Fermion Masses and Unification

Lecture I Fermion Masses and MixingsLecture II UnificationLecture III Family Symmetry and UnificationLecture IV SU(3), GUTs and SUSY Flavour

Steve KingUniversity of Southampton

Lecture IFermion Masses and Mixings

1. The Flavour Problem and See-Saw

2. From low energy data to high energy data

3. Textures in a basis

Appendix 1 References

Appendix 2 Basis Changing

1.The Flavour Problem and See-Saw

The Flavour Problem

1. Why are there three families of

quarks and leptons?

Generations of Generations of

matter matter

III III II II I I

tau

-neutrino

bbottom

ttop

muon

-neutrino

sstrange

ccharm

eelectron

ee-neutrino

Lep

tons

L

epto

ns

ddown

upu

Qua

rks

Qua

rks

Horizontal

Ver

tica

l

b

cs

ud

e

1

2

3

310110

210110

110

2101210

310

410

1210

1110

Family symmetry e.g. SU(3)

GUT symmetry e.g. SO(10)

1

t

The Flavour Problem2. Why are quark and charged

lepton masses so peculiar?

The Flavour Problem3. Why is lepton mixing so large?

c.f. small quark mixing

3

2

3 2

1

1

1ijV

Harrison, Perkins, Scott

e.g.Tri-bimaximal

CP

CP

The Flavour Problem4. What is the origin of CP violation?

Lepton CP Violation?

Normal Inverted

The Flavour Problem5. Why are neutrino masses so small?

See-saw mechanism is most elegant solution

The See-Saw Mechanism

Light neutrinos

Heavy particles

Type I see-saw mechanism Type II see-saw mechanism (SUSY)

R

LL

2II uLL

vm Y

M

The see-saw mechanism

L L

Heavy triplet

cRR R RM

1I TLL LR RR LRm m M m

Y

Type II Type I

P. Minkowski (1977), Gell-Mann, Glashow, Mohapatra, Ramond, Senjanovic, Slanski, Yanagida (1979/1980)

Lazarides, Magg, Mohapatra, Senjanovic, Shafi, Wetterich (1981)

See-Saw Standard Model (type I)Yukawa couplings to 2 Higgs doublets (or one with )

Insert the vevs

Rewrite in terms of L and R chiral fields, in matrix notation

12 . .E c

L LR R L LR R R RR Re m e m M H c

The See-Saw Matrix

1II TLL LL LR RR LRm m m M m

cLL L Lm

II c

c LL LR LL R T

LR RR R

m m

m M

Dirac matrix

Heavy Majorana matrix

Light Majorana matrix

Diagonalise to give effective mass

Type II contribution (ignored here)

Lepton mixing matrix VMNS

†

0 0

0 0

0 0

L R

eE EE

LR

m

V m V m

m

†L LEMNSV V V

23 13 12MNSV V V V

Neutrino mass matrix (Majorana)

1

2

3

0 0

0 0

0 0

L LTLL

m

V mV m

m

Atmospheric Reactor Solar

Oscillation phase

13 12

23 12

23 13

13 13 12 12

23 23 12 12

23 23 13 13

1 0 0 0 0

0 0 1 0 0

0 0 10 0

i i

i iMNS

i i

c s e c s e

V c s e s e c

s e c s e c

13 23 12

Defined as

Can be parametrised as

Quark mixing matrix VCKM

†

0 0

0 0

0 0

L R

uU UU

LR c

t

m

V m V m

m

†LLU DCKM VV V

23 13 12CKMV R V R

†

0 0

0 0

0 0

L R

dD DD

LR s

b

m

V m V m

m

13 13 12 12

23 23 12 12

23 23 13 13

1 0 0 0 0

0 0 1 0 0

0 0 0 0 1

i

CKMi

c s e c s

V c s s c

s c s e c

Defined as

Can be parametrised as

Phase convention independent

2. From low energy data to high energy data

(From Particle Data Book)

Ross and SernaQuark data (low energy)

Neutrino Masses and Mixings

Normal Inverted

Andre de Gouvea

12

23

13

13 0.1

2.4 0.1

0.20 0.05

c.f. quark mixing angles

Renormalisation Group running

MEW

MSUSY M

1 M2 M

3M

U

1016 [GeV]102

RH neutrino masses

Parameter at MU

RG running Parameter at MEW

RGEs for gauge couplings (to

one loop accuracy)

194110 6( , , 7)ab

SM beta functions MSSM beta functions

Latest coupling constant measurements at energy scale: MZ

SM couplings at low energy

. .

Two-loop RGEs for the SM:

. .Evolution of SM couplings

Two-loop RGEs for the MSSM with 1 TeV effective SUSY threshold:

. .

. .

MSSM

Two-loop RGEs for the MSSM with 1 TeV effective SUSY threshold:

MSSM

Two-loop RGEs for the MSSM with 250 GeV effective SUSY threshold:

MSSM

RGEs for t,b, in the MSSM

RGEs for Yukawa matrices in MSSM

Wavefunction anomalous dimensionsRGEs (one-loop accuracy)

Charged fermion data (high energy)

SUSY thresholds

Ross and Serna

3. Textures in a basis

Symmetric hierarchical matrices with 11 texture zero motivated by

| | dus

s

mV

m

12

12 22

0LR

mm

m m

This motivates the symmetric down texture at GUT scale of form

223( )| |

DLR

cbb

mV

m 313( )

| |DLR

ubb

mV

m

Hierarchical Symmetric Textures

Gatto et al

3 3

3 2 2

3 2

0

1

dLRY

¼ 0.2 is the Wolfenstein Parameter

3 3

3 2 2

3 2

0

1

0.15DLR

b

m

m

3 3

3 2 2

3 2

0

0.05

1

ULR

t

m

m

Up quarks are more hierarchical than down quarks

This suggests different expansion parameters for up and down

4 2: : : :1d s bm m m 4 2: : : :1u c tm m m

Detailed fits require numerical (order unity) coefficients

Detailed fits at the GUT ScaleRoss and Serna

No SUSY thresholds

With SUSY thresholds Ross and Serna

1( ) 1, ( ) , ( ) 3

3b s d

GUT GUT GUTe

m m mM M M

m m m

Georgi-Jarlskog

Final remarks on choice of basis We have considered a particular choice of quark texture in a particular basis

But it is shown in the Appendix that all choices of quark mass matrices that lead to the same quark masses and mixing angles may be related to each other under a change of basis.

For example all quark mass matrices are equivalent to the choice

3 3

3 2 2

3 2

0

1

0.15DLR

b

m

m

3 3

3 2 2

3 2

0

0.05

1

ULR

t

m

m

0 0

0 0

0 0

dDLR CKM s

b

m

m V m

m

0 0

0 0

0 0

uULR c

t

m

m m

m

However this is only true in the Standard Model, and a given high energy theory of flavour will select a particular preferred basis. Also in the see-saw mechanism all choices of see-saw matrices are NOT equivalent.

Appendix 1 ReferencesW. De Boer hep-ph/9402266

S.Raby ICTP Lectures 1994

G.G.Ross ICTP Lectures 2001

J.C. Pati ICTP Lectures 2001

S. Barr ICTP Lectures 2003

S. Raby hep-ph/0401115

S.Raby PDB 2006

A. Ceccucci et al PDB 2006

G. Ross and M. Serna 0704.1248

D. Chung et al hep-ph/0312378

S.F. King hep-ph/0310204

Appendix 2 Basis Changing

2.1 Quark sector

2.2 Effective Majorana sector

2.3 See-saw sector

S.F. King hep-ph/0610239