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
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