Aug 29-31, 2005 M. Jezabek 1

Generation of Quark and Lepton Masses in the Standard Model

International WE Heraeus Summer School on Flavour Physics and CP Violation

Dresden, 29 Aug – 7 Sep 2005

Aug 29-31, 2005 M. Jezabek 2

Preliminaries Metric: g00 = -g11 = -g22 = -g33 = 1

Dirac field Ψ (s = ½ ){ γμ , γν } = 2gμν , { γμ , γ5 } = 0

(μ,ν = 0, 1, 2, 3)

γ5† = γ5 , Tr γ5 = 0, γ5

2 = I4 Left- (right-) handed fields:

γ5 ΨL = - ΨL, γ5 ΨR = ΨR

ΨL = ½ (1 - γ5) Ψ, ΨR = ½ (1 + γ5) Ψ

Aug 29-31, 2005 M. Jezabek 3

Weak charged current (one generation)

and weak interaction (four fermion) Hamiltonian

are invariant under chiral transformations

Mass terms

flip chirality and break the chiral invariance of the weak interaction theory

][2 LLLL dueJ

JJ

GF †W

2H

)( RLLRmm ie

Aug 29-31, 2005 M. Jezabek 4

Weyl spinors

0

0

In the chiral (Weyl) basis

with

I

Ii

0

032105

- two component Weyl spinors

iI , ,I

10

01I

01

101

10

013

0

02

i

i

RLR

L,,

Aug 29-31, 2005 M. Jezabek 5

Dirac equation (m = 0):

0p with: pEp

,

LL

RR

R

L

pE

pE

p

p

0

0

0

for m = 0 chirality ↔ helicity

Aug 29-31, 2005 M. Jezabek 6

Lorentz transformations

with

The generators of rotations Ji and of boosts Κi satisfy the commutation relations:

In a more convenient basis

'' xexSx i

][2 i

kijkji Ji ],[kijkji JiJJ ],[kijkji iJ ],[

)(2

1iii iJN )(

2

1'iii iJN

kijkji NiNN ],[ ''' ],[ kijkji NiNN

0],[ ' ii NN22 SUSU

Aug 29-31, 2005 M. Jezabek 7

For Weyl fields ( )0],[

RRR

LLL

S

S

i

R

i

L

eS

eSwith

For :RL 2

1iJ 2

ii

right – handed fermions → ( ½, 0 )left– handed fermions → ( 0, ½ )

22 SUSU of

Aug 29-31, 2005 M. Jezabek 8

Parity

iiii NNNN ','

ii

ii JJ

0

0

I

I

L

RP

R

L

For the generators of rotations and boosts

(pseudo - vectors)(vectors)

and

Under parity transformation:

0PP

Aug 29-31, 2005 M. Jezabek 9

Charge conjugationFor the Pauli matrices

and

Under a Lorentz transformation

ii 22 *

R

ii

L SeS 22222

*

*

LR SS 22 *

RRLLLLLR SSS *22*2**2*2

LLRL S *2

Aug 29-31, 2005 M. Jezabek 10

Charge conjugated bi-spinor:

with

Cc

02iC

*

*

*

*

*

*002

0

0

L

R

R

L

R

LTC

R

LC

i

i

i

i

i

Aug 29-31, 2005 M. Jezabek 11

Charge – conjugation matrix C

For a Dirac particle in em field

Complex conjugation flips the relative sign between and A

])([ meAi

*])(*[ meAi

CTTT C 1† )()( with

CC

Aug 29-31, 2005 M. Jezabek 12

If

one obtains

meAiCmeAiCTT

)(])([ 1*

CmeAi ])([

TTTT

TTC

)( 0†00*0

0*0

The charge conjugation matrix fulfills the relations:

CCCCC TT ,C, 1-†1

Aug 29-31, 2005 M. Jezabek 13

Fermion masses

1. Dirac mass termFor two Weyl spinors and

is invariant under Lorentz and parity transformations:

In bi-spinor notation

L R

)( ††RLLRD m L

)(

2

1††)(

2

1

,i

RR

i

LL ee

†

††††

0

0),(

R

LRLRLLR I

I

Aug 29-31, 2005 M. Jezabek 14

L2. Majorana mass term For a left – handed Weyl spinor

is a right –handed object and

is Lorentz invariant In bi-spinor notation:

is a right – handed field

*2LR i

)( *†TLLLLLM iiM L

)1(2

15L

CTTT

TTTL

CL

CC

CC

)1(2

1)1(

2

1)1(

2

1

)1(2

1)(

55*

5

*5

Aug 29-31, 2005 M. Jezabek 15

The Majorana mass term reads

and for a right – handed field

..C chM LLLM LL

..C chM RRRM RL

R

Aug 29-31, 2005 M. Jezabek 16

GWS Theory of Electroweak Interactions

12 USU

12

RR

L

LL e

e,

RRLL VU ,

local gauge symmetry

doublet SU2 singlet SU2

The most general unitary transformation

includes the lepton – number phase transformation

which is not a local gauge symmetry.R

iRL

iL ee ,

Aug 29-31, 2005 M. Jezabek 17

Generators:

- Pauli matrices

- em charge operator

The gauge fields interact with matter fields and through the covariant derivative

where

LR

1)(,2

1)(

);3,2,1(2

1

3

RL

iii

YY

QTQY

iT

YBigTWigD '

RRLL DiDi 'ˆˆL

BigDBg

iWigD ',2

'

2'

Aug 29-31, 2005 M. Jezabek 18

Spontaneous Symmetry Breaking

Scalar field:

with non – zero vacuum expectation value

In the unitary gauge

2

1,

0

Y

)'2

1

2

1( BgiWigD

2/

02/exp

vv

xi

2

', 0

v

Aug 29-31, 2005 M. Jezabek 19

Higgs mechanism:

...)])(()([8

1

''2

1)()(

2'3'322112

†

v

BggWBggWWWWWg

DD

22

221

2

2'223

3

4)(

2

1

4cossin

0sincos

v

v

gmiWWW

ggmWBZ

mWBA

W

Zww

Aww

2'2

'

2'2sin,cos

gg

g

gg

gww

Mass eigenstates:

with:

Aug 29-31, 2005 M. Jezabek 20

and U(1)Q gauge symmetry remains unbroken

Bg

Wg

A'

3 11

QYTYgg

gTg

3'

'3 )(1

)(1

The electromagnetic field

couples to

Aug 29-31, 2005 M. Jezabek 21

Quarks

3

1,

3

2,

6

1,

Yd

Yu

Yd

u

R

R

L

L

1

1

2

Aug 29-31, 2005 M. Jezabek 22

Fermion masses Yukawa couplings:

where:

Note: in some extended models and may be different Higgs doublets

chuduf

dduf

eef

RLu

RLd

RLeeY

.~

,

,

,

L

*0*

2

~i

~

Aug 29-31, 2005 M. Jezabek 23

YL12 USU

3

2

2

1

6

13

1

2

1

6

1

12

1

2

1

Y

RL

RL

RLe

udu

ddu

ee

~,

,

,

respectively

is invariant. For example:

for respectively

Aug 29-31, 2005 M. Jezabek 24

Under weak isospin transformations

where is a unitary matrix and detU = 1

transforms as a SU2 doublet.

For the mass terms:

RRLL UU ,,kkieU

UU

UUii

ii

2*

22*

2

2*

2**

2*

2

~~

~~U

RLRL

RLRLRL UU

~~

†

Aug 29-31, 2005 M. Jezabek 25

The vacuum expectation value of breaks chiral symmetry. For example:

Dirac masses for charged leptons and quarks

If a right – handed neutrino exists:

Problem: why ?

RLeRLeeeefeef

22/

0,

vv

2,

2,

2

vvvuuddee fmfmfm

0, YR 1

2

~,

vν fmef RL

quarksleptonschargedfff

Aug 29-31, 2005 M. Jezabek 26

Generations

Three generations of quarks and leptons :)(

?:

,,:

,,:

,,:

,,:

,,:

R

RRRR

LLL

eL

RRRR

RRRR

LLL

L

el

eL

bsdd

tcuu

t

b

c

s

d

uQ

Aug 29-31, 2005 M. Jezabek 27

Yukawa couplings + SSB

with:

The mass matrices are complex and can be diagonalised by bi – unitary transformations

LRDLR mf †

fmD

Dm

Aug 29-31, 2005 M. Jezabek 28

with D diagonal, and U and V unitary.

Any (n x n) complex matrix can be diagonalised by a bi – unitary transformation

Proof• and hermitian• The eigenvalues of and are the same

• real and non – negative

unitary U and V:

with D diagonal and real. The columns of AV and U are proportional

A

DAVU †

†AA)()()( †† AXAXAAXXAA

†AAAA†

0)()( †††† AXAXXXAXAX

2†1

2†1

DAVAV

DUAAU

DUAV

AA†

Aug 29-31, 2005 M. Jezabek 29

Diagonalisation of the mass matrices

with diagonal leads to relations between the mass eigenstates and the weak interactions eigenstates :

Weak charged current for quarks

Quark mixing matrix

(Cabibbo; Kobayashi, Maskawa)

Note: in GWS theory only is observable

DD mWmV ~1

Dm~

RL, RL,'

RRLL VW '' ,

)3/1()3/2()3/1()3/2( '' QUQQQJ LqLLL

)3/1()3/2(† QWQWU q

qU

Aug 29-31, 2005 M. Jezabek 30

For (n x n) unitary matrix : n real parameters-(2n - 1) phases of and

p = ( n – 1 )2 observable real parameters

A common convention:where

qU

LdLu

)2)(1(2

1

)1(2

1

nns

nnr

srp

rotation angles

complex phases

For n = 3 (e.g. PDG):

100

0

0

0

010

0

0

0

001

1212

1212

1313

1313

2323

2323 cs

sc

ces

esc

cs

scVi

i

CKM

withijijijij sc sin,cos

Parameters: and 132312 ,,

Aug 29-31, 2005 M. Jezabek 31

Masses of neutrinos

A. Dirac neutrinosThree right – handed sterile neutrinos

B. Dirac neutrinos in Majorana form

with

,R

..chmLDR

DL

c

R

LLn )(

R

R

R

R

Lt

L

Le

L

3

2

1

,

Cc

..)(2

1chMnn L

CL

D L with

om

moM

D

TD

LDRC

RT

DC

LLDRD mmm )(

2

1

2

1L

Aug 29-31, 2005 M. Jezabek 32

↑anticommutation of fermion fields

The matrices of Majorana masses are symmetric:

↑ and are antisymmetric,

anticommutation of fermion fields

CCCCCC TT ,, 1†1

LDRT

LDRT

RT

DT

LC

RT

DC

L

TL

TL

TTLL

TL

CL

mmCCmm

CCCCC

)()(

)()()(

1

111†*†

MMCMCM cTTc

)()( 11

C 1C

Aug 29-31, 2005 M. Jezabek 33

Neutrino masses – general case

c

R

LLn )(

Rn

R

Lt

e

L

R

and

1

h.c.nM)(n LMDC

LMD

2

1L

with

RT

RLT

L

RD

TDLMD

MMMM

Mm

mMM

,MDM is a symmetric complex matrix of

dimension (3 + nR) x (3 + nR)

Note: for quarks and charged leptons due to electric charge conservation

and

0 RL MM

Aug 29-31, 2005 M. Jezabek 34

Any symmetric complex n x n matrix can be diagonalisedby a transformation:

where U is unitary, with real and

Let

where and are real and symmetric, and

is a 2n x 2n real and symmetric matrix

)( MUUM T

)(1 n

diag i

0iiBAM

A B

AB

BAM

Aug 29-31, 2005 M. Jezabek 35

Let

and

i. e.

ni

ni

i

i

ni

i

i ,,2,1,,

,

1,

,

1,

w

w

w

ν

ν

v

i

ii

i

i

w

v

w

vM

i

ii

i

i

v

w

v

wM

iiii

iiii

AB

BA

wwv

vwv

Eigenvalues of are real and equal to and at least n of them are non - negative

M ),,1(, nii

which implies that

Aug 29-31, 2005 M. Jezabek 36

)'()( * UMU

Let where V and W are real matrices which fulfill the following system of equations:

Solution:

For k-th columns in one obtains

It follows that is fulfilled and M is diagonalised by

iWVU

)(

WAWBV

VBWAV

),,,,(

),,,,(

21

21

n

n

W

V

www

vvv

k

kk

k

k

AB

BA

w

v

w

v)(

),,,(),,,( 212211 nnn iiiU uuuwvwvwv )(

Unitarity of U:

mnn

nTmmn

Tmn

Tmnn

Tmmnm ii

w

vwvwwvvwvwvuu ,)()(†

Aug 29-31, 2005 M. Jezabek 37

Massive neutrinos in the Standard Model

Before 1998 (SuperK):

A simple extention of SM:

The right – handed neutrinos are sterile. For singlets of gauge group SU3 x SU2 x U1 explicit Majorana masses are allowed

a new mass scale|MR|

Two mass scales: ?||

102

R

D

m

GeVm v

The Majorana masses of the active neutrinos are forbidden by the electroweak SU2 x U1 gauge symmetry

ML = 0

0m

L

CR

LL

Rn

R nand

R

1

L

R

Aug 29-31, 2005 M. Jezabek 38

Seesaw Mechanism

h.c.nM)(n LMDC

LMD

2

1L

RD

TDMD

Mm

mM

0

1||/ RD

MmFor the mass spectrum splits into low and high mass parts:

with a unitary matrix

and

LLL nUnn †'

),(A

2†

AI

AIU O

with:

1|| A

,UMM MDTMD U

Aug 29-31, 2005 M. Jezabek 39

'*

' 0

RRD

T

D

T

MIA

AI

Mm

m

IA

AIM

RT

DDRR

RTT

DDT

TDRD

MmAAmMM

AMAAmmA

AmAAMm

*†'

*

DR mMA 1

with

M/ is in a block – diagonal form

if

Aug 29-31, 2005 M. Jezabek 40

RMM

0

0'

with

DRT

D mMm 1

LDRCR

CR

CRRDL

CRLL

CR

LL

mM

MmA

IA

AIn

1'

†1††'

†'

)(

Aug 29-31, 2005 M. Jezabek 41

Low mass sector

For

)(

)(

||||

'

'

O

O

RR

LL

Dm

12210||

1||

GeVm

MeVm

D

D

Aug 29-31, 2005 M. Jezabek 42