1

IX. Flavor oscillation and CP violation

1. Quark mixing and the CKM matrix

2. Flavor oscillations: Mixing of neutral mesons

3. CP violation

4. Neutrino oscillations

1. Quark mixing and CKM matrix

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛∝+

bsd

)-(1 )t,c,u(J CKM5 Vµµ γγ

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⋅⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

′′′

bsd

VVVVVVVVV

bsd

tbtstd

cbcscd

ubusud

1.1 Quark mixing:

Mass eigen-states are not equal to the weak eigen-states: Quark-mixing described by unitary CKM matrix,

CKMVFor weak charged current one obtains:

b

c

W–

µ–

ν

Vcb

Weak states u’ d‘ s‘ are orthogonal: VCKM is unitary.

1== ++CKMCKMCKMCKM VVVV

2

1.2 Cabbibo-Kobayashi-Maskawa matrix

18 parameter = 9 complex elements

-5 relative quark phases (unobservable)

-9 unitarity conditions

=4 independent parameters: 3 angles + 1 phase

Number of independent parameters:

PDG parametrization

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −

bsd

cssc

ces

esc

cssc

bsd

i

i

10000

0010

0

00

001

'''

1212

1212

1313

1313

2323

2323δ

δ

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−−−−−

−

132313231223121323122312

132313231223121323122312

1313121312

ccescsscesccsscsesssccessccsescscc

ii

ii

i

δδ

δδ

δ

ijijijij sc θθ sin,cos where ==

3 Euler angles

1 Phase

( )4

23

22

32

1)1(2

1

)(2

1

λ

ληρλ

λλλ

ηρλλλ

O

AiA

A

iA

VCKM +

⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

−−−

−−

−−

=

Wolfenstein Parametrization λ, A, ρ, η

→ hierarchy expressed by orders of λ = sinθc≈ 0.22

βitdtd eVV −=

γiubub eVV −=

Magnitude of the elements:

PDG 2004

90% C.L>

3

Komplex CKM elements and CP violation

Lid L

jujiV Rid R

ju∗jiV

Lju L

id∗jiV

CP

T

Remark: For 2 quark generations the mixing is described by the real 2x2 Cabbibo matrix → no CP violation !!. To explain CPV in the SM Kobayashi and Maskawa have predicted a third quark generation.

CP (T) violation ∗≠⇔ jiji VV

i.e. Complex elements

2. Flavor oscillation: Mixing of neutral mesons

Standard Model predicts oscillations of neutral Mesons:

b b

d db b

d dW

W

Wtcu ,, tcu ,,

tcu ,,

tcu ,,

0B0B 0B0B W

Neutral mesons:

bsBbdBcuDsuKP

bsBbdBcuDsuKP

sd

sd

====

====

00000

00000

:

:

00dd BB ⇔

Transition can be described by matrix element: 00dWd BHB

4

2.1 Phenomenological description of mixing

Schrödinger equation for unstable meson:

⇒⎟⎠⎞

⎜⎝⎛ Γ−== ψψψ

2imH

dtdi

t

timt

et

eetΓ−

Γ−−

⋅=

⋅⋅=2

0

2

21

0

)(

)(

ψψ

ψψ

Γ=Γ=Γ

====

2211

2211

2211

CPT mmmHHH

2112 CP HH =⇒

00 BB WH

M and Γ hermitian:

*1221

*1221

Γ=Γ

=mm⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

Γ−Γ−

Γ−Γ−=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛0

0

22221212

*12

*121111

0

0

0

0

2212

21110

0

0

0

22

222 P

Pimim

imim

PPi

PP

HHHH

PP

PP

dtdi ΓMH

),(),,(),,(),,( 00000000ss BBBBDDKKFor neutral mesons consider 2 compon.

HHH

LLL

mPqPpP

mPqPpP

Γ−=

Γ+=

,00

,00

with

with

122 =+ qp2112

2112

2112,

2112,

Im4

Re2

Im2

Re

HH

HHmmm

HH

HHmm

LH

LH

LH

LH

−=Γ−Γ=∆Γ

=−=∆

Γ=Γ

±=

m

Γ∆Γ

≡Γ∆

≡2

und ymx

Mass eigenstates (by diagonalizing matrix)

Heavy and light mass eigenstate:

complex coefficients

Parameters of the mass states

(ICHEP04) 01.065.0/

)%90(07.0/(PDG) ps4.14

(PDG) ps007.0502.0

25.033.0

1

1

±=Γ∆Γ

<Γ∆Γ

>∆

±=∆

+

−

−

−

ss

s

d

CLm

m

Neutral B mesons

)(21

)(21

0

0

HL

HL

PPq

P

PPp

P

−=

+=Flavor/weakeigenstates

5

( ) ( )( )

00

00

0000,

)()()(

)()(

)()(21

2,

)(

0

0

BtfqpBtft

BtfpqBtf

BqBptbBqBptbpp

BtBt

B

HLtHL

B

⋅+⋅=

⋅−⋅=

−⋅++⋅=+

=

−+

−+

ψ

ψ

timt

LHLHLHLHLHLH eetbBtbtB ,,)( mit )(, ,,,,

−Γ−==

[ ]2/2/

21)( ttimttim LLHH eeeetf Γ−−Γ−−

± ±⋅=

( )( ) 2

200

200

)(,

)(,

tfpqtBBP

tftBBP

−

+

=→

=→ ( )( ) 2

200

200

)(,

)(,

tfqptBBP

tftBBP

−

+

=→

=→0B 0B

Time evolution Generic particle (before P)

Mixing of neutral mesons

Time evolution of rates:

( )[ ]mteeeBBPBBP ttt HLHL ∆++=→=→Γ+Γ−Γ−Γ− cos2

41)()( 2/0000

( )[ ]( )[ ]mteee

qpBBP

mteeepqBBP

ttt

ttt

HLHL

HLHL

∆−+=→

∆−+=→

Γ+Γ−Γ−Γ−

Γ+Γ−Γ−Γ−

cos241)(

cos241)(

2/2

00

2/2

00

CPT

CP, T- violation in mixing: 1)()( 0000 ≠⇒→≠→pqBBPBBP

Two mixing mechanisms:• Mixing through decays

• Mixing through oscillation )1(

)1(2

Omx

Oy

≈Γ∆

=

≈Γ

∆Γ= ),(),,(

),,(),,(0000

0000

ss BBBBDDKK

show different oscillation behavior

6

2.2 Neutral kaons

-0.9966 2

007.0942.0x

=Γ∆Γ

=

±=Γ∆

=

y

m

( )( ) LightLightLightS

HeavyHeavyHeavyL

KKKKKK

KKKKKK

+=+≡=

−=−≡=

CP21""

CP21""

00

00

Observation of two neutral kaons KL (long) and KS (short) with different lifetimes:

( ) ( )

1CP 1CP 2 3

ns 001.0089.0)( ns 4.07.51)(00

00

+=−=

→→

±=>>±=

ππ

ττ

SL

SL

KK

KK

KL and KS can be identified with the mass eigenstates:

( )( )

19

12

110

s10182.11MeV 10006.049.3

s100009.05303.0

−

−

−

⋅−=∆Γ

⋅±=

⋅±=∆

h

hm

Large differences between lifetimes

Phase convention:

00

00

KKCP

KKCP

=

=

Neutral kaon system

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 5 10 15 20

)()()()(

0000

0000

KKPKKPKKPKKP

→+→→−→

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10

)( 00 KKP →

S

tτ

)( 00 KKP →

CPLEAR

ee eKeK νπνπ +−−+ →→ and 00

Initially pure K0 beam

mte t ∆Γ− cos~

Expectation

Measurement

7

K0 - ⎯K0 (strangeness) oscillation in the SM

0Kd

W

W

s

0Kd

tcu ,,s 0K 0K

+π

−π

Short range effects Long range effects

Oscillation frequency ∆m:222

22

,,

222

44~ cdcscKK

Fqdqs

tcuqqKK

F VVmfmGVVmfmGmππ

≈∆ ∑=

c quark contribution dominant: although mt2 is very large,

the factor |VtsVtd|2 ~ λ5 is very small !

2.3 Neutral B Meson 2 neutral B mesons:

Bd0 =|d⎯b> - oscillations precisely measured

Bs0 =|s⎯b> - oscillations not yet observed

Mixing mechanisms:• Mixing through decay: many possible hadronic decays → Γ is large

• Mixing through oscillation

decay via mixing noB for )1.0(

B for 0 small is

2 0s

0d ⇒

⎩⎨⎧

≈≈

=Γ

∆Γ=⇒

Oy

0dB

d

b

0dB

dt

bt

0sB

s

b

0sB

st

bt

→ Significant contribution only from top loop

)(~~ 6222 λOmVVmm ttdtbt ⋅∆ )(~~ 4222 λOmVVmm ttstbt ⋅∆

Large ∆ms,d: ∆ms~1/λ2 ∆md → Bs osc. is about 20 times faster than Bd osc.

8

0dB

0dBb

b

+l

lν

−l

lν

+e

−e )4( sΥ

−−

++

−+

→

→

→

ll

ll

ll

00

00

00

BBBBBB

00)4(:GeV58.10 atBBSee

s→Υ→

=−+

ARGUS 1987

mixed µνµ +−→ *0 DB−SD π0

−+πK

µνµ +−→ *0 DB0π−Dγγ

−−+ ππK

Discovery of B0 mixingFirst e+e- B factory at DESY:

nb1)( ≈BBσ

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10

)cos1(21)( 00 mteBBP t ∆−Γ=→ Γ−

( )mteBBP t ∆+Γ=→ Γ− cos121)( 00

-1,5

-1

-0,5

0

0,5

1

1,5

0 1 2 3 4 5 6 7 8 9 10

B

tτ

)()()()(

0000

0000

BBPBBPBBPBBP

→+→→−→

Mixing of neutral B mesons

mixedunmixedmixedunmixedA

+−

=

-1ps006.0005.0511.0 ±±=∆ dm

9

3. CP violation in the K0 and B0 system

τν−π −τ τν −π−τ

↔P

↔P

Cb Cb

τν+π +τ τν +π+τ

forbidden

forbidden

• C and P violated in weak decays

• CP conserved in weak interaction ? → No !

allowed

τνπτ −− →

3.1 Observation of CP violation (CPV) in KL decays

( ) 1CP21 00 −=−= KKKL should decay into 3π w/ CP(|3π>)= -1

• Until the year 2000 the K0 was the only system in which CPV was observed

• In 2000, CPV was observed in the B0 system

• So far the observed CPV is consistent with the SM prediction. But it is too small to explain the baryon asymmetry of the universe.

310~1CP−

−+

+=

→

BR

KL ππ

Christenson et al., 1967 and not into 2π w/ CP(|2π>)= +1

10

3.2 CP Violation in the Standard Modell

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

bsd

VVVVVVVVV

bsd

tbtstd

cbcscd

ubusud

'''

Quarks:

Antiquarks:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

bsd

VVVVVVVVV

bsd

tbtstd

cbcscd

ubusud

***

***

***

'''

Phase angle≠0: complex CKM matrix

Different mixing for quarks and anti-quarks

Origin of CP Violation (CPV)

( ) 0Im ** ≠= kjilklij VVVVJStrength of CPV: Characterized by Jarlskog invariant:

γiubub eVV −=

βitdtd eVV −=

see Wolfensteinparametrization

( ) 510262 10~)(21]Im[ −∗∗ +−== λληλ OAVVVVJ csubcbusIn SM:

3.3 Unitarity Triangle

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

tbtstd

cbcscd

ubusud

VVVVVVVVV

V

Unitary CKM matrix: VV† = 1

0=++ ∗∗∗

tbtdcbcdubud VVVVVV

0=++ ∗∗∗

ubtbustsudtd VVVVVV

6 “triangle” relations:

β

α

γ

*tbtdVV

*cbcdVV

*ubudVV

area = J/2

Row 1x2, 2x3

Column 1x2, 2x3

“Real” triangles w/ similar sides:

→ expect large CPV effects

Important for Bd and Bs decays

Degenerated triangles (same area):

→ small CPV effect (Kaon decays)

11

Rescaled Unitarity Triangle

β

α

γ*

*

cbcd

ubud

VVVV *

*

cbcd

tbtd

VVVV

η

0 1

Im

ρRe

⎥⎥⎦

⎤

⎢⎢⎣

⎡−≡ *

*

argcbcd

ubud

VVVVγ

⎥⎥⎦

⎤

⎢⎢⎣

⎡−≡ *

*

argtbtd

cbcd

VVVVβ

⎥⎥⎦

⎤

⎢⎢⎣

⎡−≡ *

*

argubud

tbtd

VVVVα

3.4 “3 Ways” of CP violation in meson decays

≠p/q

0B 0B

q/p

0B0B

f f

1)()( 0000 ≠⇒→≠→pqBBPBBP

1. CP Violation in mixing:

24

21 4 sin 5 10c

t

mqp m

π β −− ≈ ≈ ×

Standard model prediction for B

Kaon system:

( )( )000

2

0001

−21

=

+21

=

KKK

KKK CP +1

CP -1

( )

( )01

022

0

02

012

0

++1

1=

++1

1=

KKK

KKK

mm

L

mm

S

εε

εε

CP eigenstates are not identical with mass/lifetime eigenstates

( )mpq εRe21−≅

( ) 3107.12.6)Re(4:2001 CPLEAR −⋅±=mε

admixture

12

)Re()Re()()()()(

mm

m

KKPKKPKKPKKP ε

εε

4≅+1

4≅

→+→→−→

20000

0000

)()() (

)() ()(

00

00

00

00exp t

eKeK

eKeKtA

eetet

eetet

νπνπ

νπνπ+−

=

−+=

+−

=

−+=

→Γ−→Γ

→Γ−→Γ=

CPLEAR( ) 3107.12.6)Re(4 −⋅±=mε

CP (T) Violation in mixing

K0⎯K0 System:

B0⎯B0 System:

0017.00007.0)1(

Re4

0034.00013.1

0067.00026.0

2 ±−=+

±=

±−=

B

B

SL

pq

A

εε

HFAG 2004

In principal the ratio measures the T invariance:

0000 KKT

KK ←↔

→

≠A 2 2

0Bf f

0B

A

2. Direct CPV

)(Prob)(Prob fBfB →≠→

1≠AA

Standard Model:

2211

2211

210

210

)(

)(δϕδϕ

δϕδϕ

iiii

iiii

eAeAXBA

eAeAXBA+−+−

++

+=→

+=→ Two interfering amplitudes A1, A2:

−with diff. CKM phases ϕ1 , ϕ2 −with diff. strong phases δ1 , δ2

)sin()sin(4 21212122

δδϕϕ −−=− AAAA

⇔≠ 1AA ≠0 ≠0

weak strong

e.g: B0 → Kπ direct CPV possibleW

tb s

u

u

g

dd+π

−K0Bb u

u

sW

dd0B +π

−K

Bϕ δ f

weak strong

δϕ ii eeAfBA =→ )(

13

First observation of direct CPV in B decays

0

0

BB K

Kππ

+ −

− +→→

Signal (227 pairs): 1606 51M BB ±0 B K π+ −→

BABARBBAABBARAR

0.133 0.030 0.009 CPA = − ± ±

hep-ex/0408057,

4.2σ

BABARBBAABBARAR

)()()()(

00

00

+−−+

+−−+

→+→

→−→=

ππππ

KBNKBNKBNKBNACP

Summer 2004

W

tb s

u

u

g

dd+π

−K0B

Effect of “penguin” contribution

J. Ellis=

0B

0B

CPf

CPCPCP

CP

ffCP

f

η=

→

:eigenstate CPB0

3. CPV in interference between mixing and decay

δϕ iiff eeAA =

δϕ iiff eeAA −=

fCPf AA η=CPV if there is a phase difference λCP between the direct path and the path with mixing:

( )fMi

f

fCP

f

fCP e

AA

pq

AA

pq φφηλ −⋅⋅⋅=≡ 2

Miepq

pq φ2 :Mixing =

Miepq

pqMixing φ2~

=1 if no CPV in decay

If no CPV in mixing and decay:( )fMi

CPCP e ϕϕηλ −−=

2

good approximation for B0→J/ψKs

14

Calculation of the time-dependent CP asymmetry

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡∆

−−∆+

+×

+∝→Γ

⎥⎥⎦

⎤

⎢⎢⎣

⎡∆

−+∆−

+×

+∝→Γ

∆−

∆−

tmtmetfB

tmtmetfB

dCP

dCPCP

CP

t

CP

dCP

dCPCP

CP

t

CP

B

B

cos2

1sinIm

21

1))((

cos2

1sinIm

21

1))((

22

2

/0

22

2

/0

0

0

λλ

λ

λ

λλ

λ

λτ

τ

≠

( ) ( )[ ]tmCtmSftBftBftBftBtA dfdf

CPCP

CPCPCP ∆−∆=

→Γ+→Γ

→Γ−→Γ= cossin

))(())(())(())(()( 00

00

Time resolved

Interference= sin2β for e.g. B0→J/ψKS≈ sin2α(eff) for e.g. B0→π+π−

indicates direct CP violationif |q/p|=1

2

2

2 1

1

1Im2

CP

CPf

CP

CPf CS

λ

λ

λλ

+

−=

+=

3.5 SM prediction for the decay B0→J/ψKs

βi

cdcb

cdcb

tdtb

tdtb

cdcs

cdcs

cscb

cscb

tdtb

tdtb eVVVV

VVVV

VVVV

VVVV

VVVV

AA

pq 2

*

*

*

*

*

*

*

*

*

*

ψK

ψKψK

S

S

Sλ −−=−=−==

b

d b

dB0 mixing

pq /

b

d

ccsd

W+

B0 decay

*cscbVVA∝

K0 mixings

d s

d

KK pq /

Miepq φ2~

β)sin(2)Im(λ

1λ

S

S

ψK

ψK

=

=⇒≈ − βi

tdtd eVVno direct CPV, no CPV in mixing

Phase of decay amplitude ϕf=0

Mixing phase ϕM=β

1−=CPηS

ame

for a

ll cc

K0

chan

nels

Beside Vtd all other CKM elements are real

15

3.6 Time dependent ACP for B0→J/ψKs

( )( )

)sin()(2sin//

//)(

0000

0000

mtKJBNKJBN

KJBNKJBNtA fMCP

SS

SS

CP ∆−−=→+⎟

⎠⎞⎜

⎝⎛ →

→−⎟⎠⎞⎜

⎝⎛ →

= ϕϕηψψ

ψψ

&&

&&

B0→J/ψKS : =β

)sin(2sin)( mttA CPCP ∆−= βη

If ACP ≠ 0:

• CP violation

• Measurement of the angle )arg( tdV=β

Oscillation with ∆md

3: Determination of ∆t = ∆z/βγc

2: Flavour-determination of other B meson („tag“)

1: Exclusive B reconstruction: flavor or CP eigenstate

Time dependent CP asymmetry measurement for B0→J/ψKs

)(

)(0

00

tBB

tBB

rec

tag

=⇒

=

(flavor eigenstates)(CP eigenstates)00

CPrec BB =

00flavrec BB =

CP analysislifetime, mixing analyses

ϒ(4S) B0

B0

e+

J/ψ

π−

π+

Ks0

π+

νµ

D*+

µ−D0

Κ−

π+

∆z ∆t e-

recB

tagB

Invariant masses

16

CP asymmetry in B0→J/ψKs

)sin(2sin)( mttA CPCP ∆−= βη

sin2β = 0.722 ± 0.040 (stat) ± 0.023 (sys) |λ| = 0.950 ± 0.031 (stat.) ± 0.013 (sys)

ICHEP 2004

3.7 Status of the Unitarity Triangle

R.Nogowski & K.Schubert, Dresden (2004)

β

α

γ

η

0 1

Im

Re

With the current precision the CKM phases behave as predicted by the Standard Model → high precision measurements necessary