[PDF] - TUDIKTP Herbstsc h ule Maria Laac h Sep revised PDF Document

SLIDE 1 TUDIKTP Herbstsc h ule Maria Laac h Sep

revised
Lectures
n

Fla v

ur

Oscillation and CP Asymmetry in B Meson Deca ys R

land

Waldi Institut f

ur

Kern und T eilchenphysik T e chnische Universit at Dr esden

SLIDE 2 ii Con ten ts

In

tro duction

P

article An tiP art icl e Oscillatio ns and CP Violation

The

Unitary CKM Matrix

Unitarit

y T riangles

Phases

and Observ ables

Oscillation

Phenomenology

Mec

hanical Analogon

Standard

Mo del Predictions

Beha

viour

f

the F

ur

Neutral Meson An tiMeson Systems

CP

Eigenstates V ersus Mass Eigenstates

Oscillation

at the

S
Determination
f

the Mixing P arameters

f

B Mesons

Predictions

for x s

y

s and

s
CP

Violation

CP

Violation in B Deca ys

CP

Violation in Common Final States

f

B

and

B

The

B s B s Case

Final

CP Eigenstates from B

r

B s Deca ys

Mixtures
f

CP Eigenstates

NonEigenstates
The

T

tal

Deca y Rate

CP

Violation at the

S
CP

Violation in K Deca ys

Measuremen

t

f

CP Violation at B Meson F actories

B

Pro duction Cross Sections

B

Meson F ractions

Fla

v

ur

T agging

Observ

ed V ersus T rue Asymmetry

Statistical

T agging

Estimating

the P erformance

Ac

kno wledgemen ts

References

SLIDE 3

In

tro duction Our understanding

f

ph ysics in general and particle ph ysics in particular has b een mainly put forw ard b y the disco v ery

f

symmetries It is remark able that most

f

the symmetries disco v ered ha v e ho w ev er nally turned

ut

to b e

nly

almostsym m etries i e to b e more

r

less brok en The

nly

un brok en symmetries so far disco v ered are the U c hargephase symmetry and the SU colour symmetry

The

consequences are that the electric and colour c harges are exactly conserv ed in all

bserv

ed reactions and that the p

sition

in SUspace cannot b e determined e g a red and a blue quark cannot b e distinguished Eac h

f

the symmetries b et w een leptons and quarks

f

dieren t a v

ur

is brok en b y the dieren t masses and electrow eak c harges

f

these particles and is b est appro ximated in strong in teractions as isospin symmetry b et w een the u and d quark due to their almost iden tical constituen t mass Although ph ysics la ws are strictly symmetric under translation

r

rotation spacetime translational and rotational symmetry is brok en through the solutions The fact that matter is not distributed homogeneously throughout the univ erse in tro duces a lo cally asymmetric structure

f

spacetime

r

asymmetric b

undary

conditions to an y microscopic system The spatial symmetries are b est appro ximated

n

a macroscopic scalethe univ erseor for microscopic systems isolated from

ther

matter b y large distances Mirror symmetry parit y P

is

brok en in a more fundamen tal sense b y w eak in teraction whic h mak es a maxim al distinction b et w een fermions

f

left and righ t c hiralit y

First

ideas

f

this unexp ected b eha viour emerged as a solution

f

the

puzzle

the fact that the neutral k aon deca ys b

th

to P

and

P

eigenstates
and

a direct

bserv

ation as leftrigh tasymm etry in w eak b eta deca ys follo w ed so

n
It

is most pronounced in the massless neutrinos whic h are pro duced in w eak in teractions

nly

with lefthanded helicit y

r

righ thanded in the case

f

an tineutrinos th us violating the c harge conjugation symmetry

C
at

the same time The pro duct

f

b

th

discrete symmetries CP

is

almost in tact and seems to b e conserv ed ev en in w eak in teraction pro cesses A small violation has rst b een

bserv

ed in

in

K

deca

ys whic h are up to no w the

nly

system whic h do es not resp ect CP symmetry completely

The

explanation

f

this violation in the Standard Mo del will b e briey discussed in the next c hapter This is not the

nly

p

ssible

description but the

ne

with no additional assumptions A t the same time the Standard Mo del predicts CP violating eects in the deca y

f

b eaut y mesons

B
B

s

B
whic

h should b e ev en large in some rare deca y c hannels

SLIDE 4

P

article An tiP article Oscillations and CP Violation Mesons are neither particles nor an tiparticles in a strict sense since they are comp

sed
f

a quark and an an tiquark This implies the existence

f

mesons with v acuum quan tum n um b ers e g f

More

imp

rtan

t is the existence

f

pairs

f

c hargeconjugate mesons whic h can b e transformed in to eac h

ther

via a v

ur

c hanging w eak in teraction transitions These are K

K
sds
d
D
D
c
u
c

u

B
B
bdb
d
and

B s B s

bsb
s
The

Unitary CKM Matrix The c harged curren t w eak in teractions resp

nsible

for a v

ur

c hanges are describ ed b y the couplings

f

the W b

son

to the curren t J cc

e
A
e
A
X

r g b

u
c
t
A
V
d

s b

A
with

a nontrivial transformation matrix V in the quark sector the Cabibb

Koba

y ashiMask a w a CKM Matrix

V
V

ud V us V ub V cd V cs V cb V td V ts V tb

A

The quark a v

urs

in

are

dened as the mass eigenstates A completely equiv alen t picture is to use the states d

s
b
with

V

and

dene a nondiagonal mass matrix Since mass generation is accomplished in the Standard Mo del via couplings to the Higgs eld

this

mo v es the question

f

the

rigin
f

the CKM matrix elemen ts in to the realm

f

mass generation whic h b elongs still to the more m ysterious parts

f

the Standard Mo del The exploration

f

the Higgs sector is the main motiv ation for the LHC storage ring whic h is built at CERN and will start

p

eration around

The

Higgsquark couplings alone in v

lv

e

indep

enden t parameters

f

the Standard Mo del the quark masses and the parameters

f

the CKM matrix whic h are not related within the theory

Lo

cal gauge in v ariance and bary

n

n um b er conserv ation requires the CKM matrix to b e unitary

If

there w ere more than three quark famili es this w

uld

not hold for the

submatrix

but this p

ssibilit

y is unlik ely

giv

en the limit

n

neutrino a v

urs

from LEP exp erimen ts who nd n

for

neutrinos with mass m uc h b elo w the Z

mass

Th us if a fourth generation exists it m ust incorp

rate

a massiv e neutrino whic h is more than a factor

hea

vier than the tau neutrino ev en if w e assume the exp erimen tal upp er limit for the latter F rom the

real

parameters

f

a general unitary matrix

can

b e absorb ed in

global

phase

relativ

e phases b et w een u c t and

relativ

e phases b et w een d s b whic h are all sub ject to con v en tion and in principle unobserv able If t w

quarks

within

ne
f

these t w

groups

w ere degenerate in mass ev en the sixth phase could b e remo v ed b y redening the basis in their t w

dimensional

subspace Rephasing ma y b e accomplished b y applying a phase factor to ev ery ro w and column V j k

e

i j

k
V

j k

Note

that j

u

c t

k
d

s b

and

the six n um b ers

u
c
t
d
s
b

represen t

nly

v e indep enden t phases in the CKM matrix since dieren t sets

f

f j

k

g yield the same result An y pro duct where eac h ro w and column en ters

nce

as V ij and

nce

via a complex conjugate V

k

l lik e V ij V k l V

il

V

k

j is in v arian t

SLIDE 5

The

Unitary CKM Matrix

under

the transformation

This

implies that

bserv

able phases m ust alw a ys corresp

nd

to similar pro ducts

f

CKM matrix elemen ts with equal n um b ers

f

V and V

factors

and appropriate com bination

f

indices Remo ving unph ysical phases the CKM matrix is describ ed b y

real

parameters where

nly
ne

is a phase parameter while the

ther

three are rotation angles in a v

ur

space The standard parametrization

rst

prop

sed

in

notation

follo ws

uses

a c hoice

f

phases that lea v e V ud and V cb real V

c
s
s
c
A
c
s
e

i

s
e

i

c
A
c
s
s
c
A
c
c
s
c
s
e

i

s
c
c
s
s
e

i

c
c
s
s
s
e

i

c
s
s
s
c
s
c
e

i

c
s
s
s
c
e

i

c
c
A
with

c ij

cos
ij
s

ij

sin
ij
and

s ij

c

ij

ij
A

con v enien t substitution

is

s

s
A
s
sin
A
and

s

cos
A
whic

h reects the apparen t hierarc h y in the size

f

mixing angles via

rders
f

a parameter

This

leads to V

p
A
A
A
p
A
A
p
A
A
i
A
i
p
A
A
p
p
A
B
A
i
A
i
A
A
A
i
A
A
i
A
C

A

O
and

agrees to O

with

the W

lfenstein

appro ximation

V
A
i
i
i

A

A
i
A
i
A
A
A
i
A
A
i
A
A
Equation
is

more con v enien t

in

higher

rders

than the

riginal

prop

sal
f

W

lfenstein
r

an exact parametrization

using

the W

lfenstein

parameters Assuming a unitary

matrix

from exp erimen tal information these parameters are

A
p
while

the phase and therefore eac h individual v alue

f
and
is

still v ery uncertain Inserting these parameters equation

sho

ws clearly the dominance

f

the diagonal matrix elemen ts indication that transitions b et w een quarks

f

dieren t families are suppressed It is the unitarit y constrain t whic h

An

equiv alen t c hoice is

s
c
whic

h leads to the same parametrization to O

SLIDE 6

P

article An tiP article Oscillations and CP Violation mak es V tb

the

b est kno wn matrix elemen t Exp erimen tal constrain ts

n

the magnitude

CL

limits

are
A

With already

ne

more family

f

quarks w e ha v e v e additional real parameters

f

whic h t w

are

new nontrivial phases Therefore the measuremen t

f

all CKM matrix elemen ts and their relativ e phases is an imp

rtan

t test

f

the Standard Mo del

Unitarit

y T riangles If nature pro vides us with just these three famili es

f

fermions unitarit y requires the follo wing

conditions

to b e fullled jV ud j

jV

us j

jV

ub j

a

jV cd j

jV

cs j

jV

cb j

b

jV td j

jV

ts j

jV

tb j

c

jV ud j

jV

cd j

jV

td j

d

jV us j

jV

cs j

jV

ts j

e

jV ub j

jV

cb j

jV

tb j

f
V
ud

V cd

V
us

V cs

V
ub

V cb

g

V

ud

V td

V
us

V ts

V
ub

V tb

h

V

cd

V td

V
cs

V ts

V
cb

V tb

i

V ud V

us
V

cd V

cs
V

td V

ts
j

V ud V

ub
V

cd V

cb
V

td V

tb
k

V us V

ub
V

cs V

cb
V

ts V

tb
l

An arbitrary phase for the whole matrix cancels in V

V
A

phase common to all elemen ts in a line column corresp

nding

to arbitrary phases b et w een u c t

d

s b

will

v anish in eqns jl gi and b ecome a common factor in eqns gi jl Dividing k b y A

V

cd V

cb

yields the unitarit y triangle

as

sho wn in gure a In the W

lfenstein

appro ximation it corresp

nds

to

i
i
A

second

ne

from h is sho wn in gure b Dividing b y A

V
us

V ts and using the appro ximation V ud

giv

es the same triangle

A

closer lo

k

ho w ev er rev eals sligh tly dieren t lengths and angles to O

The

angles

f

the unitarit y triangles k and h in gure

are

dened b y e i

V

td V ub V

ud

V

tb

jV td V ub V ud V tb j e i

V
td

V

cb

V cd V tb jV td V cb V cd V tb j

e

i

V
td

V

us

V ts V ud jV td V us V ts V ud j e i

V
ub

V

cd

V cb V ud jV ub V cd V cb V ud j

e

i

V
ub

V

ts

V us V tb jV ub V ts V us V tb j

this

geometric in terpretation has b een p

in

ted

ut

b y Bjork en

its

rst do cumen tation in prin ted form is in ref

and

more general in ref

in

the complex plane the angle

b

et w een t w

v

ectors A

ae

i and B

be

i is dened b y e i

AB
jAB

j and sin

I

mAB

jAB

j

AB
A
B

ijAB j

SLIDE 7

The

Unitary CKM Matrix

a
V

cd V

cb
A
V

td V

tb
A
i

V ud V

ub
A
i

b

V
us

V ts

A
V
ud

V td

A
i

V

ub

V tb

A
i

Fig

Unitarit

y triangles in the complex plane corresp

nding

to ak and bh Up to corrections

f

O

the

top p

in

ts are

in

b but

in

a and the righ tmost p

in

ts are

in

a but

in

b The angles are related via

These

are rephasing in v arian t expressions hence the angles resem ble ph ysical quan tities indep enden t

f

the CKM parametrization It w as rst emphasized b y Jarlsk

g
that

CP violation can b e describ ed via a rephasing in v arian t quan tit y J

I

m V ij V k l V

il

V

k

j

A
whic

h is up to a sign indep enden t

f

i j k

l
pro

vided i

k
j
l
J
I

m V ud V cs V

us

V

cd
I

m V ud V cb V

ub

V

cd
I

mV ud V ts V

us

V

td
I

m V ud V tb V

ub

V

td
I

m V us V cd V

ud

V

cs
I

m V us V cb V

ub

V

cs
I

m V us V td V

ud

V

ts
I

m V us V tb V

ub

V

ts
I

mV ub V cd V

ud

V

cb
I

m V ub V cs V

us

V

cb
I

m V ub V td V

ud

V

tb
I

m V ub V ts V

us

V

tb
I

m V cd V ts V

cs

V

td
I

mV cd V tb V

cb

V

td
I

m V cs V td V

cd

V

ts
I

m V cs V tb V

cb

V

ts
I

mV cb V td V

cd

V

tb
I

mV cb V ts V

cs

V

tb

SLIDE 8

P

article An tiP article Oscillations and CP Violation These terms are all pro ducts

f

the t yp e I m AB

jAjjB

j I m e iarg Aarg B

jAjjB

j sin arg A

arg

B

whic

h is t wice the area

f

a triangle with sides A and B

and

A and B are sides

f

a unitarit y triangle The areas

f

all six unitarit y triangles are equal and ha v e the v alue J

As

will b e sho wn b elo w CP violating

bserv

ables are t ypically prop

rtional

to the sine

f

the angles in unitarit y triangles lik e sin

I

m e i

I

m V

ub

V

cd

V cb V ud

jV

ub V cd V cb V ud j

J

jV ub V cd V cb V ud j and v anish for J

i

e if all triangles collapse in to lines If the nontrivial phase is

r
the

parameter

is
and

hence J

This

w

uld

also b e the case if t w

quarks
f

a giv en c harge had the same mass since then a rotation b et w een these t w

a

v

urs

could b e c hosen that remo v es the phase factors as can b e seen in

where
w
uld

remo v e all terms with the phase

The

area

f

all triangles dened b y gl is J

This

corresp

nds

to an area

for

the

nes

in gure

since

their sides ha v e b een reduced b y the factor A

If

J

also

the area

f

the unitarit y triangles w

uld

shrink to zero The angles

f

all six triangles gl can b e determined using the standard parametrization

in

a rewritten form V

jV

ud j jV us j jV ub je i

jV

cd je i

jV

cs je i

jV

cb j jV td je i

jV

ts je i

jV

tb j

A
with
Here

absolute v alues and phases are giv en as separate factors The angles

A
and
A
are

all p

sitiv

e and v ery small and their subscript indicates the

rder

in

f

their magnitude The unitarit y triangles in gure

ha

v e angles

In

the W

lfenstein

appro ximation the unitarit y relations read all terms giv en to

rder
r

if this is still

in

brac k ets to leading

rder
A
i
g
A
i
A
A
i
h
A
i
A
A
i
A
i
j
A
i
A
A
i
k
A
i
A
A
l
and

dene three pairs

f

unitarit y triangles

in

total

h
and
k
are

the

nes

sho wn in gure

with

three sides

f

similar length all

f
rder

A

This

is the unitarit y triangle The

ther
nes

are quite at and it will require v ery high precision to pro v e exp erimen tally that they are not degenerate to a line

i
and
l
ha

v e t w

sides
f

length A

and
ne

m uc h shorter

f
rder

A

This

limits the small angles whic h are

and
resp

ectiv ely

They

are close to the dierences

f

angles in the large triangles

g
and
j
ha

v e t w

sides
f

length

and
ne

v ery m uc h shorter

f
rder

A

with

a small angle

and
resp

ectiv ely

Both

are

f
rder

SLIDE 9

The

Unitary CKM Matrix

Tin

y dierences b et w een the t w

standard

unitarit y triangles are O

corrections

A

i
A
A
i
h
A
i
O
A
i
O
O
A
i
A
A
i
k
A
i
O
O
A
i
O
The

angles in these t w

triangles

can b e estimated from exp erimen tal constrain ts

n

a

unitary

CKM matrix leading to CL limits

All

phase angles are

nly

w eakly constrained b y these limits and

ne
f

the aims

f

exp erimen ts designed to

bserv

e CP violation in B meson deca ys is a rst measuremen t and ultimately a precise determination

f

their v alues Ho w ev er deviations from

r

extensions to the Standard Mo del ma y imply that the t w

triangles

are dissimilar

r

ev en that they are no closed triangles at all Therefore it is imp

rtan

t to distinguish measuremen ts

f

dieren t parameters ev en if they are exp ected to ha v e iden tical

r

close v alues within the three family Standard Mo del

Phases

and Observ ables The fact that phases

f

quark elds are unobserv able n um b ers has b een used to sho w that some phases in the CKM matrix are not

bserv

ables either and there remains some arbitrariness in the parametrization for this matrix The freedom to c ho

se

quark phases ma y b e extended to an tiquarks with six more phases

u
c
t
d
s
b
With

the new quark states q

j
e

i j q

q
j
e

i

j
q

j

j
u

c t d s b also the phase induced b y the CP

p

eration is c hanged The transition CP jq j i

e

i CP j j

q

j i

CP

jq

j

i

e

i

CP

j j

q
j

i requires

CP

j

CP

j

j
j

This equation lea v es

CP

j still completely undened since all three phases

n

the righ thand side are not

bserv

able and therefore sub ject to arbitrary c hanges It b ecomes meaningful ho w ev er if it is applied to

bserv

ables lik e CP eigen v alues Tw

CP

eigenstates constructed from a meson and an timeson state with eigen v alues

are

related accordingly jq j

q

k i

e

i CP j k jq k

q

j i

e

i j

k
h

jq

j
q
k

i

e

i

CP

j k jq

k
q
j

i i The new states jq

j
q
k

i

e

i

CP

j k jq

k
q
j

i ha v e the same eigen v alues and dier b y an

v

erall unobserv able phase from the

ld
nes

The CP

p

eration

n

a meson e g the pseudoscalar B

meson

j

b

di

is

CP jB

i
e

i CPB jB

i
where

the phase factor e i CPB

h

B

j

CP jB

i

dep ends

n

the parit y

f

the b

undstate

w a v e function and the c hosen quark and an tiquark phase con v en tion It is th us an unobserv able arbitrary phase

SLIDE 10

P

article An tiP article Oscillations and CP Violation Quark phase c hanges can in principle b e comp ensated b y phase c hanges

f

the CKM matrix elemen ts according to

lea

ving terms lik e hq j jV j k jq k i in v arian t Ho w ev er this is not a ph ysical requiremen t and in fact the CP transformed e i CP kj h

q

j jV

j

k j

q

k i

has

a phase whic h c hanges with the quark phases Since none

f

the t w

terms

corresp

nds

to an

bserv

able the actual c hoice

f

phases in the CKM matrix parametrization can b e made indep enden t

f

the c hoice

f

quark phases The app earance

f

an additional phase factor in

can

b e a v

ided

b y the restriction

j
j

for quark phase c hanges and an appropriate phase con v en tion whic h mak es terms related b y a CPT transformation relativ ely real If a c hoice

f

phases is p

ssible

where all CKM matrix elemen ts can b e made real also c harged curren t w eak in teractions w

uld

not violate CP symmetry

Phase

con v en tions will also en ter in to relations among deca y amplitudes An amplitude for a w eak deca y B

X

via a single w ell dened pro cess can b e written as A

hX

jHjB

i
hX

jO V jB

i
where

V is a pro duct

f

the appropriate CKM matrix elemen ts and O is an

p

erator describing the rest

f

the w eak and p

ssibly

also subsequen t strong in teraction pro cesses in v

lv

ed in the transition Since strong in teraction and also w eak in teractionexcept for non trivial phases in V are CP in v arian t the c harge conjugate mirror pro cess B

X

has an amplitude A

hX

jHjB

i
e

i CP X hX j CP O V CP

e

i CP B jB

i
e

i CP X

CP

B

hX

jO V

jB
i
e

i CP X

CP

B

V
V

A

where

also V

V
e

i arg V is just a phase Esp ecially

if

X is a CP eigenstate with eigen v alue

X
A
X

e i CPB

arg

V

A
relates

the t w

amplitudes

and the ratio A A ips sign with the CP eigen v alue All ph ysical

bserv

ables m ust b e indep enden t

f

the c hoice

f

phases This is the case if

nly

absolute v alues

f

amplitudes are in v

lv

ed but for in terference terms the phase con v en tion cancels

ften

in a more subtle w a y

Some

examples will b e sho wn in the follo wing c hapters On the

ther

hand expressions where the arbitrary phases are still presen t cannot b e

bserv

ables

SLIDE 11

Oscillation

Phenomenology

Oscillati
n

Phenomenology An unstable meson can b e describ ed b y the nonrelativistic Sc hr

dinger

equation i t

m
i
with

the solution j i

j
ie

imt e

t
whic

h repro duces the exp

nen

tial la w

f

radioactiv e deca y

since

j

j

ij

e

t

The

four meson pairs K

K
D
D
B
B
and

B s B s can b e describ ed as deca ying t w

comp
nen

t quan tum states

b

eying the Sc hr

dinger

equation i t

H

with a general Hamiltonian H

M
i
m
i
m
i
m
i
m
i
where

M and

are

hermitian

but

H is not

If

the B

B
system

is tak en as a represen tativ e to illustrate the b eha viour

f
scillating

meson pairs the indices

and
corresp
nd

to base v ectors jB

i

and jB

i
resp

ectiv ely

These

states are assumed to b e normalized i e hB

jB
i
hB
jB
i
CPT

in v ariance requires m

m
m

and

reducing

the n um b er

f

real parameters

f

the Hamiltonian to six H

H

H

H
H
m
i
m
i
m
i
m
i
CPT

in v ariance is

ne
f

the indisp ensable premises

f

an y relativistic eld theory within

r

b ey

nd

the Standard Mo del

The

generalized phenomenology including CPT violation will therefore not b e considered here but can b e found in textb

ks
The

parametrization

f

the

diagonal

elemen ts is con v enien t for calculation but it is still the most general case since

real

parameters su!ce to describ e an y H

and

H

m
H
H
R

e m

R

e H

R

e H

I

m m

I

m H

I

m H

iH
H
R

e

I

m H

I

m H

I

m

R

e H

R

e H

R

e H

R

e m

I

m

R

e H

R

e m

I

m

I

m H

I

m m

R

e

I

m H

I

m m

R

e

Solving

the eigen v alue problem det H

a
H
a
H
H
ne
btains

t w

eigenstates

with eigen v alues a

H
p

H

H
explicitly

a L

m

L

i
L
m
i
r
m
i
m
i
a

H

m

H

i
H
m
i
r
m
i
m
i

SLIDE 12

P

article An tiP article Oscillations and CP Violation where L H stands for ligh t and hea vy It is imm ediately seen that m and are the a v erage mass

m

H

m

L

and

width

H
L
The

dierences are "m

m

H

m

L

R

e r

m
i
m
i
"
H
L
I

m r

m
i
m
i
"a
a

H

a

L

r
m
i
m
i
q

jm

j
j
j
i

R e m

The

connection b et w een mass and lifetime width dierences and the

diagonal

elemen ts in the mass matrix are sho wing up in these equations esp ecially "m

if

m

and

"

if
Squaring

the last line leads to the useful relation "m

"
R

e m

whic

h relates the sign

f

"m and " with the

diagonal

elemen ts m

and
It

is con v enien t to dene the dimensionless parameters x

"m
y
"
H
L

H

L
L
H
L
H
where

x is a nonnegativ e real n um b er and y ma y

nly

assume v alues b et w een

and
It

is an asymmetry parameter in the widths

r

equiv alen tly

in

the lifetimes

L
H
The

eigen v ectors jB LH i

p

q

are

found b y inserting

in

to HjB LH i

a

LH jB LH i

giving

the ratio

m
q

p

r

H

H
m
i
"m
i
"
and
m
m
p

H

p

H

p

H

p

H

H
H
H
H
p

H

H
Normalization

requires jpj

jq

j

i

e p

p
jj
p
j

m j

q
p
jj
m

p

j

m j

and

single particle eigenstates are describ ed b y

ne

complex parameter

m
This

parameter

is

dened

nly

up to an arbitrary phase and

nly

j m j is a measurable quan tit y

The

v alue

f

the phase dep ends

n

con v en tions

ne
f

them is the denition

f

the phase

CPB
argh

B

j

CP jB

i
This

mak es also

sometimes

also denoted

e

g in

an

arbitrary quan tit y

The

standard c hoice

f

the CKM matrix

and
CPK
mak

e jj small in the K

K
system

but a consisten t con v en tion

CP

B

lea

v es it at O

in

the B

B
system

A dieren t denition

f
for

the k aon system giv en in

is

indep enden t

f

arbitrary phases In general con v en tion indep enden t parameters can b e dened if deca ys are in v

lv

ed They can usually b e expressed via the unitarit y angles see g

and

will b e giv en for the B and K systems at the appropriate places b elo w

m
r
m

is sometimes called

in

the literature e g in

SLIDE 13

Oscillation

Phenomenology

The
riginal

Hamiltonian can b e rewritten using the parameter

m

as H

m
i
m

i

m
m

m i

m
i
m
i
m

x

iy
m

x

iy
m
i
and

the mass and a v

ur

eigenstates are related b y the equations jB L i

pjB
i
q

jB

i

jB H i

pjB
i
q

jB

i

jB

i
p
jB

L i

jB

H i jB

i
q

jB L i

jB

H i The eigenstates for nonhermitian H

i

e

are

not

rthogonal
hB

H jB L i

jpj
jq

j

j

m j

j

m j

R

e

jj
In

con trast to

the

real n um b er

is

an

bserv

able The deviation

f

j m j from

ne

called d

in
is

j m j

r
F
r

an arbitrary initial state j i

b

H jB H i

b

L jB L i

ajB
i
ajB
i

where the amplitudes are related via b LH

a

p

a

q

a
a

m p a

p

b L

b

H

a
q

b L

b

H

its

time ev

lution

ma y b e describ ed using a scaled time v ariable T

t
where

is the a v erage width

f

the eigenstates B H and B L

These

states ha v e a simple exp

nen

tial dev elopmen t with time Their masses m HL

m
x
and

widths HL

y
can

b e expressed with the dimensionless parameters x and y dened in

j

ti

b

H e i m H i H

t

jB H i

b

L e i m L i L

t

jB L i

e

imtT

e

i xiy

T
e

i xiy

T
ajB
i
ajB
i
e

imtT

e

i xiy

T
e

i xiy

T
a
m

jB

i
a

m jB

i
a
e

imtT

ajB
i
ajB
i

cosx

iy
T
i
a
m

jB

i
a

m jB

i
sinx
iy
T
b

Starting with pure B

mesons

at t

corresp
nds

to

a
and

j ti

ae

imtT

cos

x

iy
T
jB
i
i

m sin x

iy
T
jB
i

SLIDE 14

P

article An tiP article Oscillations and CP Violation Starting with pure B

mesons

at t

is

describ ed b y replacing

m
m
This

case corresp

nds

to a

and

j ti

ae

imtT

cos

x

iy
T
jB
i
i
m

sinx

iy
T
jB
i
The

n um b ers

f

B

and

B

at

time T for N

pure

B

mesons

at T

are
N

B

t
N
jhB
j

t

a
a
ij
N
e

T

cosh

y T

cos

xT

N

B

t
N
jhB
j

t

a
a
ij
N
j

m j

e

T

cosh

y T

cos

xT

These

n um b ers ho w ev er can not b e

bserv

ed What is accessible b y exp erimen t is

nly

the rate

f

deca ys to a v

ur

sp ecic nal states X and X at a giv en time T

These

deca y mo des are

ften

called tagging mo des since they serv e as a tag to indicate the a v

ur
f

the mother particle at deca y time The rates can b e

btained

from

b

y m ultiplying with hX jH

r

h X jH

resp

ectiv ely

to
btain

the amplitudes They are con v erted in to rates # N B

X

t

N
Z

dPS jhX jHj t

a
ij
N
e

T X cosh y T

cos

xT

#

N B

X

t

N
Z

dPS jhX jHj t

a
ij
N
j

m j

e

T X cosh y T

cos

xT

where

X

Z

dPS jhX jHjB

ij
Z

dPS jhX jHjB

ij
is

the partial width for a nonoscillating meson It agrees in v alue for the t w

CP

conjugate pro cesses if the amplitudes dier

nly

b y phases In tegrating

v

er all times the total n um b er

f

deca ys are N B

X
Z
#

N B

X

t dt

N
X
y
x
N

B

X
Z
#

N B

X

t dt

N
X
j

m j

y
j

m j

x
The

corresp

nding

n um b ers for initial B

mesons

are

btained

with the replacemen t

m
m
If

w e ignore CP violating eects in the

scillation

i e for j m j

w

e can dene a meaningful branc hing fraction as B B

X
N
Z
#

N B

X

t

#

N B

X

t dt

X
y
X

H

X

L whic h agrees with B

B
X
dened

accordingly for the same n um b er N

f

B

mesons

at t

for

example cos u

e

iu

e

iu

cos

u

e

iu

e

iu

therefore

j cos uj

e

iuu

e

iuu

e

iu

u
e

iuu

cos
R

e u

cosh
I

m u

SLIDE 15

Oscillation

Phenomenology

Mec

hanical Analogon Equation

c

haracterizes also the mec hanical system

f

t w

coupled

p endula

f

the same length Without coupling they are b

th

describ ed b y an

scillation

frequency m and a damping constan t

They

corresp

nd

to the meson X

and

its an tiparticle X

If

they are coupled b y a spring whose elasticit y is prop

rtional

to a nonnegativ e real n um b er m

and

a nonnegativ e real damping constan t

the

solutions corresp

nd

to a longliv ed

lo

w damping ligh t

lo

w frequency eigenstate where the p endula

scillate

strictly in phase and a shortliv ed

high

damping hea vy

high

frequency eigenstate where

ne

p endulum

scillates

as a mirror image

f

the

ther

i e with phase dierence

The

dierences in frequency and damping are "m

m
and

"

resp

ectiv ely

When
ne

p endulum is excited it will slo wly transfer its energy to the

ther

and bac k This b eating corresp

nds

to the

scillation

b et w een a meson X

and

its an tiparticle X

The

b eat frequency is "m

While

the

scillating

part e imt in

is

an unobserv able phase factor in meson an timeson

scillation

a mass dierence can actually b e

bserv

ed as a frequency$ Due to the restriction

f

m

and
to

nonnegativ e real v alues this system has alw a ys "m"

in

con trast to the

scillating

mesons and can also not sim ulate CP violation since there are no nontrivial phases

Standard

Mo del Predictio ns The Hamiltonian

can

b e

btained

using H

H
H

w where H

is

the strong and electromagnetic Hamiltonian H

E
E
whic

h has the stable a v

ur

eigenstates B

and

B

and

H w is the w eak in teraction p erturbation The WignerW eissk

pf

appro ximation for small H w leads to

H

j k

H
j

k

hj

jH w jk i

X

X P Z dPS hj jH w jX ihX jH w jk i

E
E

X

i
E
E

X

where

the sum runs

v

er all m ultiparticle states X whic h are eigenstates

f

H

and

P denotes the principal v alue

f

the in tegral The mass hermitian and deca y an tihermitian parts dened b y

are

m j k

H

j k

H
k

j

E
j

k

hj

jH w jk i

X

X P Z dPS hj jH w jX ihX jH w jk i E

E

X and j k

iH

j k

H
k

j

X

X Z dPS hj jH w jX ihX jH w jk i

E
E

X

The
diagonal

elemen ts H

ha

v e nonzero con tributions in the sum from states X whic h can b e reac hed in w eak deca ys

f

b

th

B

and

B

In

con trast to the neutral k aon system for B

B
these

are

nly

a small fraction

f

all B deca ys and they con tribute with alternating signs Therefore H

are

dominated b y the leading term hB

jH

w jB

i

whic h corresp

nds

to the b

x

diagrams b W d u c t

d

W

b
b

u c t d W W

d

u c t

b

SLIDE 16

P

article An tiP article Oscillations and CP Violation They giv e appro ximately

H
hB
jHjB
i
m
G
F
e

i CPB V

tb

V

td
m
W

m B f

B

B B

S

m

t

m

W
QCD
The

CP phase is in tro duced during the ev aluation

f

the hadronic part

f

the matrix elemen t The InamiLim function S x

x
x
x
x
ln

x

x
from

the lo

p
is

to lo w est

rder

a factor m

t

m

W
An

ev aluation

f

the pro duct S m

t

m

W
and
QCD

within a consisten t renormalization sc heme yields S

QCD
The

hadronic part

f

the matrix elemen t is appro ximated b y hB

jJ
J
jB
i
X

X hB

jJ
jX

i hX jJ

jB
i
B

B

hB
jJ
ji

hjJ

jB
i
B

B f

B

p

p
where

f B is the B deca y constan t and B B accoun ts for the corrections to the v acuum insertion appro ximation A big uncertain t y is the pro duct f

B

B B

where

the most reliable calculations no w come from lattice gauge theory

with

v alues around f B p B B

Me

V

In

this appro ximation w e ha v e for the B system "m

jm
j

whic h can b e used to determine jV td j since V tb

from

exp erimen tal results

n

B

B
mixing

The eigenstates are determined b y

m
m
jm
j
e

i CPB V

tb

V

td

jV

tb

V

td

j

e

i CPB

with
arg

V

tb

V td

This

phase dep ends

n

the CKM parametrization and islik e the CP phasenot an

bserv

able The arbitrariness cancels

nly

in ph ysical

bserv

ables whic h include deca y amplitudes with further CKM elemen ts and a CP phase The corresp

nding
i

sin arg

m
cos

arg

m

is purely imaginary

i

e R e

and

therefore

Within

the same framew

rk

for the B s B s system

ms
e

i CP B s

It

m ust b e emphasized ho w ev er that there exist common nal states for all four meson pairs and

nev

er v anishes completely

lea

ving alw a ys a small

and

also a small "

Within

the Standard Mo del

can

b e appro ximated b y the absorptiv e part

f

the b

x

diagram corresp

nding

to a quark represen tation

f

the nal states This is a p

r

appro ximation to ligh t hadronic nal states whic h are dominating in the K K system and ma y still c hange the prediction for B

B

considerably

The

b

x

calculation yields

m
S

m

t

m

W
m
b

m

W
m
c

m

b

V cb V

cd

V tb V

td
O
m
c

m

b
and

" and "m ha v e

pp
site

signs The ratio can b e estimated using

to

b e " "m

y

x

m
b

m

t

SLIDE 17

Oscillation

Phenomenology

This

ratio applies to b

th

the B

and

B s systems T

leading
rder

in

m
equation
yields

j m j

I

m

m
S

m

t

m

W
m
c

m

W

I m V cb V

cd

V tb V

td
from
This

leads to a rough estimate

f

the con v en tionindep enden t n um b er

j

m j

m
c

m

t

jV cb jjV cd j jV tb jjV td j sin

sin
with

j j

where
is

the CKM unitarit y angle in gure a Since this result is based

n

a leading

rder

quark diagram the n um b er should b e tak en

nly

as an

rder
f

magnitude In particular at this lev el

f

precision it can not b e used to measure

Beha

viour

f

the F

ur

Neutral Meson An tiMeson Systems All four meson pairs K

K
D
D
B
B
and

B s B s sho w a dieren t

scillation

b eha viour since they ha v e all dieren t relations

f
"
and

"m

The

same sym b

ls

will b e used for all four systems Only when t w

sp

ecic systems shall b e compared their parameters will b e distinguished b y the subscripts K

D
d
and

s

resp

ectiv ely

The

dimensionless parameters x and y giv e the ratios

f

time constan ts in v

lv

ed

is

the harmonic a v erage

f

the lifetimes t

sc
"m
x

is the p erio d

f

the

scillation

and t rel

"
y

is the lifetime

f

the

scillation

amplitude i e the damping time constan t

f

a relaxation pro cess Numerical v alues are summarized in table

T

able

P

arameters

f

the four neutral

scillatin

g meson pairs

description

K

K
D
D
B
B
B

s

B

s

ps
%
s
y
"
jy

j

jy

j

"m
s
"m
e

V

x
"m
j

j

j

m j

Standard

Mo del exp ectation

While

the parameters

f

the K

K
system

are w ell measured

theoretical

assumptions en ter in to the B meson columns Man y precise lifetime measuremen ts for neutral B mesons ha v e b ecome a v ailable last y ear All lifetime measuremen ts are summarized in table

and

a v erage to

d
ps
Figures
sho

w the n um b er

f

mesons and an timesons as a function

f

the scaling lifetime v ariable T

t

and the asymmetry aT

#

N X

X
#

N X

X
#

N X

X
#

N X

X
T
j

m j

cosh

y T

j

m j

cos

xT

j

m j

cosh

y T

j

m j

cos

xT

for

a meson pro duced at T

as

a a v

ur

eigenstate X

and

deca ying to a a v

ursp

ecic nal state as X

r

X at a later time T

Expressed

via the small real parameter

instead
f

j m j this reads aT

cos

xT

cosh

y T cosh y T

cos

xT

SLIDE 18

P

article An tiP article Oscillations and CP Violation T able

B
lifetime

measuremen ts Measuremen ts whic h ha v e b een replaced b y more recen t

nes

are not included

d
ps
exp

erimen t

OP

AL

DELPHI
DELPHI
ALEPH
DELPHI
SLD
prel
SLD
prel
CDF
CDF
a

v erage F

r

an an timeson pro duced at T

as

a a v

ur

eigenstate X

and

deca ying to a a v

ursp

ecic nal state as X

r

X at time T

w

e

btain

a similar expression where

nly

the cos xT part c hanges sign

aT
#

N

X
X
#

N X

X
#

N

X
X
#

N X

X
T
cos

xT

cosh

y T cosh y T

cos

xT The appro ximation j m j

corresp
nding

to

leads

to a simpler expression aT

cos

xT cosh y T

aT
where

x is clearly seen as the

scillation

parameter and y as the damping parameter The k aon has b

th

x

and

y

i

e the longliving state is the hea vier mass eigenstate With these parameters

ne

half

f

a sample

f

k aons

f

either a v

ur

deca ys rapidly

mainly

in to t w

pions

with CP

and

the

ther

half transforms to a sample

f

the longliving K

L

states whic h deca y aside from the small CP violation to CP

eigenstates

and to a v

ursp

ecic states The ratio

f

lifetimes

f

the t w

states

table

is

appro ximately

The

time ev

lution
f

an initially pure K

a

v

ur

eigenstate is sho wn in gure

The

upp er diagram sho ws the n um b er

f

remaining K

and

K

after

a scaled time T

t
where
S
S
is

the a v erage width

f

the short and longliving state The deca y rate in to a v

ursp

ecic nal states is prop

rtional

to these n um b ers while the dominan t deca ys to CP eigenstates follo w dieren t ev

lution

functions due to CP violation and will b e discussed b elo w The D

meson

deca ys mainly to a v

ur

sp ecic states with w ell dened strangeness with

nly

a few deca ys to CP

eigenstates

as

K

K

K
L
and

CP

states

as K

S
r

K

S
This

leads to equal lifetimes for the t w

eigenstates

i e y

The

corresp

nding

b

x

graph has a b quark as the hea viest particle in the lo

p

whic h is accompanied b y the small CKM elemen ts V cb and V ub

The

mass dierence induced that w a y b y the Standard Mo del is v ery small corresp

nding

to x

Therefore

almost no asymmetry is visible in gure

although

the n um b er x

used

for the plot is a factor

higher

The v alue x

corresp
nds

to a total mixed fraction

f

initially pure D

states

giv en b y

N

D

X

N D

X
N

D

X
x
x
as
The

parameters

f

the B

B
system

ha v e b een in tro duced ab

v

e A go

d

appro ximation is y

and
whic

h leads for N

pure

B

at

t

to

N B

T
N
e

T

cos

xT

N

B

T
N
e

T

cos

xT

SLIDE 19

Oscillation

Phenomenology

T
NN
T
a
Fig
K
K
mixing

is determined b y the parameters x

y
and

j m j

T
t
is

the lifetime in units

f
S
the

in v erse

f

the a v erage width

f

K

L

and K

S
The

upp er diagram sho ws the n um b er

f

K

solid

and K

dotted

as a function

f

T for a sample starting with

K
mesons

The lo w er diagram sho ws the asymmetry a

N

K

N

K N K

N

K

The

relaxation pro cess so

n

dominates lea ving

nly

K

L

after not m uc h more than

ne
scillation

as sho wn in gure

The

deca y rate for a v

ursp

ecic nal states whic h are the ma jorit y

f

B

deca

ys follo ws the same time ev

lution

The asymmetry function is simply aT

cos

xT

This

asymmetry can b e

bserv

ed using a a v

urtagging

deca y

lik

e B

D
l
The

rate

f

mesons deca ying at time T in to the c hannel X are giv en b y

where

y

mak

es cosh y T

leading

to the same asymmetry function aT

cos

xT

In

tegrating

v

er all times the

bserv

ed n um b ers are N B

X
Z

# N B

X

T

dt
N
X
x
x
N

B

X
Z

# N B

X

T

dt
N
X

x

x
Their

asymmetry b ecomes a in t

N

B

X
N

B

X

N B

X
N

B

X
x

SLIDE 20

P

article An tiP article Oscillations and CP Violation

T
NN
T
a
Fig
D
D
scillation

s ha v e not y et b een

bserv

ed and will b e hardly visible ev en with x

whic

h is ab

ut
times

the exp ected v alue The

ther

parameters in this plot are y

and

j m j

and

the mixing probabilit y is as in

N

B

X

N B

X
N

B

X
x
x
It

w as this net eect whic h ga v e the rst pro

f

for a sizeable mixing parameter x

in

the B

meson

system in

The

timedep enden t particle an tiparticle

scillations
f

the neutral B meson ha v e b een rst seen six y ears later b y exp erimen ts at LEP

With

x

ab
ut
ne

p erio d is visible b efore most

f

the mesons are deca y ed If w e assume the Standard Mo del predictions to b e true the B s meson is a v ery in teresting case There will b e a small y and a v ery large x

Figure
is

plotted with x s

whic

h is close to the lo w er limit

f

the theoretical range The timein tegrated mixing probabilit y is for j m j

x
y
x

SLIDE 21

Oscillation

Phenomenology

T
NN
T
a
Fig
B
B
ev
lution

is dominated b y the

scillati

ng part with the parameters x

y
and

j m j

The

ratio

f

the areas under the dotted and solid curv e in the upp er plot is the mixing probabilit y

The

zero transition in the asymmetry

whic

h marks the crosso v er p

in

t in the upp er plot is at T

x
F
r

large x s

this

approac hes its maxim um v alue

f
where

a measuremen t

f

this quan tit y has no sensitivit y

n

x an y more T

bserv

e the rapid

scillations

a v ery go

d

lifetime resolution will b e required Exp erimen tally

a

lo w er limit x s

has

b een found at LEP see b elo w In the general case j m j

the

in tegrated mixing probabilit y dep ends

n

the initial a v

ur

It is

j

m j

x
y
x
j

m j

y
j

m j

x
y
x
y
a

for an initial B and

x
y
j

m j

x
j

m j

y
j

m j

x
y
x
y
b

for an initial B whic h is

with

j m j replaced b y j m j

r
b

y

This

exhibits already CP violation since the probabilities P X

X
and

P X

X
are

dieren t It is also T violation since the transition X

X

is the time rev ersed pro cess X

X

SLIDE 22

P

article An tiP article Oscillations and CP Violation

T
NN
T
a
Fig
B

s

B

s is exp ected to b e the most rapidly

scillatin

g system with a longer relaxation time This plot assumes x s

y

s

and

j m j

CP

Eigenstates V ersus Mass Eigenstates The follo wing discussion will again use B

B
as

an example but is applicable to eac h

f

the four systems accordingly

The

standard phase con v en tion requires all J P C

mesons

to ha v e CP jX i

jX

i

xing
CPB
Indep

enden t

f

an y con v en tion t w

rthogonal

CP eigenstates jB

i
p
jB
i
CP

jB

i
jB
i
p
jB
i
CP

jB

i
with

CP jB

i
jB
i

and CP jB

i
jB
i

can b e dened If a state agrees with

ne
f

these except for a phase factor it will b e a CP eigenstate The mass eigenstates

f

the B

B
system

are not CP eigenstates Using CP jB

i
e

i CP B jB

i
they

SLIDE 23

Oscillation

Phenomenology

are

transformed b y a CP

p

eration as CP jB L i

e

i CP B

m
e

i CPB

m
jB

L i

e

i CP B

m
e

i CPB

m
jB

H i

cos
jB

L i

sin
jB

H i CP jB H i

e

i CP B

m
e

i CP B

m
jB

H i

e

i CPB

m
e

i CPB

m
jB

L i

cos
jB

H i

sin
jB

L i Where the appro ximation for the B

B
system

dep ends

n

the phase con v en tion for the CKM matrix whic h determines the angle

If
is

c hosen b y an appropriate phase redenition e g

f

the b eld these states w

uld

b e eigenstates with CP

resp

ectiv ely

Still

there w

uld

b e CP violation in their deca y

and

the CP eigen v alue

f

the nal state w

uld

b e dieren t Therefore the question

f

whic h

f

the mass eigenstates is closest to whic h CP eigenstate has no con v en tion indep enden t answ er Only the CP eigen v alue

f

a deca y pro duct

f
ne
f

these states is an

bserv

able and the w eak in teraction do es not conserv e CP

A

meaningful question is whic h

f

B H

r

B L deca ys more

ften

in to CP

eigenstates

In con trast to the neutral k aon system most nal states from B deca ys are a v

ursp

ecic and b

th

mass eigenstates deca y in to them via either their B

r

their B

comp
nen

t The small fraction

f

states that can b e reac hed b

th

b y B

and

B

includes

the con tribution from CP eigenstates whic h app ear mainly through three pro cesses On the tree lev el there are t w

main

deca y c hannels that can pro duce CP eigenstates b

c
cd

with the c

cd
d

nal state and b

u
u

d with the quark con ten t u

ud
d

A state

f

the rst kind will ha v e deca y amplitudes A

hX

c

c

jHjB

i
V
cb

V cd A

A
hX

c

c

jHjB

i
X

e i CP B V cb V

cd

A

where
X
is

the CP eigen v alue

f

the state The corresp

nding

deca y amplitudes

f

B H and B L are A LH

hX

c

c

jHjB LH i

pA
q

A

pA
m
X

e i CP B V cb V

cd

V

cb

V cd

pA
j

m j X e i

The

deca y ratio is then in the appro ximation j m j

jA

H j

jA

L j

j
X

e i j

j
X

e i j

X

cos

X

cos

whic

h is for

less

than

for
X
and

vic e versa

In

this case the hea vier state B H will deca y more

ften

in to states with negativ e CP eigen v alue

X
Accordingly
for

the u

u

d

d

states A LH

hX

u

u

jHjB LH i

pA
j

m j X e i

and

the ratio jA H j

jA

L j

j
X

e i j

j
X

e i j

X

cos

X

cos

dep

ends

n

the angle

whic

h is lik ely to b e larger than

This

w

uld

giv e the

pp
site

answ er i e the hea vier state B H will deca y more

ften

in to states with p

sitiv

e CP eigen v alue

X

SLIDE 24

P

article An tiP article Oscillations and CP Violation Some deca ys with an in termediate state c

cd
s
r

c

c
d

s pro ceed in to K

r

K

whic

h nally result in c

cd
d

via a K

L
r

K

S

sequen tial deca y

Among

those is the goldplated deca y B

J
K
S
The

total deca y c hain in v

lv

es almost the same CKM elemen t phase factors as the direct b

c
cd

deca y

leading

to the same answ ers as for this deca y mo de a more detailed discussion follo ws b elo w in section

F
r

deca ys via W exc hange lik e b

d
c
c
r

b

d
u
u
the

same CKM elemen ts are in v

lv

ed and the same argumen ts lead to the same answ ers as ab

v

e Also the fa v

ured

p enguint yp e transition b

s

with subsequen t hadronization in to a K

L
r

K

S

has a net phase close to

leading

to the ratio

CP

eigenstates with quark con ten t d

d

can b e reac hed via CKMsuppressed p enguint yp e lo

ps

Due to the top quark dominance the amplitudes are A

hX

d

d

jHjB

i
V
tb

V td A

A
hX

d

d

jHjB

i
X

e i CPB V tb V

td

A

and

the CKM elemen t phases cancel whic h giv es jA H j jA L j

X
X

i e B H deca ys exclusiv ely in to states with negativ e CP eigen v alue

X
and

B L in to states with

X
All

these results receiv e corrections from nonleading terms lik e c quark lo

ps

in the last case

r

b

d

p enguin corrections to the b

u

transition nal states Since systems with a c

c

pair probably constitute the ma jor part for b

th

CP eigen v alues the hea vy mass eigenstate can b e said to b e the

ne

whic h deca ys more

ften

in to nal states with CP

and

the ligh t

ne

in to those with CP

but

b

th

ha v e also substan tial branc hing fractions in to nal states with the

pp
site

CP v alue This is a consequence

f

the CP violating phase in the CKM matrix but is not a CP violating deca y

since

none

f

the t w

B

mass eigenstates w as a CP eigenstate b efore it deca y ed The situation w

uld

b e dieren t for a purely real CKM matrix up to phases that can b e remo v ed b y quark phase c hanges In this case all unitarit y triangles w

uld

b e degenerate to lines and their angles w

uld

b e

r
Therefore

cos

cos
and

the hea vier state w

uld

b e the

nly

to deca y to CP

while

CP

nal

states w

uld

b e reac hed exclusiv ely via deca ys

f

B L

F
r

deca y pro ducts whic h are CP eigenstates this situation w

uld

corresp

nd

to a p erfectly predictable CP eigen v alue corresp

nding

to the mass eigenstate B L

r

B H

A

natural c hoice

f

phases in this case w

uld

force all terms

f

the w eak in teraction Hamiltonian to b e real corresp

nding

to

m
e

i CPB

Then

CP jB L i

jB

L i

jB
i

and CP jB H i

jB

H i

jB
i
and

CP is conserv ed in deca ys where this quan tum n um b er is meaningful Exactly this situation is almost true for the K

K
system

The ligh t K

S

deca ys to ab

ut
in

to the CP

eigenstates
and
while

the K

L

deca ys to

ne

third in to a CP

eigenstate

with

pions

the rest b eing mainly a v

ur

sp ecic semileptonic deca ys and

nly
are

to the CP

t

w

pion

state

Therefore

a parametrization is c hosen where K

S
K
and

K

L
K
If

w e ha v e a K

S

as deca y pro duct

f

the B

w

e are used to assign it a CP

eigen

v alue con tribution to the whole nal state T

b

e precise this is

nly

correct if the K

S

deca ys in to a CP

nal

state In this case also a K

L
will

b e assigned the same CP

eigen

v alue i e the

K
S
denotes

its nal state rather than the undeca y ed particle and a K

S
l
as

a a v

ur

sp ecic state is not included in this use

f

the lab el K

S

SLIDE 25

Oscillation

Phenomenology

Oscillatio

n at the

S

The B B system from strong in teraction

S

deca y is in an

dd

C and P eigenstate with angular momen tum L

retaining

the quan tum n um b ers J P C

f

the mother particle This system has to b e treated as a coheren t quan tum state The time ev

lution
f

a state with

dd

symmetry is dieren t from that

f
ne

with ev en symmetry

This

is due to the fact that

nly
ne

an tisymmetric X X state j X X i

jX
X

i is p

ssible

so it has to sta y constan t There are ho w ev er three symmetric states j X X i

jX
X

i jX X i j X X i and their relativ e amplitudes ma y c hange with time The quan tum n um b ers c haracterizing the t w

dieren

t mesons whic h are represen ted b y

and
here

can b e though t

f

as the spatial w a v e functions

x
and
x
r

alternativ ely the states in momen tum space jp i and jpi

Explicitly
for

initial B B states

f

w ell dened symmetry

jB
B
i
jB
B
i

the time ev

lution

from b translates in to

t
e

imt e T

c
jB
B
i
i

m scjB

B
i
i
m

scjB

B
i
s
jB
B
i
c
jB
B
i
i

m scjB

B
i
i
m

scjB

B
i
s
jB
B
i
for
t
e

imt e T

jB
B
i
jB
B
i
a

for

t
e

imt e T

cos

x

iy

T

jB
B
i
jB
B
i
i

sinx

iy

T

m

jB

B
i
m

jB

B
i
b

where the shorthand notation s

sinx
iy

T

c
cos

x

iy

T

has

b een used in

This

means the an tisymmetric state sta ys alw a ys a

correlated

B B

as

long as none

f

them has deca y ed This is a t ypical example

f

a coheren t quan tum state where b

th

mesons alw a ys ha v e exactly

pp
site

a v

ur

although none

f

the single mesons is in a a v

ur

eigenstate Only when

ne

deca ys in to a state rev ealing its a v

ur

not necessarily the rst

ne

that deca ys the state collapses and the second

ne

con tin ues as a

neparticle

state ev

lving

in time according to

The

second case b

f

an ev en w a v e function leads to a probabilit y

scillating

with t wice the single B frequency b et w een a lik esign

B

B

r

B B

and
pp
sitesign
B

B

a

v

ur

state F

r

dieren t times T

and

T

f

B meson

and
e

g the times

f

deca y

f

the t w

B

mesons w e ha v e for the an tisymm etric state j

T
T
i
e

im

T
T
cosx
iy
T
T
jB
B
i
jB
B
i
i

sinx

iy
T
T
m

jB

B
i
m

jB

B
i
a
e

im

T
T
pq
cosx
iy
T
T
jB

L B H i

jB

L B H i

i

sinx

iy
T
T
jB

L B H i

jB

L B H i

b

SLIDE 26

P

article An tiP article Oscillations and CP Violation Again it is seen that for T

T
nly

the an tisymmetric state is presen t and mixed states i e t w

nal

states indicating the same b eaut y a v

ur

will sho w up

nly

at T

T
The

mixing probabilit y

B

and B denote the a v

ur

at deca y time N B B

B

B

N

B B

B

B

B

B

B

B

is

iden tical to that for a single B meson This can b e understo

d

from the fact that the second B meson is in a a v

ur

eigenstate exactly when the rst

ne

deca ys in to a tagging mo de and then ev

lv

es in time as a single

scillating

B meson un til it deca ys itself The probabilit y can also b e

btained

from equation a using N B B

N

R

hB
B
j
T
T
i
dT
dT
and

N B B

N

B B

N

B B

accordingly
The

normalization factor N dep ends

n

the branc hing fractions in to tagging mo des and in general

n

the parameters y

and

x

but

cancels an yw a y in the ratio F

r

incoheren t B

B
pair

pro duction e g in b

b

jet fragmen tation the in tegrated mixedrate is determined b y t w

indep

enden t mixing probabilities N B B

B

B

N

B B

B

B

B

B

B

B

Equation

b is an expansion in the t w

mass

eigenstates The an tisymmetric w a v e function is alw a ys comp

sed
f

t w

dieren

t states there will b e nev er B H B H

r

B L B L

ev

en at dieren t deca y times The question

f

CP eigenstates can

nly

b e answ ered after b

th

B mesons ha v e deca y ed This in v

lv

es the phases in deca y amplitudes and includes all eects

f

CP violation whic h will b e discussed in detail b elo w F

r

the symmetric state the w a v e function is j

T
T
i
e

im

T
T
cos

x

iy
T
T
jB
B
i
jB
B
i
i

sinx

iy
T
T
m

jB

B
i
m

jB

B
i
e

im

T
T
pq
cos

x

iy
T
T
jB

L B L i

jB

H B H i

i

sinx

iy
T
T
jB

L B L i

jB

H B H i

This

is v ery similar to the function

f

an an tisymmetric state but the

scillation

is in the sum

f

the t w

lifetimes

instead

f

the lifetime dierence In the appro ximation j m j

and

y

the

in tegrated mixedrate is N B B

B

B

N

B B

B

B

B

B

B

B

x
x
x
In

the general case it is N B B

B

B

N

B B

B

B

B

B

B

B

x
y
y
x
y
x
y
x
y
but

cannot b e related to the mixing probabilities

f

single mesons

and
The

expansion in mass eigenstates sho ws that the symmetric w a v e functions consists alw a ys

f

t w

eigenstates

with the same mass i e B H B H

r

B L B L

SLIDE 27

Oscillation

Phenomenology

Determinati
n
f

the Mixing P arameters

f

B Mesons Only a small fraction

f

B meson deca ys has b een fully reconstructed Ho w ev er the a v

ur
f

a B meson can b e iden tied b y v arious tags The rst

bserv

ation

f

a then unexp ected large B

B
mixing

b y AR GUS in

used

the b est a v

ur

tags a v ailable In m ultihadron ev en ts

n

the

S

resonance

lik

esign lepton pairs w ere

bserv

ed whic h could not b e attributed to

ther

sources but semileptonic B deca ys The c harge

f

the lepton from

b
l
c

in these deca ys is iden tical to the b eaut y

r

b

ttomness

quan tum n um b er

f

the meson and these ev en ts had to b e attributed to B

B
and

B

B
nal

states from

S

deca ys The mixing probabilit y

can

b e calculated from the n um b er

f

lik e and

pp
sitesign

dilepton ev en ts as

N

l

l
N

l

l
f
N

l

l
N

l

l
N

l

l
where

the ratio

f

semileptonic branc hing fractions

f

neutral and c harged B mesons and

f

their pro duction rates en ters as f

B

B

l
X
B
S
B
B
B

B

l
X
B
S
B
B
In

addition this

bserv

ation w as supp

rted

b y four ev en ts with

ne

fully reconstructed B meson plus a lepton

f

wrong sign and

ne

exclusiv e ev en t

S
B
B
with

b

th

B

mesons

reconstructed Results

n

mixing

btained
n

the

S

are summarized in table

In

addition to leptons and fully reconstructed B mesons as a v

ur

tags also fully reconstructed D

mesons

partially reconstructed D

D
D
partially

reconstructed B

D
l
and

c harged k aons ha v e b een used T able

B
mixing

parameter

using

f

tags

exp erimen t

l

l AR GUS

CLEO
l

l

D
l
B
l

AR GUS

l

l CLEO

B
l

CLEO

D
l

l AR GUS

D
K

AR GUS

B
K

AR GUS

all
a

v eraged While

nly

the in tegrated eect can b e

bserv

ed

n

the

S

at presen tly existing symmetric colliders an

bserv

ation

f

the

scillating

b eha viour w as p

ssible

at the Z

where

the lifetime can b e measured This yields directly the frequency "m from the asymmetry at

#

N B

#

N

B
#

N B

#

N

B
t
cos

"m t Results are summarized in table

The

mixing parameter x can b e calculated from the mixing probabilit y using

and

from the

scillation

frequency as x

"m
whic

h requires also precise kno wledge

n

the a v erage lifetime

d
ps
f

the B

meson

table

x
"m
LEPSLDCDF

a vg

x
r
AR

GUS CLEO a vg

x
common

a v erage

SLIDE 28

P

article An tiP article Oscillations and CP Violation T able

B
B
eigenstate

mass dierence from the

scillati
n

frequency

"m
ps
tags

exp erimen t

D
l

ALEPH

l

l ALEPH

D
l

jet c harge ALEPH

DELPHI
l

K jet c harge DELPHI

D
l

jet c harge DELPHI

jet

c harge OP AL

D
l

OP AL

jet

c harge l OP AL

l

l OP AL

l

l L

K

a SLD

prel
jet

c harge a SLD

prel
l

a SLD

prel
jet

c harge l a SLD

prel
B
CDF
D

l CDF

l

l CDF

all
a

v eraged a using the Z

p
larization

asymmetry at SLC The t w

indep

enden t metho ds agree v ery w ell and yield a common v alue

f

the scaled mass dierence with

precision

A v alue x

compatible

with the presen t b est estimate will b e used within this pap er T able

B

s B s eigenstate mass dierence from the

scillation

frequency

"m
ps
exp

erimen t

CL
ALEPH
CL
ALEPH
jet

c harge l

CL
ALEPH
CL
OP

AL

CL
ALEPH
CL
LEP

com bined

F

rom a v erage b

b

mixing at e

e
annihilation

and Z

deca

y

a

v alue

s
can

b e inferred with large errors

due

to the small B s fraction in b jets Direct non

bserv

ations

f

the

scillation

leads to more stringen t limits

n

the frequency

summarized

in table

F
r

the B s B s system using the latest lo w er limit

n

the

scillation

frequency "m s

ps

and the B s lifetime v alue

s
ps

table

the

presen t lo w er limit is x s

whic

h is already close to the lo w est exp ected v alues It corresp

nds

to a mixing probabilit y

s

SLIDE 29

Oscillation

Phenomenology

T

able

B

s lifetime measuremen ts Measuremen ts whic h ha v e b een replaced b y more recen t

nes

are not included

s
ps
exp

erimen t

D

s l

OP

AL

D

s l

CDF
J
CDF
D

s h

DELPHI
D

s

DELPHI
l
DELPHI
D

s l

DELPHI
prel
D

s h

ALEPH
D

s l

ALEPH
D

s

OP

AL

a

v erage

Prediction

s for x s

y

s and

s

Since the lifetimes

f

B

and

B s agree within presen t precision a rst appro ximation using

is

x s x d

jV

ts j

jV

td j

whic

h giv es an estimate

f

the exp ected x s range b et w een

and
It

suers from the p

r

kno wledge

n

V td

whic

h has to b e

btained

from the measured x d

With

the top mass kno wn and lattice calculations giving more reliable n um b ers for f B

f

B s

B

B and B B s

theoretical

predictions for x s b ecome more precise The Standard Mo del no w fa v

urs

n um b ers b et w een

and
Recen

tly rep

rted

ranges are

and
The

B s meson eigenstates are exp ected to ha v e also dieren t widths A v alue

f

y s

"
f

B s

Me

V

is

predicted from quark lev el QCD calculations

a

similar n um b er y s

is
btained

using exclusiv e deca y c hannels

In

the naiv e quark mo del a larger v alue around

is

exp ected and a recen t QCD ev aluation giv es y s

The

rened ratio

is

x s y s

m
t

m

b
m
c

m

b
and

corresp

nds

to a lo w est

rder

estimate

neglecting

QCD corrections The range giv en reects

nly

a v ariation

f

quark masses Finally

replacing

d with s equation

can

b e used to estimate

s
sin
Again

large corrections to this simple calculation ma y b e exp ected

SLIDE 30

P

article An tiP article Oscillations and CP Violation

CP

Violation Standard Mo del w eak in teractions are long kno wn to violate parit y and c harge conjugation symmetries in most cases ev en maxim all y

Ho

w ev er the com bined symmetry

p

eration CP leads generally to transitions iden tical to the

riginal
nes

i e CP symmetry is conserv ed A t ypical example is the w eak deca y

f

a

lepton

in to

as

sho wn in gure

While

a

lepton

can deca y in to a lefthanded neutrino and a pion the c hargeconjugate deca y

f

a

in

to a lefthanded an tineutrino is forbidden Ho w ev er if

ne

lo

ks

at the mirrorimag e i e

ne

applies a parit y transformation at the same time the deca y is allo w ed and ev en more the amplitudes for b

th

deca ys are equal If w e extend

ur

denition

f

an tiparticle to mean not

nly

signip

f

all c hargelik e quan tities but also

f

the spin w e ha v e the CP

p

eration and a p erfect symmetry ev en for most w eakin teraction pro cesses CP violation

n

the con trary

is

a true violation

f

particle an tiparticle symmetry

whic

h can not b e restored b y a mirror P

P
l

C l C

fo

rbidden fo rbidden

Fig
P

arit y

P
and

c harge conjugation

C
p

erations

n
The

upp er righ t and lo w er left pro cesses are forbidden If all in teractions w ere CP symmetric w e had no w a y to distinguish left&righ thandedness p

si

tiv e&negativ e c harge etc P arit y and C violation in w eak in teraction connects handedness with c harge but still do es not allo w a distinction b et w een the t w

mem

b ers

f

a pair CP violating K

deca

ys ho w ev er pro vide a dieren t deca y rate function

f

time for K

and

K

whic

h could b e used to explicitly distinguish them b y a dip

r

bump in this function Although w e are presen tly not able to

bserv

e the dierence

f

matter and an timatter at far regions

f

the univ erse the absence

f

regions

f

matter an timatter annihilation b

undaries

suggests that the whole univ erse is made

f

matter violating CP asymmetry to a large exten t Small asymmetries

f

the

rder
at

the early univ erse are su!cien t to explain this presen t situation ho w ev er it is di!cult to create these from the CP violation in the Standard Mo del whic h has particle an tiparticle asymmetry

nly

in mesons whereas bary

n

n um b er violation is

bserv

ed in the univ erse If

ne

assumes bary

n

n um b er violating pro cesses at phase b

undaries
f

the early univ erse e g at the symmetry breaking phase transition to the electro w eak in teraction in the Standard Mo del still the CP asymmetry via the CKM phase is man y

rders
f

magnitude smaller than the

bserv

ed n um b er

f

bary

ns

p er bac kground photon

Th

us CP violating mec hanisms b ey

nd

the Standard Mo del are lik ely to exist and w e will p

ssibly
bserv

e them as small deviations from the Standard Mo del predictions The

rigin
f

CP violation in K and B mesons ma y b e

nly

within the Standard Mo del but

ther

p

ssibilities

are not ruled

ut

The

bserv

ation

f

CP violation in B mesons ma y either yield a consisten t picture with

ne

set

f

Standard Mo del parameters

r

pro duce con tradictory results making extensions

r

an alternativ e theory una v

idable

SLIDE 31

CP

Violation

Complemen

tary searc hes for CP violation will giv e additional constrain ts Only small CP violating eects are predicted b y the Standard Mo del in w eak deca ys

f
ther

particles lik e D

mesons
r

strange bary

ns
CP

asymmetries in the y et unobserv ed neutrino

scillations
r

CP violation in lepton deca y

are

p

ten

tial windo ws to alternativ e mo dels The searc h for magnetic monop

les
r

electric dip

le

momen ts

in

p

in

tlik e

r

spherically symmetric particles e g in leptons quarks

r

the neutron is another w a y to nd nonstandard CP violation Ho w ev er in con trast to CP violation in w eak in teraction these w

uld

not b e examples

f

brok en symmetry b et w een particles and an tiparticles but rather

f

mirror symmetry

while

the c harge conjugation symmetry is conserv ed A t presen t

nly

upp er limits

n

these eects exist and no glimpse b ey

nd

the Standard Mo del has b een

btained
CP

Violation in B Deca ys The B

B
meson

system has a simple description in the Standard Mo del One parameter x is su!cien t to parametrize the

scillation

since y

and

j m j

are

go

d

appro ximations CP violation in B deca ys

ccurs

alw a ys via in terference in three dieren t w a ys

Direct

CP violation B

X
B
X
can

b e

bserv

ed b y nal state coun ting exp erimen ts An example is the B deca y to K

where

an asymmetry N B

K
N

B

K
N

B

K
N

B

K
is

exp ected Deca ys with CP asymmetries lik e this example require in the Standard Mo del at least t w

in

terfering c hannels with dieren t CKM phase

and

a strong phase dierence

This

denes the amplitudes AB

X
A
e

i

A
e

i

e

i

A

B

X
A
e

i

A
e

i

e

i

where

A

and

A

e

i

is

unc hanged due to CP in v ariance

f

the strong in teraction They con tribute to the rates as jAj

jA
j
jA
j
R

eA

A
cos
jAj
jA
j
jA
j
R

eA

A
cos
whic

h are dieren t if cos

cos
This

is not the case if

r

if

In

the example B

K
the

rst amplitude is from a b

s

p enguin diagram whic h has a dominan t con tribution from the t quark in the lo

p

with a CKM phase arg V

tb

V ts

and

a K

state

with isopsin

The

second amplitude

ccurs

via a tree diagram b

u
us

transition with argV

ub

V us

and

isospin

and
amplitudes

The in terference terms are prop

rtional

to cos

Asymmetries
f

this t yp e can also b e

bserv

ed in c harged B deca ys e g B

K
CKM

unitarit y angles can b e extracted from those asymmetries using a v

ur

SU and isospin relations to constrain the strong phase dierence

CP

violation induces also a small asymmetry in the

scillation

probabilit y P B

B
P

B

B
due

to j m j

This

is due to the in terference

f
ther

amplitudes with the leading b

x

diagram

f

B

B

mixing with a t quark in the lo

p

The

scillation

asymmetry

starting

with an initial B

meson

can b e expanded in

as

aT

#

N B

#

N B

#

N B

#

N B

T
cos

xT cosh y T

cos
xT

cosh

y

T

O

SLIDE 32

P

article An tiP article Oscillations and CP Violation a

d

b W B

d

s c

u
K
S
D
b

d

b

W B

d
s
u

c

K
S
D
Fig
Diagrams

for B

D
K
S

a and B

D
K
S

b whic h is for y

aT
cos

xT

sin
xT
cos

xT

cos

xT

Starting

with a B

at

T

giv

es for the same asymmetry

a

T

#

N B

#

N

B
#

N B

#

N

B
T
cos

xT

cos

xT

i

e if

as

indicated b y

there

are alw a ys more B

B

than B

B
scillations

Using leptons as a v

ur

tag the net asymmetry can b e

bserv

ed at the

S

as N B B

l
l
N

B B

l
l
N

B B

l
l
N

B B

l
l
j

m j

j

m j

and

should b e v ery small in the Standard Mo del where

predicts

j j

The

rst upp er limit

n

this asymmetry

f
CL
w

as still far ab

v

e exp ectation A recen t measuremen t at LEP

giv

es a limit

f
CL
assuming

CPT symmetry

The

in terference

f
scillation

and deca y leads to lifetime dep enden t dierences B

j

t

X

j t

B
j

t

X

j t

for

a common nal state

f

B and B with asymmetry amplitude mo dulation

sin

"m t

The

nal state X can b e a CP eigenstate lik e J

K
S
CP
r
CP
r

a state that can b e reac hed from b

th

B

mesons

via dieren t pro cesses lik e B

D
K
S

and B

D
K
S

gure

In

the Standard Mo del CP violating in terference can lead to almost maxim um asymmetries In man y cases large v alues are exp ected and the timedep endence is a further handle to a v

id

misin terpretation

f

data Therefore all prop

sed

exp erimen ts fo cus

n

these eects whic h will b e describ ed in the next section A unique case

f

this in terference can b e

bserv

ed in coheren t an tisymm etric B B states i e in

S
B
B
as

a single ev en t

S
B

B

X

Y with CPX Y

CPX
CPY
L
CP

S

Here

X and Y m ust b e CP eigenstates and the exp ected rate for suc h ev en ts v aries with the lifetime dierence

f

the t w

B

mesons

SLIDE 33

CP

Violation

CP

Violation in Common Final States

f

B

and

B

The

most pronounced manifestation

f

CP violation in the B

B
system

is exp ected in in terference

f
scillation

and deca y to nal states common to B

and

B

The

eect is largest for CP eigenstates but ma y

ccur

at an y nal state where the amplitudes

f

the mixed and unmixed deca y can in terfere B

B
X
The

simplest situation is the ev

lution
f

an isolated B

meson

pro duced e g incoheren tly in b

b

fragmen tation at t

as

a a v

ur

eigenstate An unam biguous a v

ur

tag for the state at pro duction time ma y b e a c harged state from the second b

whic

h cannot mix The amplitude for B j t

X

j tt is deriv ed from

as

MB

X
e

imtT

A

n cos x

iy
T
ir

sinx

iy
T
using

the ratio

f

the upp er and lo w er path's amplitudes in

r
m

A A where A

AB
X
and

A

A

B

X
are

the deca y amplitudes

f

the a v

ur

eigenstates The cos term describ es the lo w er path with pure bB

scillation

while the sin term is a true in terference term that v anishes if r

The

amplitude for B

t
X

j tt is MB

X
e

imtT

A

n cos x

iy
T
i

r sinx

iy
T
If

X is a CP eigenstate the ratio A A is just a phase whic h includes the sign

f

the CP eigen v alue

f

X

The

phase

f

the pro duct r is indep enden t

f

con v en tions and is in fact an

bserv

able as will b e sho wn b elo w More general A and A can ha v e also dieren t magnitudes Figure

sho

ws an example for this case where the diagrams for B

D
K
S

and B

D
K
S

are dieren t Another example is a mixture

f

CP eigenstates as in the nal state D

D
whic

h is CP

for

L

and

CP

for

L

r
The

corresp

nding

deca y rates are prop

rtional

to jMj

e

T

jAj
cos

x

iy
T
jAj
jr

j

sinx
iy
T
i

sin x T

cos

x T

A
A

m

AA
m
sinh

y T

cosh

y T

A
A

m

A

A

m
e

T jAj

n
jr

j

cosh

y T

jr

j

cos

xT

cosarg

r

sinh

y T

sinarg

r

sin

xT

e

T jAj

n
jr

j

cosh

y T

jr

j

cos

xT

R

e r sinh y T

I

m r sin xT

a

jM j

e

T jAj

n
jr

j

jr

j

cosh

y T

jr

j

jr

j

cos

xT

cosarg

r

jr

j sinh y T

sinarg

r

jr

j sin xT

e

T jAj

j

m j

n
jr

j

cosh

y T

jr

j

cos

xT

R

e r sinh y T

I

m r sin xT

b

where the

scillating

in terference term is prop

rtional

to

A
A

m

AA
m
I

mA

A

m

jAj
jr

j sin arg r

jAj
I

m r

SLIDE 34

P

article An tiP article Oscillations and CP Violation and the relaxation part

f

the in terference is prop

rtional

to jr j cos arg r

R

e r

F
r

deca ys to CP eigenstates jr j

j

m j

and

the amplitude

f

the asymmetry

scillation

is the sine

f

the phase angle arg r

F
r

deca ys lik e B

B
D
K
S

where jAj

jAj

and consequen tly jr j

the

asymmetry

scillation

gets a ph ysical dilution factor D P

jr

j

jr

j

jAjj

Aj jAj

jAj
to

the sin xT term and a mixing con tribution as an additional cos xT term In the appro ximation j m j

and

y

the

rates are giv en b y jMj

e

T n jAj

jAj
jAj
jAj
cos

xT

jAjj

Aj sin arg r

sin

xT

jM

j

e

T n jAj

jAj
jAj
jAj
cos

xT

jAjj

Aj sin arg r

sin

xT

This

leads to an

scillating

asymmetry as a function

f

the prop er lifetime

f

the signal B aT

#

N

B
X
#

N B

X
#

N

B
X
#

N B

X
T
cos

xT

sin

xT

where

the B a v

urs

are tak en at T

and

the amplitudes are

jr

j

jr

j

I

m r

jr

j

D

P sin arg r If in addition jAj

jAj
esp

ecially if X is a CP eigenstate this simplies further to jr j

jMj
e

T jAj

n
sinarg

r

sin

xT

jM

j

e

T jAj

n
sinarg

r

sin

xT

The

corresp

nding

rates # N B

j

T

X
jMj
and

# N B

j

T

X
jM

j

are

illustrated in gure

They

sho w a timedep enden t asymmetry aT

#

N B

X
#

N B

X
#

N B

X
#

N B

X
T
sin

xT

with
sin

arg r

The

scaling time v ariable is T

t

s

dened

b y the signal B lifetime t s for incoheren t b

b

pro duction It has to b e replaced b y T

t

s

t

t

for

coheren t B B pro duction

n

the

S
where

t t denotes the second B meson in the

S

deca y in a a v

ur

tagging deca y mo de This is in full analogy to the mixing situation describ ed in section

and

will b e discussed in more detail b elo w

SLIDE 35

CP

Violation

T
t
N
N
T
a
Fig
Time

dep enden t rate

f

B

J
K
S
and

B

J
K
S
for
sin
and

x

The

lo w er plot sho ws the asymmetry aT

The

B s B s Case F

r

the B s meson y is not negligible and the asymmetry

is

mo dulated aT

#

N

B

s

X
#

N B s

X
#

N

B

s

X
#

N B s

X
T
cos

xT

sin

xT cosh y T

sinh

y T

where
s
is

used whic h is b eliev ed to b e a v ery go

d

appro ximation and

jr

j

jr

j

I

m r

jr

j

D

P sin arg r

R

e r

jr

j

D

P cos arg r with

F
r

the simpler case jr j

the

asymmetry is aT

#

N

B

s

X
#

N B s

X
#

N

B

s

X
#

N B s

X
T
sin

xT cosh y T

sinh

y T

SLIDE 36

P

article An tiP article Oscillations and CP Violation

T
NN
Fig
Time

dep enden t rate

f

B s

B

s

X

for arg r

r
for

arg r

r
and

for arg r

using

y

with
sin

arg r and

cos

arg r

This
p

ens an alternativ e w a y to measure CP violation parameter arg r

The

time ev

lution
f

an equal un tagged mixture

f

B s and B s mesons is giv en b y jMj

jM

j

e

T jAj

jr

j

n

cosh y T

sinh

y T

F
r

deca ys in to CP eigenstates jr j

and

the time dep endence

f

the deca y probabilit y is simply jMj

jM

j

e

T jAj

f

cosh y T

cos

arg r

sinh

y T g

jAj
n
e

t L

e

t H

i

e it is a sum

f

t w

exp
nen

tial distributions with w eigh ts

cos

arg r

This

case is illustrated in gure

n

a logarithmic scale The upp er and lo w er solid curv e corresp

nd

to the CP conserving case r

where

e g B sL deca ys in to either CP

r

CP

eigenstates

exclusiv ely

and

B sH in to the

pp
site
ne

The cen tral curv e corresp

nds

to maxim um CP violation i e R e r

where

b

th

B sL and B sH deca y in to CP

and

CP

eigenstates

with the same probabilit y

The
ther

t w

curv

es corresp

nd

to cos arg r

If

arg r

CKM

is a large angle a measuremen t

f

cos

CKM

via the mixture

f

short and long liv ed states is complemen tary to a measuremen t

f

sin

CKM

via an

scillating

asymmetry function

Due

to the large v alue

f

x s

the

latter requires a v ery precise measuremen t

f

the individual lifetimes and a v

ur

tagging while the decomp

sition
f

the short and long liv ed fractions can b e done with un tagged ev en ts and a mo dest resolution but requires a large data sample

Final

CP Eigenstates from B

r

B s Deca ys W eak deca y amplitudes can b e describ ed b y

and
F
r

CP eigenstates with eigen v alue

X

their ratio is then giv en b y A A

X

e i CP B

arg

V

where

arg V is the phase angle

f

the CKM elemen ts in v

lv

ed in A i e the B

r
b

deca y amplitude Using the CKM represen tation

with
m
e

i CP B

results

in r

X

e iarg V

SLIDE 37

CP

Violation

whic

h is a con v en tion indep enden t phase factor

f

a pro duct

f

four

r

more CKM elemen ts It can b e transformed in to an

parameter
X
r
r
whic

h is in con trast to

dened

b y

con

v en tion indep enden t but is sp ecic to a nal state X

A

measuremen t

f

the angle

arg

V

td

V tb V cd V

cb
in

the CKM triangle gure a requires a deca y with

b
c
c
d
Examples

for nal states with this quark con ten t are J

r

D

D
In

these deca ys

ne

has r

X

e i

Similarly
for
ther

nal states common to B and B the rephasing in v arian t net pro duct

f

CKM elemen ts in the com bined amplitude&mi xing phase ratio r is easily extracted using

A

summary

f

common nal states

f

B

and

B

is

giv en in table

Since

the sp ectator with a b quark is a

d

quark the deca y pro ducts

f

the b m ust ha v e the net a v

ur
f

a d quark accompanied b y

ne
r

more quark an tiquark pair T able

Examples
f

CP eigenstates as nal states

f

B

and

B

and

their sensitivit y to the CKM phases The asymmetry

has

an amplitude

X

sin

CKM
b

deca y Q CKM elemen ts angle

CKM

some nal states b

c
cd

V

tb

V td V

cd

V cb

J
c
D
D
b
c
cs

s

u
ud

V

tb

V td V

cs

V cb V

ud

V us

J
c
K
S
K
S
b
c
cs

s

d
K
L

V

tb

V td V

cs

V cb V

cd

V cs

J
c
K
L

b

u
u

d V

tb

V td V

ud

V ub

b
d

V

tb

V td V

tb

V tb

b
s

s

u
u

d V

tb

V td V

ts

V tb V

ud

V us

K
S
K
S
b
s

s

d
K
L

V

tb

V td V

ts

V tb V

cd

V cs

K
L
Only

the dominan t con tribution to the b

x

diagram for B

B
mixing

and to the p enguin transitions b

d

and b

s

are considered in table

V

ery small mo dications to the phase angles

CKM

will emerge from corrections to this appro ximation A sp ecial case are nal states with a K

S
r

K

L
If

it deca ys subsequen tly in to a CP

eigenstate

the whole system can still b e tak en as a CP eigenstate and used to extract unitarit y angles In the tables the tree lev el deca y s

u
ud

is used to determine the CKM phase angles There ma y b e p enguin con tributions as w ell whic h mo dify the asymmetry

F
r

a c quark in the lo

p

the relev an t unitarit y angle for B

c
cK

SL is exactly

for

a u quark it is the same as for the tree diagram and for a t quark it is the small angle

A

more precise treatmen t

f

the whole system includes

scillation

and deca y

f

the k aons as w ell leading to four amplitudes whic h all in terfere A

AB
B

jt B

AB
c
cK
AK
K

jt K

AK
A
AB
B

jt B

AB
c
cK
AK
K

jt K

A

K

A
AB
B

jt B

A

B

c
cK
AK
K

jt K

A

K

A
AB
B

jt B

A

B

c
cK
AK
K

jt K

AK
Here

the

scillation

amplitudes AK

K
etc

dep end

n

the k aon lifetime t K

F
r

t K

w

e ha v e A K

K
AK
K
and

an

scillation
sin

"m B t B with an amplitude

sin
as

giv en in table

F
r
ther

times t K

the

argumen t

c

hanges b y a phase angle

f

O

This

is small compared to

so

it still measures

to

that precision for an y k aon lifetime If the nal k aon is a K

L
it

will usually b e detected via its strong in teractionof a strangeness a v

ur

comp

nen

twith the detector material Since the cross section for its K

part

is considerably larger

SLIDE 38

P

article An tiP article Oscillations and CP Violation than that

f

the K

part
nly

t w

amplitudes

need to b e considered A

AB
B

jt B

AB
c
cK
AK
K

jt K

A
AB
B

jt B

A

B

c
c

K

AK
K

jt K

Their

ratio is A

A
AB
B

jt B

AB
B

jt B

V

cb V

cs

V

cb

V cs e i CPK

mK

The last factor for the K

K
system

is

btained

using the limits T

f
as

AK

K
e

imtT

i

mK sinx

iy
T
e

imt e T

i

mK e ixy T

e

ixy T

i
e

imt

mK
e

ixy T

and
as

AK

K
e

imtT

cosx
iy
T
e

imt

e

ixy T

F
r

the leading term in the b

x

diagram it is giv en b y

mK
e

i CPK

arg

V

cs

V

cd
whic

h yields A

A
AB
B

jt B

AB
B

jt B

V

cb V

cd

V

cb

V cd

i

e it b eha v es indeed as a b

c
cd

state with CP

The

angles to b e measured via

scillation&deca

y in terference include all the factor V

tb

V td from mixing whic h is

ne

side in the triangle k Hence

nly

the adjacen t angles

and
can

b e measured this w a y

Similarly
in

terference in B s

scillation

can b e used to measure the angles in the at triangle l see table

One
f

them

is

iden tical to an angle in h and diers from the third angle

in

k

nly

at

rder
This

w

uld

allo w a test

f
to

O

T

able

Examples
f

CP eigenstates as nal states

f

B s and B s

and

their sensitivit y to the CKM phases The asymmetry

has

an amplitude

X

sin

CKM
b

deca y Q CKM elemen ts angle

CKM

some nal states b

c
cs

V

tb

V ts V

cs

V cb

J
c
D
s

D

s

b

u
us

V

tb

V ts V

us

V ub

K
K
b
u
ud
s
uu
d

V

tb

V ts V

ud

V ub V

us

V ud

K
S
K
S
b
u
ud
s

d

K
L

V

tb

V ts V

ud

V ub V

cs

V cd

K
L

b

s

V

tb

V ts V

ts

V tb

b
d
s
uu
d

V

tb

V ts V

td

V tb V ud V

us
K
S
K
S
b
d
sd
K
L

V

tb

V ts V

td

V tb V cd V

cs
K
L

SLIDE 39

CP

Violation

T

able

Examples
f

CP eigenstates relev an t for B

deca

ys dep enden t

n

their relativ e

rbital

angular momen tum L

c

hannel L CP remarks J

c
K
S
K
L
K
K
S
K
K
L
J
K
S
J
K
K
S
helicities

J z

J
K
K
S
helicities

J z

c

K

K
S
D
D
D
D
D
D
If

the giv en quark lev el transitions are the

nly

con tributions to a nal state the asymmetry with time is a simple sin

CKM

sin xT b eha viour Ho w ev er man y

f

the nal states can b e reac hed via lo

p

graphs as w ell

ften

with dieren t CKM elemen ts in v

lv

ed Man y details can b e found in a recen t review

In

this case b

th

direct CP violation via the in terfering amplitudes and the

scillation&deca

y in terference lead to more complex asymmetries with the matrix elemen ts

and

b

th

a cos xT and sin xT term The most promising examples are for CP X

B
J
K
S

with

sin
and

and for CPX

B
with
sin
up

to corrections from the p enguin amplitude The CP eigen v alues

f

related c hannels can b e constructed from the data listed in table

States

with sev eral p

ssible

angular momen ta lik e v ector v ector nal states are t ypically a mixture

f

CP

and
eigenstates

Helicit y

dominance

w

uld

simplify these analyses since it is forbidden for L

nal

states and hence indicates a pure CP eigenstate In general though these states ha v e to b e decon v

luted

via a partial w a v e analysis

Mixtures
f

CP Eigenstates An example for a mixture

f

CP eigenstates is the nal state D

D
whic

h is CP

for

L

and

CP

for

L

r
Other

v ector an tiv ector deca ys lik e B s

D
s

D

s
r

B

r

v ector v ector deca ys with t w

CP

eigenstates lik e B s

J
r

B

J
K
S
K
sho

w the same prop erties In these cases the amplitude is A

V
A
A
where

V

denotes

the common CKM factors and the subscripts

f

the residual factors are the CP eigen v alues e g AB

D
D
L
V
cb

V cd A

and

AB

D
D
L
V
cb

V cd A

They

corresp

nd

to dieren t helicities

f

the v ector mesons therefore the factors A

and

A

ha

v e dieren t phases c hanging with the angles

f

the deca y pro ducts e g D

D
r

J

l
l

SLIDE 40

P

article An tiP article Oscillations and CP Violation Their ratio r A

A
A
dep

ends

n

the deca y angles con v enien tly describ ed as

and
in

the t w

helicit

y frames The amplitude ratio

bserv

able in CP violation exp erimen ts is r

e

i

r

A

r

A where

is

the in v arian t phase from the CKM elemen ts in mixing and deca y

whic

h leads for B

D
D
to

e i

m

V cb V

cd

V

cb

V cd

e

i and the co e!cien ts in the time ev

lution
are
jr

j

jr

A j

jr

A j

R

e r A

jr

j

R

e r A

jr

A j

R

e r A R e r

jr

j cos arg r

jr

A j

cos
I

m r A sin

jr

A j

R

e r A I m r

jr

j sin arg r

jr

A j

sin
I

m r A cos

jr

A j

R

e r A It has b een emphasized in

that

in terference in mixed CP eigenstates lik e these can b e used to

bserv

e a small phase angle

in

un tagged B s deca ys This is p

ssible

since the sum

f

b

th

initial a v

urs

is giv en b y

b

eing prop

rtional

to jMj

jMj
e

T jAj

n
jr

j

cosh

y T

R

e r sinh y T

and

the co e!cien t

f

the sinh y T term R e r

has

a comp

nen

t I m r A sin

whic

h b ecomes dominan t in regions

f

deca y angle space where I m r A

jr

A j

While

helicit y

corresp
nds

to a pure CP eigenstate the helicities

are

mixtures

f

b

th

eigenstates Ho w ev er states

f

denite transv erse spin pro jections with resp ect to the deca y plane

f

the second particle are pure CP eigenstates to

and

ma y b e used to decomp

se

the angular distribution in to states with ev en and

dd

CP eigen v alue

NonEigenstates

CP violation in

scillation&deca

y in terference can also b e

bserv

ed in nal states that are not CP eigenstates as long as they can b e reac hed b y b

th

B and B

As

an example

the

c hannel sho wn in gure

has

the deca ys b

cs
u

leading to the nal state D

K
S

from a B

meson

a and

b
u
s

c leading to the same nal state from a B

meson

b The amplitudes are A

V
ub

V cs

A
i
A
V

cb V

us
A
whic

h are

f

the same

rder
f

magnitude Using V

cs

V cd for the K

K
S

amplitude and ignoring small CKM phase angles O

the
bserv

able ratio is r

m

A

A
m

exp

i

arg V cb V

us

V cs V

cd

V

ub

V cd

e

i AA

jA
j

jAj

e

i

AA
jA
j

jAj

SLIDE 41

CP

Violation

where
AA
is

the strong in teraction phase dierence Similarly

for

the nal state D

K
S

r

m

A A

e

i

AA
jAj

jA

j

F rom an

bserv

ation

f

the timedep enden t asymmetry

b
th

the absolute v alues and the phases

f

these n um b ers can b e determined and the three unkno wn parameters

AA
and

jAjjA

j

b e extracted Measuring separately D

D
deca

ys in to CP eigenstates lik e

r

K K

where

b

th

amplitudes in terfere the angle

can

b e extracted from the rate measuremen t whic h b ecomes simpler if the nonmixing c harged B deca y

D
D
K
r

the selftagging deca y B

D
D
K
K
are

used A similar selftagging deca y B s

D
s

K

has

as w eak phase

and

has b een suggested

to

determine this angle

The

T

tal

Deca y Rate The total deca y rate

f

an initially pure B

sample

can b e calculated as dN dt

N
X

X Z dPS jMj

m

B

E

X

using

jMj

from

equation

with

A

AB
X
This

results in an expression lik e

with

the replacemen ts P X jAj

P

X jAj

jr

j

j

m j

and

P X A

A
P

X hB

jH

w jX ihX jH w jB

i
F

rom

ne

can write X X

m

A

A
m
i

m H

H
ix
j

m j

y
j

m j

ix
y
where

the last line is an appro ximation for

whic

h is go

d

for all four meson pairs This yields a total rate dN dt

N
j

m j

e

T n cosh y T

cos

xT

y

sinh y T

x

sin xT

In

the appro ximation y

this

is dN dt

N
e

T

cos

xT

x

sin xT

O
with

R dN dt dt

N
An

initially pure B

sample

giv es the same rates with the replacemen t

m
m

and

The

asymmetry where the B a v

ur

is understo

d

as the

ne

at T

is

therefore aT

#

N B

an

ything

#

N B

an

ything # N B

an

ything

#

N B

an

ything

T
cos

xT

x

sin xT

x
sin

xT

sin
xT
A

rst measuremen t

f

the total asymmetry at LEP ga v e the preliminary result

This

is an alternativ e w a y to measure CP violation in B

B
scillation

and giv es a limit j j

CL
but

it implies the danger

f

a p

ssible

bias due to the B

ev

en t selection

SLIDE 42

P

article An tiP article Oscillations and CP Violation

CP

Violation at the

S

B meson pairs from

S

deca y are initially in a CP

P
C
eigenstate

jB

B
i
jB
B
i

with angular momen tum L

Their

time ev

lution

is describ ed b y a where the t w

scaled

times T

and

T

ma

y b e tak en as the deca y times

f

the t w

mesons

By m ultiplying this state function with hX X jHH where hX X jHHjB

B
i
hX

jHjB

i

hX jHjB

i
A
A
are

the deca y amplitudes

f

t w

B
mesons

and amplitudes

f
ther

mixtures

f

B i and B j

yield

pro ducts

f

A i and A j accordingly

ne
btains

an amplitude M

e

i

e
T
T
C
cos

x

iy
T
T
iS
sinx
iy
T
T
where
is

a common unobserv able phase including the imt phases

f

t w

free

B mesons and the co e!cien ts are C

A
A
A
A
S
m

A

A
A
A
m

There can b e alw a ys t w

nonzero

amplitude factors separated lea ving co e!cien ts lik e C

A
A
A
A
A
A
S
A
A
m

A

A
A
m

A

These

co e!cien ts are con v en tion indep enden t factors similar to r

CP

phases common to AA and

m

cancel and the exc hange

f

quarks with an tiquarks ensure that the pro duct

f

CKM elemen ts has eac h quark index in as man y V as V

r

with the same phase V

factors

F rom the general amplitude w e can deriv e v arious sp ecial cases listed in table

The

square jM

j
leads

to a general form ula

whic

h reads

n

the

S

with the nal states X

X
from

the t w

B
mesons

# N B B

X
X
e

t

e

T g

cosh

y T

g
sinh

y T

h
cos

xT

h
sin

xT

with

T

T
T
t
t
and

g

jC
j
jS
j
g
R

eS

C
h
jC
j
jS
j
h
I

mS

C
F
r

B

mesons

w e can assume

and

y

whic

h corresp

nds

to the simpler equation # N B B

X
X
e

t

e

T g

h
cos

xT

h
sin

xT

This

includes the case

f

B B

scillation

if w e set e g A

A
Then

C

A
A
S
g
h
jA
A
j
and

g

h
F
r

a mixed mo de w e use e g A

A
Then

C

S
A
A
m
g
h
jA
A
j
and

g

h
The

corresp

nding

asymmetry is aT

#

N X

X
#

N X

X
#

N X

X
#

N X

X
#

N

X
X
#

N

X
X
#

N

X
X
#

N

X
X
cos

xT

SLIDE 43

CP

Violation

T

able

Co

ecien ts for B B

X
X
deca

ys from

S
are

the CP eigen v alues

f

nal states

resp

ectiv ely

and
d
d

denotes a p enguin mo de with a t quark in the lo

p

com denotes a nal state that can b e reac hed from b

th

B

and

B

and

needs not to b e a CP eigenstate In this case also strong phases are in v

lv

ed whic h can b e resolv ed using the c harged conjugate nal state to

The

co ecien ts

f

the timedep enden t rate according to

are

also giv en for

X
X
AA

C

AA

S

AA

g

jAAj
h
jAAj
h
jAAj
B

tag B tag A

A
m
B

tag B tag A

A
m
B

tag B tag A

A
B

tag B tag A

A
com

B tag A

A
r
A
m

A

jr

j

jr

j

jr

j

jr

j

I

m r jr j

com

B tag A

A
r
m

A

A
jr

j

jr

j

I

m r c

c

B tag A

A
e

i

sin
c
c

B tag A

A
e

i

sin
u
u

B tag A

A
e

i

sin
u
u

B tag A

A
e

i

sin
d
d

B tag A

A
d
d

B tag A

A
c
c

c

c

A

A
e

i

e

i

cos
sin
c
c

u

u

A

A
e

i

e

i

e

i

cos
cos
sin
sin
c
c

d

d

A

A
e

i

e

i

cos
corresp
nding

to gure

for

p

sitiv

e T as w ell as for negativ e T with T

T
F
r

a CP eigenstate with eigen v alue

as

meson

A
A
e

i

where
is

the phase

f

the CKM matrix elemen ts in v

lv

ed If the state

is

a tagging mo de e g A

A
w

e ha v e the situation

f

a tagged deca y with C

A
A
S
A
A
e

i

g
jA
j
jA
j
h
h
jA
j
jA
j
sin
whic

h is the t ypical situation

f

CP violation in the

scillation&deca

y in terference with a T dep endence as sho wn in gure

The

dotted curv e corresp

nds

to a B tag A

A
with

C

A
A
S
A
A
e

i

g
jA
j
jA
j
h
h
jA
j
jA
j
sin
and

the asymmetry

f

b

th

is describ ed b y equation

with
sin
The

a v

ur
f

the signal B meson

is

uniquely dened at the time

f

deca y t

f

the second B meson since the a v

ur
f

the latter can b e iden tied b y its nal state If b

th

nal states are CP eigenstates with eigen v alues

A
A
e

i

where
are

the phases

f

the CKM matrix elemen ts in v

lv

ed the co e!cien ts are C

A
A
e

i

SLIDE 44

P

article An tiP article Oscillations and CP Violation

T
t
t
NN
Fig
Time

dep enden t rate

f
S
B
J
K
S
and
S
B
J
K
S
for

sin

and

x

S
A
A
e

i

e

i

g
jA
j
jA
j
cos
cos
h
jA
j
jA
j
sin
sin
h
whic

h leads to a rate # N B B

X
X
jA
j
jA
j
e

t

e

T

cos
cos
sin
sin
cos

xT

where

the

bserv

able phase angles are

and
If

CP w ere conserv ed throughout the deca y c hain the eigen v alues are

since

the total CP eigen v alue

is

ac hiev ed via another factor

from

the relativ e L

angular

momen tum

f

b

th

B mesons This corresp

nds

to # N B B

X
X
jA
j
jA
j
e

t

t
cos
cos
sin
sin
cos

xT

while

the forbidden rate is # N B B

X
X
jA
j
jA
j
e

t

t
cos
cos
sin
sin
cos

xT

If

b

th

deca ys pro ceed via the same a v

ur

c hanging transitions the CKM angles are the same for b

th

B mesons

and

the forbidden rate is # N B B

X
X
jA
j
jA
j
e

t

e

T sin

cos

xT

This

rate should b e

It

do es actually v anish for T

i

e equal deca y times

f

b

th

Only if their lifetimes dier CP violation builds up in the in terference term

f
scillation

and deca y pro cesses F

r

dieren t CKM angles ev en at T

there

is a CP violating rate

cos
whic

h can b e called consisten tly a sp ecial case

f

direct CP violation The in terference term is here in con trast to the example

not

b et w een t w

amplitudes

for

ne

B deca y

but

b et w een t w

amplitudes

for t w

dieren

t B deca ys whic h are in a coheren t B B state All these in teresting cases will ho w ev er not b e

bserv

ed in the rst generation exp erimen ts

n

the

S

since they in v

lv

e a pro duct

f

t w

small

branc hing ratios whic h are t ypically b elo w

and

corresp

nds

to less than

ne

ev en t p er y ear at the presen tly en visaged luminosities

SLIDE 45

CP

Violation

CP

Violation in K Deca ys CP Violation has rst b een

bserv

ed in

in

K

deca

ys As sho wn in gure

the

neutral k aon system is c haracterized b y all parameters x

y

and j m j ha ving nontrivial v alues This mak es it m uc h more complicated than the B meson system The ligh t K

S

deca ys to ab

ut
in

to the CP

eigenstates
and

can therefore b e iden tied with a CP

eigenstate

K

while

K

L
K
Using

the con v en tion K

p
K
K
i

e

CP

K

this

can b e written as K

S
p
jj
K
K
K
L
p
jj
K
K
up

to an arbitrary common phase factor F

r

the standard parametrization

f

the CKM matrix

the

leading term for m

from

the b

x

graph for K

K
scillation

with a c quark in the lo

p

yields

m
e

i CPK V

cs

V

cd

jV

cs

V

cd

j

e

i CP K e i

whic

h is for this c hoice

f

the CP phase

m
i
and

corresp

nds

to

i
with

jj

Ho

w ev er to this

rder

con tributions from

b

ecome imp

rtan

t and ev en tually dominate the v alue

f
Within

the same con v en tion the CKM elemen ts in the dominan t tree deca y s

u
u

d are real hence the in v arian t phases including this deca y are iden tical to the phases put in to

m
A

con v en tion indep enden t denition

f
similar

to

is
m

A

A
m

A

A
where

A

h
I
jHjK
i

A

h
I
jHjK
i

are the deca y amplitudes in to the

nal

state with isospin

This

quan tit y is iden tical to

h
I
jHjK
L

i h

I
jHjK
S

i dened as

K

in

There

are again three p

ssibilities

for CP violation

Direct

CP violations ha v e

nly

b een

bserv

ed for neutral k aon deca ys to

where

they constitute a comp

nen

t in the asymmetries

bserv

ed in in terference

f
scillation

and deca y

The

eects are m uc h smaller than the exp ected

nes

in B deca ys The phenomenology is in complete analogy with equation

and

will b e discussed in more detail b elo w

CP

violation in the

scillation

can b e

bserv

ed using a v

ur

sp ecic deca ys lik e K

l
The
scillation

asymmetry

starting

with an initial K

meson

is dominated b y the damp ed cos xT

scillation

term seen in the lo w er diagram in gure

The

expansion

f

the asymmetry in

according

to

approac

hes aT

at

large times T

where

the longliv ed state K

L

is the

nly

remaining

ne

It is the same for an initial K

Hence

there are more K

K
scillations

than K

K
lea

ving a net excess

f

K

mesons

a

whic

h has rst b een

bserv

ed in

A

big fraction

f

neutral k aons deca y not as a a v

ur

sp ecic state but as

ne
f

the CP eigenstates e g K

and

K

The

asymmetry from in terference

f
scillation

and deca y has rates determined b y

and

has b een

bserv

ed in all these mo des

SLIDE 46

P

article An tiP article Oscillations and CP Violation The CP asymmetry in the

nal

state is a complex in terpla y b et w een CP violation in

scillation
jr

j

and

a small phase angle arg r

Therefore

a more pragmatic terminology has b een in tro duced based

n

the amplitudes for the t w

eigenstates

using the complex parameter

h
jHjK
L

i h

jHjK
S

i with a phase

arg
If

the formalism used for the B system is applied dening r

m

A

A

b y the K

and

K

amplitudes

as ab

v

e this parameter is an

parameter

for the nal state

according

to

and

w e get h

jHjK
S

i

pA
r
and

h

jH

jK

L

i

pA
r
and
r
r
r
Since

r

is

an

bserv

able this holds also for

The

co e!cien ts for

are
jr
j
R

e

j
j
jr
j
j
j
j
j
R

e r

jr
j

cosarg r

j
j
j
j
I

m r

jr
j

sinarg r

I

m

j
j
F

urthermore w e ha v e direct CP violation from A

h
jHj

K

i
h
j

CPCP H CP

CP

jK

i
e

i CP K h

jCP

H CPjK

i
e

i CPK A

where
describ

es the small dierence in the amplitudes Its

rigin

within the Standard Mo del lies in the in terference

f

tree and p enguin diagrams where hadronic p enguins con tribute

nly

to the isospin

nal

state while the tree graph and electromagnetic p enguins ha v e also a small isospin

comp
nen

t The t w

isospin

amplitudes ha v e dieren t strong phases and p enguin and tree ha v e a dieren t w eak phase Therefore there is a small direct CP violation according to

whic

h w

uld

b e

bserv

ed ev en without

scillation

If w e c ho

se

an

v

erall phase to mak e the isospin

amplitude

A

real

w e ma y write the isospin

amplitude

as A

e

i e i

with
b

eing the strong phase dierence and

the

w eak phase dierence Then the ratio

f

amplitudes is AK

r
A
e

i e i

r
A
AK
e

i CP K

r
A
e

i e i

r
A
A

A

e

i CP K

r
A
A
e

i

e

i

e

i

e

i CP K

p
A
A
e

i

i

sin

and
r
i

A

A
e

i

sin
Corresp
ndingly

w e

btain

r

m

A A

SLIDE 47

CP

Violation

T
a
Fig
Time

dep enden t rate asymmetry aT

f

K

and

K

T
t
is

the lifetime in units

f
S
the

in v erse

f

the a v erage width

f

K

L

and K

S
and
Using

the presen t w

rld

a v erage

these

n um b ers are

j
j
jr
j
sinarg

r

r
i

Expressed in these v ariables the squared matrix elemen ts are jMj

e

T jAj

jpj
j
j
n
j
j
cosh

y T

j
j
sinh

y T

R

e

cos

xT

I

m

sin

xT

jM

j

e

T jAj

jq

j

j
j
n
j
j
cosh

y T

j
j
sinh

y T

R

e

cos

xT

I

m

sin

xT

whic

h corresp

nds

to the asymmetry function aT

#

N K

#

N K

#

N K

#

N K

j
j

cosxT

j
j
e

y T

e

y T

j
j
e

y T

e

y T

j
j

cosxT

j
j

cosxT

j
j
e

y T

e

y T

sho

wn in gure

where

the K a v

ur

is understo

d

to b e the

ne

at T

The

appro ximation in

is

for j m j

SLIDE 48

P

article An tiP article Oscillations and CP Violation The phenomenology for the

state

is similar with a small c hange due to the dieren t in terference b et w een AK

r
A
e

i e i

r
A
A

K

e

i CP K

r
A
e

i e i

r
A
A

A

e

i CP K

p
A
A
e

i

e

i

e

i

e

i CPK

p
A
A
e

i

i

sin

and
p

i A

A
e

i

sin
whic

h nally leads to

A

more general description

f

the k aon system w aiving CPT in v ariance can b e found in the literature

and

is discussed in

Recen

t results

n

the k aon system

ha

v e reac hed an amazing precision

n

man y parameters but are all w ell compatible with the Standard Mo del

SLIDE 49

Measuremen

t

f

CP Violation at B Meson F actories There are t w

exp

erimen tal en vironmen ts suitable to measure CP violation

e
e
storage

rings

p

erating at a cms energy

f

the

S

resonance mass where ab

ut
n

b cross section is a v ailable for exclusiv e B B pro duction almost at rest A t luminosities b et w een

and
cm
s
they

accum ulate roughly

B
mesons

and the same n um b er B

mesons

p er y ear There ha v e b een man y places prop

sed

for suc h mac hines in the past

f

whic h recen tly the storage ring PEPI I at SLA C USA and the storage ring KEKB in the TRIST AN tunnel at KEK Japan ev

lv

ed in to real pro jects whic h b

th

aim at data taking early in

with
ne

detector eac h

Hadron

collision exp erimen ts at energies far ab

v

e the b

b

threshold whic h pro vide a substan tially higher rate

f

incoheren t b

b

pairs but need dedicated triggers to select those from a tremendous bac kground The most promising place is LHC whic h pro vides b

th

a high B pro duction cross section and a high luminosit y

There

ha v e b een sev eral prop

sals

for dedicated exp erimen ts whic h are either xed target exp erimen ts at LHC

ras

a sp ecial caseuse the pp collider mo de with a detector sensitiv e in the v ery forw ard direction

where

the cross section has a pronounced maxim um see gure

This

setup selects parton in teractions where

ne

parton has m uc h higher momen tum than the

ther

and comes therefore close to a xed target setup with the adv an tage

f

using the high luminosit y pro vided in b eamb eam collisions In all prop

sals

the setup has b een c hosen to pro vide a large b

st

whic h allo ws the separation

f

B mesons via their lifetime

0.5 1 1.5 2 2.5 3 1 2 3 200 400 600 800 1000

Fig

b
b

cross section d

d

B d B at

Te

V pp collision s v ersus

B

and

B

cross section scale arbitrary

angles

in rad

The

distribution sho ws that B meson pro duction p eaks in the v ery forw ard direction and that B mesons are accompanied b y their partner b jet at close angular distance there The n um b er

f

pro duced B mesons within a reasonable data taking p erio d is substan tially lo w er at an y en visaged e

e
collider

than at hadronic B factories On the

ther

hand there is a m uc h b etter signal tobac kground ratio and without an y sophisticated trigger ev ery B meson can b e recorded Exp erimen ts at the

S

with

nly

B and B in

ne

ev en t mak e also m uc h b etter use

f

statistical metho ds compared to an already restrictiv e ev en t selection used at hadron mac hines Therefore the e

e
colliders

will most lik ely b e able to establish CP violating eects in the B system unam biguously

giv

en their clean

SLIDE 50

Measuremen

t

f

CP Violation at B Meson F actories systematics An high energy hadron exp erimen t will b e required ho w ev er to supply the statistics needed to determine the parameters with precision While the LHC will start as a B factory not b efore

there

are also hadron exp erimen ts at lo w er energies foreseen in near future HERAB

a

xed target exp erimen t using the halo

f

the HERA

Ge

V proton storage ring at DESY and the exp erimen ts at F ermilab the most promising b eing an upgraded CDF exp erimen t

with

impro v ed p erformance in B ph ysics for the next run at the T ev atron a

Te

V pp collider All exp erimen ts are summarized in table

The

asymmetric e

e
storage

rings PEPI I and KEKB are b

th

designed to run with t w

separate

rings and man y bunc hes

and
p

er ring resp ectiv ely A t KEKB the bunc hes cross at an angle

f
mrad

in parallel

rien

tation crab crossing while the more conserv ativ e PEPI I design aims at headon collision This allo ws a closer bunc h spacing and sligh tly higher fo cussing at KEKB resulting in a gain in luminosiy b y a factor

f
If

this

ption

pro v es viable also PEPI I can b e upgraded to nonzero crossing angle Another factor

f
seems

p

ssible

if

ne

go es more close to the limits

f

mac hine design so that b y the start

f

the second generation CP exp erimen ts at LHCB a luminosit y

f
cm
s

at e

e
B

factories will b e reac hed T able

Prop
sed

B factories most

f

them b eing rejected the appro v ed

nes

are t yp ed in b

ld
The

cms energy

f

b

b
median

B momen tum hpB i

estimated

b

b

cross section and exp ected b

b

pair rate are compared mac hine t yp e E b

b

CMS hpB i

N

b

b

a b eam energies

Ge

V

Ge

V

Ge

V

CESR

Cornell e

e
BET

A PSI symm e

e
ISRB

CERN symm e

e
CESR

I I I Cornell e

e
BET

A PSI asym e

e
VEPP

No v

sibirsk

e

e
ISRB

CERN asym e

e
CESRB

Cornell e

e
KEKB

KEK e

e
PEP

I I SLA C e

e
HELENA

DESY e

e
HERAB

DESY p

C
HERAB
p
C
LHB

CERN p

Be
GAJET

CERN p

H
BTEV

FNAL pp forw

COBEX

CERN pp forw

LHCB

CERN pp forw

CDF

Run

FNAL

pp

D

Run

FNAL

pp

A

TLAS CMS LHC pp

SSC

pp

SLIDE 51

Ge

V

nb
ee
ke

V

ee
ke

V

Fig
P

eak cross section

f

the

S
using
tot
Me

V

and

v arious

ee

v alues from

ke

V to

ke

V

v

ersus the mac hine resolution

E

CMS

The

crosses are the AR GUS DORIS data righ t and CLEO CESR left errors guessed

B

Pro duction Cross Sections The cross section

n

top

f

the

S

resonance is prop

rtional

to the width ee

f
S
e
e
and

v aries with the mac hine energy spread and the total width

f

the

S

Using the AR GUS result tot

Me

V

the

predicted p eak cross section is giv en b y the diagram in gure

for

dieren t ee

Unfortunately
the

resonance parameters ha v e big errors Previous measuremen ts b y CUSB and CLEO ha v e used a parametrization v alid for narro w resonances and can therefore not b e com bined with the AR GUS results The appro ximation

tot

used for its deriv ation is not v alid for the broad

and

increases the resulting width articially

F
r

the mac hine spread

f

a t ypical B factory

f
Me

V

the

range is in the

n

b region A b etter estimate is the

bserv

ed p eak cross section at CESR whic h has a similar width in cms energy

E
Me

V

CLEO
bserv

ed a cross section

n

b

therefore

a v alue

f
n

b seems to b e a conserv ativ e n um b er for rate estimates F

r

pp scattering the cross section

f

b

b

pro duction is dominated b y gluon gluon fusion and dep ends

n

the gluon structure function

f

the proton QCD calculations are v ery uncertain close to threshold but more reliable at higher energies Precise data exist at

Te

V from CDF

for

a limited phase space region and can b e used to c hec k QCD calculations F

r

the prop

sed

mac hines and detector acceptance v

lumes

the uncertain ties are still large Figure

giv

es an idea

f

b

b

pro duction compared to the pp total cross section The lo w fraction

f

b

b

ev en ts from all in teractions at the lo w cms energies

f

xed target exp erimen ts can b e somewhat comp ensated b y the use

f

hea vy n uclear targets While the rare b

b

pro duction cross section p er n ucleon remains ab

ut

constan t and is

A
the

n um b er

f

n ucleons p er n ucleus the total cross section feels shielding b y the densely pac k ed n ucleons and scales appro ximately with the cross section area

f

the n ucleus

A
This

brings a gain in the ratio

f

cross sections

A
The

scaling

f
b
b

with A is supp

rted

b y measuremen ts

f

the c

c

pro duction cross section at high b eam energies

A

disadv an tage

f

hea vy elemen ts is ho w ev er the increased m ultiplicit y due to secondary in teractions in the n ucleus and exp erimen ts ha v e to balance the increase in relativ e rate with the increase in detector

ccupancy
The

asymmetric g g collisions dominate the b

b

pro duction pro cess pro ducing jets p eak ed b

th

in forw ard

SLIDE 52

Measuremen

t

f

CP Violation at B Meson F actories

p

s

Ge

V

nb
nb
nb
nb
nb
b
b
b
mb
mb
mb
p

b eam

Te

V

SSC
LHC
T

evatron

LHB
HERAB
b
b
tot

Fig

Estimated

range for the cross section

f

pN

b
b

X b et w een dotted curv es using the PYTHIA

parametrization
as

a lo w er limit and an estimated upp er limit

times

larger compared to the total pp cross section solid line

r

in bac kw ard direction and the b and

b

jets are close together in these ev en ts This is illustrated in gure

where

d

d

b d

b
sin
b

sin

b

d

d

b d

b

is sho wn v ersus

b

and

b
This

fact is exploited b y LHCB

whic

h is designed to

p

erate at an LHC pp collision p

in

t with an acceptance limited to the forw ard region but coming as close as some

mrad

to the b eam axis

B

Meson F ractions A t the

S
the
nly

nal states are B

B
and

B

B
whic

h are pro duced at the same rate i e

eac

h While unequal fractions due to phase space w ere discussed in the s with more precise mass measuremen ts

mo

ving the w

rld

a v erage mass dierence mB

mB
Me

V c

close

to zero and with new theoretical insigh ts

n

the size

f

the Coulom b correction for c harged B mesons

the

pro duction ratio is narro w ed do wn to f

f
B
S
B
B
B
S
B
B

SLIDE 53

A

t hadron mac hines the B meson fractions dep end

n

fragmen tation and v ary with target and cms energy

An

estimate for

Te

V protons

n

a p

r

n target using the Lund fragmen tation mo del

is

giv en in table

Exp

erimen tal ratios

btained

at LEP are included in this table and indicate an underestimated bary

n

pro duction in the Mon te Carlo mo del Little is kno wn ho w ev er

n

the eects

f

the initial u and d quarks in pp

r

pn collisions Consequen tly

the

mo del predictions should b e tak en as nothing more but an educated guess whic h should b e correct within b etter than

T

able

b

hadron pro duction rates p er ev en t predicted b y the Lund Mon te Carlo

Te

V p

pn
The

fractions

n

the Z

giv

e the b est presen t exp erimen tal kno wledge for b

b

jets without further quarks from the initial state

b

pp pn b pp pn Z

B
B
B
B
B

s

B

s

b
b
ther
ther
F

rom these estimates the exp ected fraction

f

B

p

er

b

jet is ab

ut
while

studies at the B s meson m ust build

n

samples whic h are almost a factor four smaller The exp ected rates p er y ear are estimated for v e exp erimen ts in table

Although

the LHB pro ject is no longer pursued it is included in the table as an example for a true xed target exp erimen t with the adv an tage

f

a v ery close v ertex detector

SLIDE 54

Measuremen

t

f

CP Violation at B Meson F actories a

The HERA-B Experiment at DESY

Ring Imaging Cherenkov Counter

250 mrad 220 mrad 160 mrad

Magnet Si-Strip Vertex Detector TRD Calorimeter Muon Detector Target Wires 0 m 5 10 15 20

Photon Detector Planar Mirrors

Top View Side View

Proton Beam Electron Beam Proton Beam Electron Beam

Spherical Mirrors

Vertex Vessel Inner/Outer Tracker

C4 F10 Beam Pipe

b

2 m 2 m

LHC-B

Top view

p → ← p Yoke Coil RICH-2 E-cal H-cal Muon system Tracking chambers Mirror plate Si vertex detector RICH-1

Fig

The

prop

sed

HERAB a and LHCB b detectors

SLIDE 55

Fla

v

ur

T agging

drift

chamb er cry

stat
coil

IFR DIRC e

Ge

V

e
Ge

V

Fig
Sc

hematic view

f

the BABAR detector cut along the b eam axis The fo cusing quadrup

les

and the b eam separation dip

les

are inside the detector lea ving

nly

limited space for the silicon v ertex detector Both are surrounded b y a cylindrical drift c ham b er follo w ed in radial direction b y the DIR C the Cs I calorimeter the sup erconducting magnet coil and the instrumen ted !ux return IFR

Fla

v

ur

T agging An y

scillation
r

CP asymmetry

bserv

ation requires a tag

f

the initial a v

ur

at T

Starting

with the most p

pular

example the initial a v

ur

is tagged with a c harged lepton

e
r
from

the semileptonic deca y

f

the partner b hadron The nal states are b

B
J
K
S
b
l
X
b
B
J
K
S
b
l
X

Similarly

for
ther

tags there exists alw a ys a correlation b et w een a c harge Q in most cases the electric c harge

f

the tagging particle but e g for

h

yp erons the bary

n

n um b erand the b eaut y a v

ur
f

the tag The asymmetry a is in an ideal case where the a v

ur

B

r

B

at

T

is

kno wn unam biguously from the lepton c harge aT

#

N

B
J
K
S
#

N B

J
K
S
#

N

B
J
K
S
#

N B

J
K
S
#

N J

K
S
l
#

N J

K
S
l
#

N J

K
S
l
#

N J

K
S
l

SLIDE 56

Measuremen

t

f

CP Violation at B Meson F actories

Observ

ed V ersus T rue Asymmetry In the real exp erimen t there are additional asymmetries in v

lv

ed These are few at

S

deca ys and man y more at hadron B factories using pp collisions

The

n um b er

f

B

and

B

pro

duced in pp collisions are not equal due to the fact that an excess

f
u

quarks and

d

quarks is presen t from the b eginning There are four dieren t fragmen tation probabilities f

N

B

N
b
f
N
B
N

b f s

N

B s

N
b
f

s

N
B

s

N

b This in tro duces an in trinsic asymmetry

whic

h is not presen t at e

e
colliders
p

erating at the

S
The

second b hadron used for tagging can ha v e

scillated

in to its an tiparticle with probabilit y

N
b
l
N
b
l
f
f

s

s
N

b

l
N

b

l
f
f

s

s

Here

is

the a v erage probabilit y for

b
b
and
for

b

b

through mixing%

and
s

denote the mixing probabilities

f

B

and

B s

resp

ectiv ely

Due

to the coheren t B B state in

S

deca ys this eect is absen t at B factories

p

erating at the

S
The

lepton can b e from semileptonic c harm deca y in the b

c

cascade Lik ewise almost all tags except fully reconstructed b hadronsha v e a c hance to

ccur

at the wrong c harge P art

f

this eect is ev en due to wrong particle iden tication

f

b deca y pro ducts This mistag probabilit y w is presen t at all exp erimen ts but is reduced if determined as a function

f

discriminating v ariables as describ ed b elo w

The

lepton can b e fak ed b y

r

K

with

absolute m ultipliciti es m

and

m

for

fak ed p

sitiv

e and negativ e tag leptons with m

N
N

K

K
K
m
N
N

K

K
K
Here
X

is the kinematic tagging acceptance and

X

the misiden tication probabilit y for hadron X

Tw
sources
f

hadrons con tribute in t w

dieren

t w a ys

A

t hadronic mac hines the c harged hadron pro duction through fragmen tation yields m

m
due

to the initial quarks from the pp

r

pn state with no correlation to the b a v

ur
The

c harged hadron pro duction through b hadron deca ys ma y sho w a substan tial c harge asymmetry

whic

h is correlated to the b a v

ur

and therefore has also eectiv e tagging p

w

er and mistag probabilit y

This

case is absorb ed in the mistag probabilit y w

whic

h includes mistags from true and fak ed tag leptons

Tw
b

hadrons with the same b eaut y ma y ha v e b een pro duced sim ultaneously

A

t LHB the rate

f

b

b

ev en ts is appro ximately

p

er bunc h crossing i e

in
ev

en ts ha v e t w

separate

in teractions leading b

th

to a b

b

pair According to the Lund mo del as sim ulated in the PYTHIA program single in teractions with pp

b
bb
bX

ha v e a frequency

compared

to all pp

b
b

X ev en ts The total rate

f
can

safely b e neglected The rate gets sligh tly higher with increasing cms energy

but

sta ys alw a ys b elo w the lev el to b ecome a signican t source

f

mistag

A

c

cX

ev en t

ccurs

together with a b

b

ev en t pro ducing additional leptons from c harmed hadron deca ys with no relation to the b eaut y a v

ur

A t LHB this

ccurs

with a frequency

SLIDE 57

Fla

v

ur

T agging

Although

it ma y lead to a small amoun t

f

additional uncorrelated mistags whic h are included in Mon te Carlo sim ulations it is negligible compared to the con tribution from secondary c harmed particles in b

b

X ev en ts F

r

b and

b

a v

urs

as tags at T

w

e ha v e t w

ev

en t rates # N

dN
dT

as a function

f

the prop er scaling lifetime T # N J

K
S
b
f
a
N

# N J

K
S
b
f
a
N

for giv en true asymmetry a

aT
T

agging the b

r
b

with an electron

r

m uon at a hadron mac hine the eects men tioned ab

v

e giv e the follo wing rates # N

#

N J

K
S
l
a
f
B
C
m
N
a
f
B
C
m
N

# N

#

N J

K
S
l
a
f
B
C
m
N
a
f
B
C
m
N
with

the visible branc hing fractions B for b

l
X

and C for b

l
X

b eing B

B

b

l
X
B

B

X
B

B

D
s

X

B
B

C

B

b

cX
B

c

l
X
C
C

Here

B

is the tagging acceptance for righ t sign leptons from the second b hadron

C

the tagging acceptance for wrong sign leptons mainly from secondary c harmed hadron deca ys The giv en branc hing fractions illustrate the main con tributions to b

th

classes Both fractions get additional con tributions from righ t and wrongsign pions and k aons with an amoun t dep ending

n

the lepton iden tication capabilit y

f

the detector Note that the resulting ev en t n um b ers are ab

ut

equal if the e!ciency is uniform

v

er the whole phase space and tagging p

w

er has to come from the use

f

kinematic dierences

f

b

th

lepton samples m

and

m

are

the fak e rates from misiden tication

f

hadrons

riginating

not from b deca ys Double tags e g b y a true plus a fak e lepton ha v e b een ignored in these form ulae The normalization constan t is N

N

b

b
B

B

J
K
S
l
l
T
e

jT j where

is

the reconstruction and trigger e!ciency

It

will cancel in all ratios The

bserv

ed asymmetry a

bs

is then a

bs
#

N

#

N

#

N

#

N

I
w
w
I

m

w
m
m
B
I
a
a
w
e
I
m
w
m
m
B
I
a
with

w

C

B

C
f
f
f
f
e
f
f
f
f
I

m

m
m
B
w
I
f
f
f
f
r

appro ximately

ignoring

terms O a

a
bs
I
D
a

SLIDE 58

Measuremen

t

f

CP Violation at B Meson F actories

a
a
bs
Fig
Observ

ed asymmetry a

bs

v ersus ph ysical asymmetry a for a cut

n

the tag lepton at p

Ge

V c

using

f

f
f

s

f

s

N

K

N

K

N
and

N

without

those from K

S
from

the PYTHIA JETSET Mon te Carlo programs B b

l
X
and

a lepton tagging acceptance

f
B

b

c
l
X
and

a tagging acceptance for leptons from c harm

f
corresp
nding

to p

Ge

V c

a

misiden tica tion probabilit y

f
for

K and

for
and

a tagging acceptance for fak e pions

f
and

for k aons

f
K
and
K
In

the linear appro ximation

these

parameters giv e I

and

D

The

deviation from linearit y is b elo w visibili t y in this plot with an in trinsic asymmetry I

I
I

m and a dilution factor D

w
I
w
m
m
B
D

t

D

m where the dilution factor has b een split in to a tagging comp

nen

t D t and a mixing comp

nen

t D m

The

linear appro ximation holds v ery w ell for all practical purp

ses

as can b e seen in g

from

a Mon te Carlo sim ulation using the ev en t parameters for the LHB exp erimen t The next

rder

appro ximation is a

bs
I
D

a

D

I

a
whic

h will

nly

b e imp

rtan

t at large pro duction asymmetries b et w een B

and

B

The

rate n um b er

f

reconstructed tagged ev en ts is # N tot

#

N

#

N

B
C
m
m
f
f
a
f
f
N

SLIDE 59

Fla

v

ur

T agging

On

the

S
where

a B B pair is pro duced exclusiv ely and T

is

the deca y time

f

the tag B

the

relation

simplies

considerably

with

m

m
and

f

f
to

a

bs
w

a

There

is no in trinsic asymmetry

I
unless

the detector has dieren t acceptances for p

sitiv

e and negativ e particles whic h is not considered in the relations ab

v

e there is no mixing dilution D m

and

the dilution factor is related to the mistag probabilit y simply as D

D

t

w
This

tagging dilution whic h is also a go

d

rst appro ximation in more complicated jet en vironmen ts can b e expressed in the simple form D t

w
righ

tsign

wrongsign

righ tsign

wrongsign
where

righ tsign and wrongsign refers to the n um b er

f

correct and wrong tags resp ectiv ely

As

will b e sho wn in detail b elo w the error

n

the

bserv

ed asymmetry amplitude is

a
p

N

N
Therefore

the error

n

the asymmetry amplitude

is

appro ximately

D

p N

N
p
t

D

N

s where N s is the n um b er

f

signal ev en ts and

t

is the tagging e!ciency

i

e

t

N s is the fraction

f

signal ev en ts with a a v

ur

tag The p erformance

f

tagging can therefore b e dened b y the factor

t

D

t
e

whic h giv es the eectiv e reduction in n um b er regarding statistical precision It is also called separation since it is

if

b and

b

can b e separated p erfectly ev en t b y ev en t and

if

they cannot b e distinguished

Statistical

T agging Fla v

ur

tagging exploits alw a ys a correlation b et w een the b eaut y a v

ur
f

the paren t b hadron and a c harge Q in most cases the electric c harge

f

the tagging particle but e g for

h

yp erons the bary

n

n um b er

r

strangeness and for D

mesons

the c harm This correlation is p erfect for fully reconstructed b eaut y mesons

r

bary

ns

Ho w ev er a complete reconstruction

f

B deca ys will

nly

b e p

ssible

in a v ery limited n um b er

f

ev en ts A more univ ersal approac h is to collect this information via certain c haracteristics

f

the particles whic h are able to iden tify the a v

ur

In the example ab

v

e

nly

leptons ha v e b een used with a c harge correlated to the b eaut y a v

ur

via semileptonic deca ys b

l
c
This

tagging metho d suers from t w

problems
nly

ab

ut
f

all b hadrons deca y semileptonically

and
a

substan tial fraction

f

leptons from

ther

sources ha v e the wrong c harge A more complete exploitation

f

the secondaries from B deca ys can b e ac hiev ed in a statistical analysis The a v

ur

tagging p

w

er

f

these deca y pro ducts lik e c harged leptons is determined b y the v alues

f

a small n um b er

f

discriminating v ariables X

X
whic

h help to iden tify their role in the deca y pro cess

Th

us instead

f

tting a

bs

T

D
sin

xT

I
w

e ma y t a t w

dimensional

distribution a

bs

T

D
D
sin

xT

I
to
btain

the parameter

Here

the mistag con tribution to D is not a constan t a v erage dilution factor but a function

f

the discriminating v ariables X

X
and

v aries b et w een

and
Negativ

e v alues mean that in the kinematic range dened b y fX

X
g

there are more wrongsign tags with

pp
site

sign

f

c harge and a v

ur

than righ tsign

r

samesign tags In

ther

w

rds

the correlation b et w een the c harge and the b eaut y a v

ur

has ipp ed D is ev aluated ev en t b y ev en t in addition to the scaled lifetime dierence T

SLIDE 60

Measuremen

t

f

CP Violation at B Meson F actories

p
Ge

V c

n

dn dp

a
p
Ge

Vc

D

t b Fig

Distribution

s

f

m uons in the

S

cms momen tum for righ tsign

and

wrongsign

tags

a and the corresp

nding

dilution factor b The sim ulation has b een done for the BABAR detector the p eak at lo w momen tum is due to K and

deca

ys and particle misiden ticati

n
While

this concept should b e applied to all dilution factors e g to mixing dilution as a function

f

the lifetime

f

a B meson in the second jet

r

to the dilution due to limited precision in the determination

f

the

scillation

time T as a function

f

v ertex precision in the follo wing sections

nly

the application to the tagging dilution will b e discussed The relev an t information for the t is the tagging dilution co e!cien t D t X

X
itself

It is giv en b y all tag particles i e particles with a nonzero c harge lik e quan tum n um b er Q that can b e correlated to the b eaut y a v

ur

The discriminating v ariables for eac h particles ha v e to b e c hosen in a w a y to dieren tiate maxim al ly b et w een kinematic situations with dieren t correlation strength An

b

vious v ariable is the momen tum

f

the particle in the paren t's rest frame if this can b e reconstructed This is illustrated in gure

where

the momen tum

f

m uons in the

S

rest framewhic h comes close to the B rest frameis used as discriminator Ab

v

e a v alue

f

ab

ut
Ge

V c

almost

all m uons are from semileptonic B deca ys and sho w therefore an almost p erfect correlation with the b eaut y a v

ur

This is indicated b y a dilution factor close to

in

gure b whic h is calculated from the distributions in gure a using

A

t v alues b elo w

Ge

V c there is an

pp
site

correlation due to b

c
l
X
though

diluted to ab

ut
ne

fth The ip

f

the correlation is at

Ge

V c

where

the mistag probabilit y is w

and

no a v

ur

information is

btained

Other discriminating v ariables can b e e g the impact parameter to the b eam axis

r

the angle to the closest trac k from the same B

Often

information

n

the en vironmen t ma y ha v e ev en b etter

r

at least complemen tary discriminating p

w

er e g the energy

f

a neutrino as missing momen tum in the case

f

a c harged lepton tag In the statistical approac h

ne

considers the distributions

f

the v ariables X

X
f

daugh ter particle

with

c harge Q

the

v ariables X

X
f

daugh ter particle

with

c harge Q

and

so

n

i e f B X

X
X
X
jQ
Q
if

the paren t w as a B meson

r
b

hadron and f B X

X
X
X
jQ
Q
if

the paren t w as a B meson

r

b hadron The mostly con tin uous v ariables X ij and the c harges Q i ma y b

th

b e considered random v ariables The a v

ur
f

this ev en t is assigned b y maxim um lik eliho

d

SLIDE 61

Fla

v

ur

T agging

i

e if f B

f

B the b eaut y

f

the ev en t is tak en to b e

and

vice v ersa This assignmen t is unique and dep ends

n

the X and Q v ariables together The signed dilution factor is then D t X

Q
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
where

the sign giv es the estimated b eaut y quan tum n um b er and its absolute v alue giv es the dilution factor corresp

nding

to this a v

ur

assignmen t F

r
nedimensional

distributions the true a v

ur

is statistically related to Q b y the densities f B X j jQ

f

tags with a

b

quark and f B X j jQ

f

tags with a b quark The a v erage b eaut y a v

ur
f

a sample with a certain v alue X j and Q is then

B

i

f

B X j jQ

f

B X j jQ f B X j jQ

f

B X j jQ

F
r

the

pp
site

c harge the densities are f B X j j

Q
f

B X j jQ and f B X j j

Q
f

B X j jQ up to negligible explicit CP violations This reduces four functions in the general ansatz to

nly

t w

functions

Instead

f

treating Q

and
as

t w

v

alues

f

an additional random v ariable the a v

ur
f

the paren t b hadron ma y b e a priori assigned to b e Q

This

assignmen t is arbitrary

and

could b e

pp
site

as long as it is uniquely dened Then the a v

ur

estimator can b e written as

B

i

D

t X j

Q
The

absolute v alue

f

the signed factor D t X j

f

B X j j

f

B X j j f B X j j

f

B X j j

f

B X j j

f

B X j j f B X j j

f

B X j j is the tagging dilution factor for all tags with a v alue X j

It

is jD t j

if

the a v

ur

is p erfectly correlated with Q

The

sign is a a v

ur

corrector It is negativ e if the arbitrary a priori assignmen t from Q is more

ften

wrong than righ t D t

righ

tsign

wrongsign

righ tsign

wrongsign

It is t ypically a smo

th

function

f

X j

and

can b e

btained

from a Mon te Carlo sim ulation

r

from real data measuring the B B

scillation

amplitude as describ ed b elo w The a v

ur

estimator

B

dened in

is

the a v erage a v

ur
f

ev en ts in a sample with tagging particles

f

c harge Q and discriminating v ariable X j

e

g

f

p

sitiv

e leptons at a giv en cms momen tum

B
f

B X j jQ f B X j jQ

f

B X j jQ

f

B X j jQ f B X j jQ

f

B X j jQ The co e!cien ts are the fractions

r

a p

steriori

probabilities

f

ev en ts from B and B

resp

ectiv ely

This

v ariable is itself a discriminating v ariable comprising all used information in the ev en t It can b e split in to its signused as a v

ur

guessand its absolute v alue j

B

j

whic

h is the tagging dilution factor D t

B
j
B

j All informations X ij from

ne
r

sev eral tags in an ev en t are com bined to estimate the a v

ur
f

the tag hadron more reliably

A

full exploitation

f

this tagging metho d is ho w ev er imp

ssible

since the detailed information required to determine the inn umerable m ultidim ensional functions cannot b e ac hiev ed within the statistical precision

f

an y exp erimen t Sev eral appro ximations are prop

sed

to

v

ercome the tec hnical problems

SLIDE 62

Measuremen

t

f

CP Violation at B Meson F actories One approac h is to use

nly
nedimensional

distributions and to assume factorization

f

B X

X
X
jQ
Q
f

B X

jQ
f

B X

jQ
f

B X

jQ
With

this ansatz

ne

can add all a v

ur

estimators

f

the same ev en t lik e relativistic v elo cities F

r

t w

tagging

particles with v ariables X

Q
and

X

Q
inserting
in

to the result

B
B
B
B
B
B
B
leads

immediately to

B
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
f

B X

jQ
Rep

eated addition

f

the estimators from all tagging particles and discriminating v ariables yields the total a v

ur

estimator

f

the ev en t Although factorization is not v alid the appro ximation is usually quite go

d

and can b e tested b y determining D t

B
The

deviation D t

B
j
B

j is a measure

f

the correlations% if it is small the appro ximation is a go

d
ne

if it is large a dieren t w a y to com bine tagging information ma y impro v e the result A v ery promising approac h for this impro v emen t is to com bine man y discriminating v ariables for

ne

particle in to

ne
utput

v ariable whic h is an

ptim

um represen tation

f

the a v

ur

information including the correlations A linear com bination

f

carefully c hosen v ariables is used in a Fisher discriminan t analysis Ev en b etter results are ac hiev ed with the help

f

learning neural net w

rks
A

com bination

f

b

th

metho ds

r

more exible neural net w

rks

whic h allo w input

f

a v ariable n um b er

f

tag particles and their corresp

nding

parameters can nally giv e a maxim um

f

information In a statistical tagging metho d the eectiv e p erformance

r

separation

t

D

t
e

can no longer b e split in to t w

factors

In fact

t

is

ne

in the ultimate realization where ev ery ev en t is used as a tag The pro duct

n

the

ther

hand can still b e ev aluated as

t

D

t
e
Z
d

t ds D

t

s ds where d t ds

f

s is the probabilit y densit y

f

s

j
B

j

In

the general concept a a v

ur

estimator is constructed for ev ery ev en twith a v alue close to

if

no go

d

tagging information is a v ailableand the p erformance n um b er

t

D

t
e
hD
t

i

hs
i

is the a v erage tagging dilution

f

all ev en ts The idea to calculate a a v

ur

estimator for ev ery ev en t is esp ecially promising

n

the

S
where

B

B
are

pro duced exclusiv ely

After

remo ving the signal CP c hannel e g J

K
S
the

whole residual ev en t is from the

ther

B

so

ev ery c harged particle can b e used for statistical tagging

SLIDE 63

Estimating

the P erformance

Estimating

the P erformance The p erformance

f

an exp erimen t can b e estimated b y the exp ected error within

ne

y ear

f

data taking Complications due to p enguin amplitudes p enguin p

llution

are shortly discussed b elo w The error

n

an asymmetry amplitude

r

lik ewise

is

appro ximated as

S
p

N e

where

the eectiv e n um b er

f

ev en ts is N e

D
r

D c D

m
t

D

t
e
s

B B

X
f
b
b
L
s

with the follo wing factors

s

is the assumed dut y time

f

an y dedicated exp erimen t at design luminosit y in a y ear L is the luminosit y

f

the mac hine

b
b

is the cross section for b quark pair pro duction f

is

the fraction

f

b quarks turned in to B

mesons

B B

X
is

the pro duct branc hing fraction

f

B

in

to the nal state in v estigated e g B B

J
K
S
B

K

S
B

J

e
e
B

J

s

is the signal reconstruction e!ciency

including

the geometric acceptance and the trigger e!ciency

This

ma y b e correlated with

t
if

a trigger uses information from the tag

t

is the tagging e!ciency

It

go es alw a ys with the tagging dilution factor D t as the pro duct

t

D

t
Exploiting

statistical tagging to its fullest

t
while

all loss from am biguous tags is mo v ed in to the eectiv e a v erage hD

t

i

t

D

t
e
D

m is the mixing dilution factor D m

at

the

S
D

m

for

incoheren t b

b

pro duction and tags from the

ther

b hadron Samejet tagging tec hniques lik e using a jet c harge

r

the pion from B

B
are

not sub ject to mixing dilution D c

c

is a dilution factor induced b y bac kground to the signal c hannel The total n um b er

f

ev en ts is increased b y bac kground b y a factor D c

and

at the same time the asymmetry amplitude is decreased b y a factor D c

giving

rise to a reduction factor D

c

for the eectiv e n um b er This is the simplest w a y to incorp

rate

bac kground eects A detailed study

f

the time ev

lution
f

bac kground can further reduce its inuence

n

the error D r is the dilution factor arising from lifetime resolution Comparing the v arious exp erimen tal prop

sals

requires detailed information since they use dieren t assumptions for these n um b ers and also dieren t lev els

f

realism in detector sim ulation It is ho w ev er fair to predict thatif the Standard Mo del is righ t CP violation will b e disco v ered around the y ear

SLIDE 64 Ac kno wledgemen ts My understanding

f

hea vy a v

ur

ph ysics and B B mixing has ev

lv

ed during m y long sta y with the AR GUS collab

ration

and I

w

e thanks to man y colleagues there for ha ving impro v ed m y kno wledge in theory

data

analysis and exp erimen tal tec hniques During m y w

rk

for LHB and BABAR I ha v e proted greatly from the common w

rk

with h undreds

f

ph ysicists in these collab

rations

all around the w

rld

Man y ideas written in this pap er ha v e b een ev

lv

ed

r

claried in v ery fruitful discussions with

ther

colleagues among them Gio v anni Carb

ni

Rob ert Fleisc her Joac him Graf F rank Krauss Christof Kreuter F rancois Le Dib erder Ralph M

uller

Dominic P

tsc

hk e Helen Quinn Klaus R Sc h ub ert Bernhard Spaan and J

rg

Urban I thank the

rganizers
f

the Herbstsc h ule Maria Laac h for arranging this in teresting sc ho

l

SLIDE 65 References

T

D Lee C N Y ang Ph ys Rev

C

S W u et al Ph ys Rev

%

R L Garvin et al Ph ys Rev

%

J J F riedman V L T elegdi Ph ys Rev

J

H Christenson JW Cronin VL Fitc h R T urla y

Ph

ys Rev Lett

N

Cabibb

Ph

ys Rev Lett

M

Koba y ashi T Mask a w a Progr Theor Ph ys

see

e g c hapter

in

M E P eskin D V Sc hro eder An In tro duction to Quan tum Field Theory AddisonW esley Publ Reading

C

Jarlsk

g

Mysteries in the Standard Mo del Pro c

f

the In t Symp

n

Pro duction and Deca y

f

Hea vy Fla v

urs

Heidelb erg

p
The

Large Hadron Collider Conceptual Design CERN&A C&% C H Llew ellyn Smith Pro c

f

the th In t Symp

n

LeptonPhoton In teractions Beijing

eds

ZP

Zheng

HS Chen W

rld

Scien tic Publ Singap

re
p
The

P article Data Group Ph ys Rev D

LL

Chau WY Keung Ph ys Rev Lett

H

Harari M Leurer Ph ys Lett B

L

W

lfenstein

Ph ys Rev Lett

Z

Z Xing Ph ys Rev D

M

Koba y ashi Progr Theor Ph ys

K

R Sc h ub ert Pro c

f

the In t Europh ysics Conf

n

High Energy Ph ysics Uppsala Sw eden ed O Botner June

v
l

I I p

C

Hamzaoui J L Rosner A I Sanda Pro c

f

the W

rkshop
n

High Sensitivit y Beaut y Ph ysics at F ermilab Bata via IL USA eds N Lo c ky erd J Slaugh ter No v

p
C

Jarlsk

g

Ph ys Rev Lett

%

Z Ph ys C

A

Ali D London Nucl Ph ys Pro c Suppl A

T

T W u C N Y ang Ph ys Rev Lett

G

L

uders

Dan Mat Fys Medd

%

Ann Ph ys NY

e

g E D Commins P

H

Buc ksbaum W eak In teractions

f

Leptons and Quarks Cam bridge Univ ersit y Press

T

Nak ada PSIPR % T Nak ada Pro c

f

the XVIth In t Symp

n

LeptonPhoton In teractions Ithaca

eds

P

S

Drell D L Rubin AIP

New

Y

rk
p
A

J Buras R Fleisc her TUMHEP& to app ear in Hea vy Fla v

urs

I I ed b y A J Buras M Lindner W

rld

Scien tic

V

F W eissk

pf

E P

Wigner

Z Ph ys

and

Z Ph ys

A

detailed discussion is in O Nac h tmann Elemen tarteilc henph ysik

Ph

anomene und Konzepte in German View eg Braunsc h w eig % see also

A

J Buras W S( lominski H Steger Nucl Ph ys B

T

Inami C S Lim Progr Theor Ph ys

and

erratum Progr Theor Ph ys

SLIDE 66

References
A

J Buras M Jamin P

H

W eisz Nucl Ph ys B

for

recen t reviews see e g H Wittig DESY

%

A P atel eprin t hepph& % V Gim ) enez G Martinelli C T Sac hra jda Nucl Ph ys Pro c Suppl

Y

Nir Ph ys Lett B

OP

AL Coll Z Ph ys C

DELPHI

Coll Z Ph ys C

DELPHI

Coll Z Ph ys C

ALEPH

Coll CERNPPE&

DELPHI

Coll CERNPPE&

SLD

Coll SLA CPUB

SLD

Coll SLA CPUB

T

Miao CDF Coll FERMILABCONF&E

H

Albrec h t et al AR GUS Ph ys Lett B

ALEPH

Collab Ph ys Lett B

M

Artuso et al CLEO Ph ys Rev Lett

H

Albrec h t et al AR GUS Z Ph ys C

J

Bartelt et al CLEO Ph ys Rev Lett

H

Albrec h t et al AR GUS Ph ys Lett B

H

Albrec h t et al AR GUS Ph ys Lett B

ALEPH

Collab Ph ys Lett B

ALEPH

Collab CERNPPE&

DELPHI

Collab Ph ys Lett B

DELPHI

Collab Z Ph ys C

DELPHI

Collab CERNPPE&

OP

AL Collab Ph ys Lett B

OP

AL Collab CERNPPE&

OP

AL Collab CERNPPE&

OP

AL Collab CERNPPE&

L

Collab Ph ys Lett B

SLD

Collab SLA CPUB

SLD

Collab SLA CPUB

SLD

Collab SLA CPUB

CDF

Collab FERMILABCONF&E con tributed pap er pa to the th In t Conf

n

High Energy Ph ysics W arsa w

CDF

Collab FERMILABPUB&E

F

DeJongh G Mic hail CDF FERMILABCONF&E

ALEPH

Collab Ph ys Lett B

ALEPH

Collab Ph ys Lett B

ALEPH

Collab con tributed pap er pa to the th In t Conf

n

High Energy Ph ysics

SLIDE 67

W

arsa w

O

Sc hneider presen tation at the th In t Symp

n

LeptonPhoton In teractions Ham burg

OP

AL Coll Ph ys Lett B

CDF

Coll FERMILABCONF&E con tributed pap er pa to the th In t Conf

n

High Energy Ph ysics W arsa w

CDF

Coll Ph ys Rev Lett

DELPHI

Coll Z Ph ys C

DELPHI

Coll DELPHI

CONF
con

tributed pap er pa to the th In t Conf

n

High Energy Ph ysics W arsa w

ALEPH

Coll Z Ph ys C

OP

AL Collab CERNPPE&

H

Wittig OUTPP

subm

to J Mo d Ph ys A

I

Bigi et al CERNTH&

published

in

B

Deca ys ed S Stone W

rld

Scien tic Publ

R

Aleksan Ph ys Lett B

M

Benek e G Buc halla I Dunietz Ph ys Rev D

I

Dunietz FERMILABPUB&T % T Bro wder S P akv asa UH

see

e g H Quinn SLA CPUB

an

exp erimen t to measure CP violation in

deca

ys is E J An tos et al FERMILAB Prop

sal

P

for

a recen t review

n

neutrino

scillations

see e g A Y u Smirno v talk giv en at the th In t Conf

n

High Energy Ph ysics W arsa w

eprin

t hepph& % CP and T violation is discussed in M T animoto Ph ys Rev D

%

J Arafune J Sato Ph ys Rev D

U

Kilian et al Z Ph ys C

%

S Y Choi et al Ph ys Rev D

%

J H K

uhn

E Mirk es TTP % Y S Tsai Ph ys Lett B

for

reviews see e g W Grim us F

rtsc

hr Ph ys

%

W Bernreuther M Suzuki Rev Mo d Ph ys

%

J Bernab ) eu et al FTUV& and Pro c

f

the Ringb erg W

rkshop
n

P ersp ectiv es for Electro w eak In teractions in e

e
Collisions

F eb

ed

b y B A Kniehl W

rld

Scien tic

M

Gronau et al Ph ys Rev D

%

Ph ys Rev D

%

Ph ys Lett B

%

Ph ys Rev Lett

%

L W

lfenstein

Ph ys Rev D

%

N G Deshpande X G He Ph ys Rev Lett

%

Ph ys Rev Lett

%

A J Buras R Fleisc her Ph ys Lett B

%

Ph ys Lett B

%

G Kramer W F P almer Ph ys Rev D

%

B Grinstein R F Leb ed Ph ys Rev D

A

B Carter A I Sanda Ph ys Rev Lett

%

Ph ys Rev D

I

I Bigi A I Sanda Nucl Ph ys B

%

Nucl Ph ys B

R

Fleisc her I Dunietz Ph ys Lett B

%

Ph ys Rev D

I

Dunietz et al Ph ys Rev D

M

Gronau D London Ph ys Lett B

M

Gronau D Wyler Ph ys Lett B

SLIDE 68

References
I

Dunietz Ph ys Lett B

R

Aleksan I Dunietz B Ka yser Z Ph ys C

DELPHI

Collab M F eindt et al DELPHI

CONF
con

tributed pap er to the th In t Symp

n

LeptonPhoton In teractions Ham burg

ZZ

Xing Ph ys Rev D

D

E Dorfan et al Ph ys Rev Lett

%

S Bennett et al Ph ys Rev Lett

for

a recen t review see T Ruf CERNPPE&

Prop
sal

for an Electron P

sitron

Collider for Hea vy Fla v

ur

P article Ph ysics and Sync hrotron Radiation PR PSI Villigen

F

easibilit y Study for a B Meson F actory in the CERN ISR T unnel ed T Nak ada CERN

HELENA
A

Beaut y F actory in Ham burg DESY

PEP

I I Conceptual Design Rep

rt

SLA C % BABAR Letter

f

In ten t SLA C % J T Seeman SLA CPub % see also

BELLE

Collab Letter

f

In ten t for a Study

f

CP Violation in B Meson Deca ys KEK Rep

rt
%