July 1-2, 2004 1

The Origin of CP Violationin the Standard Model

Topical Lectures

July 1-2, 2004

Marcel Merk

July 1-2, 2004 2

Contents

• Introduction: symmetry and non-observables• CPT Invariance• CP Violation in the Standard Model Lagrangian• Re-phasing independent CP Violation quantities• The Fermion masses• The matter anti-matter asymmetry

Theory Oriented!

July 1-2, 2004 3

Literature

• C.Jarlskog, “Introduction to CP Violation”, Advanced Series on Directions in High Energy Physics – Vol 3: “CP Violation’, 1998, p3.

• Y.Nir, “CP Violation In and Beyond the Standard Model”, Lectures given at the XXVII SLAC Summer Institute, hep-ph/9911321.

• Branco, Lavoura, Silva: “CP Violation”, International series of monographs on physics, Oxford univ. press, 1999.

• Bigi and Sanda: “CP Violation”, Cambridge monographs on particle physics, nuclear physics and cosmology, Cambridge univ. press, 2000.

• T.D. Lee, “Particle Physics and Introduction to Field Theory”, Contemporary Concepts in Physics Volume 1,Revised and Updated First Edition, Harwood Academic Publishers, 1990.

• C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”, Frontiers in Physics, Benjamin-Cummings, 1983.

• H. Fritsch and Z. Xing, “Mass and Flavour Mixing Schemes of Quarks and Leptons”, hep-ph/9912358.

• Mark Trodden, “Electroweak Baryogenesis”, hep-ph/9803479.

References:

July 1-2, 2004 4

Introduction: Symmetry and non-Observables

T.D.Lee:“The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the non-observables”

There are four main types of symmetry:• Permutation symmetry:

Bose-Einstein and Fermi-Dirac Statistics• Continuous space-time symmetries:

translation, rotation, acceleration,…• Discrete symmetries:

space inversion, time inversion, charge inversion • Unitary symmetries: gauge invariances:

U1(charge), SU2(isospin), SU3(color),..

If a quantity is fundamentally non-observable it is related to an exact symmetry

If a quantity could in principle be observed by an improved measurement; the symmetry is said to be broken

Noether Theorem: symmetry conservation law

July 1-2, 2004 5

Symmetry and non-observables

1r��������������

2r��������������

d��������������

Simple Example: Potential energy V between two particles:

Absolute position is a non-observable:The interaction is independent on the choice of 0.

Symmetry: V is invariant under arbitrary space translations:

2 2r r d ������������������������������������������

1 1r r d ������������������������������������������

1 2V V r r ����������������������������

1 2 1 2 1 2 0d

p p F F Vdt

������������������������������������������������������������������������������������

Consequently: Total momentum is conserved:

00’

July 1-2, 2004 6

Symmetry and non-observablesNon-observables Symmetry Transformations Conservation Laws or Selection

Rules

Difference between identical particles

Permutation B.-E. or F.D. statistics

Absolute spatial position Space translation momentum

Absolute time Time translation energy

Absolute spatial direction Rotation angular momentum

Absolute velocity Lorentz transformation generators of the Lorentz group

Absolute right (or left) parity

Absolute sign of electric charge charge conjugation

Relative phase between states of different charge Q

charge

Relative phase between states of different baryon number B

baryon number

Relative phase between states of different lepton number L

lepton number

Difference between different co- herent mixture of p and n states

isospin

r r

e e

t t r r ������������� �

iQe iNe iLe

p pU

n n

r r

July 1-2, 2004 7

Parity ViolationBefore 1956 physicists were convinced that the laws of naturewere left-right symmetric. Strange?

A “gedanken” experiment: Consider two perfectly mirror symmetric cars:

“L” and “R” are fully symmetric,Each nut, bolt, molecule etc.However the engine is a black box

Person “L” gets in, starts, ….. 60 km/h

Person “R” gets in, starts, ….. What happens?

What happens in case the ignition mechanism uses, say, Co60 decay?

“L” “R”

Gas pedaldriver

Gas pedal driver

July 1-2, 2004 8

CPT InvarianceLocal Field theories always respect:

• Lorentz Invariance• Symmetry under CPT operation (an electron = a positron travelling back in time)=> Consequence: mass of particle = mass of anti-particle:

• Question 1:

The mass difference between KL and KS: m = 3.5 x 10-6 eV => CPT violation?

• Question 2: How come the lifetime of KS = 0.089 ns while the lifetime of the KL = 51.7 ns?

• Question 3:

BaBar measures decay rate B-> J/ KS and Bbar-> J/ KS. Clearly not the same: how can it be?

† 1

1

M p p H p p CPT CPT H CPT CPT p

p CPT H CPT p p H p M p

=> Similarly the total decay-rate of a particle is equal to that of the anti-particle

Answer 3:Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so) is the same for B and Bbar. (Amazing! – at least to me)

Answer 1 + 2: A KL ≠ an anti-KS particle!

(Lüders, Pauli, Schwinger)

(anti-unitarity)

July 1-2, 2004 9

LSM contains:LKinetic : fermion fieldsLHiggs : the Higgs potentialLYukawa : the Higgs – Fermion interactions

Plan:• Look at symmetry aspects of the Lagrangian• How is CP violation implemented?→ Several “miracles” happen in symmetry breaking

(3) (2) (1) (3) (1)SM C L Y C EMG SU SU U SU U

CP in the Standard Model Lagrangian(The origin of the CKM-matrix)

Standard Model gauge symmetry:

Note Immediately: The weak part is explicitly parity violating

Outline:

• Lorentz structure of the Lagrangian

• Introduce the fermion fields in the SM

• LKinetic : local gauge invariance : fermions ↔ bosons

• LHiggs : spontaneous symmetry breaking

• LYukawa : the origin of fermion masses

• VCKM : CP violation

July 1-2, 2004 10

Lagrangian Density

, , , ,j jx t x t x t

L L

4 ,A d x x t

L

, , 1, 2,...,j x t j N

Local field theories work with Lagrangian densities:

The fundamental quantity, when discussing symmetries is the Action:

If the action is (is not) invariant under a symmetry operation then the symmetry in question is a good (broken) one

=> Unitarity of the interaction requires the Lagrangian to be Hermitian

with the fields taken at ,x t

July 1-2, 2004 11

Structure of a Lagrangian

2

5

, , , , ,

1, , ,

2, , ,

x t i x t x t m x t x t

x t x t V x t

x t a ib x t x t

L

( ) 0i m

Example:Consider a spin-1/2 (Dirac) particle (“nucleon”) interacting with a spin-0 (Scalar) object (“meson”)

Meson potential

Nucleon field

Nucleon – meson interaction

Lorentz structure: a Lagrangian in field theory can be built using combinations of:S: Scalar fields : 1P: Pseudoscalar fields : 5

V: Vector fields : A: Axial vector fields : 5

T: Tensor fields : Dirac field

2( ) 0i m

Exercise:What are the symmetries of this theory under C, P, CP ? Can a and b be any complex numbers?Note: the interaction term contains scalar and pseudoscalar parts

Scalar field

Violates P, conserves C, violates CPa and b must be real from Hermeticity

July 1-2, 2004 12

1 1 2 2 1 22 2 1 1 2 1

1 1 2 2 1 25 2 5 2 5 1 5 1 5 2 5 1

1 1 2 2 1 22 2 1 1 2 1

1 1 2 2 1 25 2 5 2 5 1 5 1 5 2 5 1

1 1 22 2 1

:

:

:

:

:

P C CP T CPT

S

P

V

A

T

1 2 22 1 1

Transformation Properties

Transformation properties of Dirac spinor bilinears:

†

2 00

†

†

( , ) ( , ) ( , )

: ( , ) ( , ) ( , )

: ( , ) ( , ) ( , )

: ( , ) ( , ) ( , )

: ( , ) ( , ) ( , )

T

P C

x t x t x t

Scalar Field x t x t x t

Dirac Field x t x t i x t

Vector Field V x t V x t V x t

Axial Field A x t A x t A x t

00

Feynman Metric:

, kkQ Q Q Q

(Ignoring arbitrary phases)

c→c* c→c*

July 1-2, 2004 13

The Standard Model Lagrangian

SM Kinetic Higgs Yukawa L L L L

• LKinetic : •Introduce the massless fermion fields

•Require local gauge invariance => gives rise to existence of gauge bosons

• LHiggs : •Introduce Higgs potential with <> ≠ 0

•Spontaneous symmetry breaking

• LYukawa : •Ad hoc interactions between Higgs field & fermions

• LYukawa → Lmass : • fermion weak eigenstates:

-- mass matrix is (3x3) non-diagonal • fermion mass eigenstates: -- mass matrix is (3x3) diagonal

• LKinetic in mass eigenstates: CKM – matrix

(3) (2) (1) (3) (1)SM C L Y C QG SU SU U SU U

The W+, W-,Z0 bosons acquire a mass

=> CP Conserving

=> CP Conserving

=> CP violating with a single phase

=> CP-violating

=> CP-conserving!

=> CP violating with a single phase

July 1-2, 2004 14

Fields: Notation

(3,2,1 6)(3,2,1 6)

(3,2,1 6)I

I

I

LiLi

Qd

u

SU(3)C SU(2)L

YLeft- handed

generationindex

Interaction rep.Quarks:

Leptons:

Scalar field:

(1,2, 1 2)(1,2, 1 2)

(1,1, 1)

II

I LiLi

Ll

(3,1,2 3)IRiu (3,1, 1 3)I

Rid

(1,1, 1)IRil

0(1, 2,1 2)

•

• •

•

• •

•

Q = T3 + Y

Under SU2:Left handed doubletsRight hander singlets

Note:Interaction representation: standard model interaction is independent of generation number

IRi

5 51 1;

2 2L R

Fermions: with = QL, uR, dR, LL, lR, R

July 1-2, 2004 15

Fields: Notation

Explicitly:

3

3

( )1 61 2

1 2

, , , , , ,(3,2,1 6) , ,

, , , , , ,

I I Ir r r

I Ig gI

Li

L L L

I I

I I Ib b b

I I Ib b b

I I Ir r

g

g g grI

IT

YT

u c t

d s b

u c t

d

c t

d sQ

bb

u

s

• Similarly for the quark singlets:

2 3

1 3

(3,1, 2 3) , , , , , , , ,

(3,1, 1 3) , , , , , , , ,

IRi R R R

IR

I I Ir r r

I I Ir

I I Ir r r

I I Ir r

I I Ir r r

I I Ii Rr r r r Rr R r

y

y

t

d s b

u c tu

d d s b

cuu c t

d s b

3

3

1 2

1 2(1,2, 1 2) , ,

II IeI

Li I IIL LL

TL

Te

• And similarly the (charged) singlets: (1,1, 1) , ,I I I IRi R R Rl e

• The left handed leptons:

• The left handed quark doublet :

Q = T3 + Y

July 1-2, 2004 16

Intermezzo: Local Gauge Invariance in a single transparancy Basic principle: The Lagrangian must be invariant under local gauge transformations

Example: massless Dirac Spinors in QED: i L

“global” U(1) gauge transformation: ix x e x

“local” U(1) gauge transformation: i xx x e x

Is the Lagrangian invariant?

;i x i x

i x i x

x e x x e x

x e x ie x x

Then: i i x

Not invariant!

=> Introduce the covariant derivative: D ieA

and demand that transforms as: 1A A A x

e

Then it turns out that:

L L L

• Introduce charged fermion field (electron)• Demand invariance under local gauge transformations (U(1))• The price to pay is that a gauge field A must be introduced at the same time (the photon)

is invariant!

Conclusion:

July 1-2, 2004 17

KineticL : Fermions + gauge bosons + interactions

Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant

Start with the Dirac Lagrangian: ( )i L

Replace: s a a b big G igW T ig YL BD

Fields:

Generators:

Gaa : : 8 gluons

Wb: weak bosons: W1, W2, W3

BB: : hypercharge boson

La : Gell-Mann matrices: ½ a (3x3) SU(3)C

Tb : Pauli Matrices: ½ b (2x2) SU(2)L

Y : Hypercharge: U(1)Y

:The Kinetic PartSM Higgs YukKinetic awa L L LL

For the remainder we only consider Electroweak: SU(2)L x U(1)Y

July 1-2, 2004 18

: The Kinetic PartSM Higgs YukKinetic awa L L LL

Exercise:Show that this Lagrangian formally violates both P and CShow that this Lagrangian conserves CP

: ( ) ( )

, , , ,

kinetic

I I I I ILi Ri Ri Li Ri

i i D

with Q u d L l

L

For example the term with QLiI becomes:

( )2 2

(

6

)I I Ikinetic Li Li Li

I ILi b La a b is

i i ig G gW g

Q iQ D Q

iQ B Q

L

and similarly for all other terms (uRiI,dRi

I,LLiI,lRi

I).

Writing out only the weak part for the quarks:

1 1 2 2 3 3( , ) ,

.

2

..2 2

IIWeak I

kinetic L LL

I I I I I I I IL L L L L L L L

uu d i u d

d

g giu u id d u d d u

W

W

W

W

ig W

L

uLI

dLI

gW+

LKin = CP conserving

W+ = (1/√2) (W1+ i W2)W- = (1/√ 2) (W1 – i W2)

L=JW

July 1-2, 2004 19

: The Higgs PotentialSM Kinetic Hig Yg ukawas LL L L

†

22 † †1

2

Higgs Higgs

Higgs

D D V

V

L

2 0 :

0

→Note LHiggs = CP conserving

V

V()

Symmetry BrokenSymmetry

2 0 :

0

2v

~ 246 GeV

Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value

Procedure:0 0 0

e i m

e i m

Substitute: 0

0

2

v He

And rewrite the Lagrangian (tedious): 1. 2. The W+,W-,Z0 bosons acquire mass3. The Higgs boson H appears

: (3) (2) (1) (3) (1)SM C L Y C EMG SU SU U SU U

“The realization of the vacuum breaks the symmetry”

(The other 3 Higgs fields are “eaten” by the W, Z bosons)

2v

July 1-2, 2004 20

: The Yukawa PartSM Kinetic Hig Y kawags u L L L L

, ,d u lij ij ijY Y Y

Since we have a Higgs field we can add (ad-hoc) interactions between and the fermions in a gauge invariant way.

. .LiYukawa ij Rj h cY L

L must be Her-mitian (unitary)

. .

I I I ILi Rj Li Rj

d ui

I ILi

j ij

lj Ri j

Y Y

Y

Q d Q u

L l h c

The result is:

are arbitrary complex matrices which operate in family space (3x3)=> Flavour physics!

doubletssinglet

* *2

0 1

1 0i

With:

To be manifestly invariant under SU(2)

July 1-2, 2004 21

: The Yukawa Part

0 0 0

0 0 0

0

11 12 13

21 22 13

31 32 0 33 0

, , ,

, , ,

, , ,

I I I I I IL L L L L L

I I I I I IL L L L L L

I

d d d

d

I I I I

d d

d IL L L L

dL L

d

u d u s u b

c d c s

Y Y Y

Y Y Y

Y

c

Y Y

b

t d t s t b

IR

IR

IR

d

s

b

Writing the first term explicitly:

0( , )I I

L LIR

dij jiY u dd

SM Kinetic Hig Y kawags u L L L L

Question:In what aspect is this Lagrangian similar to the example of the nucleon-meson potential?

July 1-2, 2004 22

: The Yukawa Part

†*Li Rj Rji Lij ijY Y

†Li Rj Rj Li

SM Kinetic Hig Y kawags u L L L L

Exercise (intuitive proof) Show that:

• The hermiticity of the Lagrangian implies that there are terms in pairs of the form:

• However a transformation under CP gives:

and leaves the coefficients Yij and Yij* unchanged

CP is conserved in LYukawa only if Yij = Yij

*

. .LiYukawa ij Rj h cY L

In general LYukawa is CP violating † †det , 0d d u um Y Y Y Y

Formally, CP is violated if:

July 1-2, 2004 23

: The Yukawa PartSM Kinetic Hig Y kawags u L L L L

There are 3 Yukawa matrices (in the case of massless neutrino’s):

, ,d u lij ij ijY Y Y

Each matrix is 3x3 complex:• 27 real parameters• 27 imaginary parameters (“phases”)

many of the parameters are equivalent, since the physics described by one set of couplings is the same as another It can be shown (see ref. [Nir]) that the independent parameters are:

• 12 real parameters• 1 imaginary phase

This single phase is the source of all CP violation in the Standard Model

……Revisit later

July 1-2, 2004 24

: The Fermion MassesYukawa MassL L

0

( , ) ... ...I I IYuk

d u lij ij jL Rj iL iu Yd YdY

L

0. . . :2

v HS S B e

S.S.B

Start with the Yukawa Lagrangian

After which the following mass term emerges:

. .

I d I I u IYuk Mass Li ij Rj Li ij Rj

I l ILi ij Rj

d M d u M u

l M l h c

L L

with , ,2 2 2

d d u u l lij ij ij ij ij ij

v v vM Y M Y M Y

LMass is CP violating in a similar way as LYuk

July 1-2, 2004 25

: The Fermion MassesYukawa MassL L

., , , ,, , .

I I

I I I I I I I I

L LI

I

I I

I

R

I

L

R

I

R

I

ed u

s u c t c e

b t

hs cd bMassd u lM M M

L

†f f fdiagonaL R l

f MV M V

S.S.B

Writing in an explicit form:

The matrices M can always be diagonalised by unitary matrices VLf and VR

f such that:

Then the real fermion mass eigenstates are given by:

dLI , uL

I , lLI are the weak interaction eigenstates

dL , uL , lL are the mass eigenstates (“physical particles”)

I ILi Lj Ri Rj

I ILi Lj Ri Rj

I ILi Lj R

d dL Rij ij

u uL Rij ij

l lL R Rjiiij j

d d d d

u u u

V V

V V

V V

u

l l l l

† †, ,

I

I I I I

I

f f f fL R

fL RV

d

d MV V Vs b s

b

July 1-2, 2004 26

: The Fermion MassesYukawa MassL L

,

., , .

, , ,L

d u

sL

R R

c

e

R

L

b t

Mass

m m

m m

h

m m

m

m

d s b

d u

s u c t c

b t

e

e

m

c

L

S.S.B

In terms of the mass eigenstates:

Mass u c t

d s b

e

uu cc tt

dd ss bb

m m m

m m m

m ee m m

L= CP Conserving?

In flavour space one can choose:Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are non-diagonalMass basis: The fermion masses are diagonal, but some gauge currents (charged weak interactions) are not diagonal in flavour space

In the weak basis: LYukawa = CP violatingIn the mass basis: LYukawa → LMass = CP conserving

=>What happened to the charged current interactions (in LKinetic) ?

July 1-2, 2004 27

: The Charged CurrentCKMWL L

The charged current interaction for quarks in the interaction basis is:

The charged current interaction for quarks in the mass basis is:

, ,2

CKMLW

L

d

u c t V s

b

gW

L

†

2u

L L LW

dLi iu V

gV d W

L

The unitary matrix: †u dCKM L LV V V

is the Cabibbo Kobayashi Maskawa mixing matrix:

† 1CKM CKMV V

2I ILi LW i

gWu d

L

With:

Lepton sector: similarly †lMNS L LV V V

However, for massless neutrino’s: VL = arbitrary. Choose it such that VMNS = 1

=> There is no mixing in the lepton sector

July 1-2, 2004 28

Flavour Changing Neutral Currents

3 3( )2

( )6

I I INC Li Li Li

g gQ iQ QW B

- L

To illustrate the SM neutral current take the W3 and B term of the Kinetic Lagrangian:

In terms of physical fields no non-diagonal contributions occur for the neutral Currents. => GIM mechanism

21 1sin( )

co 2 3sI I I

Z Li Li LiW

W

gQ QZQ

- L

2

2 †

1 1( ) sin

cos 2 3

1 1sin

cos 2 3

I I IZ Li W Li Li

W

d dW Li L L Li

W

gd d Z d

gd V V d Z

- L

And consider the Z-boson field: 3cos sinW WZ W B

Take further QLiI=dLi

I

† †( 1)u u d dR R R RV V V V

21 1( ) sin ...

cos 2 3Z Li W Li Li Li LiW

gQ d d Z u u Z

- L

Use:

Standard Model forbids flavour changing neutral currents.

tan W g g and

July 1-2, 2004 29

Charged Currents

5 5 5 5*

5 * 5

2 2

1 1 1 1

2 2 2 22 2

1 12 2

I I I ICC Li Lj Lj Li CC CC

i ij j j ij i

i ij j j ji i

g gu W d d W u J W J W

g gu W V d d W V u

g gu W V d d W V u

L

5 * 51 12 2

CP iCC j ij i i ij k

g gd W V u u W V d

L

A comparison shows that CP is conserved only if Vij = Vij*

(Together with (x,t) -> (-x,t))

The charged current term reads:

Under the CP operator this gives:

In general the charged current term is CP violating

July 1-2, 2004 30

Charged Currents

5 5 5 5*

5 * 5

2 2

1 1 1 1

2 2 2 22 2

1 12 2

I I I ICC Li Lj Lj Li CC CC

i ij j j ij i

i ij j j ji i

g gu W d d W u J W J W

g gu W V d d W V u

g gu W V d d W V u

L

5 * 51 12 2

CP iCC j ij i i ij k

g gd W V u u W V d

L

A comparison shows that CP is conserved only if Vij = Vij*

(Together with (x,t) -> (-x,t))

The charged current term reads:

Under the CP operator this gives:

In general the charged current term is CP violating

July 1-2, 2004 31

Where were we?

July 1-2, 2004 32

The Standard Model Lagrangian (recap)

SM Kinetic Higgs Yukawa L L L L

• LKinetic : •Introduce the massless fermion fields

•Require local gauge invariance => gives rise to existence of gauge bosons

• LHiggs : •Introduce Higgs potential with <> ≠ 0

•Spontaneous symmetry breaking

• LYukawa : •Ad hoc interactions between Higgs field & fermions

• LYukawa → Lmass : • fermion weak eigenstates:

-- mass matrix is (3x3) non-diagonal • fermion mass eigenstates: -- mass matrix is (3x3) diagonal

• LKinetic in mass eigenstates: CKM – matrix

(3) (2) (1) (3) (1)SM C L Y C QG SU SU U SU U

The W+, W-,Z0 bosons acquire a mass

=> CP Conserving

=> CP Conserving

=> CP violating with a single phase

=> CP-violating

=> CP-conserving!

=> CP violating with a single phase

July 1-2, 2004 33

Quark field re-phasing

u d

c s

t b

ud us ub

cd cs cb

td ts tb

e V V V e

V e V V V e

e V V V e

uiiLi Liu e u dii

Li Lid e dUnder a quark phase transformation:

and a simultaneous rephasing of the CKM matrix:

expj j jV i V or

the charged current CC Li ij LjJ u V d is left invariant

Degrees of freedom in VCKM in 3 N generationsNumber of real parameters: 9 + N2

Number of imaginary parameters: 9 + N2

Number of constraints (VV† = 1): 9 - N2

Number of relative quark phases: 5 - (2N-1) -----------------------Total degrees of freedom: 4 (N-1)2

Number of Euler angles: 3 N (N-1) / 2Number of CP phases: 1 (N-1) (N-2) / 2

No CP violation in SM. This is the reason Kobayashi and Maskawa first suggested a third family of fermions!

cos sin

sin cosCKMV

2 generations:

July 1-2, 2004 34

The LEP collider @ CERN

Maybe the most important result of LEP: “There are 3 generations of neutrino’s”

L3

Aleph

OpalDelphi

Geneva Airport “Cointrin”MZ

Light, left-handed, “active”

July 1-2, 2004 35

The lepton sector (Intermezzo)

• N. Cabibbo: Phys. Rev.Lett. 10, 531 (1963)– 2 family flavour mixing in quark sector (GIM mechanism)

• M.Kobayashi and T.Maskawa, Prog. Theor. Phys 49, 652 (1973)– 3 family flavour mixing in quark sector

• Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. 28, 870 (1962)– 2 family flavour mixing in neutrino sector to explain neutrino oscillations!

• In case neutrino masses are of the Dirac type, the situation in the lepton sector is very similar as in the quark sector: VMNS ~ VCKM.

– The is one CP violating phase in the lepton MNS matrix

• In case neutrino masses are of the Majorana type (a neutrino is its own anti-particle → no freedom to redefine neutrino phases)

– There are 3 CP violating phases in the lepton MNS matrix• However, the two extra phases are unobservable in neutrino oscillations

– There is even a CP violating phase in case Ndim = 2

July 1-2, 2004 36

Lepton mixing and neutrino oscillations

• In the CKM we write by convention the mixing for the down type quarks; in the lepton sector we write it for the (up-type) neutrinos. Is it relevant?

– If yes: why? – If not, why don’t we measure charged lepton oscillations rather then neutrino

oscillations?

lLI

νLI

W+ 2 2

CC i ij j i ij j

g gJ l V lV

However, observation of neutrino oscillations is possible due to small neutrino mass differences.

Question:

e

W

July 1-2, 2004 37

Rephasing Invariants

• Simplest: Ui = |Vi|2 is independent of quark re-phasing

• Next simplest: Quartets: Qij = Vi Vj Vj* Vi

* with ≠ and i≠j

– “Each quark phase appears with and without *”

• V†V=1: Unitarity triangle: Vud Vcd* + Vus Vcs* + Vub Vcb* = 0

– Multiply the equation by Vus* Vcs and take the imaginary part:

– Im (Vus* Vcs Vud Vcd

*) = - Im (Vus* Vcs Vub Vcb

*)

– J = Im Qudcs = - Im Qubcs

– The imaginary part of each Quartet combination is the same (up to a sign)– In fact it is equal to 2x the surface of the unitarity triangle

• Im[Vi Vj Vj* Vi*] = J ∑ ijk where J is the universal Jarlskog invariant

• Amount of CP Violation is proportional to J

The standard representation of the CKM matrix is:

12 13 12 13 13

12 23 12 23 13 12 23 12 23 13 23 13

12 23 12 23 13 12 23 12 23 13 23 13

cos

sin

iud us ub

ij iji icd cs cb

ij iji itd ts tb

V V V c c s c s ec

V V V V s c c s s e c c s c s e s cs

V V V s s c c s e c s s c s e c c

However, many representations are possible. What are the invariants under re-phasing?

July 1-2, 2004 38

unitarity:

The Unitarity Triangle

*

*

*

*

*

*

arg arg

arg arg

arg arg

td tbubtd

ud ub

cd cbtbcd

td tb

ud ubcbud

cd cb

V VQ

V V

V VQ

V V

V VQ

V V

* * * 0ud ub cd cb td tbV V V V V V

Vcd Vcb*

Vtd Vtb*Vud Vub

*

Under re-phasing: expj j jV i V the unitary angles are invariant

(In fact, rephasing implies a rotation of the whole triangle)

Area = ½ |Im Qudcb| = ½ |J|

The “db” triangle: VCKM† VCKM = 1

July 1-2, 2004 39

Wolfenstein Parametrization

2 3

2 2 4

3 2

1 / 2

1 / 2

1 1

A

V A O

A A

i

i

iud us ub

cd cs cb

td ts ti

b

V V V

V V V

V

e

Ve V

Wolfenstein realised that the non-diagonal CKM elements are relatively small compared to the diagonal elements, and parametrized as follows:

Normalised CKM triangle:

(0,0) (1,0)

July 1-2, 2004 40

CP Violation and quark masses

Note that the massless Lagrangian has a global symmetry for unitary transformations in flavour space.

Let’s now assume two quarks with the same charge are degenerate

in mass, eg.: ms = mb

Redefine: s’ = Vus s + Vub b

Now the u quark only couples to s’ and not to b’ : i.e. V13’ = 0

Using unitarity we can show that the CKM matrix can now be written as:

cos sin 0

sin cos cos cos sin

sin sin cos sin cosCKMV

CP conserving

Necessary criteria for CP violation:, , ,

, ,u c c t t u

d s s b b d

m m m m m m

m m m m m m

July 1-2, 2004 41

The Amount of CP Violation

12 13 12 13 13

12 23 12 23 13 12 23 12 23 13 23 13

12 23 12 23 13 12 23 12 23 13 23 13

cos

sin

i

ij iji i

ij iji i

c c s c s ec

V s c c s s e c c s c s e s cs

s s c c s e c s s c s e c c

2 512 23 13 12 23 13 sin 3.0 0.3 10J c c c s s s

However, also required is:

2 2 2 2 2 2 2 2 2 2 2 2 0t c c u t u b s s d b dm m m m m m m m m m m m

All requirements for CP violation can be summarized by:

(The maximal value J might have = 1/(6√3) ~ 0.1)

Using Standard Parametrization of CKM:

† † 2 2 2 2 2 2

2 2 2 2 2 2

5 10 12

det , 2

6 10 4 10 (GeV ) 0 CP Violation

d d u u t c c u u t

b s s d d b

m M M M M J m m m m m m

m m m m m m

Is CP violation maximal? => One has to understand the origin of mass!

(eg.: J=Im(Vus Vcb Vub* Vcs

*) )

July 1-2, 2004 42

Mass Patterns

Mass spectra ( = Mz, MS-bar scheme)

mu ~ 1 - 3 MeV , mc ~ 0.5 – 0.6 GeV , mt ~ 180 GeV md ~ 2 - 5 MeV , ms ~ 35 – 100 MeV , mb ~ 2.9 GeV

4

2

,u c

c t

d s

s b

m m

m m

m m

m m

Why are neutrino’s so light? Related to the fact that they are the only neutral fermions?See-saw mechanism?

me = 0.51 MeV , m = 105 MeV , m = 1777 MeV

• Do you want to be famous?• Do you want to be a king?• Do you want more then the nobel prize?

- Then solve the mass Problem – R.P. Feynman

Observe:

July 1-2, 2004 43

Matter - antimatter asymmetryIn the visible universe matter dominates over anti-matter:•There are no antimatter particles present in cosmic rays•There are no intense -ray sources in the universe due to matter anti-matter collisions

Hubble deep field - optical

July 1-2, 2004 44

Big Bang Cosmology

Equal amounts of matter & antimatter

q+q⇄ +

Matter Dominates !+ CMB

July 1-2, 2004 45

The matter anti-matter asymmetry

0.4 100.3(6.5 ) 10baryons

photons

N

N

Almost all matter annihilated with antimatter, producing photons…

, ,lm lmaT Y

WMAP satellite

1 1,T 2 2,T

2.7248K 2.7252K

Cosmic Microwave Background

Angular Power Spectrum

l

July 1-2, 2004 46

A matter dominated universe can evolve in case three conditions occur simultaneous:

1) Baryon number violation: L(B)≠02) C and CP Violation: (N→f) ≠ (N→f) 3) Thermal non-equilibrium:

otherwise: CPT invariance => CP invariance

The Sakharov conditions

910

B B

B B

B B

N N

N N

N N

N

Sakharov (1964)

Convert 1 in 109 anti-quarks into a quark in an early stage of universe:

Anti-

July 1-2, 2004 47

0

1

3q Xr

n

n nr

n

Baryogenesis at the GUT Scale

duu

Xe+

uu-

proton

0

GUT theories predict proton decay mediated by heavy X gauge bosons:

X boson has baryon number violating (1) couplings: X →q q, X→q l

Proton lifetime: s

Decay process

Decay fraction

B

X → q q r 2/3

X → q l 1-r 1/3

X → q q r -2/3

X → q l 1-r -1/3

2 / 3

1

1 1/ 3

2 / 3 1 1/ 3

/ 3

X

X

X X

B r

B r

r

r

B r

r

B

Efficiencyof Baryon asymmetry build-up:A simple Baryogenesis model:

CP Violation (2) : r ≠ r

Assuming the back reaction does not occur (3):

Initial X number density

Initial light particle number density

Conceptually simple

July 1-2, 2004 48

Baryogenesis at Electroweak Scale

SM Electroweak Interactions:1) Baryon number violation in weak anomaly: Conserves “B-L” but violates “B+L”2) CP Violation in the CKM3) Non-equilibrium: electroweak phase transition

Conceptually difficult

Electroweak phase transition wipes out GUT Baryon asymmetry!

Problems:1. Higgs mass is too heavy. In order to have a first order phase transition: Requirement: mH < ~ 70 GeV/c2 , from LEP mH > ~ 100 GeV/c2

2. CP Violation in CKM is not enough:

9Requirement: 10N B

N

Leptogenesis:• Uses the large right handed majorana neutrino masses in the see-saw mechanism to generate a lepton asymmetry at high energies (using the MNS equivalent of CKM).• Uses the electroweak sphaleron (“B-L” conserving) processes to communicate this to a baryon asymmetry, which survives further evolution of the universe.

Can it generate a sufficiently large asymmetry?

2012

from CKM: 10B CKMB

B cB

N N J

N N T

July 1-2, 2004 49

Conclusion

Biertje?

Key questions in B physics:• Is the SM the only source of CP Violations?• Does the SM fully explain flavour physics?