Niels Tuning (1)

Particle Physics II – CP violation (also known as “Physics of Anti-matter”) Lecture 3

N. Tuning

Plan 1)  Mon 5 Feb: Anti-matter + SM

2)  Wed 7 Feb: CKM matrix + Unitarity Triangle

3)  Mon 26 Feb: Mixing + Master eqs. + B0→J/ψKs

4)  Wed 28 Feb: CP violation in B(s) decays (I)

5)  Mon 12 Mar: CP violation in B(s) and K decays (II)

6)  Wed 14 Mar: Rare decays + Flavour Anomalies

7)  Wed 21 Mar: Exam

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Ø  Final Mark: §  if (mark > 5.5) mark = max(exam, 0.85*exam + 0.15*homework)

§  else mark = exam

Ø  In parallel: Lectures on Flavour Physics by prof.dr. R. Fleischer

Diagonalize Yukawa matrix Yij –  Mass terms –  Quarks rotate

–  Off diagonal terms in charged current couplings

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Recap SM Kinetic Higgs Yukawa= + +L L L L

0( , ) ...I I

Yuk Li L Rjd Ij id dY u

ϕ

ϕ

+⎛ ⎞− = +⎜ ⎟⎜ ⎟

⎝ ⎠L

...2 2Kinetic Li Li

I I ILi L

Ii

g gu W d d W uµ µµ µγ γ− += + +L

( ) ( )5 5*1 1 ...2 2ij iCKM i j j j ig gu W d d uV VWµ µ

µ µγ γ γ γ− += − + − +L

( ) ( ), , , , ...d u

s cL L

b tR R

Mass

m d m ud s b m s u c t m c

m b m t

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟− = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

g g g gL

I

ICKM

I

d ds V sb b

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟→⎜ ⎟ ⎜ ⎟

⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

SM CKM Higgs Mass= + +L L L L

uI

dI

W

u

d,s,b

W

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Charged Currents

( ) ( )

†

*

5 5 5 5

5 5

2 21 1 1 12 2 2 22 2

1 12 2

I I I ICC Li Li Li Li CC

ij ji

ij i

CC

i j j i

i j j ij

g gu W d d W u J W J W

g gu W d d W uV V

Vg gu W d d W uV

µ µ µ µµ µ µ µ

µ µµ µ

µ µµ µ

γ γ

γ γ γ γγ γ

γ γ γ γ

− + − − + +

− +

− +

= + = +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − − −= +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= − + −

L

( ) ( )5 * 51 12 2

CP iCC j i i jij ij

g gd W u u WV V dµ µµ µγ γ γ γ+⎯⎯→ − + −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

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CKM-matrix: where are the phases?

u

d,s,b

W

•  Possibility 1: simply 3 ‘rotations’, and put phase on smallest:

•  Possibility 2: parameterize according to magnitude, in O(λ):

This was theory, now comes experiment

•  We already saw how the moduli |Vij| are determined

•  Now we will work towards the measurement of the imaginary part –  Parameter: η

–  Equivalent: angles α, β, γ .

•  To measure this, we need the formalism of neutral meson oscillations…

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Neutral Meson Oscillations

Why?

•  Loop diagram: sensitive to new particles

•  Provides a second amplitude Ø  interference effects in B-decays

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b s

s

b

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Dynamics of Neutral B (or K) mesons…

00 ( )( ) ( ) ( )

( )a t

t a t B b t Bb t⎛ ⎞

Ψ = + ≡ ⎜ ⎟⎝ ⎠

i Ht∂Ψ = Ψ

∂

H = M 00 M

!

"##

$

%&&

hermitian! "# $#

No mixing, no decay…

H = M 00 M

!

"##

$

%&&

hermitian! "# $#

−i2

Γ 00 Γ

!

"##

$

%&&

hermitian! "# $#

No mixing, but with decays… (i.e.: H is not Hermitian!)

( ) ( )( ) ( ) ( )( ) ( )( )

2 2 * * 00

a td a t b t a t b tb tdt⎛ ⎞Γ⎛ ⎞

+ = − ⎜ ⎟⎜ ⎟Γ⎝ ⎠⎝ ⎠

è With decays included, probability of observing either B0 or B0 must go down as time goes by:

0⇒Γ >

Time evolution of B0 and B0 can be described by an effective Hamiltonian:

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Describing Mixing…

00 ( )( ) ( ) ( )

( )a t

t a t B b t Bb t⎛ ⎞

Ψ = + ≡ ⎜ ⎟⎝ ⎠

i Ht∂Ψ = Ψ

∂

H = M 00 M

!

"##

$

%&&

hermitian! "# $#

−i2

Γ 00 Γ

!

"##

$

%&&

hermitian! "# $#

Where to put the mixing term?

H =M M12

M12* M

!

"

##

$

%

&&

hermitian! "## $##

−i2

Γ Γ12Γ12* Γ

"

#

$$

%

&

''

hermitian! "## $##

Now with mixing – but what is the difference between M12 and Γ12?

M12 describes B0 ↔ B0 via off-shell states, e.g. the weak box diagram

Γ12 describes B0↔f↔B0 via on-shell states, eg. f=π+π-

Time evolution of B0 and B0 can be described by an effective Hamiltonian:

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Solving the Schrödinger Equation

( ) ( )12 12

12 12

2 2

2 2

i iM Mi t t

i it M Mψ ψ

∗ ∗

⎛ ⎞− Γ − Γ⎜ ⎟∂= ⎜ ⎟

∂ ⎜ ⎟− Γ − Γ⎜ ⎟⎝ ⎠

Eigenvalues: –  Mass and lifetime of physical states: mass eigenstates

12 12 12 1222 2i im M M ∗ ∗⎛ ⎞⎛ ⎞Δ = ℜ − Γ − Γ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

12 12 12 1242 2i iM M ∗ ∗⎛ ⎞⎛ ⎞ΔΓ = ℑ − Γ − Γ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

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Solving the Schrödinger Equation

( ) ( )12 12

12 12

2 2

2 2

i iM Mi t t

i it M Mψ ψ

∗ ∗

⎛ ⎞− Γ − Γ⎜ ⎟∂= ⎜ ⎟

∂ ⎜ ⎟− Γ − Γ⎜ ⎟⎝ ⎠

Eigenvectors: –  mass eigenstates

H

L

B p B q B

B p B q B

= +

= − 12 12 12 122 2i iq p M M∗ ∗⎛ ⎞ ⎛ ⎞= − Γ − Γ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Time evolution

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•  With diagonal Hamiltonian, usual time evolution is obtained:

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B Oscillation Amplitudes

( )

( )

( )

0 0 0

0 0 0

:

( ) ( )

( ) ( )

t

qB t g t B g t BppB t g t B g t Bq

ψ

+ −

+ −

= +

= +

( )2

i t i te eg tω ω+ −

±

− −±=

For B0, expect: ΔΓ ~ 0, |q/p|=1

( )1 12 2

/ 2

2

i mt i mtimt t e eg t e e

− Δ + Δ

− −Γ±

⎡ ⎤±⎢ ⎥

⎢ ⎥⎢ ⎥⎣ ⎦

;( )

( )

/ 2

/ 2

cos2

sin2

imt t

imt t

mtg t e e

mtg it e e

− −Γ+

− −Γ−

Δ=

Δ=

( )

( )

0

0

1212

H L

H L

B B Bp

B B Bq

= +

= −

For an initially produced B0 or a B0 it then follows: (using:

with

~

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Measuring B Oscillations

( )g t+

( )q g tp −

0B

0B

0B

Xν+ −ll

Xν− +ll

Dec

ay p

roba

bilit

y

( )g t+

( )p g tq −

0B

0B

0B

Xν+ −ll

Xν− +ll

B0àB0

B0àB0

Proper Time à

0mx Δ≡ =

Γ1mx Δ

≡Γ?1mx Δ

≡ ≈Γ

For B0, expect: ΔΓ ~ 0, |q/p|=1

( ) ( )2

1 cos2

teg t m t−Γ

± ± Δ ⋅⎡ ⎤⎣ ⎦;Examples:

~

>

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Compare the mesons:

P0àP0

P0àP0

Probability to measure P or P, when we start with 100% P

Time à

Pro

bab

ility

à

<τ> Δm x=Δm/Γ y=ΔΓ/2Γ K0 2.6 10-8 s 5.29 ns-1 Δm/ΓS=0.49 ~1

D0 0.41 10-12 s 0.001 fs-1 ~0 0.01

B0 1.53 10-12 s 0.507 ps-1 0.78 ~0

Bs0 1.47 10-12 s 17.8 ps-1 12.1 ~0.05

By the way, ħ=6.58 10-22 MeVs

x=Δm/Γ: avg nr of oscillations before decay

Summary (1)

•  Start with Schrodinger equation:

•  Find eigenvalue:

•  Solve eigenstates:

•  Eigenstates have diagonal Hamiltonian: mass eigenstates!

( )( )

( )a t

tb t

ψ⎛ ⎞

= ⎜ ⎟⎝ ⎠

(2-component state in P0 and P0 subspace)

pq

ψ ±

⎛ ⎞= ⎜ ⎟±⎝ ⎠

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Summary (2)

•  Two mass eigenstates

•  Time evolution:

•  Probability for |P0> à |P0> !

•  Express in M=mH+mL and Δm=mH-mL à Δm dependence

0 ( )P t

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Summary 00

00

H

L

B p B q B

B p B q B

= +

= −

•  p, q:

•  Δm, ΔΓ:

•  x,y: mixing often quoted in scaled parameters:

12 12 12 1222 2i im M M ∗ ∗⎛ ⎞⎛ ⎞Δ = ℜ − Γ − Γ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

12 12 12 1242 2i iM M ∗ ∗⎛ ⎞⎛ ⎞ΔΓ = ℑ − Γ − Γ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

( )

( )

0 0 0

0 0 0

( ) ( )

( ) ( )

qB t g t B g t BppB t g t B g t Bq

+ −

+ −

= +

= +

12 12

12 12

/ 2/ 2

M iqp M i

∗ ∗− Γ=

− Γ

q,p,Mij,Γij related through:

( )

( )

/ 2

/ 2

cos2

sin2

imt t

imt t

mtg t e e

mtg it e e

− −Γ+

− −Γ−

Δ=

Δ=

with

Time dependence (if ΔΓ~0, like for B0):

2

mx yΔ ΔΓ= =Γ Γ

cos( ) cos = cosm t tmt xτ τ

Δ⎛ ⎞ ⎛ ⎞Δ = ⎜ ⎟ ⎜ ⎟Γ⎝ ⎠ ⎝ ⎠

Personal impression:

•  People think it is a complicated part of the Standard Model (me too:-). Why?

1) Non-intuitive concepts? §  Imaginary phase in transition amplitude, T ~ eiφ

§  Different bases to express quark states, d’=0.97 d + 0.22 s + 0.003 b

§  Oscillations (mixing) of mesons: |K0> ↔ |�K0>

2)  Complicated calculations?

3)  Many decay modes? “Beetopaipaigamma…”

–  PDG reports 347 decay modes of the B0-meson:

•  Γ1 l+ νl anything ( 10.33 ± 0.28 ) × 10−2

•  Γ347 ν ν γ <4.7 × 10−5 CL=90%

–  And for one decay there are often more than one decay amplitudes… Niels Tuning (20)

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )( )

2 2 220

20 2 2

2 2

2

1 2

f

f

B f A g t g t g t g t

B f A g t g t g t g t

λ λ

λλ λ

∗+ − + −

∗ ∗+ − + −

⎡ ⎤Γ → ∝ + + ℜ⎣ ⎦

⎡ ⎤Γ → ∝ + + ℜ⎢ ⎥

⎢ ⎥⎣ ⎦

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Describing Mixing

M12 describes B0 ↔ B0 via off-shell states, e.g. the weak box diagram

Γ12 describes B0↔f↔B0 via on-shell states, eg. f=π+π-

Time evolution of B0 and�B0 can be described by an effective Hamiltonian:

Box diagram and Δm

0 00 0 0 0

H LH H L LP P

m m m P H P P H PΔ = − = −

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Inami and Lim, Prog.Theor.Phys.65:297,1981

Box diagram and Δm

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Box diagram and Δm: Inami-Lim

C.Gay, B Mixing, hep-ex/0103016

•  K-mixing

Box diagram and Δm: Inami-Lim

C.Gay, B Mixing, hep-ex/0103016

•  B0-mixing

Box diagram and Δm: Inami-Lim

C.Gay, B Mixing, hep-ex/0103016

•  Bs0-mixing

Next: measurements of oscillations

1.  B0 mixing: Ø  1987: Argus, first

Ø  2001: Babar/Belle, precise

2.  Bs0 mixing:

Ø  2006: CDF: first

Ø  2010: D0: anomalous ??

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B0 mixing

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B0 mixing

•  What is the probability to observe a B0/B0 at time t, when it was produced as a B0 at t=0? –  Calculate observable probility Ψ*Ψ(t)

•  A simple B0 decay experiment. –  Given a source B0 mesons produced in a flavor eigenstate |B0>

–  You measure the decay time of each meson that decays into a flavor eigenstate (either B0 or�B0) you will find that

( )

( ))cos(12

)|)((

)cos(12

)|)((

/00

/00

mteBtBprob

mteBtBprob

t

t

Δ−∝

Δ+∝

−

−

τ

τ

)cos()()()()(

0000

0000 tmtNtNtNtN

BBBB

BBBB ⋅Δ=+

−

→→

→→

B0 oscillations:

–  First evidence of heavy top

–  à mtop>50 GeV

–  Needed to break GIM cancellations

NB: loops can reveal heavy particles!

B0 mixing: 1987 Argus

Phys.Lett.B192:245,1987

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B0 mixing pointed to the top quark:

B0 mixing: t quark GIM: c quark

ARGUS Coll, Phys.Lett.B192:245,1987

b d

d b

d s

μμ

K0→µµ pointed to the charm quark: GIM, Phys.Rev.D2,1285,1970

…

…

…

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B0 mixing: 2001 B-factories

•  You can really see this because (amazingly) B0 mixing has same time scale as decay –  τ=1.54 ps

–  Δm=0.5 ps-1

–  50/50 point at πΔm ≈ τ

–  Maximal oscillation at 2πΔm ≈ 2τ

•  Actual measurement of B0/�B0 oscillation –  Also precision measurement

of Δm!

)cos()()()()(

0000

0000 tmtNtNtNtN

BBBB

BBBB ⋅Δ=+

−

→→

→→

Bs0 mixing

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Bs0 mixing: 2006

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Bs0 mixing (Δms): SM Prediction

)(1)1(

2/1)(2/1

4

23

22

32

λ

ληρλ

λλλ

ηρλλλ

OAiA

AiA

VVVVVVVVV

V

tbts

cbcscd

ubusud

CKM

td

+⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−−

−−

−−

=⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

=

Vts

Vts

Vts

Vts

CKM Matrix Wolfenstein parameterization

2

22

2

2

2

2

td

ts

Bd

Bs

td

ts

BdBd

BsBs

Bd

Bs

d

s

VV

mm

VV

BfBf

mm

mm

ξ==Δ

Δ

Ratio of frequencies for B0 and Bs

ξ = 1.210 +0.047 from lattice QCD-0.035

Vts ~ λ2

Vtd ~λ3 à Δms ~ (1/λ2)Δmd ~ 25 Δmd

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Bs0 mixing (Δms): Unitarity Triangle

0*** =++ tbtdcbcdubud VVVVVVCKM Matrix Unitarity Condition

cdts

td

cbcd

td

VVV

VVVV

tb 1*

*

×=

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Bs0 mixing (Δms)

0 00 0

0 00 0

( ) ( )cos( )

( ) ( )B B B B

B B B B

N t N tt

N t N tm→ →

→ →+

Δ−

= ⋅

Δms=17.77 ±0.10(stat)±0.07(sys) ps-1

cos

(Δm

st)

Proper Time t (ps)

hep-ex/0609040

Bs b

b

s

s t

t W W Bs

g̃ Bs Bs

b

s

s

b

x

x

b̃

b̃

s ̃

s ̃

g̃

- - - -

- - -

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Bs0 mixing (Δms): New: LHCb

0 00 0

0 00 0

( ) ( )cos( )

( ) ( )B B B B

B B B B

N t N tt

N t N tm→ →

→ →+

Δ−

= ⋅

Bs b

b

s

s t

t W W Bs

g̃ Bs Bs

b

s

s

b

x

x

b̃

b̃

s ̃

s ̃

g̃

LHCb, arXiv:1304.4741

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Bs0 mixing (Δms): New: LHCb

0 00 0

0 00 0

( ) ( )cos( )

( ) ( )B B B B

B B B B

N t N tt

N t N tm→ →

→ →+

Δ−

= ⋅

LHCb, arXiv:1304.4741

Ideal

Tagg

ing,

re

solu

tion

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Mixing à CP violation?

•  NB: Just mixing is not necessarily CP violation!

•  However, by studying certain decays with and without mixing, CP violation is observed

•  Next: Measuring CP violation… Finally

Meson Decays

•  Formalism of meson oscillations:

•  Subsequent: decay

0 ( )P t

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Notation: Define Af and λf

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Some algebra for the decay P0 à f

0 ( )P t

Interference

P0 àf P0àP0 àf

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Some algebra for the decay P0 à f

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The ‘master equations’ Interference (‘direct’) Decay

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The ‘master equations’ Interference (‘direct’) Decay

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Classification of CP Violating effects

1.  CP violation in decay

2.  CP violation in mixing

3.  CP violation in interference

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What’s the time?

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