niels tuning (1) particle physics ii – cp violation lecture 3 n. tuning

Niels Tuning (1)

Particle Physics II – CP violation

Lecture 3

N. Tuning

Niels Tuning (2)

Outline• 1 May

– Introduction: matter and anti-matter

– P, C and CP symmetries

– K-system• CP violation

• Oscillations

– Cabibbo-GIM mechanism

• 8 May– CP violation in the Lagrangian

– CKM matrix

– B-system

• 15 May– B-factories

– BJ/Psi Ks

– Delta ms

Niels Tuning (3)

Literature

• Slides based on courses from Wouter Verkerke and Marcel Merk.

• W.E. Burcham and M. Jobes, Nuclear and Particle Physics, chapters 11 and 14.

• Z. Ligeti, hep-ph/0302031, Introduction to Heavy Meson Decays and CP Asymmetries

• Y. Nir, hep-ph/0109090, CP Violation – A New Era

• H. Quinn, hep-ph/0111177, B Physics and CP Violation

Niels Tuning (4)

: The Kinetic Part

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

uiu d i u d g W W W

d

g giu u id d u W d d W u

L

: ( ) ( )

, , , ,

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

Writing out only the weak part for the quarks:

dLI

gW+

uLI

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

L=JW

1

2

3

0 1

1 0

0

0

1 0

0 1

i

i

SM Kinetic Higgs Yukawa L L L L

Recap from last week

Niels Tuning (5)

: The Higgs Potential

2† 2 † †1

2Higgs Higgs HiggsD D V V L

2 0 :

0

V()

Symmetry

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

V

BrokenSymmetry

2 0 :

0

2v

~ 246 GeV2v

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



Niels Tuning (6)

: The Yukawa PartSM Kinetic Higgs Yukawa 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

. .I I I I I ILi Rj L

d u lij i Rj Liij j RjiY Y YQ d Q u L l h c

The result is:

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

doubletssinglet

0* *

2

0 1

1 0i

With:

(The CP conjugate of To be manifestly invariant under SU(2) )

i, j : indices for the 3 generations!


Niels Tuning (7)

: The Fermion Masses

., , , ,, , .

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

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

LI

f f f fL R

f

R

L RV V

d

d s b sV

b

VM

Yukawa MassL LS.S.B


Niels Tuning (8)

: The Charged CurrentCKMWL LThe 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


Niels Tuning (9)

The Standard Model Lagrangian (recap)


• 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


Niels Tuning (10)

Exploit apparent ranking for a convenient parameterization

• Given current experimental precision on CKM element values, we usually drop 4 and 5 terms as well

– Effect of order 0.2%...

• Deviation of ranking of 1st and 2nd generation ( vs 2) parameterized in A parameter

• Deviation of ranking between 1st and 3rd generation, parameterized through |-i|

• Complex phase parameterized in arg(-i)

23

22

3 2

12

12

1 1L L

A id d

s A s

b bA i A


Niels Tuning (11)

Deriving the triangle interpretation

• Starting point: the 9 unitarity constraints on the CKM matrix

• Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

* * * 0ub ud cb cd tb tdV V V V V V

* * *

* * *

* * *

1 0 0

0 1 0

0 0 1

ud cd td ud us ub

us cs ts cd cs cb

ub cb tb td ts tb

V V V V V V

V V V V V V V V

V V V V V V


Niels Tuning (12)

Visualizing arg(Vub) and arg(Vtd) in the () plane

• We can now put this triangle in the () plane

)(*

*

iVV

VV

cdcb

udub

)1(*

*

iVV

VV

cdcb

tdtb


Niels Tuning (13)

Dynamics of Neutral B (or K) mesons…

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

00 ( )( ) ( ) ( )

( )

a tt a t B b t B

b t

i H

t

hermitian

0

0

MH

M

No mixing, no decay…

hermitian hermitian

0 0

0 02

M iH

M

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

2 2 * * 0

0

a tda t b t a t b t

b tdt

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

0

Niels Tuning (14)

Describing Mixing…

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

00 ( )( ) ( ) ( )

( )

a tt a t B b t B

b t

i H

t

hermitian hermitian

0 0

0 02

M iH

M

Where to put the mixing term?

12 12* *12 12

hermitian hermitian

2

M M iH

M M

Now with mixing – but what is the difference between M12 and 12?

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

12 describes B0fB0 via on-shell states, eg. f=

For details, look up “Wigner-Weisskopf” approximation…

Niels Tuning (15)

Solving the Schrödinger Equation

H

L

B p B q B

B p B q B

12 12

12 12

2 2

2 2

i iM M

i t ti it

M M

12 0 0, 1q

p if: 12 12 12 122 2

i iq p M M

i tH H

i tL L

B t B e

B t B e

1

2m M m

2

im

12 12 12 1222 2

i im M M

12 12 12 1242 2

i iM M

From the eigenvalue calculation:

Eigenvectors:

m and follow from the Hamiltonian:

H Lt B t B t Solution:

( and are initial conditions):

Niels Tuning (16)

B Oscillation Amplitudes

0 0 0

0 0 0

:

( ) ( )

( ) ( )

t

qB t g t B g t B

p

pB t g t B g t B

q

( )2

i t i te eg t

For B0, expect: ~ 0, |q/p|=1

1 1

2 2/ 2

2

i mt i mt

imt t e eg t e e

/ 2

/ 2

cos2

sin2

imt t

imt t

mtg t e e

mtg it e e

0

0

1

2

1

2

H L

H L

B B Bp

B B Bq

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

with

Niels Tuning (17)

Measuring B Oscillations

( )g t

( )q

g tp

0B

0B

0B

X

X

Dec

ay p

rob

ab

ilit

y

( )g t

( )p

g tq

0B

0B

0B

X

X

B0B0

B0B0

Proper Time

0m

x

1m

x

1m

x

For B0, expect: ~ 0, |q/p|=1

21 cos

2

teg t m t

Examples:

Niels Tuning (18)

Measuring 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 *(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

mte

BtBprob

mte

BtBprob

t

t

)cos()()(

)()(

0000

0000 tmtNtN

tNtN

BBBB

BBBB

Niels Tuning (19)

Measuring B0 mixing

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

– m=0.47 ps-1

– 50/50 point at m

– Maximal oscillation at 2m 2

• Actual measurementof B0/B0bar oscillation– Also precision measurement

of m!

)cos()()(

)()(

0000

0000 tmtNtN

tNtN

BBBB

BBBB

Niels Tuning (20)

Back to finding new measurements

• Next order of business: Devise an experiment that measures arg(Vtd)and arg(Vub).

– What will such a measurement look like in the () plane?

β

-i

-i

γ1 1

1 1 1

1 1

e

e

CKM phasesFictitious measurement of consistent with CKM model

)(*

*

iVV

VV

cdcb

udub

)1(*

*

iVV

VV

cdcb

tdtb

Niels Tuning (21)

The B0 mixing formalism and the angle b

• Reduction to single (set of 2) amplitudes is major advantage in understanding B0 mixing physics

• A mixing diagram has (to very good approximation) a weak phase of 2– An experiment that involves interference between an amplitude

with mixing and an amplitude without mixing is sensitive to the angle !

• Small miracle of B physics: unlike the K0 system you can easily interpret the amount of observable CP violation to CKM phases!

Niels Tuning (22)

Find the right set of two amplitudes

• General idea to measure b: Look at interference between B0 fCP and B0 B0 fCP

– Where fCP is a CP eigenstate (because both B0 and B0 must be able to decay into it)

• Example (not really random): B0 J/ KS

B0 f B0 B0 f

Niels Tuning (23)

Back to business – Measuring with B0 J/ KS

• Were going to measure arg(Vtd2)=2 through the

interference of these two processes

• We now know from the B0 mixing formalism that the magnitude of both amplitudes varies with time

B0 f B0 B0 f

)cos(12

)(/(

)cos(12

)/(

/00

/0

mte

tKJBBprob

mte

KJBprob

t

S

t

S

Niels Tuning (24)

How can we construct an observable that measures b

• What do we know about the relative phases of the diagrams?

B0 f B0 B0 f

(strong)= (strong)=

(weak)=0 (weak)=2

(mixing)=/2

)2/sin()2/cos(),( 002/

0 mtBp

qimtBetx timt

There is a phase difference of i between the B0 and B0bar

Decays are identical

K0 mixing exactlycancels Vcs

0 0 0

0 0 0

:

( ) ( )

( ) ( )

t

qB t g t B g t B

p

pB t g t B g t B

q

/ 2

/ 2

cos2

sin2

imt t

imt t

mtg t e e

mtg it e e

Niels Tuning (25)

Measuring ACP(t) in B0 J/ KS

• What do we need to observe to measure

• We need to measure

1) J/ and KS decay products

2) Lifetime of B0 meson before decay

3) Flavor of B0 meson at t=0 (B0 or B0bar)

• First two items relatively easy– Lifetime can be measured from flight length if B0 has momentum

in laboratory

• Last item is the major headache: How do you measure a property of a particle before it decays?

)sin()2sin()(00

00

mtNN

NNtA

fBfB

fBfBCP

Niels Tuning (26)

B0(t) B0(t) ACP(t) = sin(2β)sin(mdt)

sin2

Dsin2

Putting it all together: sin(2) from B0 J/ KS

• Effect of detector imperfections

– Dilution of ACP amplitude due imperfect tagging

– Blurring of ACP sine wave due to finite t resolution

Imperfect flavor tagging

Finite t resolution

t t

Niels Tuning (27)

Combined result for sin2

sin2β = 0.722 0.040 (stat) 0.023 (sys)

J/ψ KL (CP even) mode(cc) KS (CP odd) modes

hep-ex/0408127

ACP amplitudedampened by (1-2w)w flav. Tag. mistake rate

Niels Tuning (28)

4

Consistency with other measurements in (,) plane

Method as in Höcker et al, Eur.Phys.J.C21:225-259,2001

Prices measurement ofsin(2b) agrees perfectlywith other measurementsand CKM model assumptions

The CKM model of CP violation experimentallyconfirmed with high precision!

4-fold ambiguity because we measure sin(2), not

1

2

3

without sin(2)

Niels Tuning (29)

Back to business – Measuring with B0 J/ KS

• Were going to measure arg(Vtd2)=2 through the

interference of these two processes

• We now know from the B0 mixing formalism that the magnitude of both amplitudes varies with time

B0 f B0 B0 f

)cos(12

)(/(

)cos(12

)/(

/00

/0

mte

tKJBBprob

mte

KJBprob

t

S

t

S

Niels Tuning (30)


• The easiest case: calculate (B0 J/ KS) at t=m

– Why is it easy: cos(mt)=0 both amplitudes (with and without mixing) have same magnitude: |A1|=|A2|

– Draw this scenario as vector diagram

– NB: Both red and blue vectors have unit length

+ =/2+2

1-cos()

sin

()

cos()N(B0 f) |A|2 (1-cos)2+sin2 = 1 -2cos+cos2+sin2 = 2-2cos(/2+2) 1-sin(2)

Niels Tuning (31)


• Now also look at CP-conjugate process

• Directly observable result (essentially just from counting) measure CKM phase directly!

)2sin(00

00

fBfB

fBfBCP NN

NNACP

+ =/2+2

+ = /2-2

N(B0 f) |A|2 (1-cos)2+sin2 = 1 -2cos+cos2+sin2 = 2-2cos(/2+2) 1-sin(2)

N(B0 f) (1+cos)2+sin2 = 2+2cos(/2-2) 1+sin(2)

1-cos()

sin

()

sin

()

1+cos()

)2sin()2/(00

00

fBfB

fBfBCP NN

NNmtA

Niels Tuning (32)

Bs mixing

• Δms has been measured at Fermilab 4 weeks ago!

Niels Tuning (33)

Standard Model Prediction

)(

1)1(

2/1

)(2/14

23

22

32

O

AiA

A

iA

VVV

VVV

VVV

V

tbts

cbcscd

ubusud

CKM

td

Vts

Vts

Vts

Vts

CKM Matrix Wolfenstein parameterization

2

2

22

2

2

2

td

ts

Bd

Bs

td

ts

BdBd

BsBs

Bd

Bs

d

s

V

V

m

m

V

V

Bf

Bf

m

m

m

m

Ratio of frequencies for B0 and Bs

= 1.210 +0.047 from lattice QCD-0.035

Vts ~ 2, Vtd ~3, =0.224±0.012

(hep/lat-0510113)

Niels Tuning (34)

Unitarity Triangle

0*** tbtdcbcdubud VVVVVVCKM Matrix Unitarity Condition

cdts

td

cbcd

td

VV

V

VV

VVtb

1*

*

Niels Tuning (35)

Before the measurement: Unitarity Triangle Fit

• CKM Fit result: ms: 18.3+6.5 (1) : +11.4 (2) ps-1

from md

from md/msLower limit on ms

-1.5 -2.7

Niels Tuning (36)

Measurement .. In a Perfect World

what about detector effects?

“Rig

ht

Sig

n”

“Wro

ng

Sig

n”

Niels Tuning (37)

Hadronic Bs Decays

• relatively small signal yields (few thousand decays)

• momentum completely contained in tracker

• superior sensitivity at higher ms

0sB

sD

W

b cd

s

u

s

ss DB0

Niels Tuning (38)

Semileptonic Bs Decays

• relatively large signal yields (several 10’s of thousands)

• correct for missing neutrino momentum on average

• loss in proper time resolution

• superior sensitivity in lower ms range

lss lDB 0

0sB

sD

W

b cl

s

l

s

Niels Tuning (39)

Tagging the B Production Flavor

vertexing (same) side

“opposite” side

• use a combined same side and opposite side tag!

• use muon, electron tagging, jet charge on opposite side

• jet selection algorithms: vertex, jet probability and highest pT

• particle ID based kaon tag on same side

e,

Niels Tuning (40)

Combined Amplitude Scan

A/A (17.25 ps-1) = 3.5

How significant is this result?

Preliminary

25.3 ps-1

Niels Tuning (41)

Conclusions

• found signature consistent with Bs - Bs oscillations

• probability of fluctuation from random tags is 0.5%

ms = 17.33 +0.42 (stat) ± 0.07 (syst) ps-1

|Vtd / Vts| = 0.208 +0.008 (stat ± syst)

-0.21

-0.007

Niels Tuning (42)

Outline• 1 May

– Introduction: matter and anti-matter

– P, C and CP symmetries

– K-system• CP violation

• Oscillations

– Cabibbo-GIM mechanism

• 8 May– CP violation in the Lagrangian

– CKM matrix

– B-system

• 15 May– B-factories

– BJ/Psi Ks

– Delta ms

Niels Tuning (43)

Remember the following:

• CP violation is discovered in the K-system

• CP violation is naturally included if there are 3 generations or more

• CP violation manifests itself as a complex phase in the CKM matrix

• The CKM matrix gives the strengths and phases of the weak couplings

• CP violation is apparent in experiments/processes with 2 interfering amplitudes

• The angle β is measured through B0 J/ KS

• Mixing of neutral mesons happens through the “box” diagram

niels tuning (1) particle physics ii – cp violation lecture 3 n. tuning

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