lecture ii - det matematisk-naturvitenskapelige fakultet, uio · for decays via the quark...

ITRG lecture week Oslo 06-03-2006ITRG lecture week Oslo 06-03-2006 G. Eigen, IFT BergenG. Eigen, IFT Bergen

Lecture II:Lecture II: The CKM matrix

Types of CP violation

Time-dependent CP violation in B0 decay

Reconstruction of B mesons

The Golden Mode B J/ K0S

Strategy for CP Asymmetry Measurements

B-Factory Experiments


The The CCabibbo abibbo KKobayashi obayashi MMaskawa askawa MatrixMatrix

In the Standard ModelStandard Model CPCP violation violation results from Yukawa couplings of the HHiggs field to quark fields

quark mass mixing, since HH couples to weak eigenstates that differ from mass eigenstates

Weak eigenstates are related to mass eigenstates by the unitary CCabibbo-abibbo-KKobayashi-obayashi-MMaskawaaskawa Matrix:

The The CKMCKM matrix is unitary and is matrix is unitary and is completely determined by completely determined by 44 parameters: parameters: 3 angles & 1 phase3 angles & 1 phase

W boson couplings W boson couplings

At least At least 22 CKMCKM matrix elements matrix elements occur in each weak occur in each weak interaction process interaction process

d'

s'

b'

=

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

d

s

b

u c t

d s b

VVCKMCKM

u c t

d’ s’ b’

weak weak eigenstateseigenstates mass mass eigenstateseigenstates


Suitable representation of CKMCKM matrix is small-angle Wolfenstein parametrization:

VCKM

=

1 1

2

2 A 3( i )

1 1

2

2 A 2

A 3(1 i ) A 2 1

+ O( 4)

4 parameters:4 parameters:=0.22, =0.22, AA, , , ,

CPCP

VCKM

=

0.9739 0.9751 0.221 0.227 0.0029 0.0045)

0.221 0.227 0.973 0.9744 0.039 0.044

0.0048 0.014 0.037 0.043 0.9990 0.9992

Present knowledge of magnitude of CKM matrix

The The CCabibbo abibbo KKobayashi obayashi MMaskawa askawa MatrixMatrix

= 1 1

2

2( )

= 1 1

2

2( )withwith


II.2 II.2 Unitarity Unitarity Conditions of CKM MatrixConditions of CKM Matrix

Vud Vus VubVcd Vcs VcbVtd Vts Vtb

**

VVududVVubub++VVcdcdVVcbcb++VVtdtdVVtbtb = 0 = 0**

The unitarity conditions ofThe unitarity conditions of CKMCKM matrix yieldmatrix yield 66 triangulartriangular relations relations

Physicswise most interesting is the relation: Physicswise most interesting is the relation:

The graphical representation of this relation The graphical representation of this relation yields the so-calledyields the so-called UUnitarity nitarity TTriangleriangle!!

All triangles have the same areaAll triangles have the same area AA J = J = m( Vm( VijijVVklklVV**

ililVV**kjkj) ) 0 0

for any ifor any i kk and j and j ll ( (Jarlskog determinantJarlskog determinant))

J=(3.0±0.9)J=(3.0±0.9) 1010-5-5 CPCP

****

1 -sin1 -sin c c 1 1


Measurements in Measurements in Unitarity Unitarity TriangleTriangle BABAR measures BABAR measures 44 CKM elements & CKM elements & 33 phases phases

overconstrains unitarity overconstrains unitarity triangletriangle test Standard Model test Standard Model

VVtdtdVV**tbtb

VVcd cd VV**cbcb

mmdd,, mmss

VVud ud VV**ubub

VVcd cd VV**cbcb

BB(B(B XXuull ))

BB(B(B ,, ll ))

BB(B(B XXccll ))

BB(B(B DD**ll )) 11

A

CP(B + , ± , + )

ACP(B J / K

S

0 )

ACP(B K

S

0 )

B(B± [D0,D0]K± )


Measurements of the Measurements of the -- Plane Plane

There are many useful constraints from B and K decays


So a measurement of the CP asymmetry will determine Im

If Im just depends only on electroweak parameters

Measuring Angles of Measuring Angles of Unitarity Unitarity TriangleTriangle

In general A & A have contributions from several processes

A = Aie

iie

+ ii

i

If all amplitudes to a particular decay all have the same CKM phase D, the strong phases cancel in the ratio and we obtain

A = A

For 12«M12, we can parameterize q/p in terms of a CKM mixing phase M by

A

A= 1

A = Aie

iie

ii

i

&&

where Ai are real, i are strong phases and i are weak phases

A

A= e

2iD&&

q

p=

M12

M12

*= e

2iM


So in this case we get

Note that here Im depends on CKM parameters only this allows for precise predictions

Lets now focus on processes that in the Standard Model are dominated by amplitudes having a single CKM phase

Lets first express q/p in terms of CKM matrix elements:

Bd

0 : q

pBd

=

Vtb

*Vtd

VtbV

td

*

= e

2iM+

D( ) m =-sin2

M+

D( )

Bs

0 : q

pBs

=

Vtb

*Vts

VtbV

ts

*



For decays via the quark subprocesses where q is a Q=2/3 quark (u,c) and q’ is a Q=-1/3 quark (d’,s’) the amplitude ratio in terms of CKM matrix elements becomes

If a single K0S or K0

L is present in the final state, K0-K0 mixing is essential since B0 K0 while B0 K0

interference is only possible via K0-K0 mixing, yielding

K 0 : q

pK0

=

Vcd

*Vcs

VcdV

cs

*

A

A=

VcbV

qq '

*

Vcb

*Vqq '

b cqq ' and b uqq'

Thus, for these modes is given by

=q

pB

A

A

q

pK0

So sign(Im ) depends on CP of CP eigenstate, =1 for CP-even =-1 for CP-odd states



CP asymmetries in B0 decays into CP eigenstates provide a good way to measure the 3 angles of the Unitarity Triangle, defined by

Lets consider some examples

ArgV

tdV

tb

*

VudV

ub

*, Arg

VcdV

cb

*

VtdV

tb

*, Arg

VudV

ub

*

VcdV

cb

*

B J / K

s

0:

=

Vtb

*V

td

VtbV

td

*

Vud

*V

ub

VudV

ub

*m = sin 2

KS

0=

Vtb

*V

td

VtbV

td

*

Vcs

*V

cb

VcsV

cb

*

Vcd

*V

cs

VcdV

cs

*m = sin 2

B+

:

0B

J/J/

KK00SS

VV**cscs

VV**udud

VVcbcb

Measurement of is more complicated (see later)

The goal is to make many independent measurements of both sides and angles of the UnitarityTriangle to overconstrain it test SM


||VVtdtdVV**tbtb||

||VVcdcdVV**cbcb||

||VVududVV**ubub||

VVcbcb


New Sources of CP in Early UniverseNew Sources of CP in Early Universe In Standard ModelStandard Model CPCP violation violation arises from the single phase of

CKMCKM matrix

The CKMCKM phase is way too small to explain baryon-antibaryon asymmetry in our universe

baryon/photonbaryon/photon ratio: SM prediction: nnBB/n/n 10 10-20-20

observation: nnBB/n/n 1010-11-11

99 orders of magnitude difference! need new sources of CPCP violationviolation

New sources of CPCP violationviolation can arise from simple extensions of the Standard ModelStandard Model, e.g.: supersymmetrysupersymmetry

allow a first order phase transition and new sources of CPCP

It is not unlikely that new sources of CPCP exist that already contribute in low energy processes such as neutral B decays


Epoch

1.5x1010 a

5 x109 a

109 a

1013 s

10-10 s

10-34 s

10-43 s

2x102 s

Time

QuantumGravity

GUT

EW

Quark

Lepton

Photon

CosmicInflation

H

He

You are here

1032

1027

1015

109

6x103

18

3

massless q, l, g, X, Y bosons

quark-gluon plasma

qq => e+e-

e+e- decay

p, nformation

p-n bound

’sdecouple

heavy X, Y decay

Theory ofEverything

SuperSymmetry

SU(3) QCD

weakforce

nuclearforce

dd

Energy(GeV)

Heavy Star

Black HoleProtogalaxy

6

1019

1014

102

10-4

10-9

10-12

10-13

ppnn

pn

n

n p

p

Temperature [K]

’s decouple

q

q g10-8 q excessdue to CP violation

quarkconfinement

3o K back-groundradiation

SU(2)xU(1)

5x10-13

atoms form

1 s

10-4 s

10-32 s

10-35 s

BIG BANG

Composition of the universe

q

qq qg g q

uu u

q qg q

Galaxy

Symmetries

pn

strings

p

EVOLUTIONEVOLUTION

OF OUROF OUR

UNIVERSEUNIVERSE


Mixing is governed by q/p where

Types of CP Violation: in MixingTypes of CP Violation: in Mixing

When CP is conserved mass eigenstates are CP eigenstates The relative phase between M12 & 12 vanishes

Thus, if

This type of CP violation is called “CP violation in mixing” or indirect CP violation it results from mass eigenstates being different from CP eigenstates

One way to observe this type of CP violation is to explore decays where the b quark decays as B Xl

q p 1 CP violation

q

p

2

=

M12

* i2 12

*

M12

i2 12

Asl=

Bphys

0 (t) X +( ) Bphys

0 (t) X( )B

phys

0 (t) X +( ) + Bphys

0 (t) X( ) which follows from

X H B

phys

0 (t) = q p( ) g (t)A*

=

1 q p4

1 + q p4

O (10 2)

+ X H B

phys

0 (t) = p q( ) g (t)A&&


For any final state f the quantity is independent of phase conventions

There are 2 types of phases that may appear in 1. Weak phases: complex parameters in the Lagrangian that

contribute to the decay amplitude Conjugate complex parameter enters CP conjugated amplitude Phase in has opposite sign In SM these parameters are CKM matrix elements (note: weak phase in single amplitude is convention dependent while phase difference of 2 different amplitudes is convention independent) 2. Strong phases: phases appearing in scattering or decay

amplitudes These are CP conserving entering with same sign in amplitude and CP conjugated amplitude They typically originate from intermediate on-shell states dominant rescattering is typically caused by strong interaction

Types of CP Violation: in DecayTypes of CP Violation: in Decay

A

fA

f

A

f & A

f

A

f & A

f


It is useful to write amplitudes in terms of the magnitude Ai, the strong-phase contribution ei i and the weak-phase contribution ei i

Types of CP Violation: in DecayTypes of CP Violation: in Decay

A

f= A

iexp i(

i+

i){ }

i

, Af= exp 2i(

f B){ } A

iexp i(

i i){ }

i

where f is flavor dependent strong phase & B is CP phase of the B meson if f is CP eigenstate, e2i B =±1 is CP eigenvalue

So the amplitudes ratio here is

Af

Af

=

Aiexp i(

i i){ }

i

Aiexp i(

i+

i){ }

i When CP is conserved the weak phases i

are all equal if

This type is called “CP violation in decay” or direct CP violation

Note, that direct CP violation will not occur unless two terms with different weak phases have also different strong phases

A

fA

f1 CP violation

A

2

A2

= 2 AiA

jsin

i j( ) sin i j( )i,j

Any CP asymmetries in charged B decays result from direct CP violation

Af=

(B+ f ) (B f )

(B+ f ) + (B f )=

1 A A2

1 + A A2


Consider neutral B decays into CP eigenstates fCP

The quantity of interest here is

When CP is conserved |q/p|=1, & relative phase between |q/p| & vanishes

Thus, if | |=1 but Im 0, we have “CP violation in the interference between decays with and without mixing”

The CP asymmetry here is given by

Types of CP Violation: from InterferenceTypes of CP Violation: from Interference between Decays with & without Mixingbetween Decays with & without Mixing

=fCP

q

p

AfCP

AfCP

A

fCP

AfCP

= 1

A

fCP

AfCP

AfCP

(t) =

(B0 (t) fCP) (B0 (t) f

CP)

(B0 (t) fCP) + (B0 (t) f

CP)=

(1fCP

2

) cos mBt 2 m

fCP

sin mBt

1 +fCP

2

For | fCP |=1 this simplifies to

A

fCP

(t) = 2 mfCP

sin mt


CPCP Violation Violation from Interferencefrom Interference between Decays with & without Mixing between Decays with & without Mixing

Process CP conjugated process

CPCP Violation Violation is is caused bycaused by interference effect interference effect between mixing & decaybetween mixing & decay

~ ei

0B

0B

CP

mix

ing

decay

CPfA

CPfA

~ ei

0B

0B

CPm

ixin

gdecay

CPfA

CPfA ~ ei

~ ei

~ ei

~ ei

Decay rate Decay rate CPCP-conjugated -conjugated decay ratedecay rate

t = 0 t = 0

fCP

t t fCP


by H. Miyake

Mixing-induced CP Direct CPV

AfCP

( t) B0( t)

B0( t)

B0( t) +

B0( t)

S =2 Im( )

1 +

2

, C =

1

2

1 +

2

=q

p

Af

Af

= 1 S = m ( ) , C = 0

A

fCP

( t) = m( ) sin( m t) A

fCP

( t) = S sin m t C cos m t

For single decay amplitude: |q/p|=1

mixingmixing decaydecay

Relevant parameterRelevant parameter-C-C

Time-dependent CP Violation in B Time-dependent CP Violation in B00 decays decays


Reconstruction of B MesonsReconstruction of B Mesons

mES

Esidebands

signal

region

beam energysubstituted mass

energy difference

J/J/ K KSS

e+e (4S) BB

Event shapesEvent shapes

(4S)(4S) BBBB ee++ee-- qqqq

m

ES= E

beam

*2 pB

*2

E = E

B

*E

Beam

*

In center-of-massIn center-of-mass frameframe B-mesons B-mesons are are ~at rest~at rest useful kinematic constraintsuseful kinematic constraints


Golden Mode: BGolden Mode: B00 J/J/ KK00

A

J/ KS,L

0( t) =

J/ KS,L

0sin 2 sin ( m

Bd

t)

|

J/ KS,L

0| = 1 Single weak phase Single weak phase no direct no direct CPCP

BCP = 1

0 (J/ ) KS

0

BCP = +1

0 (J/ ) KL

0

KK00 mixing is requiredmixing is required

Quark processQuark process bb ccsccs

J/ K0= e

2i

is determined by phase of is determined by phase of decay amplitude and decay amplitude and BB00BB00 & & KK00KK00 mixing phases mixing phases

B0

ccK0

B0

ccK0

Theoretically cleanest way to measure Theoretically cleanest way to measure sin sin 22

0B

0BBB00 W W

d

b

J/J/

KK00SS


Linac Linac and PEP II at SLACand PEP II at SLAC

LinacLinac

BABARBABAR

PEP IIPEP IIe-

e+

LL design: design: 3x1033 cm-2s-1

135 pb-1/day

LL reached: reached: >1x1034 cm-2s-1

730 pb-1/day

beam energies ee--: 9 : 9 GeVGeV ee++: 3.1 : 3.1 GeVGeV

Beam Beam currentscurrents

L

1nb=101nb=10--3333cmcm22


The BABAR ExperimentThe BABAR Experiment

(4) Electromagnetic

Calorimeter(6) Instrumented

Iron Yoke(3) Cherenkov-

Detector

(2) Drift Chamber

(1) Silicon

Vertex Detector

ee--

ee++

(5) 1.5 T Solenoid

BABAR has recorded over 300 Million BB pairs


e- e+

ee--(8 (8 GeVGeV))

ee++(3.1 (3.1 GeVGeV))

The The B-Meson B-Meson Factory Factory KEK BKEK B

LL=1.58=1.58 10103434 cm cm-2-2ss-1-1 and 1178.2 pb1178.2 pb-1-1/day/day

Beam energy


The BelleThe Belle DetectorDetector


Logged Integrated LuminosityLogged Integrated Luminosity

~0.8 ab ~0.8 ab-1-1 in total (sum)

Peak luminosity > 1034cm-2s-1

in both experiments !

lecture ii - det matematisk-naturvitenskapelige fakultet, uio · for decays via the quark...

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