flavour oscillation and cp violation flavour oscillation: mixing

Content

Flavour Oscillation and CP Violation

• Quarks Mixing and the CKM Matrix

• Flavour Oscillation: Mixing of Neutral Mesons

• CP violation

• Neutrino Mixing

Stephanie Hansmann-Menzemer 1

Quark Mixing in SMstrong, elm, weak NC conserve flavour and have identical coupling (constants) for

all up-type and down-type quarks, all leptons and all neutrinos:

• uγµu, cγµc, tγµt

• eγµe, µγµµ, τγµτ

• dγµd, sγµs, bγµb

• νeγµνe, νµγµνµ, ντγ

µντ

these interactions leave abmiguity for definition of quark and lepton eigenstates

e.g. rotational freedom in space of up-type quark:

x = 1√2(u + c) would still have the same form of Lagrangian with same coupling:

xγµx

weak CC couples up and down type quarks: uγµ(1 − γ5)d, ...


Quark Mixing in SMYukawa term (coupling of Higgs to fermions) introduces:

“

u, c, t”

mu

0

B

B

@

u

c

t

1

C

C

A

+“

d, s, b”

md

0

B

B

@

d

s

b

1

C

C

A

; mu, md (3x3) mass matrices

choice of mass eigenstates for representation of Lagrangien introduceds a (3x3)

rotation matrix for u-type quarks and one for d-type quarks.

U†muU =

0

B

B

@

mu 0 0

0 mc 0

0 0 mt

1

C

C

A

V †mdV =

0

B

B

@

md 0 0

0 ms 0

0 0 mb

1

C

C

A

U, V are unitary matrices, q are mass eigenstates

→

0

B

B

@

u

c

t

1

C

C

A

= U

0

B

B

@

u

c

t

1

C

C

A

0

B

B

@

d

s

b

1

C

C

A

= V

0

B

B

@

d

s

b

1

C

C

A

All terms of the Lagrangian but CC are invariant under this rotation.

LCC =“

u c t”

U†Uγµ(1 − γ5)V †V

0

B

B

@

d

s

b

1

C

C

A

=“

u c t”

Uγµ(1 − γ5)V †

0

B

B

@

d

s

b

1

C

C

A


Quark Mixing in SM

“

u c t”

Uγµ(1 − γ5)V †

0

B

B

@

d

s

b

1

C

C

A

=“

u c t”

γµ(1 − γ5)UV †

0

B

B

@

d

s

b

1

C

C

A

0

B

B

@

d′

s′

b′

1

C

C

A

= UV †

0

B

B

@

d

s

b

1

C

C

A

=

0

B

B

@

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

1

C

C

A

×

0

B

B

@

d

s

b

1

C

C

A

flavour CKM matrix mass

JCCµ ∝

“

u, c, t”

γµ“

1 − γ5”

VCKM

0

B

B

@

d

s

b

1

C

C

A

for quarks: flavour eigenstates 6= mass eigentstates!direct result of coupling of Higgs to quark flavours!


CKM Matrix I

d′

s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

d

s

b

flavour CKM matrix mass

18 parameters (9 complex elements)-5 relative quark phases (unobservable)-9 unitarity conditions————————-= 4 independent parameters 3 Euler angles and 1 Phase

4 fundamental Standard Model Parameters (out of ∼18 (28))


CKM Matrix II

Lagrangian insensitive to phases of left-handed fields:possible redefinition:

uL → eiφ(u)uL cL → eiφ(c)cL tL → eiφ(t)tL

dL → eiφ(d)dL sL → eiφ(s)sL bL → eiφ(b)bL

φ(q): real numbers

V =

e−iφ(u) 0 0

0 e−iφ(c) 0

0 0 e−iφ(t)

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

eiφ(d) 0 0

0 eiφ(s) 0

0 0 eiφ(b)

5 unobservable phase differences:

Vlj → ei(φ(j)−φ(l))VljStephanie Hansmann-Menzemer 6

CKM Matrix III

u c t

d s b

u

d ’

c

s ’

t

b ’

Diagonal elements of CKM matrix are close to one.Only small of diagonal contributions.Mixing between quark families is “CKM suppressed”.


Definition of CKM Angles

VCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

=

1 − λ2/2 λ Aλ3e−iγ

−λ 1 − λ2/2 Aλ2

Aλ3e−iβd −Aλ2 1

+ O(λ4)

cos θC = λ ∼ 0.22;

one unitarity relation: VudV∗ub + VcdV

∗cb + VtdV

∗tb = 0

Bd triangle:α = arg

(

− VtdV ∗tb

VudV ∗ub

)

;

β = arg(

−VcdV ∗cb

VtdV ∗tb

)

;

γ = arg(

−VudV ∗ub

VcdV ∗cb

)

;

area of triangle 6= 0 → complex contributions of CKM elementsStephanie Hansmann-Menzemer 8

Unitarity Triangle

γ

α

α

dm∆

Kε

Kεsm∆ & dm∆

ubV

βsin 2(excl. at CL > 0.95)

< 0βsol. w/ cos 2

α

βγ

ρ-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

η

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

excl

uded

are

a ha

s C

L >

0.95

Winter 12

CKMf i t t e r

Current status of knowledge on “the” (Bd) CKM triangle.Sofar all measurements consistent with each other.


Complex Elements and CP Violation


Neutral Meson Mixing


Phenomenology of Mixing ISchrödinger equation for unstable mesons (at rest):

i ddt|Ψ >= H|Ψ >= (m − i

2Γ)|ψ >

→ |Ψ(t) >= |Ψ0 > e−imte−1

2Γt

→ ||Ψ(t) > |2 = ||Ψ0 > |2e−Γt

For neutral mesons, consider 2 components (formulas equivalent valid for K0, D0, B0s mesons):

id

dt

0

@

B0

B0

1

A = H

0

@

B0

B0

1

A =

0

@

H11 H21

H12 H22

1

A

0

@

B0

B0

1

A

=“

M − i2Γ

”

0

@

B0

B0

1

A =

0

@

m11 − i2Γ11 m21 − i

2Γ21

m12 − i2Γ12 m22 − i

2Γ22

1

A

0

@

B0

B0

1

A

CPT theorem:

m11 = m22 = m(B0) = m(B0)

Γ11 = Γ22 = Γ

= 1τ(B0)

= 1

τ(B0)

off-diagonal elements ⇒ mixing

M , Γ hermetic:

m12 = m∗21, Γ12 = Γ∗

21


Mass Eigenstates

diagonlizing matrix → mass eigenstates x with eigenvalues mx + iΓx2

BL = p|B0 > +q|B0 > with mL, ΓL

BH = p|B0 > −q|B0 > with mH , ΓH

|p2| + |q2| = 1, complexe coefficients

Flavour eigenstates:

B0 = 12p(|BL > +|BH >)

B0 = 12q (|BL > −|BH >)

Parameters of mass states:

mH,L = m ± Re√

H12H21

ΓH,L = Γ ∓ 2Im√

H12H21

∆m = mH − MK = 2Re√

H12H21

∆Γ = ΓH − ΓL = −4Im√

H12H21

x ≡ ∆mΓ ; y ≡ ∆Γ

2Γ


Phenomenology of Mixing II

H =

m − i2Γ ∆m

2 − i2∆Γ

∆m2 − i

2∆Γ m − i2Γ

Two mixing mechanisms:

• mixing through decay: y = ∆ΓΓ ∼ O(1)

• mixing through oscillation: x = ∆mΓ ∼ O(1)

(K0K0, B0B0, B0s B0

s and D0D0 show different behavior)

long distant, on shell states

important for K not for B mesons

→ ∆Γ

short distant, virtual states

→ ∆m Stephanie Hansmann-Menzemer 14

Time Evolution I|BH/L(t) >= bH/L(t)|BH,L > with bH/L(t) = e−imH/Lte−ΓH/Lt/2

|B0(t) > =|BL(t) > +|BH(t) >

2p

=1

2p(bL(t)(p|B0 > +q|B0 >) + bH(t)(p|B0 > −q|B0 >))

= f+(t)|B0 > − q

pf−(t)|B0 >

|B0(t) > = f+(t)|B0 > +p

qf−(t)|B0 >

f±(t) = 12

“

eimH te−ΓH t/2 ± e−imLte−ΓLt/2”

CP-violation in mixing: P (B0 → B0) 6= P (B0 → B0) ⇔ | qp| 6= 1


Time Evolution II

f±(t) = 12

(

eimH te−ΓH t/2 ± e−imLte−ΓLt/2)

P (B0 → B0) = P (B0 → B0) = |f+(t)|2 =14

(

e−ΓLt + e−ΓH t + 2e−(ΓL+ΓH)t/2 cos ∆mt)

P (B0 → B0) = | qp |2|f−(t)|2 =

14 |

qp |2

(

e−ΓLt + e−ΓH t − 2e−(ΓL+ΓH)t/2 cos ∆mt)

P (B0 → B0) = |pq |2|f−(t)|2 =

14 |

pq |2

(

e−ΓLt + e−ΓH t − 2e−(ΓL+ΓH)t/2 cos ∆mt)

Mixing asymmetry :P (B0→B0)−P (B0→B0)

P (B0→B0)+P (B0→B0)= 2e−(ΓH+ΓL)t/2 cos ∆mt

e−ΓHt+e−ΓLt

assume | qp | = 1Stephanie Hansmann-Menzemer 16

Phenomenology of Mixing IIITwo mixing mechanisms:

• mixing through decay: y = ∆ΓΓ ∼ O(1)

• mixing through oscillation: x = ∆mΓ ∼ O(1)

K0/K0 D0/D0 B0/B0 Bs/Bs

τ [ps]∗ 89 0.4 1.6 1.5

51700

Γ [ps−1] 5.6× 10−3 2.4 0.64 0.62

y= ∆Γ2Γ

-0.997 0.01 |y|<0.01 0.03±0.03

∆m [ps−1] 5.3 × 10−3 0.02 0.5 17.8

x= ∆mΓ

0.95 0.01 0.8 26

xxxxxxxx∗) at LHCb energies lifetime in ps ∼ decay length in cm

K0: mixing in decay and mixing in oscillation (medium) -

D0: very slow mixing - 2008

Bd: dominantly mixing in oscillation (medium) - 1987

Bs: dominantly mixing in oscillation (very fast) - 2006Stephanie Hansmann-Menzemer 17

How to Measure Mixing?

• identify initial flavour of meson

• identify final flavour of meson→ identify Meson as mixed or unmixed

• if mixing is fast ... need to measured decay time:

decay length L+ momentum p → t = mLp

• if mixing is slow ... can deduce information from time integratedasymmetry


Neutral Kaonsassume no CP violation, Ks, KL correspond to mass eigenstates:

K short

• τ(K0L) = 51.7 ± 0.44 ns

• typical decay length: 5-20 m

• KL → 3π BR > 99%

• CP = -1

|KL >= 1√

2|K0 > −|K0 >

K long

• τ(K0S) = 0.089 ± 0.001 ns

• typical decay length: few cm

• KS → 2π BR > 99%

• CP = +1

|KS >= 1√

2|K0 > +|K0 >


CPLEAR Experiment


CPLEAR Experiment

A = 2e−(ΓS+ΓL)t/2 cos ∆mte−ΓSt+e−ΓLt


B-Mixing

B mesons are produced in high energetic collitions:

• B factories:e+e− → Y (4S) → B0B0

e+e− → Y (4S) → B+B−

50% : 50% for charged and neutral B meson productionm(Y(4S)) = 10.58 GeV; not sufficient energy to produce Bse−

e+

b

b

Y(4S)γ

b

b

uu

B−

B+

e.g. ARGUS, BELLE, BABAR

• pp, pp collidere.g. TEVATRON, LHC


ARGUS 1987

e+e− → Y (4S) → B0B0

√s = 10.58 GeV

unmixed:

B0B0 → ℓ+ℓ−

mixed:

B0B0 → ℓ+ℓ+

B0B0 → ℓ−ℓ−

time integrated measurement:

∼ 18% of B0 mix before they decay

decay of other B meson is used to tagged production

tt

d

d

b

b

Bd Bd

V V

VV

tb td

td tb*

*

(u,c) (u,c)


B0s

−

¯B0

sMixing Analysis

xxx t = L Bm

p

L−D

−π

+K−

K

+π

b hadronPV

Bs0

s

signal sideopposite side

1) Bs selection & reconstruction2) Measurement of proper decay time ct

3) Flavor tagging (main challenge at hadron colliders)

Time dependent asymmetry measurement:

A(t) ≡ N(t)mixed−N(t)unmixed

N(t)mixed+N(t)unmixed∝ cos(∆mst)


Time Dependent Asymmetries

decay time, ps0.0 0.5 1.0 1.5 2.0

prob

abili

ty d

ensi

ty

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

totalunmixedmixed

decay time, ps

asym

met

ry

��

��

A = #unmixed − #mixed#unmixed + #mixed

0 2.5 5 7.5 10proper decay time, t [ps]

asym

met

ry Bd mixing ∆md = 0.5 ps−1

Bs mixing ∆ms= 18 ps−1

A = 2e−(ΓH+ΓL)t/2 cos ∆mte−ΓHt+e−ΓLt ∼ cos ∆m

ΓH ∼ ΓL


Time dependent Asymmetry

???Stephanie Hansmann-Menzemer 27

Tagging Dilution

Measured Asymmetrie:

Ameasured(t) =N(B0)

′

(t) − N(B0)′

(t)

N(B0)′

(t) + N(B0)′

(t)

=N(B0)(t)(1 − Pmt) + N(B0)(t)Pmt − N(B0)(t)(1 − Pmt) − N(B0)(t)Pmt

NRS(t) + NWS(t)

= (1 − 2Pmt)NB0

(t) − NB0(t)

NB0(t) + NB0

(t)= (1 − 2Pmt)A(t) = DA(t)

N′

B0/N

′

B0: as B0/B0 tagged decays

Pmt : Mistag-Probability

D : Tagging Dilution


Auswirkung der Tagging Dilution

Mis-tag dilutes the observed Oszillation!

decay time, ps0.0 0.5 1.0 1.5 2.0

prob

abili

ty d

ensi

ty

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

perfect tagging

totalunmixedmixed

decay time, ps0.0 0.5 1.0 1.5 2.0

prob

abili

ty d

ensi

ty

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P(mistag) = 0.4

totalunmixedmixed

Amplitude: 1-2P(mistag)


Asymmetry Measurement

Tagging dilution D = 1 − 2Pmt

Always true ↔ 100% dilution; radom Tag (Pmt = 50%) ↔ 0% Dilution;

Tagging efficiency ǫ = Ntagged

Nall

effective statist. size of sample

→ Neff = Nall × ǫD2.

[%] ǫD2 Reduktion des Datensatzes

D0/CDF 2.5 - 5.0 × 20-50

BABAR/BELLE ≈ 30 × 3-4

LHCb 3.0 × 30

e+e− experiments factor of 10 better!


Flavour Tagging Methoden

K+

sBbl−

Opposite Side Tagging

bb Paarproduktion→ korrelierte ProduktionsflavourInklusive Reko. des OS B

xxxxxx Same Side Tagging

xb

u

u

Bb

s

K +

s

s

Nicht bei Y (4S) möglich!(keine Fragmentation)Stephanie Hansmann-Menzemer 31

Mixing @ Babar & CDF

B0B0

x

B0s B

0s (2006)

[ps]sm∆/πDecay Time Modulo 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Fitt

ed A

mpl

itude

-2

-1

0

1

2

data

cosine with A=1.28

CDF Run II Preliminary -1L = 1.0 fb


Bs − Bs Mixing at LHCb

∼ 9.250 Bs candidates in 3 channels

Proper time resolution: σt = 45 fs

∆ms = 17.725 ± 0.041 ± 0.026 ps−1

world best measurement!

]2 mass [MeV/csB5400 5600 5800

2#

even

ts /

15 M

eV/c

0

2000

datafitsignalmisid. bkg.

Ks D→sB

comb. bkg.

LHCb preliminary

= 7 TeVs

-1341 pb

[ ps ]sm∆ / πt modulo 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35

mix

A

-0.4

-0.2

0

0.2

0.4

LHCb preliminary

= 7 TeVs

-1341 pb

Aufl

ösun

gre

d.A

mpl

itude

LHC

b-C

ON

F-2

011-

050


flavour oscillation and cp violation flavour oscillation: mixing

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