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) etc

If these are gauged and broken at high energies then no direct low energy signatures

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

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

Froggatt-Nielsen Mechanism

What is the origin of the higher order operators?

To answer this 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

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 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)

The motivation for 123 from tri-bimaximal neutrino mixing

3

3 3

3 3

0 0

0

1LRY

3

0

0

1

3

23

0

1

1

3123

1

1

1

For tri-bimaximal neutrino mixing we

need

A Realistic 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

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

• But also the soft masses

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

Typical upper bounds on

Clearly off-diagonal elements 12 must be very small

Quarks

Leptons

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

Solving the SUSY Flavour Problem with SU(3) Family Symmetry

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

Solving the SUSY CP Problem with SU(3) Family Symmetry

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

31 511 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

Conclusions• SU(3) gauged family symmetry, when spontaneously

broken by particular flavon vevs, provides an explanation of tri-bimaximal neutrino mixing

• When combined with SUSY it gives approx. universal squark and slepton masses, suppressing SUSY FCNCs

• It also suppresses SUSY contributions to EDMs by an extra order of magnitude compared to mSUGRA or CMSSM remaining phase must be <0.1

• Maybe SUGRA can help with this remaining 10% tuning

problem – work in progress