DSpace at VNU: Prospects for the Measurement of the CP Asymmetry inBMeson Decay

Fortschr. Phys. 47 (1999) 7 ± 8, 707±±853

Prospects for the Measurement
of the CP Asymmetry in B Meson Decays
Roland Waldi
Institut fuÈr Kern- und Teilchenphysik
Technische UniversitaÈt Dresden

Abstract
The expected effects from CP violation in neutral B decays in the framework of the Standard Model
are reviewed. Time dependent rates and asymmetries are presented with emphasis on their observability
at recently proposed B factories. Detectors and methods to extract CP asymmetry parameters are presented, including techniques for flavour tagging and data fits. The expected performance of an e‡ eÀ
and a hadron beam experiment is illustrated with the most promising final states JawKs0 and p‡ pÀ .

Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

709

2. Particle Anti-Particle Oscillations and CP Violation . . . . . . . .
2.1. The Unitary CKM Matrix . . . . . . . . . . . . . . . . .
2.1.1. Unitarity Triangles . . . . . . . . . . . . . . . . .
2.1.2. Phases and Observables . . . . . . . . . . . . . . .
2.2. Oscillation Phenomenology . . . . . . . . . . . . . . . . .
2.2.1. Standard Model Predictions . . . . . . . . . . . . . .
2.2.2. Behaviour of the Four Neutral Meson Anti-Meson Systems .
2.2.3. CP Eigenstates Versus Mass Eigenstates . . . . . . . . .
2.2.4. Oscillation at the U(4S) . . . . . . . . . . . . . . . .
2.2.5. Determination of the Mixing Parameters of B Mesons . . .
2.2.6. Predictions for xs Y ys and ds . . . . . . . . . . . . . .
2.3. CP Violation . . . . . . . . . . . . . . . . . . . . . . .

2.3.1. CP Violation in B Decays . . . . . . . . . . . . . . .
2.3.2. CP Violation in Common Final States of B0 and B"0 . . . .
2.3.2.1. The Bs aB"s Case . . . . . . . . . . . . . . .
2.3.2.2. CP Violation at the U(4S) . . . . . . . . . . .
2.3.2.3. Time Integrated Asymmetries . . . . . . . . .
2.3.2.4. Final CP Eigenstates from B0 or Bs Decays . . . .
2.3.2.5. The B 3 pp Decay . . . . . . . . . . . . .
2.3.2.6. Mixtures of CP Eigenstates . . . . . . . . . .
2.3.2.7. Non-Eigenstates . . . . . . . . . . . . . . .
2.3.2.8. The Total Decay Rate . . . . . . . . . . . . .
2.3.3. CP Violation in K Decays . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

710
710
712
715
717
721
724
730
732
735
737
738

739
741
744
745
750
750
754
758
759
759
760

3. Measurement of CP Violation at B Meson Factories .
3.1. B Meson Factories. . . . . . . . . . . . . . .
3.1.1. B Production Cross Sections. . . . . . . .
3.1.2. B Meson Fractions. . . . . . . . . . . .
3.1.3. B Meson Yields . . . . . . . . . . . . .
3.2. Two Typical Detectors . . . . . . . . . . . . .
3.2.1. The LHB Detector at a Fixed Target Hadronic

.
.
.
.
.
.
.

.
.

.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.

.

764
765
767
769
770
770
771

.
.
.
.
.
.
B

. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Factory .

.
.
.
.

.
.
.


708

R. Waldi, Prospects for the Measurement of the CP Asymmetry
3.2.1.1. Vertex Region . . . . . . . . . . . . . . . . . . . .
3.2.1.2. Tracking and Particle Identification . . . . . . . . . . .
3.2.1.3. Trigger . . . . . . . . . . . . . . . . . . . . . . .
3.2.1.4. Other Experiments . . . . . . . . . . . . . . . . . .
3.2.2. The BABAR Detector at the PEP II e‡ eÀ Storage Ring . . . . . .
3.2.2.1. Vertex Detector . . . . . . . . . . . . . . . . . . . .
3.2.2.2. Tracking and Particle Identification . . . . . . . . . . .
3.2.2.3. Electromagnetic Calorimeter . . . . . . . . . . . . . .
3.2.2.4. Muon and Neutral Hadron Detector . . . . . . . . . . .
3.2.2.5. Trigger . . . . . . . . . . . . . . . . . . . . . . .
3.2.2.6. Other Expriments . . . . . . . . . . . . . . . . . . .
3.2.3. U (4S) Factories and Hadron Colliders: Two Complementary Concepts

.
.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

. 772
. 773
. 774
. 774
. 774
. 777
. 778
. 780
. 780
. 781
. 781
. 781

4. Analysis Techniques and Tools to Estimate Experimental Performance.
4.1. Flavour Tagging . . . . . . . . . . . . . . . . . . . . . . .
4.1.1. Observed Versus True Asymmetry . . . . . . . . . . . .
4.1.2. Statistical Tagging . . . . . . . . . . . . . . . . . . .

4.1.3. Specific Tags . . . . . . . . . . . . . . . . . . . . .
4.1.3.1. Lepton Tags . . . . . . . . . . . . . . . . . .
4.1.3.2. Lepton Tags at LHB. . . . . . . . . . . . . . .
4.1.3.3. Kaon Tags . . . . . . . . . . . . . . . . . . .
4.1.3.4. Kaon Tags at LHB . . . . . . . . . . . . . . .
4.1.3.5. Charm Tags . . . . . . . . . . . . . . . . . .
4.1.3.6. Other Tags. . . . . . . . . . . . . . . . . . .
4.1.4. Combined Tagging Results at the U(4S) . . . . . . . . . .
4.1.5. Special Tags at Hadron Machines . . . . . . . . . . . . .
4.1.5.1. Vertex Reconstruction and Charged B Tags . . . . .
4.1.5.2. Tag Jet Charge
. . . . . . . . . . . . . . . . .
‡
4.1.5.3. The B** 3 B0 p‡ Cascade . . . . . . . . . . .
4.1.5.4. Same Jet Charge . . . . . . . . . . . . . . . .
4.1.6. Combined Tagging for Hadron Machines . . . . . . . . .
4.1.7. Determining I and D from Data . . . . . . . . . . . . .
4.2. Fitting CP Asymmetries . . . . . . . . . . . . . . . . . . . .
4.2.1. Fit to the Time Dependent Asymmetry. . . . . . . . . . .
4.2.1.1. Fit of L . . . . . . . . . . . . . . . . . . . .
4.2.1.2. Fit of L and I . . . . . . . . . . . . . . . . .
4.2.1.3. Fit of Q . . . . . . . . . . . . . . . . . . . .
4.2.1.4. Fit of Q and I . . . . . . . . . . . . . . . . .
4.2.1.5. Fit of L and Q . . . . . . . . . . . . . . . . .
4.2.2. Using Time Integrated Numbers . . . . . . . . . . . . .
4.2.2.1. Background . . . . . . . . . . . . . . . . . .
4.2.3. Improved Fit Procedure for Small Data Samples. . . . . . .
4.2.3.1. Background . . . . . . . . . . . . . . . . . .

.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

782
783
784
787
791
791
795
798
800
802
804
806
807
808
808
809
810
810
810
813
814
817
818

820
821
822
823
823
824
827

5. Performance for the Key Final States . . . .
5.1. B0 3 p‡ pÀ . . . . . . . . . . . . .
5.2. B0 3 p‡ pÀ Reconstruction at LHB . . .
5.2.1. Mass Resolution . . . . . . . . .
5.2.2. Trigger Efficiency and Reconstruction
5.2.3. Backgrounds . . . . . . . . . .
5.2.3.1. Background from Bs Decays
5.2.3.2. Combinatoric Background .
5.2.4. Acceptance . . . . . . . . . . .
5.3. B0 3 p‡ pÀ Reconstruction at BABAR . .
5.3.1. Backgrounds . . . . . . . . . .
5.3.2. The Lifetime Measurement . . . .
5.4. B0 3 Jaw Ks0 . . . . . . . . . . . . .
5.5. B0 3 Jaw Ks0 Reconstruction at LHB . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

827
828
829
832
833
834
836

837
839
840
840
841
842
842

. . . .
. . . .
. . . .
. . . .
Losses .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.


709

Fortschr. Phys. 47 (1999) 7±±8
5.6. B0 3 JawKs0 Reconstruction at BABAR . . .
5.7. Present Experimental Information . . . . . .
5.8. Comparison of Future Experiments . . . . . .
5.8.1. B0 3 JawKs0 and the Determination of b
5.8.2. B0 3 p‡ pÀ and the Determination of a.

.
.
.
.
.

845
845
846
846
847

6. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

848

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

848

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

849

1.

.
.
.
.
.

.
.
.
.
.

.
.
.
.

.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.

.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.

.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

Introduction

Our understanding of physics in general and particle physics in particular has been mainly
put forward by the discovery of symmetries. It is remarkable, that most of the symmetries
discovered have, however, finally turned out to be only ``almost-symmetriesº, i.e. to be
more or less broken.
The only unbroken symmetries so far discovered are the U(1) charge-phase symmetry
and the SU(3) colour symmetry. The consequences are, that the electric and colour charges
are exactly conserved in all observed reactions, and that the position in SU(3)-space cannot
be determined, e.g. a ``redº and a ``blueº quark cannot be distinguished.

Each of the symmetries between leptons and quarks of different flavour is broken by the
different masses and electro-weak charges of these particles, and is best approximated in
strong interactions as isospin symmetry between the u and d quark due to their almost
identical constituent mass.
Although physics laws are strictly symmetric under translation or rotation, space-time
translational and rotational symmetry is broken through the solutions: The fact that matter
is not distributed homogeneously throughout the universe introduces a locally asymmetric
structure of space-time, or asymmetric boundary conditions to any microscopic system. The
spatial symmetries are best approximated on a macroscopic scale ±± the universe ±± or for
microscopic systems isolated from other matter by large distances.
Mirror symmetry (parity P) is broken in a more fundamental sense by weak interaction,
which makes a maximal distinction between fermions of left and right chirality. First ideas
of this unexpected behaviour emerged as a solution of the ``Q t puzzleº, the fact that the
neutral kaon decays both to P ˆ ‡1 and P ˆ À1 eigenstates [1], and a direct observation
as left-right-asymmetry in weak beta decays followed soon [2]. It is most pronounced in
the massless neutrinos, which are produced in weak interactions only with lefthanded helicity, or righthanded in the case of anti-neutrinos, thus violating the charge-conjugation symmetry (C) at the same time.
The product of both discrete symmetries, CP, is almost intact, and seemed to be conserved even in weak interaction processes. A small violation has first been observed in
1964 [3] in K 0 decays, which are up to now the only system which does not respect CP
symmetry completely. The explanation of this violation in the Standard Model will be
briefly discussed in the next chapter. This is not the only possible description, but the one
with no additional assumptions. At the same time, the Standard Model predicts CP violating effects in the decay of beauty mesons (B0 , Bs , B‡ ), which should be even large in some
rare decay channels.
This paper will describe these effects, and discuss techniques and the prospects of their
measurement within the next few years. After a discussion of meson flavour oscillations
and CP violation in the Standard Model, the concept of typical experiments at B meson
factories are presented. In section 4, analysis techniques and in particular methods for
flavour tagging are introduced, and information factors for different fit situations are dis-


710

R. Waldi, Prospects for the Measurement of the CP Asymmetry

cussed. Section 5 presents experimental performance for two examples and in the last section an outlook to the next few years is based on extrapolating these studies to the most
promising proposed experiments.
2.

Particle Anti-Particle Oscillations and CP Violation

Mesons are neither particles nor anti-particles in a strict sense, since they are composed of
a quark and an anti-quark. This implies the existence of mesons with vacuum quantum
numbers (e.g. f0 ). More important is the existence of pairs of charge-conjugate mesons,
which can be transformed into each other via flavour changing weak interaction transitions.
" D0 aD"0 (c"
" d),
" and Bs aB"s (bsab"
" s).
These are K 0 aK"0 ("
sdasd),
ua"
cu), B0 aB"0 (bdab
2.1.

The Unitary CKM Matrix

The charged current weak interactions responsible for flavour changes are described by the
couplings of the W boson to the current
H I
H I
H I

H I
e
d
n"e
u"
1 À g5 f g € f g
1 À g5
f g
f g
"
Jmcc ˆ d n"m e gm
m
V
Á
‡
…2X1†
c
g
d e
dse
d e m
2
2
rY gY b
"t
n"t
t
b
with a non-trivial transformation matrix V in the quark sector, the Cabibbo±Kobayashi±
Maskawa (CKM) Matrix [4, 5]:

H
I
Vud Vus Vub
V ˆ d Vcd Vcs Vcb e X
Vtd Vts Vtb
The quark flavours in (2.1) are defined as the mass eigenstates. A completely equivalent
picture is to use the states …dH Y sH Y bH † with V  1, and define a non-diagonal mass matrix.
Since mass generation is accomplished in the Standard Model via couplings to the Higgs
field [6], this moves the question of the origin of the CKM matrix elements into the realm
of mass generation, which belongs still to the more ``mysteriousº parts [7] of the Standard
Model. The exploration of the Higgs sector is the main motivation for the LHC storage
ring, which is built at CERN and will start operation around 2005 [8]. The Higgs-quark
couplings alone involve 10 independent parameters of the Standard Model, the quark
masses and the parameters of the CKM matrix, which are not related within the theory.
Local gauge invariance and baryon number conservation requires the CKM matrix to be
unitary. If there were more than three quark families, this would not hold for the 3 Â 3
submatrix, but this possibility is unlikely, given the limit on neutrino flavours from LEP
experiments, who find nn ˆ 2X991 Æ 0X016 [9] for neutrinos with mass much below the Z 0
mass. Thus, if a fourth generation exists, it must incorporate a massive neutrino which is
more than a factor 1000 heavier than the tau neutrino, even if we assume the experimental
upper limit for the latter.
From the 9 real parameters of a general unitary matrix, 5 can be absorbed in 1 global
phase, 2 relative phases between uY cY t and 2 relative phases between dY sY b which are all
subject to convention and in principle unobservable. If two quarks within one of these two
groups were degenerate in mass, even the sixth phase could be removed by redefining the
basis in their two-dimensional subspace.


711

Fortschr. Phys. 47 (1999) 7±±8

Rephasing may be accomplished by applying a phase factor to every row and column:
Vjk 3 ei…fj Àfk † Vjk X

…2X2†

Note that j ˆ uY cY t, k ˆ dY sY b, and the six numbers fu Y fc Y ft Y fd Y fs Y fb represent only
five independent phases in the CKM matrix, since different sets of ffj Y fk g yield the same
result. Any product where each row and column enters once as Vij and once via a complex
conjugate Vkl* like Vij Vkl Vil*Vkj* is invariant under the transformation (2.2). This implies that
observable phases must always correspond to similar products of CKM matrix elements
with equal numbers of V and V * factors and appropriate combination of indices.
Removing unphysical phases, the CKM matrix is described by 4 real parameters, where
only one is a phase parameter, while the other three are rotation angles in flavour space.
The standard parametrization [9] (first proposed in [10], notation follows [11]) uses a
choice of phases, that leave Vud and Vcb real:
I
H
IH
IH
c12 s12 0
1
0
0
0 s13 eÀid13
c13
g
f
gf

gf
V ˆ d 0 c23 s23 ed
0
1
0
ed Às12 c12 0 e
H

0 Às23

c23

Às13 eid13

c12 c13
f
ˆ d Às12 c23 Àc12 s13 s23 eid13
s12 s23 Àc12 s13 c23 e

id13

0

0

c13

0

s13 eÀid13

s12 c13

g
c13 s23 e

c12 c23 Às12 s13 s23 eid13
id13

Àc12 s23 Às12 s13 c23 e

1
I
…2X3†

c13 c23

with cij ˆ cos qij , sij ˆ sin qij , and sij b 0, cij b 0 (0 qij pa2).
A convenient substitution1 ) is s12 ˆ l, s23 ˆ Al2 , s13 sin d13 ˆ Al3 h, and
s13 cos d13 ˆ Al3 r, which reflects the apparent hierarchy in the size of mixing angles via
orders of a parameter l. This leads to
H

l2 l4
À
1
À
f
2
8

f


f
1
f
2 5
l
r
‡
ih
À
Àl
À
A
f
Vˆf
2
f
4
2
35
f
d 3
l2
Al 1 À …r ‡ ih† 1 À
2

l



l2
1 A 4
‡
1À À
l
8 2
2


1
ÀAl2 À Al4 r ‡ ih À
2

3

I

Al …r À ih† g
g
g
g
2
Al
g‡ o…l6 †
g
g
g
1 2 4e
1À A l

2
…2X4†

3

and agrees to o…l † with the Wolfenstein approximation [12]:
H
I
l2
3
l
Al
…r
À
ih†
1
À
f
g
2
f
g
f
g
2
Vˆf
gX
l
2
f

g
Al
Àl
1
À
d
e
2
3
2
Al …1 À r À ih† ÀAl
1

…2X5†

Equation (2.4) is more convenient [13] in higher orders than the original proposal of Wolfenstein, or an exact parametrization [14] using the Wolfenstein parameters.
1

) An equivalent choice is l ˆ s12 c13 which leads to the same parametrization to o…l5 †.


712

R. Waldi, Prospects for the Measurement of the CP Asymmetry

Assuming a unitary 3 Â 3 matrix, from experimental information these parameters are [9]
l ˆ 0X2205 Æ 0X0018 Y
A ˆ 0X80 Æ 0X08 Y
p
r2 ‡ h2 ˆ 0X36 Æ 0X08

while the phase and therefore each individual value of r and h is still very uncertain.
Inserting these parameters, equation (2.5) shows clearly the dominance of the diagonal
matrix elements, indicating that transitions between quarks of different families are suppressed. It is the unitarity constraint which makes Vtb ˆ 0X9992 Æ 0X0002 the best known
matrix element. Experimental constraints on the magnitude (90% CL limits [9]) are:
H
I
0X9745 F F F 0X9757
0X219 F F F 0X224
0X002 F F F 0X005
d 0X218 F F F 0X224
0X9736 F F F 0X9750
0X036 F F F 0X046 e X
0X004 F F F 0X014
0X034 F F F 0X046
0X9989 F F F 0X9993
With already one more family of quarks, we have five additional real parameters, of which
two are new non-trivial phases. Therefore, the measurement of all CKM matrix elements
and their relative phases is an important test of the Standard Model.

2.1.1.

Unitarity Triangles

If nature provides us with just these three families of fermions, unitarity requires the following 12 conditions to be fulfilled:
jVud j2 ‡ jVus j2 ‡ jVub j2 ˆ 1 Y

…2X6a†

jVcd j2 ‡ jVcs j2 ‡ jVcb j2 ˆ 1 Y

…2X6b†

jVtd j2 ‡ jVts j2 ‡ jVtb j2 ˆ 1 Y

…2X6c†

jVud j2 ‡ jVcd j2 ‡ jVtd j2 ˆ 1 Y

…2X6d†

jVus j2 ‡ jVcs j2 ‡ jVts j2 ˆ 1 Y

…2X6e†

jVub j2 ‡ jVcb j2 ‡ jVtb j2 ˆ 1 Y

…2X6f†

V *ud Vcd ‡ V *us Vcs ‡ V *ub Vcb ˆ 0 Y

…2X6g†

V *ud Vtd ‡ V *us Vts ‡ V *ub Vtb ˆ 0 Y

…2X6h†

V *cd Vtd ‡ V *cs Vts ‡ V *cb Vtb ˆ 0 Y

…2X6i†

Vud V *us ‡ Vcd V *cs ‡ Vtd V *ts ˆ 0 Y

…2X6j†

Vud V *ub ‡ Vcd V *cb ‡ Vtd V *tb ˆ 0 Y

…2X6k†

Vus V *ub ‡ Vcs V *cb ‡ Vts V *tb ˆ 0 X

…2X6l†


Fortschr. Phys. 47 (1999) 7±±8

713

Dividing (2.6k) by Al3 % ÀVcd V *cb yields the unitarity triangle2 † as shown in figure 2.1a.
In the Wolfenstein approximation, it corresponds to
…r ‡ ih† À 1 ‡ …1 À r À ih† ˆ 0 X

…2X7†

A second one from (2.6h) is shown in figure 2.1b. Dividing by Al3 % ÀV *us Vts and using
the approximation Vud % 1 gives the same triangle (2.7). A closer look, however, reveals
slightly different lengths and angles to o…l2 †.
The angles of the unitarity triangles (2.6k and h) in figure are defined by
Vtd Vub V *ud V *tb
Y
jVtd Vub Vud Vtb j

H
V *td V *cb Vcd Vtb
V *td V *us Vts Vud
% eib ˆ À
Y
eib ˆ À
jVtd Vcb Vcd Vtb j
jVtd Vus Vts Vud j
V *ub V *cd Vcb Vud
V *ub V *ts Vus tb
H
% eig ˆ À
X
eig ˆ À
jVub Vcd Vcb Vud j
jVub Vts Vus Vtb j

eia ˆ À

These are rephasing invariant expressions, hence the angles resemble physical quantities
independent of the CKM parametrization. It was first emphasized by Jarlskog [17], that CP
violation can be described via a rephasing invariant quantity
J ˆ Æ Im Vij Vkl V *il V *kj % A2 l6 h
which is up to a sign independent of iY jY kY l, provided i Tˆ k, j Tˆ l. The areas of all six
unitarity triangles defined by (2.6g±l) are equal and have the value Ja2. This corresponds
to an area % ha2 for the ones in figure 2.1, since their sides have been reduced by the
factor Al3. As will be shown below, CP violating observables are typically proportional to
the sine of the angles in unitarity triangles, like
sin g ˆ Im eig ˆ À

Im …V *ub V *cd Vcb Vud †
J
ˆÀ
jVub Vcd Vcb Vud j
jVub Vcd Vcb Vud j

and vanish for J ˆ 0, i.e. if all triangles collapse into lines. If the non-trivial phase is 0 or
p, the parameter h is 0 and hence J ˆ 0. This would also be the case if two quarks of a
given charge had the same mass, since then a rotation between these two flavours could be
chosen that removes the phase factors, as can be seen in (2.3) where q13 ˆ 0 would remove
all terms with the phase d13 .
The angles of all six triangles (2.6g±l) can be determined using the standard parametrization (2.3) in a rewritten form
H
I
jVud j
jVus j
jVub j eeÀi~g
jVcb j e
V ˆ d ÀjVcd j eif4 jVcs j eeÀif6
…2X8†
Àib~
if2
jVtd j ee
ÀjVts j ee
jVtb j
with g~  d13 . Here, absolute values and phases are given as separate factors. The angles
f2 % hl2 , f4 % hA2 l4 , and f6 % hA2 l6 are all positive and very small and their subscript
2
† This geometric interpretation has been pointed out by Bjorken $ 1986; its first documentation in
printed form is in ref. 15 and more general in ref. 16.

714

R. Waldi, Prospects for the Measurement of the CP Asymmetry

Fig. 2.1: Unitarity triangles in the complex plane, corresponding to a: (2.6k) and b: (2.6h). Up to
2
2
corrections of o…l4 † the top points are …rY h† in (b), but …‰1 À l2 Š rY ‰1 À l2 Š h† in (a), and the right2 1
2
most points are …1Y 0† in (a), but …1 À l ‰2 À rŠY l h† in (b). The angles are related via
g À gH ˆ bH À b % l2 h.

indicates the order in l of their magnitude. The unitarity triangles in figure 2.1 have angles
b ˆ b~ ‡ f4 Y
H

g ˆ g~ À f2 Y

bH ˆ b~ ‡ f2 Y
a ˆ p À b~ À g~ X

g ˆ g~ À f4 Y

In the Wolfenstein approximation, the unitarity relations read (all terms given to order l3
or, if this is still 0, [in brackets] to leading order)
Àl ‡

1 3

1
l ‡ l À l3 ‡ ‰A2 l5 …r ‡ ih†Š ˆ 0 Y
2
2

…2X6gH †


715

Fortschr. Phys. 47 (1999) 7±±8

Al3 …1 À r À ih† À Al3 ‡ Al3 …r ‡ ih† ˆ 0 Y

…2X6hH †

‰ÀAl4 …1 À r À ih†Š À Al2 ‡ Al2 ˆ 0 Y

…2X6iH †

lÀ

1 3
1
l À l ‡ l3 À ‰A2 l5 …1 À r À ih†Š ˆ 0 Y
2
2

…2X6jH †

Al3 …r ‡ ih† À Al3 ‡ Al3 …1 À r À ih† ˆ 0 Y

…2X6kH †

‰Al4 …r ‡ ih†Š ‡ Al2 À Al2 ˆ 0

…2X6lH †

and define three pairs of unitarity triangles, 6 in total:
· (2.6hH ) and (2.6kH ) are the ones shown in figure 2.1 with three sides of similar length, all
of order Al3. This is ``the unitarity triangleº. The other ones are quite flat, and it will
require very high precision to prove experimentally that they are not degenerate to a line.
· (2.6iH ) and (2.6lH ) have two sides of length Al2 and one much shorter of order Al4. This
limits the small angles, which are f2 ‡ f6 and f2 À f6 , respectively. They are close to
the differences of angles in the large triangles g À gH ˆ bH À b ˆ f2 À f4 .
· (2.6gH ) and (2.6jH ) have two sides of length l and one very much shorter of order A2 l5,
with a small angle f4 À f6 and f4 ‡ f6 , respectively. Both are of order l4.
Tiny differences between the two standard unitarity triangles are o…l2 † corrections,
Al3 …1 À r À ih† ‡


1
5
‡Al r ‡ ih À
‡ o…l7 †
2

‡ Al5



Al3 …r ‡ ih† ‡
À

ÀAl3

‡ Al3 …r ‡ ih† ˆ 0



1
À r À ih ‡ o…l7 †
2
ÀAl3 ‡

1
Al5 …r ‡ ih† ‡ o…l7 †
2

‡ o…l7 † Y

Al3 …1 À r À ih† ˆ 0

‡ o…l7 †

‡

…2X6hHH †

…2X6kHH †

1
Al5 …r ‡ ih† ‡ o…l7 † X
2

The angles in these two triangles can be estimated from experimental constraints on a 3 Â 3
unitary CKM matrix, leading to 95% CL limits [18]
25

a

125 Y

11

b

35 Y

40

g

145 X

All phase angles are only weakly constrained by these limits, and one of the aims of
experiments designed to observe CP violation in B meson decays is a first measurement,
and ultimately a precise determination of their values. However, deviations from or extensions to the Standard Model may imply that the two triangles are dissimilar, or even that
they are no (closed) triangles at all. Therefore, it is important to distinguish measurements
of different parameters, even if they are expected to have identical or close values within

the three family Standard Model.
2.1.2.

Phases and Observables

The fact that phases of quark fields are unobservable numbers has been used to show that
some phases in the CKM matrix are not observables either, and there remains some arbitrariness in the parametrization for this matrix. The freedom to choose quark phases may be


716

R. Waldi, Prospects for the Measurement of the CP Asymmetry

"c Y f
"t Y f
"d Y f
"s Y f
"b . With the new quark
extended to antiquarks, with six more phases f"u Y f
states
"

qHj ˆ eifj qY q"Hj ˆ eifj q"j Y

j ˆ uY cY tY dY sY b

also the phase induced by the CP operation is changed. The transition
H

CP jqj i ˆ eifCP j j"

qj i 3 CP jqHj i ˆ eifCP j j"
qHj i
requires
"j X
fHCP j ˆ fCP j ‡ fj À f
This equation leaves fHCP j still completely undefined, since all three phases on the righthand side are not observable, and therefore subject to arbitrary changes. It becomes meaningful, however, if it is applied to observables, like CP eigenvalues. Two CP eigenstates
constructed from a meson and anti-meson state with eigenvalues Æ1 are related accordingly:
Ã
H
" Â
jqj q"k i Æ eifCP jk jqk q"j i ˆ eÀi…fj ‡fk † jqHj q"Hk i Æ eifCP jk jqHk q"Hj i X
H

The new states jqHj q"Hk i Æ eifCP jk jqHk q"Hj i have the same eigenvalues, and differ by an overall
unobservable phase from the old ones.
" is
The CP operation on a meson, e.g. the pseudoscalar B0 meson jbdi,
CP jB0 i ˆ eifCP B jB"0 i Y

…2X9†

where the phase factor eifCP B ˆ hB"0 j CP jB0 i depends on the parity of the bound-state wave
function, and the chosen quark and antiquark phase convention. It is thus an unobservable,
arbitrary phase.
Quark phase changes can in principle be compensated by phase changes of the CKM
matrix elements according to (2.2), leaving terms like
hqj j Vjk jqk i
invariant. However, this is not a physical requirement, and in fact the CP transformed
qj j V *jk j"
qk i Y

eifCP kj h"

…2X10†

has a phase which changes with the quark phases. Since none of the two terms corresponds
to an observable, the actual choice of phases in the CKM matrix parametrization can be
made independent of the choice of quark phases.
The appearance of an additional phase factor in (2.10) can be avoided by the restriction
"j ˆ Àfj for quark phase changes, and an appropriate phase convention which makes
f
terms related by a CPT transformation relatively real. If a choice of phases is possible
where all CKM matrix elements can be made real, also charged current weak interactions
would not violate CP symmetry.
Phase conventions will also enter into relations among decay amplitudes. An amplitude
for a weak decay B0 3 X via a single well defined process can be written as
A ˆ hXj H jB0 i ˆ hXj OV jB0 i Y

…2X11†


717

Fortschr. Phys. 47 (1999) 7±±8

where V is a product of the appropriate CKM matrix elements and O is an operator describing
the rest of the weak and possibly also subsequent strong interaction processes involved in the
transition. Since strong interaction and also weak interaction ±± except for nontrivial phases
in V ±± are CP invariant, the charge conjugate mirror process B"0 3 X has an amplitude
A ˆ hXj H jB"0 i ˆ eifCP X hXj CP OV CP+ eÀifCP B jB0 i
ˆ ei…fCP X ÀfCP B † hXj OV * jB0 i

V*
AY
ˆ ei…fCP X ÀfCP B †
V

…2X12†

where also
V*
ˆ eÀ2i arg V
V
is just a phase. Especially, if X is a CP eigenstate with eigenvalue hX ˆ Æ1,
A ˆ hX eÀi…fCP ‡2 arg V† A

…2X13†

relates the two amplitudes, and the ratio AaA flips sign with the CP eigenvalue.
All physical observables must be independent of the choice of phases. This is the case if
only absolute values of amplitudes are involved, but for interference terms the phase convention cancels often in a more subtle way. Some examples will be shown in the following
chapters. On the other hand, expressions where the arbitrary phases are still present cannot
be observables.
2.2.

Oscillation Phenomenology

The four meson pairs K 0 aK"0 , D0 aD"0 , B0 aB"0 , and Bs aB"s can be described as decaying twocomponent quantum states obeying the SchroÈdinger equation
i dt w ˆ Hw
with a general Hamiltonian
H

i
f m11 À 2 G 11
i
HˆMÀ G ˆf
d
i
2
m*12 À G *12
2

I
i
G 12 g
2
gY
e
i
m22 À G 22
2
m12 À

…2X14†

where M and G are hermitian, but H is not [19]. If the B0 aB"0 system is taken as a representative to illustrate the behaviour of oscillating meson pairs, the indices 1 and 2 correspond to base vectors jB0 i and jB"0 i, respectively.
CPT invariance requires m11 ˆ m22 Xˆ m and G 11 ˆ G 22 Xˆ G, reducing the number of
real parameters of the Hamiltonian to six.
H
I
i
i



À
G
m
G
m
À
12
12 g
f
H H12
2
2
gX
Hˆ
…2X15†
ˆf
d
e
i
i
H21 H
m*12 À G *12
mÀ G
2
2


718

R. Waldi, Prospects for the Measurement of the CP Asymmetry

CPT invariance is one of the indispensable premises of any relativistic field theory within
or beyond the Standard Model [20]. The generalized phenomenology including CPT violation will therefore not be considered here, but can be found in textbooks [21].
Solving the eigenvalue problem det …H À a Á 1† ˆ …H À a†2 À H12 H21 ˆ 0, one obtains
p
two eigenstates with eigenvalues a ˆ H Æ H12 H21 , explicitly
s



i
i
i
i
…2X16†
m12 À G 12
aL ˆ mL À G L ˆ m À G À
m*12 À G *12 Y
2
2
2
2
s



i
i

i
i
aH ˆ mH À G H ˆ m À G ‡
m12 À G 12
m*12 À G *12 Y
2
2
2
2
where LY H stands for ``lightº and ``heavyº. It is immediately seen that m and G are the
average mass 12…mH ‡ mL † and width 12…G H ‡ G L †. The differences are

s


Dm mH À mL
i
i
ˆ Re
m12 À G 12
ˆ
m*12 À G *12 Y
…2X17†
2
2
2
2
DG G H À G L
ˆ À2 Im
ˆ

2
2


s


i
i
m12 À G 12
m*12 À G *12 Y
2
2

s



i
i
m12 À G 12
m*12 À G *12
2
2
r
1
ˆ jm12 j2 À jG 12 j2 À i Re …m12 G *12 † X
4

Da aH À aL

ˆ
ˆ
2
2

The connection between mass and lifetime (width) differences and the off-diagonal elements in the mass matrix are showing up in these equations, especially Dm ˆ 0 if m12 ˆ 0
and DG ˆ 0 if G 12 ˆ 0. Squaring the last line leads to the useful relation
Dm Á DG ˆ 4 Re …m12 G *12 †

…2X18†

which relates the sign of Dm and DG with the off-diagonal elements m12 and G 12 . It is
convenient to define the dimensionless parameters
xˆ

Dm
Y
G

yˆ

DG G H À G L tL À tH
ˆ
Y
ˆ
G H ‡ G L tL ‡ tH
2G

…2X19†

where x is a non-negative real number, and y may only assume values between À1 and 1.
It is an asymmetry parameter inthe widths
or, equivalently, in the lifetimes tL Y tH .

p
The eigenvectors jBLY H i ˆ
are found by inserting (2.16) into H jBLY H i ˆ
Æq
aLYH jBLY H i, giving the ratio
i
r
m*12 À G *12
q 1ÀE
H21
2
hm Xˆ ˆ
ˆÀ
ˆ À2
i
H12
p 1‡E
Dm À DG
2

…2X20†


719

Fortschr. Phys. 47 (1999) 7±±8

and
p p
1 À hm
H12 À H21
H12 ‡ H21
p X
Eˆ
ˆ p p ˆ
1 ‡ hm
H12 À H21 H12 ‡ H21 À 2 H12 H21
Normalization requires jpj2 ‡ jqj2 ˆ 1, i.e.
1‡E
1
p ˆ q ˆ q Y
2
2…1 ‡ jEj †
1 ‡ jhm j2
1ÀE
hm

q ˆ q ˆ q
2
2
2…1 ‡ jEj †
1 ‡ jhm j
and single particle eigenstates are described by one complex parameter hm. This parameter3 † is defined only up to an arbitrary phase, and only jhm j is a measurable quantity.
The value of the phase depends on conventions, one of them is the definition of the
phase fCP B ˆ arg hB"0 j CP jB0 i. This makes also E (sometimes also denoted "E, e.g. in [23])
an arbitrary quantity. The standard choice of the CKM matrix (2.3) and fCP K ˆ 0 make

jEj small in the K 0 aK"0 system, but a consistent convention fCP B ˆ 0 leaves it at
o…0X1 F F F 1† in the B0 aB"0 system. A different definition of E for the kaon system given in
[22] is independent of arbitrary phases. In general, convention independent parameters
can be defined if decays are involved. They can usually be expressed via the unitarity
angles (see fig. 2.1) and will be given for the B and K systems at the appropriate places
below.
The original Hamiltonian can be rewritten using the parameter hm as
H
i
f
mÀ G
f
2
f
Hˆf
i
f
Dm À DG
d
2
Àhm
2

I
i
H
DG
i
g
2

À
mÀ G
g f
2
2hm
g f
gˆd G
g
À hm …x À iy†
e
i
2
mÀ G
2
Dm À

À

I
G 1
…x À iy† g
2 hm
g
e
i
mÀ G
2
…2X21†

and the mass and flavour eigenstates are related by the equations

jBL i ˆ p jB0 i ‡ q jB"0 i Y
jB0 i ˆ

1
…jBL i ‡ jBH i† Y
2p

jBH i ˆ p jB0 i À q jB"0 i Y
jB"0 i ˆ

1
…jBL i À jBH i† X
2q

The eigenstates for non-hermitian H, i.e. G 12 Tˆ 0, are not orthogonal:
d Xˆ hBH j BL i ˆ jpj2 À jqj2 ˆ
3

1 À jhm j2
1 ‡ jhm j2

ˆ

2 Re E
1 ‡ jEj2

X

) hm or Àhm is sometimes called a in the literature, e.g. in [22, 23].

…2X22†


720

R. Waldi, Prospects for the Measurement of the CP Asymmetry

In contrast to E the real number d is an observable. The deviation of jhm j from one (called
da in [22]) is
r
1Àd
À 1 % Àd X
jhm j À 1 ˆ
1‡d
For an arbitrary initial state
jw…0†i ˆ bH jBH i ‡ bL jBL i ˆ a jB0 i ‡ a" jB"0 i Y
where the amplitudes are related via
bLY H ˆ

1
2



a
a"
Æ
p
q

a ˆ p …bL ‡ bH †Y


ˆ

a Æ a"ahm
Y
2p

a" ˆ q …bL À bH †

its time evolution may be described using a scaled time variable
T Xˆ Gt Y

…2X23†

where G is the average width of the eigenstates BH and BL . These states have a simple
exponential development with time. Their masses mHY L ˆ m Æ x G2 and widths
G HY L ˆ G…1 Æ y† can be expressed with the dimensionless parameters x and y defined in
(2.19).
jw…t†i ˆ bH eÀi …mH ÀiG H a2† t jBH i ‡ bL eÀi …mL ÀiG L a2† t jBL i
ei …xÀiy† Ta2 ‡ eÀi …xÀiy† Ta2
…a jB0 i ‡ a" jB"0 i†
2


i …xÀiy† Ta2
À eÀi …xÀiy† Ta2 a"
ÀimtÀTa2 e
0

0
"
jB i ‡ ahm jB i
‡e
2
hm
T
ˆ eÀimtÀTa2 …a jB0 i ‡ a" jB"0 i† cos …x À iy†
2


!
T
a" 0
jB i ‡ ahm jB"0 i sin …x À iy†
X
‡i
2
hm
ˆ eÀimtÀTa2

…2X24a†

…2X24b†

Starting with pure B0 mesons at t ˆ 0 corresponds to a" ˆ 0 and
jw…t†i ˆ a eÀimtÀTa2 cos …x À iy†

!
T 0

T "0
jB i ‡ ihm sin …x À iy†
jB i X
2
2

…2X25†

Starting with pure B"0 mesons at t ˆ 0 is described by replacing hm 6 1ahm . The numbers
of B0 and B"0 at time T for N0 pure B0 mesons at T ˆ 0 are
eÀT
…cosh yT ‡ cos xT† Y
2
eÀT
NB"0 …t† ˆ N0 jhB"0 j w…tY a" ˆ 0Y a ˆ 1†ij2 ˆ N0 jhm j2
…cosh yT À cos xT† X …2X26†
2
NB0 …t† ˆ N0 jhB0 j w…tY a" ˆ 0Y a ˆ 1†ij2 ˆ N0


721

Fortschr. Phys. 47 (1999) 7±±8

These numbers, however, can not be observed. What is accessible by experiment is only
the rate of decays to flavour specific final states X and X at a given time T. These decay
modes are often called tagging modes, since they serve as a ``tagº to indicate the flavour
of the mother particle at decay time. The rates can be obtained from (2.25) by multiplying
with hXj H or hXj H, respectively, to obtain the amplitudes. They are converted into rates
…

1
N• B0 3X …t† ˆ N0 dPS jhXj H jw…tY a" ˆ 0†ij2 ˆ N0 eÀT G X …cosh yT ‡ cos xT† Y
2
…
1
N• B"0 3X …t† ˆ N0 dPS jhXj H jw…tY a" ˆ 0†ij2 ˆ N0 jhm j2 eÀT G X …cosh yT Àcos xT† Y
2
…2X27†
where
GX ˆ

„

dPS jhXj H jB0 ij2 ˆ

„

dPS jhXj H jB"0 ij2

is the partial width for a non-oscillating meson. It agrees in value for the two CP conjugate
processes if the amplitudes differ only by phases. Integrating over all times the total number of decays are
I
…

NB0 3X ˆ
0
I
…

NB"0 3X ˆ

0

!
GX
1
1
•
‡
NB0 3X …t† dt ˆ N0
Y
G 2…1 À y2 † 2…1 ‡ x2 †
4
5
2
2
G
jh
j
jh
j
X
m
m
À
N• B"0 3X …t† dt ˆ N0
X
G 2…1 À y2 † 2…1 ‡ x2 †

The corresponding numbers for initial B"0 mesons are obtained with the replacement
hm 3 1ahm . If we ignore CP violating effects in the oscillation, i.e. for jhm j ˆ 1, we can

define a meaningful branching fraction as
1
f…B 3 X† ˆ
N0
0

I
…

‰N• B0 3X …t† ‡ N• B"0 3X …t†Š dt ˆ

0

GX
1 GX
1 GX
ˆ
‡
G…1 À y2 † 2 G H 2 G L

which agrees with f…B"0 3 X† defined accordingly for the same number N0 of B"0 mesons
at t ˆ 0.
2.2.1.

Standard Model Predictions

The Hamiltonian (2.15) can be obtained using
H ˆ H0 ‡ Hw Y
where H0 is the strong and electromagnetic Hamiltonian



E0 0
H0 ˆ
0 E0


722

R. Waldi, Prospects for the Measurement of the CP Asymmetry

which has the stable flavour eigenstates B0 and B"0 , and H w is the weak interaction perturbation. The Wigner-Weisskopf approximation for small Hw leads to [24]
Hjk ˆ H0 jk ‡ hjj Hw jki ‡
Â

„
€ dPS hjj H w jXi hXj H w jki

€
X

!
1
À ipd…E0 À EX † Y
E0 À EX

…2X28†

where the sum runs over all multiparticle states X which are eigenstates of H0 , and €
denotes the principal value of the integral. The mass (hermitian) and decay (anti-hermitian)
parts defined by (2.14) are

mjk ˆ

€
1
…Hjk ‡ H *kj † ˆ E0 djk ‡ hjj H w jki ‡ €
2
X

…
dPS

hjj H w jXi hXj Hw jki
E0 À EX

and
G jk ˆ i…Hjk À H *kj † ˆ 2p

€„
X

dPS hjj H w jXi hXj H w jki d…E0 À EX † X

The off-diagonal elements H12Y 21 have non-zero contributions in the sum from states X
which can be reached in weak decays of both B0 and B"0 . In contrast to the neutral kaon
system, for B0 aB"0 these are only a small fraction of all B decays, and they contribute with
alternating signs. Therefore H12Y 21 are dominated by the leading term hB0 j H w jB"0 i which
corresponds to the box diagrams

.

They give approximately [25]
H12 ˆ hB0 j H jB"0 i % m12 ˆ À

G2F ÀifCP B 2 *2 2
e
Vtb V td mW mB ‰fB2 BB Š Á ‰S…m2t am2W † Á hQCD Š X
12p2
…2X29†

The CP phase is introduced during the evaluation of the hadronic part of the matrix element. The Inami±Lim function S from the loop [26] is to lowest order a factor m2t am2W . An
evaluation of the product S…m2t am2W † and hQCD within a consistent renormalization scheme
yields S % 2X3, hQCD % 0X55 [27]. A big uncertainty is the product fB2 BB , where the most
reliable
p calculations now come from lattice gauge theory [28] with values around
fB BB % …200 Æ 40† MeV.
In this approximation, we have for the B system
Dm ˆ 2 jm12 j


Fortschr. Phys. 47 (1999) 7±±8

723

which can be used to determine jVtd j from experimental results on B0 aB"0 mixing. The
eigenstates are determined by
hm ˆ À

m*12
V *2V 2
~

ˆ eifCP B 2td 2td ˆ ei…fCP B À2b† Y
jm12 j
jVtb Vtd j

…2X30†

with Àb~ ˆ arg V *tb Vtd . This phase depends on the CKM parametrization and is ±± like the
CP phase ±± not an observable. The arbitrariness cancels only in physical observables,
which include decay amplitudes with further CKM elements and a CP phase. The corresponding
E ˆ Ài

sin arg hm
1 ‡ cos arg hm

is purely imaginary, i.e. Re E ˆ 0 and therefore d ˆ 0. Within the same framework, for the
Bs aB"s system
hms ˆ ei…fCP Bs ‡2f2 † X

…2X31†

It must be emphasized, however, that there exist common final states for all four meson
pairs, and G 12 never vanishes completely, leaving always a small d, and also a small DG.
Within the Standard Model G 12 can be approximated by the absorptive part of the box
diagram, corresponding to a quark representation of the final states. This is a poor approximation to light hadronic final states which are dominating in the KaK" system, and may
still change the prediction for BaB" considerably. The box calculation yields [25]
G 12 % Àm12 Á

 4 !
3p
m2b

8 m2c Vcb V *cd
mc
1
‡
‡
o
3 m2b Vtb V *td
m4b
2S…m2t am2W † m2W

…2X32†

and DG and Dm have opposite signs. The ratio can be estimated using (2.18) to be
DG 2y
3p m2b
1
ˆ %À
X
$À
Dm
x
2 m2t
250

…2X33†

This ratio applies to both the B0 and Bs systems.
To leading order in G 12 am12 equation (2.20) yields
jhm j À 1 ˆ À

1
G 12
2p
m2c
Vcb V *cd
Im
%
Im
Y
2
2
2
m12 S…m2t amW † mW
Vtb V *td

…2X34†

from (2.32). This leads to a rough estimate of the convention-independent number
d % 1 À jhm j % À2p

m2c jVcb j jVcd j
sin b
sin b $ À
m2t jVtb j jVtd j
2000

…2X35†

with jdj ( 1 where b is the CKM unitarity angle in figure 2.1a. Since this result is based
on a leading order quark diagram, the number should be taken only as an order of magnitude. In particular, at this level of precision it can not be used to measure b.

724

R. Waldi, Prospects for the Measurement of the CP Asymmetry
Table 2.1
Parameters of the four neutral oscillating meson pairs [9]

description

K 0 aK"0

D0 aD"0

B0 aB"0

Bs aB"s

t [ps]
G [sÀ1 ]
y ˆ DGa2G

89.3 Æ 0.1; 51700 Æ 400
5.61 Á 109
À0X9966

0.415 Æ .004
2X4 Á 1012
jyj ` 0X08

1.54 Æ 0.03
1.49 Æ 0.06
…6X41 Æ 0X16† Á 1011 …6X2 Æ 0X4† Á 1011
jyj F 0.01
À0X05 F F F À 0X15*

Dm [sÀ1 Š
Dm [eV]
x ˆ DmaG

…5X30 Æ 0X02† Á 109
…3X49 Æ 0X01† Á 10À6
0X945 Æ 0X002

`2 Á 1011
`1X3 Á 10À4
`0.09

d
jhm j2

…3X27 Æ 0X12† Á 10À3
0X99348 Æ 0X00024

%1*

…4X65 Æ 0X19† Á 1011
…3X1 Æ 0X1† Á 10À4
0X72 Æ 0X03
$ À10À3 *

1 F F F 1X002*

b7X8 Á 1012
b5X1 Á 10À3
11 F F F 40*
jdj ` 10À3 *
%1*

* Standard Model expectation [29, 18]

2.2.2.

Behaviour of the Four Neutral Meson Anti-Meson Systems

All four meson pairs K 0 aK"0 , D0 aD"0 , B0 aB"0 , and Bs aB"s show a different oscillation behaviour, since they have all different relations of G, DG, and Dm. The same symbols will be
used for all four systems. Only when two specific systems shall be compared, their parameters will be distinguished by the subscripts K, D, d, and s, respectively. The dimensionless
parameters x and y give the ratios of time constants involved: t ˆ 1aG is the harmonic
average of the lifetimes, tosc ˆ 2paDm ˆ 2ptax is the period of the oscillation, and
trel ˆ 2aDG ˆ tay is the lifetime of the oscillation amplitude, i.e. the damping time constant of a relaxation process. Numerical values are summarized in table 2.1.
While the parameters of the K 0 aK"0 system are well measured [9], theoretical assumptions enter into the B meson columns. Many precise lifetime measurements for neutral B
mesons have become available last year. All lifetime measurements are summarized in
table 2.2, and average to td ˆ …1X54 Æ 0X03† ps.
Table 2.2
B0 lifetime measurements. Measurements which have been replaced by more recent ones are
not included
td [ps]

experiment

1X53 Æ 0X12 Æ 0X08

1X61 Æ0X14
0X13 Æ 0X08
1X63 Æ 0X14 Æ 0X13
1X532 Æ 0X041 Æ 0X040
1X55 Æ 0X06 Æ 0X03
1X52 Æ 0X06 Æ 0X04
1X64 Æ 0X08 Æ 0X08
1X58 Æ 0X09 Æ 0X02
1X474 Æ 0X039Æ0X052
0X051

OPAL 95 [30]
DELPHI 95 [31]
DELPHI 95 [32]
DELPHI 96 [33]
ALEPH 96 [34]
L3 98 [35]
SLD 97 [36]
CDF 96 [37]
CDF 98 [38]

1X54 Æ 0X03

average


Fortschr. Phys. 47 (1999) 7±±8

725

Figures 2.2±±2.5 show the number of mesons and anti-mesons as a function of the scaling lifetime variable T ˆ tat and the asymmetry

2
2
• 3 X† À N…X
• 3 X†
N…X
 ˆ …1 À jhm j † cosh yT ‡ …1 ‡ jhm j † cos xT
a…T† ˆ
• 3 X† ‡ N…X
• 3 X†T …1 ‡ jh j2 † cosh yT ‡ …1 À jh j2 † cos xT
N…X
m
m
…2X36†
for a meson produced at T ˆ 0 as a flavour eigenstate X, and decaying to a flavour-specific
final state as X or X at a later time T. Expressed via the small real parameter d instead of
jhm j this reads
a…T† ˆ

cos xT ‡ d cosh yT
X
cosh yT ‡ d cos xT

…2X37†

For an anti-meson produced at T ˆ 0 as a flavour eigenstate X, and decaying to a flavourspecific final state as X or X at time T, we obtain a similar expression, where only the
cos xT part changes sign:

• 3 X†

• 3 X† À N…X
N…X
 ˆ À cos xT À d cosh yT X
a"…T† ˆ
• 3 X†T
• 3 X† ‡ N…X
cosh yT À d cos xT
N…X
The approximation jhm j ˆ 1 corresponding to d ˆ 0 leads to a simpler expression
a…T† ˆ

cos xT
ˆ À"
a…T† Y
cosh yT

…2X38†

where x is clearly seen as the oscillation parameter, and y as the damping parameter.
The kaon has both x % 1 and y % À1, i.e. the long-living state is the heavier mass eigenstate. With these parameters one half of a sample of kaons of either flavour decays rapidly,
mainly into two pions with CP ˆ ‡1, and the other half transforms to a sample of the
long-living KL0 states, which decay (aside from the small CP violation) to CP ˆ À1 eigenstates and to flavour-specific states. The ratio of lifetimes of the two states (table 2.1) is
approximately 580. The time evolution of an initially pure K 0 flavour eigenstate is shown
in figure 2.2. The upper diagram shows the number of remaining K 0 and K"0 after a scaled
time T ˆ Gt, where G % G S a2 ˆ 1a…2tS † is the average width of the short- and long-living
state. The decay rate into flavour-specific final states is proportional to these numbers,
while the dominant decays to CP eigenstates follow different evolution functions due to CP
violation, and will be discussed below.
The D0 meson decays mainly to flavour specific states with well defined strangeness,
" KL0 p0 and CP ˆ À1 states, as

with only a few decays to CP ˆ ‡1 eigenstates, as pp, K K,
KS0 p0 or KS0 w. This leads to equal lifetimes for the two eigenstates, i.e. y % 0. The corresponding box graph has a b quark as the heaviest particle in the loop, which is accompanied by
the small CKM elements Vcb and Vub . The mass difference induced that way by the Standard
Model is very small, corresponding to x F 0.002. Therefore, almost no asymmetry is visible in
figure 2.3, although the number x ˆ 0X02 used for the plot is a factor 10 higher. The value
x ˆ 0X002 corresponds to a total mixed fraction of initially pure D0 states given by
cˆ

ND"0 3X
x2
ˆ
ND0 3X ‡ ND"0 3X 2…1 ‡ x2 †

as c % 2 Á 10À6 .

…2X39†


726

R. Waldi, Prospects for the Measurement of the CP Asymmetry

Fig. 2.2: K 0 aK"0 mixing is determined by
the parameters x ˆ 0X95, y ˆ 0X996, and
jhm j2 ˆ 0X994. T ˆ ta"
t is the lifetime in
units of t" % 2tS , the inverse of the average
width of KL0 and KS0 . The upper diagram
shows the number of K 0 (solid) and K"0
(dotted) as a function of T for a sample

starting with 100% K 0 mesons. The lower
diagram
shows
the
asymmetry
a ˆ …NK À NK" †a…NK ‡ NK" †. The relaxation
process soon dominates, leaving only KL0
after not much more than one oscillation.

The parameters of the B0 aB"0 system have been introduced above. A good approximation
is y ˆ 0 and d ˆ 0, which leads for N0 pure B0 at t ˆ 0 to
1
N0 eÀT …1 ‡ cos xT† Y
2
1
NB"0 …T† ˆ N0 eÀT …1 À cos xT†
2
NB0 …T† ˆ

…2X40†

as shown in figure 2.4.. The decay rate for flavour-specific final states (which are the majority of B0 decays) follows the same time evolution. The asymmetry function is simply
a…T† ˆ cos xT X

…2X41†

This asymmetry can be observed using a flavour-tagging decay, like B0 3 DÀ l‡ n. The rate
of mesons decaying at time T into the channel X are given by (2.27) where y ˆ 1 makes

727

Fortschr. Phys. 47 (1999) 7±±8

Fig. 2.3: D0 aD"0 oscillations have not yet
been observed, and will be hardly visible
even with x ˆ 0X02, which is about 10 times
the expected value. The other parameters in
this plot are y ˆ 0 and jhm j ˆ 1.

cosh yT ˆ 1 leading to the same asymmetry function a…T† ˆ cos xT. Integrating over all
times, the observed numbers are
…

1
NB0 3X ˆ N• B0 3X …T† dt ˆ N0
2
…
1
NB"0 3X ˆ N• B"0 3X …T† dt ˆ N0
2
Their asymmetry becomes
aint ˆ

NB"0 3X À NB0 3X
1
ˆ
NB"0 3X ‡ NB0 3X 1 ‡ x2

G X 2 ‡ x2

Y
G 1 ‡ x2
G X x2
X
G 1 ‡ x2


728

R. Waldi, Prospects for the Measurement of the CP Asymmetry

Fig. 2.4: B0 aB"0 evolution is dominated by
the oscillating part, with the parameters
x ˆ 0X70, y ˆ 0, and jhm j ˆ 1. The ratio of
the areas under the dotted and solid curve in
the upper plot is the mixing probability c.
The zero transition in the asymmetry, which
marks the crossover point in the upper plot,
p
.
is at T ˆ 2x

and the mixing probability is as in (2.39)
cˆ

NB"0 3X
x2
X
ˆ
NB0 3X ‡ NB"0 3X 2…1 ‡ x2 †

…2X42†

It was this net effect which gave the first proof for a sizeable mixing parameter x % 0X7 in
the B0 meson system in 1987 [39]. The time-dependent particle anti-particle oscillations of
the neutral B meson have been first seen six years later by experiments at LEP [40]. With
x % 0X7, about one period is visible before most of the mesons are decayed.
If we assume the Standard Model predictions to be true, the Bs meson is a very interesting case. There will be a small y and a very large x. Figure 2.5 is plotted with xs ˆ 15,
which is close to the lower limit of the theoretical range. The time-integrated mixing prob-


729

Fortschr. Phys. 47 (1999) 7±±8

Fig. 2.5: Bs aB"s is expected to be the most
rapidly oscillating system, with a longer relaxation time. This plot assumes xs ˆ 15,
ys ˆ 0X10, and jhm j ˆ 1.

ability is for jhm j ˆ 1
cˆ

x 2 ‡ y2
X
2…1 ‡ x2 †

For large xs ) 1, this approaches its maximum value of 0X5, where a measurement of this
quantity has no sensitivity on x any more. To observe the rapid oscillations, a very good
lifetime resolution will be required. Experimentally, a lower limit xs b 18 has been found at
LEP (see below).

In the general case jhm j Tˆ 1, the integrated mixing probability depends on the initial
flavour. It is
cˆ

jhm j2 …x2 ‡ y2 †
2

2

2 ‡ x2 …1 ‡ jhm j † À y2 …1 À jhm j †

ˆ

…1 À d† …x2 ‡ y2 †
2‰1 ‡ x2 ‡ d…1 À y2 †Š

…2X43a†


730

R. Waldi, Prospects for the Measurement of the CP Asymmetry

for an initial B and
c" ˆ

…x2 ‡ y2 †
2

2 jhm j ‡

x2 …1

2

‡ jhm j † ‡

y2 …1

2

À jhm j †

ˆ

…1 ‡ d† …x2 ‡ y2 †
2‰1 ‡ x2 À d…1 À y2 †Š

…2X43b†

for an initial B" (which is c with jhm j replaced by 1ajhm j or d by Àd). This exhibits already
CP violation, since the probabilities P…X 3 X† and P…X 3 X† are different. It is also T
violation, since the transition X 3 X is the time reversed process X 3 X.
2.2.3.

CP Eigenstates Versus Mass Eigenstates

The following discussion will again use B0 aB"0 as an example, but is applicable to each of
the four systems accordingly.
The standard phase convention requires all J PC ˆ 0À‡ mesons to have CP jXi ˆ ÀjXi,

fixing fCP B ˆ p. Independent of any convention, two orthogonal CP eigenstates
1
jB0‡ i ˆ p …jB0 i ‡ CP jB0 i† Y
2

1
jB0À i ˆ p …jB0 i À CP jB0 i† X
2

…2X44†

with CP jB0‡ i ˆ jB0‡ i and CP jB0À i ˆ ÀjB0À i can be defined. If a state agrees with one of
these except for a phase factor, it will be a CP eigenstate.
The mass eigenstates of the B0 aB"0 system are not CP eigenstates. Using
CP jB0 i ˆ eifCP B jB"0 i, they are transformed by a CP operation as
 if

 if

e CP B eÀifCP B hm
e CP B eÀifCP B hm
‡
À
CP jBL i ˆ
jBL i À
jBH i
2hm
2
2hm
2

% cos 2b~ jBL i À sin 2b~ jBH i
 if

 if

e CP B eÀifCP B hm
e CP B eÀifCP B hm
‡
À
CP jBH i ˆ À
jBH i ‡
jBL i
2 hm
2
2 hm
2
% Àcos 2b~ jBH i ‡ sin 2b~ jBL i X
Where the approximation for the B0 aB"0 system depends on the phase convention for the CKM
~ If b~ ˆ 0 is chosen by an appropriate phase redefinition
matrix which determines the angle b.
e.g. of the b field, these states would be eigenstates with CP ˆ Æ1, respectively. Still, there
would be CP violation in their decay, and the CP eigenvalue of the final state would be different. Therefore, the question of which of the mass eigenstates is closest to which CP eigenstate
has no convention independent answer. Only the CP eigenvalue of a decay product of one of
these states is an observable, and the weak interaction does not conserve CP.
A meaningful question is which of BH or BL decays more often into CP ˆ Æ1 eigenstates. In contrast to the neutral kaon system, most final states from B decays are flavourspecific, and both mass eigenstates decay into them via either their B0 or their B"0 component. The small fraction of states that can be reached both by B0 and B"0 includes the
contribution from CP eigenstates which appear mainly through three processes. On the tree
level, there are two main decay channels that can produce CP eigenstates: b 3 c"
cd with the
" A state of the first kind will
c"

cd d" final state, and b 3 u"
ud with the quark content u"
ud d.
have decay amplitudes
* Vcd A0 Y
A ˆ hXc"c j H jB0 i ˆ V cb

A ˆ hXc"c j H jB"0 i ˆ hX eÀifCP B Vcb V *cd A0 Y


731

Fortschr. Phys. 47 (1999) 7±±8

where hX ˆ Æ1 is the CP eigenvalue of the state. The corresponding decay amplitudes of
BH and BL are
ALY H ˆ hXc"c j H jBLY H i ˆ pA Æ qA


*
ÀifCP B Vcb V cd
ˆ pA 1 Æ hm hX e
V *cb Vcd
ˆ pA…1 Æ jhm j hX eÀ2ib † X
The decay ratio is then (in the approximation jhm j ˆ 1)
jAH j2
jAL j

2

ˆ

j1 À hX eÀ2ib j2
j1 ‡ hX

eÀ2ib j2

ˆ

1 À hX cos 2b
Y
1 ‡ hX cos 2b

…2X45†

which is for b ` p4 less than 1 for hX ˆ ‡1 and vice versa. In this case, the heavier state
BH will decay more often into states with negative CP eigenvalue, hX ˆ À1.
Accordingly, for the u"
ud d" states
À
Á
ALY H ˆ hXu"u j H jBLY H i ˆ pA 1 Æ jhm j hX e2ia
and the ratio
jAH j2
jAL j

2

ˆ

j1 À hX e2ia j2
j1 ‡ hX

e2ia j2

ˆ

1 À hX cos 2a
1 ‡ hX cos 2a

…2X46†

depends on the angle a, which is likely to be larger than p4 . This would give the opposite
answer, i.e. the heavier state BH will decay more often into states with positive CP eigenvalue, hX ˆ ‡1.
" proceed into K 0 or K"0, which
Some decays with an intermediate state c"
cd"
s or c"
cds
0
0
"
finally result in c"
cd d via a KL or KS sequential decay. Among those is the gold-plated
decay B0 3 Jaw KS0. The total decay chain involves almost the same CKM element phase
factors as the direct b 3 c"
cd decay, leading to the same answers as for this decay mode (a
more detailed discussion follows below in section 2.3).
For decays via W exchange, like bd" 3 c"
c or bd" 3 u"

u, the same CKM elements are
involved, and the same arguments lead to the same answers as above. Also, the favoured
penguin-type transition b 3 s with subsequent hadronization into a KL0 or KS0 has a net
phase close to bH leading to the ratio (2.45).
CP eigenstates with quark content dd" can be reached via CKM-suppressed penguin-type
loops. Due to the top quark dominance the amplitudes are
* Vtd A0 Y
A ˆ hXdd"j H jB0 i % V tb

A ˆ hXdd"j H jB"0 i % hX eÀifCP B Vtb V *td A0

and the CKM element phases cancel, which gives
jAH j2
jAL j

2

ˆ

1 À hX
1 ‡ hX

i.e. BH decays exclusively into states with negative CP eigenvalue, hX ˆ À1, and BL into
states with hX ˆ ‡1.
All these results receive corrections from non-leading terms, like c quark loops in the last
case, or b 3 d penguin corrections to the b 3 u transition final states. The general case for