

Hera-B 00-012

Physics 00-002



Present knowledge of the Cabibbo-
Kobayashi-Maskawa matrix



M. B (1) (1) (1) (1) (1)
ARGIOTTI , A. BERTIN , M. BRUSCHI , M. CAPPONI , S. DE CASTRO ,
R. D (1) (1) (1) (1) (1)
ON , P. FACCIOLI , D. GALLI , B. GIACOBBE , U. MARCONI ,
I. M (1) (1) (2) (1) (1)
ASSA , M. PICCININI , M. POLI , N. SEMPRINI CESARI , R. SPIGHI ,
V. V (1) (1) (1) (1) (1)
AGNONI , S. VECCHI , M. VILLA , A. VITALE AND A. ZOCCOLI

(1) Dipartimento di Fisica dell'Universit di Bologna  Bologna, Italy
Istituto Nazionale di Fisica Nucleare Sezione di Bologna  Bologna, Italy

(2) Dipartimento di Energetica `Sergio Stecco' dell'Universit di Firenze  Firenze, Italy
Istituto Nazionale di Fisica Nucleare Sezione di Bologna  Bologna, Italy




Abstract

A complete review of the Cabibbo-Kobayashi-Maskawa (CKM) matrix
elements and of the experimental methods for their determination is presented.
A critical analysis of the relevant experimental results, and in particular of the
most recent ones, allows to improve the accuracies of all the matrix elements.
A 2 minimization with the three-family unitarity constraint on the CKM
matrix is performed to test the current interpretation of the CP violating
phenomena inside the Standard Model. A complete and unambiguous solution
satisfying all the imposed constraints is found. As a by-product of the fit, the
precision on the values of the matrix elements is further increased and it is
possible to obtain estimates for the important CP violation observables sin2,
sin2 and . Finally, an independent estimation of the CKM elements based
on a Bayesian approach is performed. This complementary method constitutes
a check of the results obtained, providing also the probability functions of the
CKM elements and of the related quantities.

PACS 12.15.Hh  Determination of CKM matrix elements.


Published on La Rivista del Nuovo Cimento, 23 N3 (2000), 1.




1 Introduction ............................................................................................................ 3

1.1 The CKM matrix................................................................................................ 4

1.2 The unitarity triangle......................................................................................... 6

2 Experimental determination of the CKM matrix elements................................... 8

2.1 |Vud|...................................................................................................................9
2.1.1 Super-allowed nuclear decays .................................................................... 9
2.1.2 Neutron
decay ......................................................................................... 11
2.1.3 Pion
decay............................................................................................15
2.1.4 Summary of |Vud| determinations ............................................................. 16

2.2 |Vus| ................................................................................................................. 17
2.2.1 Kaon semileptonic decays........................................................................ 17
2.2.2 Hyperon semileptonic decays................................................................... 19

2.3 |Vcd| and |Vcs|................................................................................................... 20
2.3.1 |Vcd| ......................................................................................................... 20
2.3.2 |Vcs| from neutrino charm production ....................................................... 21
2.3.3 |Vcs| from semileptonic decays of D mesons............................................. 22
2.3.4 |Vcs| from W  boson decays ..................................................................... 23
2.3.5 Determination of the best values of |Vcd| and |Vcs| .................................... 24

2.4 |Vub|................................................................................................................. 25

2.5 |Vcb|................................................................................................................. 26
2.5.1 Exclusive
decays ..................................................................................... 26
2.5.2 Inclusive
decays ...................................................................................... 28

2.6 |Vub|/|Vcb|......................................................................................................... 29

2.7 Elements of the third row................................................................................. 31
2.7.1 Experimental determination of f .......................................................... 33
Ds
2.7.2 Lattice QCD results ................................................................................. 35
2.7.3 Measurements of m and m ........................................................... 38
B B
d s

2.7.4 CP violation in the neutral kaon system; ...................................... 40
K K

2.7.5 The
constraint .................................................................................43
K

2.7.6 b s penguin decays............................................................................45
2.8 Summary ......................................................................................................... 47

3 An up-to-date profile of the unitarity triangle ..................................................... 48

3.1 Description of the fit procedure ....................................................................... 48

3.2 Constraints on the unitarity triangle in Wolfenstein's parametrization. ........... 52

3.3 Results............................................................................................................. 54

3.4 Bayesian determination of the CKM matrix ..................................................... 61

3.5 New prospects ................................................................................................. 68

Conclusions ................................................................................................................... 74

References ..................................................................................................................... 76




2


1 Introduction

One of the most interesting and mysterious aspects of nature is its behaviour under
Charge-Parity (CP) transformation. For more than 35 years after the discovery of CP
violation in the neutral kaon sector, no conclusive evidence of violation of this symmetry
has been found in other phenomena, but a huge experimental effort is being made in this
direction. The main reason for the interest in this field is related to the question whether
the Standard Model is able to account for the observed magnitude of these phenomena.
CP violation is introduced, in a general way, into the charged-current weak interactions
between quarks existing in at least three different families. In the Standard Model, it
originates from an unremovable complex phase in the Cabibbo-Kobayashi-Maskawa1,2
(CKM) matrix, which describes the flavour mixing between the quark states. With this
formulation, it is possible to foresee many different CP violating phenomena, which will
be a matter of study in the current (as well as in the next) generation of B factories. In
order to have reliable estimations for these phenomena, as well as to test this sector of
the Standard Model, a precise knowledge of all the matrix elements is necessary.
The values of the CKM matrix elements are not fixed by theory and must be evaluated
on the basis of the available experimental information. This can be performed by
studying specific classes of processes, such as, for example, the semileptonic decays of
mesons or baryons, and by exploiting the relations between the experimental
determinations of decay rates or branching ratios and the relevant matrix elements. The
theoretical uncertainties in the hadronic system description usually limit the final
precision obtained in the determination of the CKM matrix. Notwithstanding this
difficulty, which in some cases makes the related experimental information even
unserviceable, all the CKM matrix elements of the first two rows can be safely
determined.
The requirement that the CKM matrix be a unitary matrix imposes strong constraints
between its elements. This characteristic allows one to test the goodness of the values
found and of the assumptions made in the Standard Model in order to account for the
observed CP violation.

The main purposes of this work are a survey of the CKM matrix elements as
determined one by one from the experimental data and the refinement of their values by
means of a unitarity-constrained minimum-2 fit. The main results of this study already
appeared in Refs. 3 and 4. The impact of new measurements foreseen in the next years
is also emphasized.

In the following paragraphs, a short introduction to the CKM matrix and the
unitarity triangle is presented. In Section 2, the direct measurements of the CKM matrix
elements and other experimental constraints are discussed taking into account the most
recent experimental, theoretical and phenomenological achievements. In Section 3, we
present the methodology used to obtain the maximum precision for the CKM parameters
from the results presented in Section 2, through the requirement that the CKM matrix be
a 33 unitary matrix. Finally, the determination of the imaginary part of the matrix and
the best estimation of the CP violating observables are presented.





3


1.1 The CKM matrix


In the Standard Model, the quark mass eigenstates (physical states) do not take part
as pure states in weak interactions. The unitary transformation connecting the two bases
of mass and weak eigenstates is represented by the Cabibbo-Kobayashi-Maskawa (CKM)
matrix. By convention, the charge +2/3 quarks (u, c and t) are chosen to be pure states,
and flavour mixing is described in terms of a 33 matrix operating on the d, s and b
quark states:

d
d V V V
ud us ub d
s = V (1. 1-1)
CKM s = V V V
cd cs cb s


b b V V V
td ts tb b
Thus d', s' and b', instead of d, s and b, are partners of u, c and t respectively within
the weak isospin doublets.
All the properties of weak quark interaction are codified inside the CKM matrix, which
makes it possible to extend the Cabibbo1 model, preserving the weak coupling
universality while explaining the existing priority scale among the transitions occurring
inside one quark family and those connecting two neighbouring families, or the first with
the third one. Its structure incorporates the GIM5 mechanism, which suppresses the
flavour-changing neutral-current (FCNC) processes. Finally, the imaginary part of the
CKM matrix is the source of all the CP-violating phenomena which the Standard Model
is able to account for.
CP symmetry, which is equivalent, according to the CPT theorem, to time reversal
invariance T, would be conserved if the matrix were real. On the other hand, to account
for CP violation V must be complex independently of the phase convention of the
CKM

fermionic fields, which it acts on. The Lagrangian density term which represents the
( 
W -mediated) charge current processes involving quarks depends on V according to
CKM

the expression
d


/
 G -  + (1. 1-2)
F (u c t )
W (1 5)V s W .
h .
c
CKM
b

Let us consider initially the most general case. As an NN unitary matrix, V depends
CKM
2
on N real parameters, 2N - 1 of which can be reabsorbed by the quark fields which
multiply the right and left sides of the matrix, by means of a global redefinition of the
arbitrary phases. Of the remaining (N - 1)2 free parameters, N (N - 1)/ 2 are the Euler
angles, common to the real (orthogonal) matrix and to the complex (unitary) one, the
others are unremovable complex phases; they are therefore physically meaningful and
can be measured as signals of CP violation. The N = 2 matrix, which contains only one
parameter (the Cabibbo angle), is real; consequently, it cannot give rise to CP violation.
This is precisely the reason why Kobayashi and Maskawa3 conjectured the existence of
three quark doublets in 1973. The authors presented the hypothesis as one of several
possible explanations, which were able to account for CP non-conservation with a
minimal extension of the two-generation model. But the idea began to be held in high
consideration when Perl and collab. 6 discovered the third lepton () in 1975, following




4


up the original intuition and preceding experience of Zichichi and collab.7, and after the
discovery of the fifth quark (b) at Fermilab8 in 1977.
The CKM matrix (N = 3) can be parametrized in terms of three Euler angles and one
phase; the latter is entirely responsible for the CP violation in the Standard Model.
Several possible parametrizations differ both in the choice of the Euler angles and in the
positioning of the phases; the one proposed by Chau and Keung9 combining notations
already used by Maiani10 and Wolfenstein11 is adopted by the PDG12 as the `canonical'
parametrization:
-i
13
c c s c s e
12 13 12 13 13

V (1. 1-3)
CKM = - s c - i i
13
c s s e c c - 13
s s s e s c
12 23 12 23 13 12 23 12 23 13 23 13
i i
13 13
s s c c s e c s s c s e c c
12 23 - 12 23 13 - 12 23 - 12 23 13 23 13
where
c = cos , s = sin , is the mixing angle between the ith and the jth
ij ij ij ij ij

generation ( is the Cabibbo angle), and is the phase angle.
12 13

Given the experimentally observed hierarchy among couplings
V V V V , (1. 1-4)
ub < cb < ,
us cd < 1

the matrix elements can be written in terms of powers of the sine of the Cabibbo angle
= s V V .


(1. 1-5)
us cd .
0 22
12

If one defines the parameters A, , according to the relations
2 -
s = A , s e i 13 = A3 (1. 1-6)
13 ( -i)
23

and neglects, for example, terms of order ( 6
), the following expression
2 4
1- - 3

A ( - i)

2 8
2 4
V
CKM = 2 4 1 2 4
- 1-
A - + iA 1- -
(1+4 2A)
2
A
2 2 8
2 2 4
3

A ( 2 A
1- ) 2 2 1 2
+ -
i 1- - A 1- - +
i 1-

2 2 2 2

(1. 1-7)
is obtained. This parametrization, formulated by Wolfenstein11, has the advantage of
making the characters which differentiate the CKM matrix from a common unitary
matrix particularly evident: it is almost diagonal (the diagonal elements are close to
unity) and its elements decrease in magnitude with increasing distance from the

diagonal, according to a nearly symmetrical pattern ( V V at the first non-zero
ji ij

order in ). The second part of Eq. (1.1-6) yields the following relation (for 0)

tan =
(1. 1-8)
13
between the phase and the parameters and : therefore 0 is the CP symmetry-
breaking condition in the Standard Model, as expressed in Wolfenstein's
parametrization.







5


1.2 The unitarity triangle


Among the orthonormality relations of the row-vectors and those of the column-vectors
of the CKM matrix (9 independent relations in all),
2 2
V + 2
V + V = 1 i = u, c,t (1. 2-1)
id is ib

V V V V V V (1. 2-2)
ud us +
cd cs +
td ts = 0

V V V V V V (1. 2-3)
ud ub +
cd cb +
td tb = 0


(the first provides a way of testing the unitarity condition, with reference to direct
measurements of the elements of the same row), Eq. (1. 2-3) is of particular interest, as
it defines on the complex plane a triangle whose sides have dimensions of the same
order in ( ( 3
)) and thus subtend angles having comparable amplitudes. The length of
one of the sides can be normalised to the real value 1: dividing Eq. (1. 2-3) by
V V ,
cd cb

one obtains the so-called unitarity triangle, defined by the relation

1+ V V V V
td tb + ud ub = 0
, (1. 2-4)
V V V V
cd cb cd cb

and represented in Figure 1, where the angles are indicated, according to a widespread
convention, by the letters , , .

(,)
* V V *
V V td tb
ud ub
* V V *
V V cd cb
cd cb

(0,0) (1,0)
Figure 1.  The unitarity triangle


The vertex of the triangle is the complex vector

1+ V V
td tb = ( + i) 2

1- +
( 4). (1.2-5)
V V
cd cb 2
1
The correction of order 4
is equal to 4 -4
f (,, )
A 310 ; thus
8
2
2

= 1- and = 1- (1. 2-6)
2
2
are, almost exactly, the coordinates of the vertex in the complex plane. The angle is
equal to the phase of the canonical parametrization:
13

= arctan = arctan = . (1.2-7)
13




6


, and are in direct relation to the CP asymmetries which are peculiar to B meson
2
decays. Forthcoming experiments will measure sin2, sin2 and sin or sin2. The first
two observables can be expressed as functions of and in the following way:
2 ( 2 2
+ - ) 2 (1- )
sin 2 = ( ; sin 2 = . (1. 2-8)
2
+ - )2
2 2
+ (1- )2 2
+
The CP asymmetry which should be measured in the decays 0 0 0
B , B J K is defined
d d S

as
-
CP N ( 0 0
B J K N B J K
d S ) ( 0 0
d S )
a = (1. 2-9)
0
J KS N ( 0 0
B J K + N B J K
d S ) ( 0 0
d S )
(N is the number of events in a given proper time interval) and is proportional to sin2.
A very preliminary measurement of this asymmetry has been performed recently by
CDF13, but given the lack of precision it cannot be interpreted as a definitive proof of
CP violation in the b quark sector. New, precise measurements of this quantity are
expected in the forthcoming years from dedicated experiments like BaBar, Belle and
HERA-B, which should start their physics data taking since the beginning of the year
2000. Moreover, the upgraded version of the CDF detector should provide new results
from the beginning of 2001. Finally, the LHCb experiment should be operative from
year 2006. These experiments will be active at different machines ( + -
e e colliders, pN
fixed target, p
p and pp colliders) and with different experimental techniques. This will

lead to independent determinations of the angles of the unitarity triangle. The attainable
precision will also depend on the technique used and on the actual value of these
quantities.
























7


2 Experimental determination of the CKM matrix
elements

As it will become explicit in the following Subsections, the possibility of carrying
out independent measurements of the CKM matrix elements and other observables
connected with weak decay amplitudes is due to the great precision with which the
Fermi constant G was measured.
F

The measurement of the muon lifetime, by which G is determined, has been the
F

subject of nearly half a century of experimental inquiries14, which began in the early
Forties. The first accurate measurement was carried out at CERN15 in 1962 and, for the
first time, an inconsistency with the coupling constant of the nuclear transition
14 O14N + +
e + clearly emerged. One year later this evidence led Cabibbo1 to a new
e

formulation of the weak-coupling universality principle. The present level of precision
was achieved in 1984 by the Saclay-CERN-Bologna16 and TRIUMPH17 groups. The SCB
group also provided one of the most meaningful tests of CPT invariance through a
determination of the ratio
+ - . From the world average of these measurements, the
 

following value of G is obtained12 taking into account the radiative corrections:
F

G (*). (2-1)
F = ( - -
.
1 16639  ) 5 2
00001
.
0 10 GeV

Sections 2. 1 to 2. 6 describe the direct measurements of the CKM matrix elements of
the first two rows. In most cases, the main source of uncertainty in the values obtained
is the theoretical interpretation of the experimental results. In fact, detailed calculations
of hadronic terms are generally needed to relate the measured quantities to the CKM
elements. Moreover, different theoretical approaches do not always give univocal
predictions. For example, in the recent investigations of rare B decays the data analysis
is strongly dependent on the model used. The problems of theoretical interpretation are
not restricted to the latest results: the extraction of V from the hyperon decay
us

measurements, for example, still cannot rely on a solid agreement between independent
theoretical calculations. As a further example, the lack of consistency between the very
precise V value obtained from the analysis of nuclear decays and the unitarity of the
ud

CKM matrix has led to the formulation of a number of (not yet fully established)
hypotheses which can explain the discrepancy with the expected value by predicting
additional nuclear structure effects. The discussion of these aspects is essential in
obtaining reliable estimations of the CKM matrix elements.
From a different point of view, the main interest in the measurement of the
observables which are related to the elements of the third row (B meson mixing, CP
violation parameters) is the chance of revealing the existence of new physics, since the
hypotheses involved are subject to a wide range of possible modifications in the
extensions of the Standard Model. The information available on the elements of the third
row will be collectively discussed in section 2. 7, since all the measurements give
combinations rather than single CKM elements.



* Units = c = 1 are used here and throughout the text.


8


2.1 |Vud|

Super-allowed nuclear transitions, neutron decay and pion decay are the
experimental sources which provide basic information for the determination of V .
ud

Several difficulties are encountered in the theoretical description of these processes when
tracing back from the hadron to the quark level (u d) transition. The result obtained
from the study of nuclear decays is sufficient to make V the CKM matrix element
ud

known with the greatest precision; nevertheless, additional nuclear corrections, which
have been proposed in order to account for a 2 incompatibility with the value expected
assuming unitarity, may still have to be applied. The deduction of V from neutron
ud

decay has to deal with the historical inconsistency between the independent
measurements of the axial coupling constant as well as of the neutron mean-life. Pion
decay provides the cleanest way of extracting V , but a suitable level of precision has
ud

still to be attained through a refined measurement of the decay rate.


2.1.1 Super-allowed nuclear decays

The most precise and the largest amount of experimental information on V comes
ud
from the analysis of super-allowed nuclear decays, in which the spin-parity P +
J = 0
and isospin of the nuclei are preserved. Due to the energies released in these processes,
the so-called allowed approximation assumes that the emitted leptons have zero relative
angular momentum. Under this hypothesis, the + +
0 0 nuclear decays can be simply
described considering only the vector part of the weak interaction (Fermi transitions).
Moreover, the nucleus involved in a transition occurring within an isospin multiplet is
labelled with the same quantum numbers in the final state as in the initial one; then,
assuming an exact isospin symmetry and that the other nucleons behave as spectators,
the initial and final nuclear states are completely superimposed, so that the nuclear
matrix element can be assumed independent of any detail of the nuclear structure and is
given by a simple isospin coefficient. In this way the experimental values of ft  the
product of the phase space factor f, depending on the atomic number and the end-point
energy, and the half life t of the specific process  are expected to be nearly the same in
all the super-allowed transitions.
For the super-allowed transitions occurring within isospin T= 1 multiplets, ft is defined
by
= K
ft = K
2
(2. 1-1)
2 2 2
G M 2G V
F ud
V fi

2
where
M is the matrix element, G the vector coupling constant, which depends
fi = 2 V

on the V matrix element, G = G V , and the constant K is
ud V F ud

- -
K 3
= 2 ln 2 m5 10 4
= 271
.
8120  .
0 012 10 . (2. 1-2)
e ( ) GeV s





9


A more accurate description of these processes, which includes the small (of the
order of 1%) and calculable contribution of electromagnetic effects and radiative
corrections, leads to the following expression:
t ( K
ft 1 + 1- = (2. 1-3)
R )( C ) 2
2
G
2 V 1+
F ud ( R )
where and are the nucleus-dependent and nucleus-independent radiative
R R

2
corrections and is the charge-dependent correction to M due to isospin
C fi

symmetry breaking.
The nucleus independence of the t value and its direct correspondence to the V
ud

matrix element are two important theoretical aspects which make the study of super-
allowed nuclear transitions particularly useful for the determination of V . From an
ud

experimental point of view, the study of these reactions is simpler than that of neutron or
pion decays and more precise measurements can be obtained. The ft value of a super-
allowed transition A A
X + Y + + e
+ is determined by measuring the total half-life,
e
0 0

the branching fraction (R) and the end-point energy of the specific process, which can be
clearly identified by measuring the relative yield and the energy of the monochromatic

rays emitted in the Y nuclear de-excitations.
The averaged experimental results of nine super-allowed decays, including the
recent data for 10 C (particularly relevant as it is close to the Z= 0 limit where the effects
of the nuclear structure vanish) have been collected and updated by Towner and Hardy18
(see Table I). The radiative and nuclear structure corrections have been evaluated taking
into account the results of several independent calculations19. As expected, the corrected
ft-values are in fact mutually compatible. Their weighted average
t = 3072.3  0.9  1.1 = 3072.3  2.0 s (2. 1-4)
(where the second error is due to the systematic difference between two independent
computations of and has been added linearly to the statistical error) leads, with
C
= (2.40  0.08) 10-2,
(2. 1-5)
R

to the following measurement of V :
ud

V = 0.9740  0.00014
ud exp  0.00048 th = 0.9740  0.0005 (2.1-6)
(the second error represents the uncertainty on the radiative corrections).

In spite of its precision, the value of V obtained from nuclear decays has been
ud

often criticized due to the inconsistency it presents with the unitarity of the 33 CKM
matrix. Combining the result 2. 1-6 with the best values of V (2.2-16) and V (2. 4-
us ub

4), in fact, a deviation of 1. 9 from unity is found:
2 2 2
V V V . (2. 1-7)
ud + us + ub = .
0 9971 0015
.
0

This result may indicate that the value 2. 1-6 is underestimated. Various attempts have
been made to explain this inconsistency by including additional corrective factors.
On pure phenomenological grounds, Wilkinson20 admitted a residual dependence of t
on Z and tried to extract it from the experimental data. After extrapolating t to Z = 0
the value V = 0.9746  0.0006 was obtained.
ud





10


Saito and Thomas21 proposed a physical model of the infinite nuclear matter, in which
the influence of nuclear binding, mediated by meson exchange, is studied in terms of the
quark degrees of freedom. A non-negligible charge-symmetry violating effect was found
to involve the quarks inside bound nucleons, resulting in an additional correction to the
ft value, which should raise V by 0. 06 to 0. 08%,
ud

V = 0.9747  0.0006, (2. 1-8)
ud
lowering the discrepancy with the unitarity condition to 1 .
Since none of the model proposed have been clearly established yet, they are not applied
directly neither in Ref. 18 nor by the PDG12. However, we follow the PDG in enlarging
the uncertainty in the determination of V :
ud

V = 0.974  0.001. (2. 1-9)
ud nuclear

Q t1 2 R P t
EC ft
Z EC
(keV) (ms) (%) (%) (s) (s)
10 C 5 1907.77(9) 19290(12) 1.4645(19) 0.296 3038.7(45) 3072.9(48)
14 O 7 2830.51(22) 70603(18) 99.336(10) 0.087 3038.1(18) 3069.7(26)
26m Al 12 4232.42(35) 6344.9(19) 99.97 0.083 3035.8(17) 3070.0(21)
34 Cl 16 5491.71(22) 1525.76(88) 99.988 0.078 3048.4(19) 3070.1(24)
38m K 18 6044.34(12) 923.95(64) 99.998 0.082 3049.5(21) 3071.1(27)
42 Sc 20 6425.58(28) 680.72(26) 99.9941(14) 0.095 3045.1(14) 3077.3(23)
46 V 22 7050.63(69) 422.51(11) 99.9848(13) 0.096 3044.6(18) 3074.4(27)
50 Mn 24 7632.39(28) 283.25(14) 99.942(3) 0.100 3043.7(16) 3073.8(27)
54 Co 26 8242.56(28) 193.270(63) 99.9955(6) 0.104 3045.8(11) 3072.2(27)
Average
t = 3072.3(9)
2 /8 = 1.10

Table I.  Experimental results, computed electron-capture probabilities ( P ) and ft-
EC
values of nine super-allowed +
decays (see Ref. 18). The values of the total half-lives
( t1 2 ), the branching ratios (R) and the electron-capture probabilities computed for 0+
0+ transitions have been used to determine the partial half-life t = t1 2 (1+ P . f is the
EC ) R

value of the phase-space integral corresponding to the atomic number Z of the residual
nucleus and the Q-value of each reaction ( Q is the Q-value of the electron-capture
EC
process, Q Q + + 2m ). The last column lists the ft-values after correction for
EC e


nucleus-dependent effects.



2.1.2 Neutron decay

The free neutron mean life ( ) and the ratio g g between the coupling
n A V

constants of axial and vector currents (both take part in the decay of neutrons) are in this
case the experimental parameters required for the determination of V 18:
ud





11


2 K / ln 2
V = (2. 1-10)
ud 2 2
G 1+ 1+ 3 g g f 1+
F ( R )( ( A V ) ) ( R ) n
where K is the typical constant of decays (Eq. 2.1-2), the nucleus-independent
R

radiative correction (2. 1-5) and f (1 + the phase-space function (which depends on
R )
the Q-value) after radiative correction, f (1+ =1.71489  0.0000222. The
R )

experimental data concerning neutron decay are shown in Table II.

The results of several measurements, based on two different experimental
techniques, are used for the determination of the neutron mean life.
i. The method used in the first experiments consists in counting the decays which
occur in a definite portion of a continuous or pulsed neutron beam. The precision of
the result depends mainly on the accuracy with which it is possible to determine the
spatial dimensions of the observed part of the beam and its density of neutrons.
From these quantities the initial population, N 0 , can be computed; the mean life is

derived from the relation ( -
N - N N = e
0 , where N is the number of the
t ) t
0 t

neutrons which have decayed in the transit time t.
ii. The recent development of techniques for the production and the accumulation of
ultra-cold neutrons (having kinetic energy lower than -7
310 eV ) has considerably
reduced the sources of systematic errors in the measurement of the mean life. Very
slow neutrons can be confined within material walls or magnetic traps. By counting
the number of neutrons inside the cell as a function of time, can be determined
n

as the only parameter of the exponential decay law. The efficiency of the detector
(which does not affect the measured ratio between the populations at different
times), the volume of the cell and the initial density do not need to be known and it
is not necessary to detect the decay products.

The ratio g g between the axial and vector coupling constants can be determined
A V

by measuring the parameters describing the correlations between neutron spin and lepton
momenta. The decay probability per unit time has the following expression:
p p p p p p
e e ( e )
P(p ,p ,) 1 + + + + (2. 1-11)
e a A B D
Ee
E Ee
E Ee
E

where E , p and
e
E , e
p are the energies and the momenta of the electron and the

antineutrino, the neutron spin).
i. The axial part of weak interaction is responsible for an anisotropic production of the
electrons. The number of electrons emitted in the same direction as the nuclear spin
(
N ) is smaller than the number of electrons going in the opposite direction (
N ).
Consequently, when the neutrons are polarized the term which correlates neutron
spin and electron momentum is non-zero and its coefficient

N -
A N (2. 1-12)

N + N
is measurable; then = g g can be determined from the relation
A V
( + )
1
A = -2 2 .
(2. 1-13)
1+ 3


12




Experimental techniques (s) References
n
918  14 1972 Christensen et al. 23
decay count of beamed 891  9 1988 Spivak24
neutrons 876  21 1988 Last et al. 25
878  30 1989 Kossakowski et al. 26
889.2  4.8 1996 Byrne et al. 27
903  13 1986 Kosvintsev et al. 28
877  10 1989 Paul et al. 29
Accumulation of ultra-cold 887.6  3.0 1989 Mampe et al. 30
neutrons 888.4  3.3 1992 Nesvizhevskii et al. 31
882.6  2.7 1993 Mampe et al. 32
886.7  1.9
Average (2 /7 = 1.53, s =1.24)



Measured quantities g g References
A V
-1.258  0.015 1975 Krohn et al. 33
-1.261  0.012 1979 Erozolimskii et al. 34
Decay asymmetry of free -1.262  0.005 1986 Bopp et al. 35
polarized neutrons -1.2594  0.0038 1997 Yerozolimsky et al. 36
-1.266  0.004 1997 Liaud et al. 37
-1.274  0.003 1997 Abele et al. 38
Electron-neutrino correlation
coefficient (from the analysis -1.259  0.017 1978 Stratowa et al. 39
of the proton energy spectrum)
Rate of muon capture in -1.24  0.04 1984 Bertin et al. 40
hydrogen
-1.2665  0.0025
Average (2 /6 = 1.88, s =1.37)
Table II.  Neutron decay: measurement of the parameters necessary for the
determination of V (statistical and systematic errors have been added in quadrature).
ud

The results of more than twenty years of experimental investigation are listed. The
mutual determinations of and g g (obtained fixing the value of cos or V )
n A V C ud

have obviously not been taken into account. All the data have been used in the
calculation of the averages; however, given the wide range of the error values, the
compatibility test (2) has been performed without taking into account the least precise
measurements. The degrees of freedom of the measurements whose uncertainties exceed
the (arbitrarily defined) threshold value of ten times the unscaled error in the average
have not been counted. The g g value independently obtained from the measured
A V

rate of nuclear muon capture by a proton is also quoted for comparison.






13


F
V i
ud gure 2.
. The -g g
A V n
1.19 1.21 1.23 1.25 1.27 1.29
dar 840 860 880 900 920 940
 N
k bands eutr
Krohn et al. Christensen
on mean life and et al. 1972
1975
repr Kosvintsev et
es Stratowa et al. 1986
en al. 1978
t the
Spivak 1988
av Erozolimskii
g
e A
r et al. 1979 Last et al.
age g 1988
14 V
v meas
a Bertin et al.
lue Kossakowski
1984 et al. 1989
s (w urements
ith r Paul et al.
Bopp et al. 1989
1986
es us
cale Mampe et al.
ed in the deter 1989
d e Yerozolimsky
et al. 1997
rr Nesvizhevskii
or et al. 1992
s, s Liaud et al.
ee 1997 Mampe et al.
Table mination of 1993

Abele et al.
II). Byrne et al.
1997 1996



ii. In the experiments using non-polarized neutrons the coefficient
2
-1
a = - 2
(2. 1-14)
1+ 3
of the electron-neutrino angular correlation is measured, since it is the only term of
the decay probability which does not depend on neutron spin and therefore does not
vanish on the average. The parameter a is extracted by comparing the measured
energy distribution of the outgoing protons to the spectrum shape calculated as a
function of a.
iii. A determination of g g was deduced by the experimental data regarding muon
A V

capture in hydrogen40 -- a process which, according to the hypothesis of lepton
universality, is analogous to the reversed neutron decay.

To compute the average values of and g g , scale factors have been applied to
n A V

the errors because of the disagreement between independent determinations. The
dispersion of the measurements is shown in Figure 2.
Substituting the best values of and g g in Eq. (2. 1-10), the result
n A V

V (2. 1-15)
ud = 9755
.
0  0.0016g g  0010
.
0  .
0 0004
neutron = .
0 9755  0019
.
0
A V n R

is obtained. The experimental uncertainty, which is the most substantial contribution in
the total error, exceeds the one of the measurement (2. 1-6) deduced from nuclear decays
(which is mainly theoretical in origin) by more than one order of magnitude. This
determination is compatible with the nuclear result (2. 1-9) and with unitarity (1-2-1):
2 2 2
V V V . (2. 1-16)
ud + us + ub = 0000
.
1  0039
.
0

It should however be noticed that single measurements of g g would lead to different
A V

determinations of V which in some cases are incompatible with each other, with the
ud

nuclear result and with unitarity. For example, when the measurement by Abele at al. 38
is used alone (not averaged), it leads to the value V = 0.9712  0.0020, which is even
ud

lower than the result given by nuclear decays (2. 1-6) and deviate from unitarity by 2. 1
2
standard deviations ( V ). On the contrary, if the average of the
ui = 9916
.
0  .
0 0040
i=d ,s,b

remaining seven measurements is considered ( g g = -1.2621  0.0023 with
A V
2 /6 =0.26), the resulting value V = 0.9788  0.0016 actually exceeds unitarity by
ud

2 2
: V . Most likely, therefore, the determination 2. 1-15 is the
ui = .
1 0065  0033
.
0
i=d ,s,b

result of a number of accidental compensations between opposite systematic effects.


2.1.3 Pion decay

With respect to the already described methods for determining V , the advantages
ud

of studying pion decay are twofold: it is a pure vector transition and, obviously, it
does not depend on effects due to nuclear structure. Unfortunately, the small ( -8
10 )




15


branching ratio of the channel + 0 +
e has not yet been measured precisely enough
e

to provide a competitive determination of V . In the relation18
ud
2 ( + 0 +
K / ln 2)7 ( e e )
V = (2. 1-17)
ud 2
G
2 1+ f f f
1 2 1+
F ( R ) ( R )
K , G and are given by Eqs. (2.1-2), (2-1) and (2.1-5) respectively,
F R
5
, f = [(m m  - m = 
 - m 0 ) m ] where12
0 ( .45936 .0 )
0005
e 30
R = ( .
1 05 .
0 )
15 %
MeV and the corrections f (m -
1 and f 2 , calculated as functions of  m 0 ) m  , are41

f f2 = 0
1 = 0.941039 and .951439 . Using the values of the branching ratio of the
reaction and the pion mean life,
7 ( + 0 +
e )= (1 025
. [41, 12]
(2. 1-18)
e  0.034) -8
10
= (2 6033
.
 0 0005
. )-8s [42,12], (2. 1-19)
from Eq. (2. 1-17) the result
V 9670
.
0  .
0 0160  .
0 0009 = .
0 967  016
.
0
ud =
pion 7 , (2. 1-20)

is obtained, which is consistent with the other measurements of V and with unitarity:
ud

2
V . (2. 1-21)
ui = 984
.
0  .
0 031
i=d ,s,b

The experimental uncertainty should be reduced by a factor of 10 before it becomes
comparable to the theoretical error. A new, more precise measurement of the decay rate,
therefore, would be essential. It would make it possible to get around the controversial
aspects of the theoretical description of nuclear decays, and provide an unambiguous
determination of V without requiring the knowledge of additional experimental
ud

parameters such as the axial coupling constant in the case of neutron decay.


2.1.4 Summary of |Vud| determinations

The best value of V is calculated as the weighted average of the independent
ud

determinations 2. 1-9, 2. 1-15 and 2. 1-21:
V = 0.9740  0.0010. (2.1-22)
ud nuclear

V = 0.9755  0.0019. (2.1-23)
ud neutron
V 0.967  0.016. (2. 1-24)
ud =
pion

Average V = 0.9743  0.0008.

(2. 1-25)
ud












16


2.2 |Vus|

A precise determination of V can be obtained from the analysis of kaon and
us

hyperon semileptonic decays. In this Section, we will describe the results and point out
the main theoretical uncertainties connected with the two methods.


2.2.1 Kaon semileptonic decays

The + 0 -
K 0
+ K processes, called respectively +
K and 0
K ,
L +
and 3 3
are pure vector s
+
u transitions. From the observed decay rates the value of

V can be extracted according to the relation43
us

2
2
= G V
F us 5 2
m C f . (*)
(2. 2-1)
K
3 1( 2
0) (1+ )(1+ )
192 I
Here
G is the Fermi constant, I the phase-space integral, m the kaon mass, C a
F K
normalization constant ( 2
C =1/2 and 2
C =1 for the +
K and 0
K transitions respectively),
3 3
(channel-dependent) and the radiative corrections44, 45
+
0
= ( K
.
2 12  08
.
0 ) -
% = .
2
%
0 or
e3 . (2. 2-2)
0
%
5
.
0
0
or Ke3
The form factor f appearing in Eq. (2. 2-1) and the analogous f account for the
2 (t )
1(t )

dependence of the matrix element of the vector current on the square momentum transfer
2
t = ( p :
K - )

p
    
K (p = + + - .
(2. 2-3)
K ) u s (p ) C ({p p
K )f1(t ) (p p
K )f2 (t )}
To compute the integral I, the t-dependence of both form factors has to be known:
(m -m
K )2
1 m m 3m m m
3 2
2
2
2
2 2
2 K - 2
2
I = dt 1 1 f t f t (2. 2-4)
8 2
+ -
1( ) ( )
+

m t t t m
2
K m 2
(2 + 2) ( )


where
( 2
m + - - , f (t ) f t + t f t m - ,
1 ( ) 2 ( ) ( 2 2
K
m )
K
m t )2
2 2 2
4 mK
m

f t f t f , f (t ) f (t) f (0)
1( ) 1 ( ) (0
1 )
and m , m ( = ,
e  ) are the masses of the particles involved in the process. One
K
m ,

can determine the reduced form factor f from the experimental data, which can be
1 (t )

fitted with an f depending linearly on t. On the contrary, the linear form does not
1 (t )

give a good description of the data in the case of f (t). However, since f (t) is
multiplied by the square lepton mass in the expression of I (2. 2-4), its contribution can



* This formula is valid if the small (3 ) CP violating effect which differentiates the rates of
0 - 0 +
K +
e and K -
e decays is neglected. This is certainly an allowed
L e L e

approximation given the current precision (1%) of the experimental data and the presence of
larger theoretical uncertainties.



17


+ 0
be made negligible by considering only the K decay yields ( K +
e and
e3 e
0 -
K +
e
L e ). In this way, the integrals can be computed as43

I + = .
0 1605 ,
0009
.
0
Ke3 (2. 2-5)
I = 
0 .
0 1561 .
0008
.
0
Ke3
Substituting these values and the latest average determination of the rates for the +
K
e3
and 0
K processes12
e3
+ = ( -
560
.
2  .
0 ) 15
033 10 MeV
Ke3 (2. 2-6)
-
= 
0 ( 937
.
4 ) 15
053
.
0 10 MeV
Ke3

into Eq. (2. 2-1), one obtains
+ 0

f K Vus =  
1 (0) 2181
.
0 .
0 0015 0001
.
0
(2. 2-7)
0 -
f K V =  
1 (0) 2101
.
0 0013
.
0 0001
.
0
us

where the theoretical uncertainty of the radiative corrections (a relative error of the same
order as that on has been assumed for ) is quoted separately to show that it is
negligible. The two results can be combined after taking into account the effects of
isospin symmetry breaking: with43
+ 0 -
K K
f f =  (2. 2-8)
1 ( 0
0) 1 (0) .1022 0 002
.
the weighted average
0 -
f K V (2. 2-9)
us = 
1 (0) 2114
.
0 0016
.
0

is obtained (*).
Two calculations were performed to determine the form factor of the +
K decay at
e3

zero momentum transfer, giving the following results:
0 -
K
f =  43, (2. 2-10)
1 (0) .0961 008
.
0
0 -
K
f =  46. (2. 2-11)
1 (0) .0963 .0004
The reliability of the error estimate in Eq. (2. 2-10) was questioned by the authors of
Ref. 47 due to the uncertainties which are inherent in all chiral perturbation theory
computations. However, a perfectly consistent result was provided by the second,
independent calculation (2. 2-11), performed in the framework of a relativistic
constituent quark model which proved to be successful in the description of the electro-
weak properties of light mesons.
Using the result 2. 2-10, from Eq. (2. 2-9) one obtains

V = 0.2200  0.0017
us exp  0.0018th = 0.2200  0.0025. (2. 2-12)





* -
The two determinations of 0
f K , though consistent at the 11% confidence level
1 (0) Vus
(2/1 = 2.6), have been averaged, as suggested by the PDG12, after multiplying the errors by the

scale-factor s = 2 1 =1.6. This procedure is followed also in those cases where other authors
prefer not to average the results at all. Incidentally, better agreeing results would be obtained for
0 -

f K if the channel-dependent radiative corrections in Eq. 2.2-2 would not be applied.
1 (0) Vus


18


2.2.2 Hyperon semileptonic decays

The deduction of V from hyperon semileptonic decay rates requires a more
us

complex theoretical analysis. While the kaon decays are described by the vector
interaction alone and thus the SU(3) symmetry-breaking occurs at the second order only
(Ademollo-Gatto's theorem), an accurate study of hyperon decays requires the additional
computation of the more substantial (first-order) correction for the axial form factor.
Donoghue, Holstein and Klimt48 (DHK) computed SU(3)-breaking corrections up to the
second order for the vector part and to the first order for the axial one, and applied them
to the WA2 data (SPS) combined with the previous world averages (ten distinct hyperon
decay channels in all)49. Finally, they quoted the result

V = 0.220  0.001
us exp  0.003th, (2. 2-13)

where the error is evidently dominated by the theoretical uncertainty. The PDG12 do not
use the hyperon decay result to determine the best value of V . Flores-Mendieta et
us

al. 50 tested the reliability of four distinct theoretical models which provide second order
SU(3)-breaking corrections for hyperon semileptonic transitions and pointed out a strong
model-dependence of the results. In particular, using the DHK model they found a value
of V noticeably higher than the one originally determined by the authors themselves
us

(2.2-13), and much less consistent with the kaon decay data. This disagreement may be
explained as the result of a different treatment of the experimental data. For example, in
Ref. 50 only the three most precisely measured decay channels instead of the original ten
were taken into account. This reduction in the experimental basis increases considerably
the relative statistical weight of the -
e

- data, which is the one showing by far
e

the worst agreement with the DHK model, as already found by the authors themselves.
Moreover, the weighted average of two disagreeing results
( - -
e [49]
e ) = ( -
83
.
1  .
0 79) 6 1
10 s
( - -
e [51] (2. 2-14)
e ) = ( -
.
3 44  .
0 19) 6 1
10 s
average ( - -
e with 2 = 4
e ) = ( -
.
3 36  19
.
0 ) 6 1
10 s
-1
(the first of which is consistent with the DHK fitted value of 2.65106 s ) is used for the
-
e

- decay width in both the analyses, but only DHK apply a scale factor of 2
e

to the errors to make the two measurements compatible. The authors of the second
analysis50 quote the value obtained using the chiral perturbation model of Ref. 52 as the
best determination of V from hyperon decays:
us

V = 0.2176  0.0026. (2.2-15)
us

However, they also show that values ranging from 0. 21 to 0. 25 can be obtained by
applying different models.

Since apparently unresolved theoretical uncertainties still affect the description of
hyperon decays, we prefer not to use these results. Therefore, we assume the
determination obtained from kaon decays
V = 0.2200  0.0025 (2.2-16)
us

as the best value, noting however that it is in good agreement with the results of two
different analyses of hyperon semileptonic decays.



19


2.3 |Vcd| and |Vcs|

A measurement of the matrix elements V and V can be obtained from the
cd cs

study of the deep inelastic scattering of neutrinos on nucleons. For the determination of

V the experimental data concerning the D K e
+
semileptonic transitions and the
cs e
recent measurements of the 
W hadronic decays are also used.


2.3.1 |Vcd|

V and V can be extracted by comparing the cross sections of the charm
cd cs

production induced by neutrino and anti-neutrino scattering on nucleons.

-
 +
V V
( )
cd cs
W W+ 
d s
( ) c s
Figure 3.  Neutrino-induced dimuon production.


The dimuon production from neutrino-nucleon scattering can be described as a two-
stage process (see Figure 3). The charge-current interaction of the neutrino with a d or s
quark yields the first muon and changes the quark flavour into charm. The second muon,
having the opposite charge, is the result of the semileptonic decay of the c quark:
+ d (s) c + - c s + + +
 (  ) (2.3-1)
+ d (s) c + + c s + - +
 (  )

The cross-sections of the dimuon processes were measured by the collaborations
CDHS53, CCFR54 and CHARM II55. The cross-section terms containing the dependence
on V and V were extracted by fitting the neutrino and anti-neutrino data
cd cs

simultaneously:
2
CDHS V B (2. 3-2)
cd = (4 1
.
c  0.7) -3
10
2 + . + . -
CCFR V B . (2. 3-3)
cd c = ( 0 38 0 37
5 34 - . stat - . 10
0 39 0 55 syst ) 3

2 + . -
CHARM II V B (2. 3-4)
cd c = ( 0 35
42
.
4 - .  .
0 34
stat syst 10
0 34 ) 3
where B is the weighted average of the semileptonic branching ratios of the charmed
c

hadrons produced by the neutrino-nucleon interaction. The analysis performed by CHDS
made use of leading-order QCD calculations, while the CCFR and CHARM II results
were obtained in the context of the more accurate next-to-leading-order formalism.





20


Combining Eqs. (2. 3-2), (2. 3-3) and (2. 3-4) (*), the following weighted average is
obtained:
2 -3
V B .
(2. 3-5)
cd c = ( 63
.
4  .
0 )
34 10

The value of B was calculated56 using the independently measured rates of charmed
c

hadron production from neutrinos57 and the semileptonic branching ratios58 of charmed
hadrons:
B = 0.0919  0.0094. (2.3-6)
c

Thus from Eq. (2. 3-5) the value
V = 0.224  0.014 (2. 3-7)
cd

is obtained.


2.3.2 |Vcs| from neutrino charm production

From dimuon cross-section measurements, the above-mentioned CDHS, CCFR and
CHARM II experiments obtained also the following results:
2
V
CDHS cs = 9.31.6 (2. 3-8)
V
cd

CCFR 2
V B (2. 3-9)
cs c = ( + 09
.
0
.
2 00 0 10
. 10-
stat - syst
15
.
0 ) 2
+ 2
2

1 V
CHARM II cs
1 + = .
3 58 + 49
.
0
(2. 3-10)
-  0 44
.
41
.
0 stat syst
+ 2 V
cd

Here the parameter , which quantifies the relative size of the strange quark sea, is
defined as the ratio of the integrated distribution function of strange quarks to that of u
and d quarks:
1 1
= [x s(x) + x s(x)]dx [xu (x) + x d (x)]dx
(2. 3-11)
0 0
where s(x) , s (x) , u (x) and d (x) are the quark density distributions in momentum
space and x is the fraction of nucleon momentum carried by the quark. All three
analyses assumed the equality s (x) = s(x) . Although this assumption can be
contradicted59,60, CCFR verified that the removal of the s (x) = s(x) constraint does not
lead to a significant modification of the results.
A new analysis by CCFR61, which has determined the value of with an
independent measurement of the inclusive neutrino cross-section, makes it possible to
extract a measurement of V from these data. The result,
cs
+0 028
= 0.453 0 106 .
. -0 096 , (2. 3-12)
.

indicates a marked SU(3)-flavour-symmetry violation (an SU(3)-symmetric sea would
have = 1). In order to compare the CDHS, CCFR and CHARM II results for V , we
cs


* For the estimate of V (and of V in the next Subsection), we symmetrize the errors in the
cd cs

CCFR result, taking the average of the positive and negative errors (statistical and systematic
added in quadrature) while keeping the same central value.



21


multiply Eqs. (2. 3-9) and (2. 3-10) by + 2 = 2.453  0.126, Eq. (2.3-8) and the
2
determination of V V resulting from Eq. (2. 3-10) by Eq. (2. 3-5), obtaining:
cs cd
2 -
CDHS 2
V B (2. 3-13)
cs c = ( 31
.
4  )
81
.
0 10
2 -
CCFR 2
V B (2. 3-14)
cs c = ( .
4 91 )
46
.
0 10
2 -
CHARM II 2
V B
cs c = ( 60
.
3  .
0 )
79 10 (2.3-15)
2 -
Average 2
V B
cs c = ( .
4 53  .
0 )
37 10 (2.3-16)
From Eqs. (2. 3-6), (2. 3-12) and (2. 3-16) V is determined as follows:
cs

V = 1.04  0.16. (2.3-17)
cs



2.3.3 |Vcs| from semileptonic decays of D mesons


V can also be obtained by comparing the measured decay amplitude of the
cs

semileptonic processes D ( D K e
+
) to the corresponding expression in the theory
e3 e

of weak interaction. The method is very similar to the one used for the derivation of

V from K decays. The decay rate is described by the relation
us e3


(D +
Ke cs
(2. 3-18)
e ) 2
2
G V
F 5
= m f I
D +
3 1 ( 2
0) (1 )
192
where the meanings of the single quantities are the same as in Eq. (2. 2-1). The phase-
space density I is obtained by calculation of the integral 2. 2-4 with m and
K
m replaced

respectively by m and m ; in this case, the observed square-momentum spectrum can
D K

be fitted62 using the single-pole parametrization of the reduced form factor,
f t f t f m2 2 - , with m m .
2 GeV
1 . Assuming this form
pole (m t
pole )
1 ( ) = 1( ) (0
1 )= pole Ds

exactly and neglecting the term proportional to 2
m f in the integral, we get I = 0.195.
e (t)2
Although isospin symmetry requires equal transition rates for 0 -
D K e
+ and
e
+
D K 0e
+ decays, the measured values differ considerably outside their errors. In
e

averaging the available experimental results for the D branching ratios and D
e3

lifetimes12, we rescale the error by the factor 2
s = = 6
.
2 , obtaining:
( +
D Ke . (2. 3-19)
e ) = ( -
.
8 3  .
1 0) 10 1
10 s
After substituting into Eq. (2. 3-18), we get
f 0 V 0.53  0.06.
(2. 3-20)
1( )2 2
cs =
Most of the theoretical evaluations for the form factor at zero-recoil (see Ref. 62 for a
review) give results included between 0.6 and 0.8. Assuming f (0 =  , we
1 ) .07 1.
0
obtain
V = 1.04  0.16, (2.3-21)
cs

in full agreement with Eq. (2. 3-17).





22


2.3.4 |Vcs| from W  boson decays

New measurements have been obtained from the recent study of 
W boson decays at
LEP2. Their precision is already higher than that of the previous independent results;
moreover, the errors are dominated by the experimental uncertainty and are bound to be
further reduced. DELPHI63, ALEPH64, L365 and OPAL66 extracted the value of V
cs

using two independent procedures. The ratio
+
W q
c = V + V + V
W ( ) 2 2 2
R = = (2. 3-22)
c ( q d ,s,
+
W hadrons)
b cd cs cb
2 2 2 2 2 2
V + V + V + V + V + V
ud us ub cd cs cb

(in the second equality, corrections of order 2 2
m = m are neglected; the transitions
q u ,d ,c,s ,b W

with a top quark in the final state are energetically forbidden) was determined by tagging
the flavours of the hadronic jets which arise from the fragmentation of the primary
quarks coming from 
W decays. The results obtained,
+
0 46
. 18
.
0
-  0 07
. DELPHI
0 14
. stat syst

.
0 51 .
0 05 0 03
.
W  
R = ALEPH (2. 3-23)
c stat syst
0 50
.  0 11
.  .
0 04 L3
stat syst
.0470.04 0 06
. OPAL
stat syst

are consistent with the value  expected assuming the unitarity of the CKM matrix. The
ratio of hadronic 
W decays to leptonic 
W decays,
7 ( +
W )
hadrons 7 ( +
W )
hadrons 2 2 2 2 2 2 m
S W ,
7 ( + 1
W leptons) =
1-7 ( +
W )=(V + V + V + V + V + V
ud us ub cd cs cb ) ( )
+
hadrons
(2. 3-24)
where is the strong coupling constant and once again the quark phase-space
S

corrections have been neglected, was also measured. The results
( +
.
66 0 .36
-  .
0 9
7
.
3 stat syst )% DELPHI

7 ( 89
.
66 0 67
. .
0 31 % ALEPH
+
W hadrons) (  
stat syst )
= ( (2. 3-25)
69
.
68  0 68
.  0 39
.
stat syst )% L3
( 34
.
68  0 61
.  31
.
0
stat syst )% OPAL

correspond (using m 12) to the determinations
S ( W ) = 120
.
0  .
0 005
1 87
.  0 31
. DELPHI
stat + syst
2 1 946
.  0 065
. ALEPH
V =
(2. 3-26)
ij stat + syst
, 2 113
. 077
.
0
i =u c  stat + L3
syst
j =d ,s,b 2 079
.  0 066
. stat + OPAL
syst

which have to be compared with the Standard Model expectation of 2.
From the measurements of the observables 2. 3-22 and 2. 3-24 each experiment
obtained two independent determinations of V (see Table III). The values compiled by
cs

the PDG67,12 were assigned to the other CKM matrix elements. However, their
uncertainties only contribute to a small fraction of the V systematic errors (for
cs

example, 0.003 in DELPHI's measurement, to be compared with 0.05), so that the
correlation of these results with the other measurements of the CKM matrix elements is
negligible.



23


V from 7 (W q
c ) V from 7 (W hadrons)
cs cs

DELPHI63 0.94 +0 32
. 0.90  0.17
- stat  0.13syst stat  0.04syst
0 26
.
ALEPH64 1.00  0.11stat  0.07syst 0.947  0.031stat  0.015syst
L365 0.98  0.22stat  0.08syst 1.032  0.033stat  0.018syst
OPAL66 0.91  0.07stat  0.11syst 1.015  0.029stat  0.015syst

Table III.  Measurements of V obtained using W decays at LEP2. In most cases, the
cs

results are still preliminary.


The average determinations (with statistical and systematic errors added in
quadrature) are
+
DELPHI V = 0 16
.
.
0 91 , (2. 3-27)
cs -0 15
. stat + syst
ALEPH V = .
0 950  033
.
0 , (2. 3-28)
cs stat + syst
L3 V = 031
.
1  .
0 037 , (2. 3-29)
cs stat + syst

OPAL V = 009
.
1  032
.
0 . (2. 3-30)
cs stat + syst

Finally, we average them allowing for a common systematic error of  0.015 and obtain
V = 0.993  0.025 (2 = 1.4), (2. 3-31)
cs

where the scale factor s= 1. 2 has been applied to (the uncorrelated part of) the errors.


2.3.5 Determination of the best values of |Vcd| and |Vcs|

The best values of V and V are obtained as the result of an overall fit to the
cd cs

measurements mentioned in the preceding Subsections. 8 constraints (Eqs. 2. 3-
2, 3, 4, 6, 8, 9, 10, 12) and 2 direct determinations (Eqs. 2. 3-21, 31) are included in the fit
(*). The maximum likelihood estimates of the 4 parameters of the fit, V , V , B
cd cs c

and , are
+ .
0 013
V (2. 3-32)
cd = 225
.
0 -0 011
.

V (2. 3-33)
cs = .
0 996  .
0 024
+ .
0 0086 +
B .
0 067
= .
0 489 (2. 3-34)
c = .
0 0935- .00080 - .
0 059

with 2/d.o.f. = 2/6 = 0.7 and a 10.5% correlation between V and V .
cd cs

These results can be used to test the unitarity of the CKM matrix with reference to the
elements of the second row: using Eqs. (2. 3-32), (2. 3-33) and (2. 5-10) ( V ) we obtain
cb

2 2 2
V V V . (2. 3-35)
cd + cs + cb = .
1 044  048
.
0




* The following method is used to take the asymmetric errors into account. Each error is treated
as a function of the parameters, which is constant and equal to the positive or negative error
when the current value of the unknown term of the constraint falls outside the positive or the
negative margin; in the intermediate region, the function is matched continuously by a straight
line.



24


2.4 |Vub|

CP violation can be included in the framework of the Standard Model only if all
elements of the CKM quark-mixing matrix have non-zero values. This fact justified the
strong experimental and theoretical effort devoted to the determination of V , which is
ub

expected, on the basis of the unitarity condition, to be very small.
The observation of an excess of events in the lepton end-point spectra, interpreted as the
result of charmless semileptonic decays of B mesons, was made by CLEO68 at CESR
and ARGUS69 at DESY in 1990 and gave the first experimental evidence70 of a non-zero
V . The subsequent experiments were faced with the hard task of a precise
ub

determination of its value. The main difficulty in measuring signals from b
-
u
processes arises from the large background induced by the stronger b
-
c
transitions. The contribution of the charmed B decays can be suppressed by looking for
exclusive decays were charmless final states can be directly observed by means of
invariant-mass peaks. Such measurements, though easier from an experimental point of
view, involve the transition between the heavy B meson and a light daughter hadron,
which is extremely challenging to describe precisely, and lead to model-dependent
results. This approach, successfully exploited by the CLEO collaboration in the
measurement of the 0 0 - +
B -

+ B
and branching ratios, led to the

following average determination of V (see Table IV):
ub

V = ( 0
+ 21
3 25
.  0.14 .  0.55 10
-0 , (2. 4-1)
.29 ) 3-
ub CLEO exclusive

where the errors are respectively statistical, systematic and theoretical.

Exclusive measurements, feasible with the B production at the (4#), are not
effective at LEP, where the lepton momenta in the b-hadron rest frame cannot be
reconstructed with sufficient accuracy, the efficiency of the B decay vertex
reconstruction depends on the charge multiplicities of the final state and the
fragmentation of the b quarks produces different particles which dilute the signal. These
disadvantages are compensated for by the back-to-back topology of the b quark
production from Z decays, which avoids the mixing of the B decay products. This
characteristic is fully exploited by a novel technique based on the study of the shape of
the invariant mass of the hadronic system recoiling against the lepton in b
-
u
transitions. The analyses, up to now performed by the ALEPH and L3 collaborations,
have allowed the determination of the b X inclusive branching ratios quoted in
u

Table IV. The basic remark is that a smaller model-dependence is expected in predicting
the shape of this invariant-mass distribution, while the recent progress in the theoretical
calculations has succeeded in containing the uncertainties at the level of few percent71.
The weighted average of the inclusive branching ratios obtained by the LEP experiments
leads to
7 (b X 2 = 0.99 (2.4-2)
u ) = ( -
.
2 0  8
.
0 ) 3
10
(the correlations between systematic errors have been taken into account following the
indications found in Ref. 72), from which, using the average b quark lifetime12





25


= ( 564
.
1  014
.
0 )ps , the value of V can be extracted in the context of the Heavy
b ub

Quark Theory71:
V = (4.3  0.9exp  0.3th ) 10-3 = (4.3  0.9) 10-3. (2.4-3)
ub LEP inclusive

This determination can be averaged with the CLEO mean value coming from exclusive
branching-ratio measurements, giving the following reference value of V :
ub

V = (3.6  0.5) 10-3 2 = 0.92. (2. 4-4)
ub




Experiment Decays
measured 7 V
ub

0
B - +
(1 8. 0 4.  0 3. 0.2) -4
10
CLEO73 ( +0.3
3.3  0.2  0.7 10
-0
.4 ) -3
0
+ 5
0 (2 5.0 4. . 0 5. 10
-0 7
. ) 4-
B - +

0 - +
CLEO74 B + 35
.
0 -
  ( 0
+ 23
. -
3.23  0.24  0.58 10
-
.
0 26 ) 3
( .269 0 41
. 50
.
0 10
- 40
.
0 ) 4
+ + -
L375 (3 3.1 0.1 7.) 3-
10 ( 08. 14.
0
.
6  2
.
0 10
1
- 0
. 1
- 9
. ) 3
inclusive
b X 2
V
ub = ( 
u 68
.
18 94
.
5
ALEPH76 (1 73
.  0 55
.  0. ) 3
-
55 10
 5 94
. 1 ) -6
45
. 10

Table IV.  Summary of the experimental determinations of V . The branching ratios
ub

measurements have statistical and systematic errors quoted separately. Where a third
error is given, it accounts for the model-dependence of the results.




2.5 |Vcb|

The charmed semileptonic decays of B mesons constitute the experimental basis for
the direct determination of the matrix element V . In this Section, two different
cb

methods, which make use of exclusive and inclusive measurements, are described.


2.5.1 Exclusive decays

A number of research groups have carried out measurements of V studying the
cb


B D B
D
and exclusive decays in the kinematic configuration in which the
D meson is produced at rest (zero recoil) and the energy of the lepton-neutrino system is
at its maximum.
In the context of the Heavy Quark Effective Theory (HQET)77, which approximates
the masses of the heavy quarks involved in the process (b and c in this case) as infinite,
the hadronic form factors appearing in the expressions of the differential semileptonic
decay rates can be defined using one universal function ( )
w (Isgur-Wise function),

which is independent of the initial and final heavy mesons. This form factor describes
the superposition of the light quark wave functions. The maximum overlap is reached in




26


the limit of zero recoil, for which the normalization (w = )
1 = ( 2 2
q = q ) =1 is
max
adopted; here 2 2 2
w = 2
v v = m + m - q m
2 m , and q = (p p
B - D )2
B D ( B D ) ( B D) is the square
momentum transferred to the leptonic system, with 2
q m m . However, the
max = ( B - D )2

analytic form of the function cannot be predicted within HQET. Therefore, to extract the
value of V the differential decay rate
cb

2
d G 2
F 2
= f ( ,
w m ,m ) ( )
w V (2. 5-1)
3 B D
dw 48 cb

(f is a known function measuring the density of states) is measured in the limit of zero
recoil, where the matrix element is the only unknown factor. Due to the reduced
observable statistics at the point of zero recoil (w= 1), the data are collected in the range
w> 1; the value of 1
( ) V is then obtained by extrapolating to the limit w 1+. For
cb

this purpose the function ( )
w , a priori unknown, is parametrized as a Taylor

expansion around the point w = 1:
2
( )
w = 2
( )[
1 1- ^ (w - )
1 + ^c(w - )
1 + ]
(2. 5-2)
(the first derivative must be negative, since the condition w = 1 corresponds to the
maximum superposition of the wave functions and thus to the maximum of (w)). The
value of the intercept ( )
1 V is then extracted from a fit to the differential decay
cb

data. Although not all the experiments are sensitive to the quadratic term ( ( )
w is often

assumed to depend linearly on w), basic QCD considerations favour a positive value of
the curvature c^ , having a definite correlation with the slope 2
^
( ^c .
0 66 ^ 2
- .
0 11 )78.

The form factor ( )
1 , which equals 1 at the leading order in the HQ expansion, is then
determined by calculating perturbatively the effect of the finiteness of the heavy quark
masses. While the second order 1 m ( m or m ) power corrections have been
Q = m
Q b c
computed in the case of
B D
decays, for which the (1 m term is absent79, at
Q )

present the B
D decays can be described with less theoretical accuracy, since only
the first order of the expansion is known. Moreover, both the overall branching ratio
and the rate near the point of zero-recoil are smaller for the B
D mode, thus
limiting the precision of the measurements.
From the experimental analysis of the
B D exclusive decays, CLEO and the
LEP experiments obtained accurate determinations of the parameter ( )
1 V . The data
cb

are displayed in Table V. OPAL and DELPHI derived their results using a quadratic
parametrization of the Isgur-Wise function, while a linear dependence was assumed by
CLEO and ALEPH. We do not report the results of other measurements carried out by
ARGUS80 (
B D B
D
), ALEPH84 and CLEO81 ( ), which provide less precise
(though consistent) information.
Since the LEP measurements, though presumably highly correlated, are scarcely
consistent even within their total errors, we compute their weighted average separately
(in this way we avoid diluting the reduced 2 with the contribution of the independent
measurement by CLEO) and rescale the error by the factor 2
s = 2 = .
2 0 , obtaining
( ) -
1 V . (2. 5-3)
cb = (35  2) 3
10
D LEP





27


Further averaging with the CLEO measurement leads to the result given in Table V.



D ( ) 3
1 Vcb 10
0
B *+ -
D
- 0 CLEO II82 35.1  1.9  1.9
* -
B D


OPAL83
32.8  1.9  2.2
+
B 0 -
D ALEPH84 31.9  1.8  1.9
DELPHI85 37.95  1.34  1.59

Average 35.0  1.6

Table V.  1 measurements. The errors are statistical and systematic
D ( ) Vcb
respectively.



Detailed calculations of the QCD and finite-mass corrections which have to be applied to
the form factor normalization give the result78
= 0.91  0.03.
( )
1 (2. 5-4)
D

Then we can extract V from the best value of 1 :
D ( ) V
cb cb

V (38.5  1.8exp  1.3th)10-3 = (38.5  2.2)10-3. (2.5-5)
cb =
exclusive



2.5.2 Inclusive decays

The measurements of the inclusive branching ratio of B semileptonic decays to
charmed final states and of the B meson mean life provide another independent
determination of V . The relation between the matrix element and the experimental
cb

observables is calculated in the framework of the Heavy Quark Theory71:
- B X
3 ( c )
V . (2. 5-6)
cb = 1
.
41 10 
-2 1
- (1 06
.
0 )
.
6 77 10 ps

The measurements on the (4#) and those at the Z give quite different results (*):
( 0 
B B X V (2. 5-7)
cb =
S ( -
.
39 91 55
.
0 ) 3

c ) - -
= 39
.
6  18
.
0 10
ps 10
(4S ) ( ) 2 1 (4 )
(b X V (2. 5-8)
cb =
Z ( -
.
41 48  50
.
0 ) 3
10
c )Z = ( - -
90
.
6  17
.
0 ) 2 1
10 ps

(only the experimental errors are quoted in the V values). Their average (2 = 4.5,
cb

error rescaled by 2.1) leads to


* The world-average total semileptonic branching-ratios have been corrected by the b u
contribution (Eq. 2.4-2). The average of the 0
B and 
B mean lives12 has been used to calculate
(B X .
c )(4S)




28



9 = (40.8  0.8  2.4 ) 10-3 = (40.8  2.5) 10-3.
FE (2.5-9)
LQFOXVLYH
exp th


Finally, we obtain our best value of FE
9 averaging the results obtained from the
inclusive (Eq. 2.5-5) and exclusive (2.5-9) measurements:

FE
9 = (39.5  1.7) 10-3.
(2.5-10)



2.6 |V |/|V |
ub cb


The first observations of E l
-
X l transitions, carried out by the CLEO and,
subsequently, the ARGUS collaboration, were based on the gain in sensitivity obtained by
restricting the data selection to the final part of the phase-space, where the lepton
momenta are beyond the end-point of the E l
-
F l processes. An excess of leptons with
momenta in the range 3
.
2 *H9 < S < 6
.
2
l *H9 is interpreted as evidence of charmless B
decays. The drawback of this approach is represented by the limited portion of the
observed phase-space, which requires an extrapolation to the low momentum region in
order to recover the full decay rate, leading to a model-dependent measurement. This
technique led CLEO, ARGUS, and CLEO II to the direct measurement of the ratio

XE
9 FE
9 (Table VI). Recently, the DELPHI collaboration, by means of a technique based
on the study of the shape of the invariant-mass distribution of the hadronic system
recoiling against the lepton, has produced a new measurement of the XE
9 FE
9 ratio,
which takes into account the whole lepton spectrum (Table VI).
We get a further estimation of the XE
9 FE
9 ratio using our reference values for XE
9
and E
9 derived in Sections 2.4 and 2.5 (Eqs. 2.4-4 and 2.5-10):

XE
9 = 0.091  0.013.
E
9 (2.6-1)
By taking the weighted average of this value with the DELPHI determination we obtain
the following evaluation of the XE
9 E
9 ratio:

XE
9 = 0.093  0.011.
E
9 (2.6-2)

We note that the CLEO and ARGUS XE
9 E
9 measurements, quoted in the first part of
Table VI, are spread out by the considerable differences of the theoretical models
available at that time. On the contrary, the theoretical uncertainties connected to our
estimation (2.6-2) of the XE
9 E
9 ratio are at the level of few percent71. Therefore, we
can use this value to reject, by means of a 2 test, the models which give conflicting
predictions. The 2 values indicating the compatibility with the reference value 2.6-2,
2(KS)=9.4, 2(WSB)=6.9, 2(ACCMM)=2.4, 2(ISGW)=24,
(with 3 degrees of freedom) induce to reject the KS and IGSW models at the 5% level and
to recover the WSB and ACCMM predictions. In particular, the ACCMM model, which is
also able to reproduce a good agreement between the three measurements (the 2 of the
average is 2.5, see Table VI), leads to the following average determination:

XE
9 = 0.088  0.009,
E
9 (2.6-3)




29


where the error has been rescaled by the factor 5
.
2 .

This result is finally averaged with the DELPHI measurement, giving our V V
ub cb

reference value:
V V = 0.090  0.008. (2. 6-4)
ub cb





Model KS86 WSB87 ACCMM88 ISGW89
Experiment
CLEO68
2.2< p 0.095  0.011 0.114  0.018 0.089  0.011 0.148  0.020
< 2. 4 GeV
2.4< p< 2.6 GeV
ARGUS70 0.110  0.012 0.130  0.015 0.110  0.012 0.200  0.023
2.3< p< 2.6 GeV
CLEO II90
2.3< p 0.057  0.006 0.075  0.007 0.078  0.008 0.104  0.010
< 2. 4 GeV
2.4< p< 2.6 GeV
Average 0.073  0.005 0.088  0.006 0.088  0.006 0.124  0.008
(2/2) (10.3) (6.7) (2.5) (8.2)


DELPHI91 0.100  0.011stat  0.018 syst  0.009 model

Table VI.  Summary of the experimental determinations of V V . The upper part
ub cb

compares the V V measurements obtained by CLEO, ARGUS and CLEO II using
ub cb

four distinct theoretical models. The lepton momentum ( p ) ranges observed by each
experiment is also indicated. The CLEO and CLEO II results have been renormalized to
the present value of the inclusive branching ratio of the background processes,
7 (B X 7 (B
X 7 B X (obtained using
u
)- ( )= ( .
10 25  22
.
0 )%
c ) =

Eq. (2. 4-2) and the value of 7 (B X ) quoted in Ref. 12).



















30





2.7 Elements of the third row


The m
and m
meson-antimeson oscillation frequencies, the parameter
B B K
d s

measuring the mass-matrix CP violation in the neutral-kaon system and the inclusive rate
of the bs rare decays constitute the main experimental constraints on the values of
Vtd, Vts and Vtb.
All the measurements involving elements of the third row implicitly contain the
hypothesis of a three-generation CKM matrix. For example, CDF92 measured the ratio
7 (t Wb) 7 (t Wq), which can be translated into a direct measurement of
q=d ,s b
,

V ,
tb

V 7 t Wb
tb ( ) +
V , (2. 7-1)
tb = = 96
.
0 -
2 2 2 7 t Wq
V V V
td + ts + tb ( ) 0 16
.
.
0 12

q=d ,s,b

2 2 2
only if the assumption V V V is made. It has to be stressed that, if this
td + ts + tb =1
condition is relaxed, no determination is possible for V using such an experimental
tb

procedure (based on tt production) and only very loose bounds can be set ( V > 0.045
tb

at 95% C. L. )92. On the other hand, a measurement of the single top production cross
section, which is directly proportional to V 2 without any underlying hypothesis, will
tb

be possible in Runs 2 and 3 of CDF and D093.
The mixing and CP violating phenomena are directly related to the existence of new
physics. If the d, s and b quarks were allowed to couple with the members of a fourth
quark family (or with another unknown species of heavy fermions) or a different kind of
highly massive gauge bosons could replace the exchanged 
W , new, significant
contributions would have to be expected in the description of these processes. A
common property of the weak transitions of which these observables are characteristic
parameters is the ability to change the flavour of the quarks involved without changing
their charge (effective FCNC processes). The corresponding amplitudes can be obtained,
within the Standard Model, by iterating the basic, charged-current weak couplings. In
this way, the S= 2 and/or B= 2 transitions on which the mixing phenomena of neutral
mesons are based, are traced back to the elementary S= 1 and B= 1 processes, which
are represented by single CKM matrix elements. The description of mixing by means of
the so-called `box' diagrams (see Figure 4), in which two virtual 
W bosons are
exchanged between two quark lines, cannot lead to a quantification of the transition
amplitudes without the additional assumption that only three quark families exist. The
hypothesis of the 33 unitarity of the CKM matrix, which has not been used for the
deduction of the elements of the first two rows, plays an essential role in the
measurement of the elements of the third row, even if it not imposed as a direct
constraint.







31


u, c, t u, c, t
b d b s

0 W
0 W B0
B B0
B
d V d s V s
W td W ts
d b s b
u, c, t u, c, t
Figure 4.  Examples of box diagrams describing 0 0 0
B B and 0
B B mixing.
s -
d - d s



The effective Hamiltonian for the B= 2 transitions ( 0 0
B B mixing), for example,
d - d

is proportional (neglecting corrective terms) to

V V V V S(x , x ) , (2. 7-2)
id ib jd jb i j
i, j

where S (x , x ) are the Inami-Lim94 functions (apart from the sign, irrelevant for the
i j

final formulae), which depend on quark masses ( 2 2
x m m ),
i = i W
1 3 3 log x 3 1
S (x , x ) = x x i
+ - + -
2
i j i j (x x
i j )

4 (
2 1- x 4 1- - 4
1- 1-
i ) ( xi ) x x
i j ( xi )( xj) i j

(2. 7-3)
3
1 9 3 3 x
i
S (x , x ) S(x ) = x + - - log x (2. 7-4)
i i i i 4 (
4 1- x 2
2 1- x 2 1- x
i ) (
i ) i
i

and the indexes in the summation assume in turn all the values corresponding to the up-
type quarks which take part in the intermediate virtual state; therefore, in the three-
family model, i, j = u, c, t. The unitarity condition is then used again to simplify the
expression 2. 7-2. The approximate equality between orders of magnitude

V V V V V V (2. 7-5)
ud ub
cd cb
td tb = ( 3
)
applies in Wolfenstein's parametrization; the relative weight of the summation terms is
thus determined by the functions S (x , x ) , the magnitudes of which fulfil the following
i j

relation (the values of the masses and all other parameters occurring in the expressions
quoted in this paragraph will be given in Table XV):
4 1 0
S (x , x ) : S (x , x ) : S(x , x ) , (2. 7-6)
t t t c c c 10 :10 :10
while all other combinations are negligible in comparison with S (x , x ) . Therefore, the
c c

box diagram is dominated by the top quark contribution and, when all other terms are
2
neglected, the 0
B mass difference is proportional to V V S (x ) . Replacing the d
d td tb t

quark by an s quark does not alter the proportions between the members of Eq. (2. 7-6),
so that the same approximation applies to 0 0
B B mixing:
s - s

2 2
GF 2 2
m m m f B S x V V (2. 7-7)
B = ( )
d 2 W B B B B t td tb
6 d d d

2 2
GF 2 2
m m m f B S x V V (2. 7-8)
B = ( )
s 2 W B B B B t ts tb
6 s s s

Here S(x ) is the loop function (2.7-4) corrected for the effects of perturbative QCD;
B t

its value does not depend on the convention chosen for the top quark mass, provided the
same convention is used to compute ; the result95, 96 = 0. is valid within
B 55  0 01
.
B

the MS (Minimal Subtraction) renormalization scheme and, for reasons of consistency,



32


the mass m has to be computed by rescaling the pole mass7: m (m ) = 166  5 .
t t pole ( )GeV
t

2 2
f and f are the B meson decay constants; the products f B and f B ( B
B B B B B B B
d s d d s s d

and B are called `bag' factors) parametrize the matrix elements between the initial and
Bs
the final hadronic state. Their values are the main theoretical uncertainty in the relations
2. 7-7 and 2. 7-8. In order to determine these parameters, most theoretical analyses make
use of QCD computations performed on a discretized space-time (lattice-QCD). (A
comparison between some recent results will be shown in Table IX and in Table X).
Even though no direct experimental determinations of f and f exist at present,
B B
d s
a number of experiments have measured the rates of the processes +
D and
s 
+ 
+
D , leading to an estimate of the decay constant f . An indirect measurement
s
+ Ds

of f and f can then be obtained by extrapolating from D to B sector97 by means of
B B
d s

the theoretical determinations of f f and f f , which are quite accurate.
B D B D
d s s s

Moreover, the measurement of f can be used to check the results of lattice
Ds

simulations, which generally determine f and the B meson decay constants
Ds
simultaneously.


2.7.1 Experimental determination of f
Ds

According to the Standard Model, +
D semileptonic decays occur by pair
s
+

annihilation of the constituent quark (c) and antiquark ( s ) into a virtual W; the decay
constant f parametrizes the matrix element between the quark-antiquark wave
Ds

function and vacuum. The strength of the coupling is the same as in a c s charge-
current process and the decay amplitude is proportional to Vcs:

7 ( + G V m
D f m m . (2. 7-9)
s + ) 2
2
2 2

F cs 2 2
= 1

D D D -

s s s 2
8
mDs
The dependence on the square lepton mass causes the suppression of the decays having
an electron (or positron) in the final state. Table VII shows the measured values of the
relevant branching ratios.
E653 and CLEO respectively determined the ratios
7 ( +
D 7 D (2. 7-10)
s +
 s 
 ) ( + + )=0.
 16  0 06
.  0 03
.
and 7 ( +
D 7 D (2. 7-11)
s +
  ) ( +s +
)= 0.173 0 023
.  0 035
.
from which the values listed in the Table have been obtained using12
7 ( +
D , (2. 7-12)
s +
  )= ( .
2 0  .
0 5)%
7 ( +
D . (2. 7-13)
s +
)= ( .
3 6  .
0 9)%
The measurements of the channel +
D ,
s
+
L3: 7 ( +
D (2. 7-14)
s +
)= (7.4  .
2 8  .
2 4)%
DELPHI: 7 ( +
D , (2. 7-15)
s +
)= ( .
8 5  .
4 2  .
2 6)%
have been converted according to the relation




33


7 ( + +
+ 7 D
D (2. 7-16)
s + s
  ) ( )
9.743
which follows from Eq. (2. 7-9) (see Table VIII for the mass values). The same fixed
ratio between the two branching ratios was assumed by BES and ALEPH, which
determined the value of 7 ( +
D by fitting the data provided by the observation
s 
+  )
of both leptonic channels.


7 ( +
D (%)
s 
+  )
Experiment / Decays
+ + +0.18 0
+ 19
WA7598 D 0 40 .
.
s   -0 14
. -0.18
+
Ds +
  +1.3 +0
BES99 (**) 3
1.5 .
-0 6
. -0 2
.
+
Ds +

+
Ds +
 
E653100 (*) 0.32  0.12  0.10
+ +
Ds  
+ +
L3101 D (**) 0.76  0.29  0.25
s
+ +
DELPHI102 D (**) 0.87  0.43  0.27
s

+
Ds +
 
CLEO103 (*) 0.62  0.08  0.20
+
Ds +


+
Ds +
 
ALEPH104 (**) 0.64  0.08  0.26
+
Ds +

Weighted average 0.512  0.098
2 /6 = 0.93
f = 254  25
D ( )MeV
s


Table VII.  Branching ratio measurements for the +
D decay. The errors
s 
+ 
quoted are statistical and systematic respectively. See the footnote on page 24 for the
description of how the asymmetric errors are included in the average. (*) E653 and
CLEO measured the ratios between the branching fractions of the decays indicated (see
text). (**) Lepton universality has been assumed in converting the +
D result into
s
+
a 7 ( +
D measurement.
s 
+  )



From the average branching ratio, using the data given in Table VIII the result
f = 254  25 (2.7-17)
D ( )MeV
s
is obtained. The error is entirely due to the experimental determination of the decay
rate. The correlation between this result and the measurement of V is completely
cs

negligible. The value of V used for the deduction of Eq. (2.7-17) is the output value
cs

of a unitarity-constrained fit performed by the PDG12 using all the directly measured
CKM matrix elements (it is perfectly consistent with the result we will obtain in Sect. 3



34


after imposing the unitarity constraint); in this case, given its precision, it can be regarded
as a constant (*). The result is in good agreement with a value ( f = (241 36)MeV )
s
D

obtained in Ref. 105 as an application of the Isgur-Wise Theory to the measured rates of
the decays B D
D and B D
D .
s s



m (105 65838
. 9 0 0
. )
00034 MeV
+0 29
.

m (1777.05-0
.26 )MeV
m  (1968 5. 0 6.)MeV
Ds
12
-
 (0.467  0.017)10 s
Ds
V 0.9745  0.0008 90%C. .
L
cs


Table VIII.  List of the parameters used to extract f from the measurements of
s
D

%U ( +
D and %U ( +
D (see Ref. 12).
s
+ )
s 
+  )




2.7.2 Lattice QCD results

Some recent lattice computations of heavy meson decay constants and bag factors are
shown in Table IX and X. The large systematic errors are due to several types of
approximations. For example, the results have been obtained on a finite space-time lattice
and therefore are dependent on the lattice spacing. Moreover, the bag factors have been
computed in a renormalization scheme which is peculiar to the lattice gauge theory, and

have to be converted using a continuum renormalization scheme (such as the MS
scheme), within which the experimental data are analysed. The extreme SU(3) symmetry,
often assumed for the light quarks, and the use of perturbative techniques are two other
sources of uncertainty which the systematic errors usually account for. On the other hand,
the evaluation of the consequences of the so-called quenched approximation is still at a
preliminary stage. In this approximation, which is common to all the computations, the
contribution of the sea quarks in closed loops is neglected, leading to a substantial
reduction in computing time. However, only the MILC collaboration106 has performed an
estimation of the resulting systematic errors; moreover, its investigation of quenching
effects was limited to a very simplified case107, in which only two quark flavours were
considered and the extrapolations to the physical masses and to the continuum were
omitted. While the theoretical panorama is still evolving, at present a truly reliable
estimate of the uncertainties on the currently available results cannot be given.






* The correlation between two measurements can obviously be eliminated if the margin of error
corresponding to the 100% confidence level is assumed for one of them. The error in the
determination of f does not increase if that for V (already corresponding to a 90% confidence
Ds cs

level) is rescaled by a factor of 25.



35


f f f f f f f f f
B (MeV ) B (MeV )
D (MeV ) B D B D B B Ref.
s d s d s s s s d


224  2  16  11 (0.77  0.09) 173  4  9  9 (0.89  0.10) 199  3  10  10 [108]
+ 31 39
+ +4 +0
162  7  5  5 11 6 .
-8 
190 555 
13
 6 9- 0- 02
1.18  0.03 0.05-0 [109]
221  9 (0.73  0.08) 161  16 (0.86  0.06) 190  12 1.18  0.08 [110]
f f 1
213 14
+ B +0.02
d D 0
+ 04
=0 76 .
. 164 14
+ = 0.88 185 +13 +0
-8  9 1 .
11
- 8
11
- 11 0
- 05 05
. -0 03
. . [111]
f f f f -
13 004
.
D D D B
s s s

201  6  15  7 [112]
25
+ 17
+
210 +0 04
. +0.08 25
+ 23
+ +0 05
. +0 05
. +34 +27 +0 04
.
-
9 9 1- 0.75  0.03-0 157 0 85
.  0. 17110 1.11 0.02-0 [106]
.03  0.03
.02 - 0 -
11 9 -0 -
03 003
. -0 -9 -2
1.16  0.03 [113]
237  16 (0.76  0.14) 180  32 (0.87  0.17) 1.14  0.08 [114]
1.17  0.03 [115]
8
+ 7
+ 7
+ +0 04
14711 1758 11 1.20  0 04 .
12
-  9  6 10
- 7-0 . -0 [116]
+6
231 12 0
+
0 78 0
+ 12 + 0
+
.  0 04 . 0 88
.  0 03 .
-0 11
. -0 26
+
179 -
18 9 . -0 28
204 -
16 0 1 14
. 0. -
03 001 [117]
.


Table IX.  Recent lattice QCD results for the heavy-meson decay constants. The errors reflect many different causes of systematic
uncertainties which are inherent in lattice QCD calculations. However, their evaluation is incomplete in almost all cases. In particular,
the effects of the quenched approximation, in which only the valence quark contribution is taken into account, has not been estimated
except for a partial evaluation by the authors of Ref. 106 (see the last error quoted). Moreover, some results are still preliminary. Where
direct determinations are not available, the values of f f and f f obtained by combining the results listed in the same row are
B D B D
d s s s

shown (in brackets) for comparison.





36


The independent predictions for the ratios f f , f f and f f are fully
B D B D B B
d s s s s d

compatible. This evidence can be made even more significant if the values calculated
from f , f and f are indicated where direct determinations are not available.
D B B
s d s

Apparently, the systematic errors tend to cancel out in the ratios. Assuming that a
similar compensatory effect occurs between the quenching errors, the following
estimates will be used for the purposes of the present discussion:
f f , (2. 7-18)
B D = 0 76
.  0 04
.
d s

f f . (2. 7-19)
B D = 0.87  0.04
s s
They correspond to the central value f f =1.145 (ratio of 2.7-19 to 2.7-18), which
B B
s d

all the direct determinations are consistent with. Almost all predictions for f f are
B B
s d

included in the range described by the estimate
f f .
(2. 7-20)
B B = 1 15
.  0 04
.
s d

The comparison of the f predictions from all models with the experimental value
Ds

(2. 7-17) shows a systematic theoretical underestimation: this goes in the direction to
confirm the evaluation of the quenching errors made by MILC. Using the experimental
value of f and Eqs. (2. 7-18, 19) (which are valid under the assumption that the
Ds

systematic errors in the ratios f f , f f are small) the following indirect
B D B D
d s s s

measurements of the B meson decay constants are obtained:
f = 193  22 , (2. 7-21)
B ( )MeV
d
f = 221 24 . (2. 7-22)
B ( )MeV
s

The differences between the central values of these measurements and the average
quenched results f = 165MeV , f =191MeV are nearly equal to the magnitudes of
B B
d s

the quenching errors computed by MILC ( + 23 MeV and + 27 MeV respectively).
On the other and, the direct calculations based on QCD sum rules favour slightly lower
values of the decay constants: for example, the average f = is quoted
B (16030)MeV
d

in a review118 of the latest theoretical developments in this field. However, only the
theoretical determinations of the ratios between decay constants (Eqs. 2. 7-18, 19, 20),
which are expected to be less model-dependent, will be used in the present analysis.

Since no information can be obtained from experiments about the bag factors B
Bd

and B , a value of B with which all values in Table X are compatible is assumed:
B B
s d

B ;

(2. 7-23)
B =1 30
.  0 15
.
d

The error estimate is conservative enough to cover a probable discrepancy which may be
due to quenching effects (roughly 4%119).
No calculation has highlighted any difference between B and B due to SU(3)-
B B
s d

breaking effects: they will be assumed to be nearly equal:
BBs =1 00
.  0.01

(2. 7-24)
BBd
that is, using (2. 7-23),
B B B .
(2. 7-25)
B = B = B =1 30
.  0.15
s d





37


B B B
B B B Ref.
d s d

1.01  0.01 [115]
1.29  0.08  0.05  0.03 [120]
+0.04
1.40  0.06-0 0.99  0.01  0.01 [121]
.26

1.17  0.09  0.04  0.03 1.00  0.02 [113]

1. 46  0.19 1 [122]
0.99  0.03 [110]

Table X.  Lattice-QCD evaluations for the bag parameters entering into the
expressions of the 0
B and 0
B mass differences.
d s



From Eqs. (2. 7-17), (2. 7-18), (2. 7-19) and (2. 7-25) the following estimates of the
hadronic factors occurring in Eqs. (2. 7-7) and (2. 7-8) are obtained:
f B =   =  (2. 7-26)
B B 220 22 17 220 28
d d ( exp th )MeV ( )MeV
f B =   =  (2. 7-27)
B B 252 25 19 252 31
s s ( exp th )MeV ( )MeV
where the first error is due to the experimental uncertainty in f .
Ds



2.7.3 Measurements of m and m
B B
d s


The world average value of m
,
Bd
m ,
(2. 7-28)
B = ( -
.
0 473  ) 1
016
.
0 ps
d

was computed by the LEP B Oscillations Working Group123 using the results of the
measurements carried out by CLEO and ARGUS (at the (4#)), CDF, SLD, and the
LEP experiments . An accurate procedure was followed in order to take all common
systematic uncertainties into account. The same method was applied to the 0
B
s

oscillation data collected by ALEPH, DELPHI, OPAL, CDF and SLD. The fast
oscillations 0 0
B B have not been resolved yet, but a lower limit on m
,
s - s Bs
m -
> , (2. 7-29)
B 3
.
14 1
ps %
95 C. .
L
s

has been set using the amplitude method, of which a detailed theoretical account was
given by Moser and Roussarie124. The time evolution of an initially pure 0
B state is
s

described by the functions
t
0 1 -
P(B ) = e Bs 1+ cos
s ( m t
Bs )
2 Bs
(2. 7-30)
t
0 1 -
P(B ) = e Bs 1- cos
s ( m t
Bs )
2 Bs
They are the probability density functions representing the likelihood that at the time t
the meson is found in the 0
B and 0
B states respectively; the amplitude is defined as
s s
the coefficient of the oscillating term:



38


t
0 0 1 -
P(B , B ) = e Bs 1 cos
s s ( m t
Bs )
2 . (2. 7-31)
Bs

The experiments measure at each fixed value of m
; the expected physical values
Bs

are = 1 if m
is the actual oscillation frequency, otherwise = 0. The amplitude
Bs

spectrum obtained from the combined results is shown in Figure 5. The lower limit
1
-
m is the value at which is incompatible with 1 at a 95% confidence
B = 3
.
14 ps
s

level:
(m m . (2. 7-32)
B ) + 1 645
. (
B ) = 1
s s




3


data  1
2.5 1.645

Amplitude data  1.645
2


95% C.L. limit 14.3 ps-1
1.5 sensitivity 14.7 ps-1



1



0.5



0



-0.5



-1
0 2 4 6 8 10 12 14 16 18 20
( )
m ps -1
Bs
Figure 5.  Spectrum of the 0 0
B B oscillation amplitude, determined by the LEP B
s - s

Oscillations Working Group as a world average of the results achieved by ALEPH,
DELPHI, OPAL, CDF and SLD. The values of the amplitude and its error are
shown as functions of the oscillation frequency m
. The precision of the amplitude
Bs

measurement decreases as m
increases; the sensitivity -1
m is the value
B = .
14 7 ps
Bs s

at which it is no longer possible to distinguish between = 1 and = 0 because of the
increased margin of error. At -1
m the dotted line, representing the graph of
B = .
14 7 ps
s

the function 1.645 (
m , intersects the line = 1, so the values = 1 and = 0
B )
s

cannot be considered as mutually incompatible at a confidence level higher than
P(x > x + .
1 645 ) . The coloured region represents the confidence interval
x = 5%
1 645
. (90%); its upper margin is the graph of (m +1 645 m
; the value
B ) . ( B )
s s

of this function does not exceed 1 until the frequency 1
-
m is reached;
B = 3
.
14 ps
s
therefore, for -1
m the probability that equals 1 is lower than 5%.
B < 3
.
14 ps
s





39


Using the result in Eq. (2. 7-26), the measurement (2. 7-28) of the m
oscillation
Bd

frequency can be translated into the following determination of the product V V :
tb td

V V (2. 7-33)
tb td = ( -
.
7 9  4
.
1 ) 3
10
where the values of the parameters occurring in Eq. (2. 7-7) have been anticipated from
Table XV. This is actually a determination of V itself, since the unitarity condition
td

fixes V as + ( 4
1 ). However, if the result of the direct measurement 2.7-1 is
tb

assumed for V , one obtains
tb

V . (2. 7-34)
td = ( -
2
.
8  .
2 0) 3
10

The result 2. 7-29 can similarly be used to set a lower limit on the value of V .
ts

However, a more tight constraint is provided by the ratio of m (Eq. 2.7-8) to m
B B
s d

(Eq. 2. 7-7):
2
m
m f
B B B V
s s s ts
= . (2. 7-35)
m
m f V
B B B td
d d d

The limit (2. 7-29) correspond, with f f (obtained by combining
B B = .
1 149  .
0 036
s d

Eqs. 2. 7-18, 19, 20) and m given by Eq. (2. 7-28), to
Bd
V V > (2. 7-36)
ts td 4 5
. %
95 C. .
L



2.7.4 CP violation in the neutral kaon system;
K K


The CP violation measured by the parameter corresponds to an asymmetry in
K

the extent to which the neutral kaons K0 and K0 take part in the formation of the K
L

and K mass eigenstates (called long-lived and short-lived states since 600 ).
S K K
L S


In fact only the combinations in which K0 and K0 have the same weight,
0 0
K K  K , CP K =  K , (2. 7-37)

are CP eigenstates, while the S= 2 forces which cause the mixing favour the non-
homogeneous configurations
0 0 1-
K (2. 7-38)
S p K + q K K + q p
+ K-
1+ q p
1- q p
and 0 0
K . (2. 7-39)
L p K - q K K +
- K+
1+ q p
is defined in terms of the amplitudes of the processes 0 0 + -
K
L ,
,S
K

( CP = + ):

A(K
+ -
L )
+- + ,

(2. 7-40)
A(K
+ -
S ) K K
A(K
0 0
L )
00 -
2 .

(2. 7-41)
A(K
0 0
S ) K K

40


Here parametrizes the `direct' CP violation predicted by the Standard Model, which
K

involves the dynamics of S= 1 decays and thus has not the same effect on the two
channels. This asymmetry is accounted for by the different way in which
+- and 00
are dependent on . According to a class of models which refer to Wolfenstein's125
K

`superweak' theory, the observable CP asymmetry of neutral kaon decays should arise
from the S= 2 sector alone: therefore would vanish, the equality +- = 00 =
K K
would apply and the CP violation observed in 0 0 + -
K , decays should be
L
entirely attributed to a CP-even component already present in the initial state K (thus
L
0
q p 1; q p is not independent of the global phase-convention of the states K and

0
K ).

The superweak theory has been unequivocally contradicted by the recent
preliminary results of the KteV and NA48 experiments: the new measurements of the
ratio Re are incompatible with the value at a virtually 100%
K K = 0
K K ( K K )
confidence level (6.8 and 2.5 respectively):

KteV: Re K = (28.0  3 0
. stat  2 6
. syst 1 0
. MCstat ) -4
10 = (28 0
.  4. ) -4
1 10
K

NA48: K - -
Re = ( .
18 5  5
.
4
stat  .
5 8syst ) 4
10 = ( .
18 5  3
.
7 ) 4
10
K
(2. 7-42)

Experiment Re(
K K )
-
E731126 (7 4. 5 9.) 4
10
-
NA31127 (23.06 5.) 4
10
-
KteV128 (28.0 4. ) 4
1 10
-
NA48129 ( 5.
18  .
7 3) 4
10
( -
.
21 2  6
.
4 ) 4
10
Average 2/3 = 2.8, s = 1.7

Table XI.  Experimental determinations of the parameter Re( which quantifies
K K )

the direct CP violation in the neutral kaon system. The scale factor s =1.7 has been
applied to the error in the average because of the slight discrepancy between the
measurements.


The margin of certainty reduces slightly if the world average (see Table XI) is assumed
as the best value of Re( :
K K )
K -
Re = (21 2
.  6
.
4 ) 4
10
. (2. 7-43)
K

The average is, however, incompatible with the lack of direct CP violation at a
99. 9999% (4.6 ) confidence level.
It has often been argued130 that a value of greater than 210-3 cannot be
K K

accounted for within the present model of CP violation based on a single complex phase




41


in the CKM matrix. In actual fact, the evidence for direct CP violation is the only
straightforward, unambiguous conclusion which can be drawn from the experimental
result, while, at present, large theoretical uncertainties affect the Standard Model
computations of the magnitude of the CP-violating effect. Most computed values (see
Table XII) are consistent with a lower value than the measured one. However, they are
strongly dependent on the strange quark mass (roughly 2
m ), of which the
K K 1 s

most recent estimates, based on lattice-QCD calculations, give much lower values than
before. The Standard Model prediction is summarized by the following expression:131
2
K 150MeV (1 2)
= ImV V . (2. 7-44)
td ts - .
1 35 + 1
.
1 S B .
1 0 67
.
0 S B
6 + ( - ) (3 2)
[ 8 ]
K m (m )
s c

Here S represents the short-distance QCD effects, which have been calculated up to the
next-to-leading order:140,132
6.5 S 8.5;
(2. 7-45)
(1 2)
B and (3 2)
B are the hadronic matrix elements of the operators
6 8 6
O and 8
O which
contribute to the effective Hamiltonian; they are evaluated in the 0
K ( )I=0
(I= 1/2) and 0
K ( ) (I= 3/2) transitions respectively:133,134,135
I =2 (1 2)
0 8 3 2
. 6
B 1.3, ( )
0 6
. 8
B 1 0
. . (2. 7-46)

The term containing the CKM matrix elements is equal to
2 5 4
-
ImV V (2. 7-47)
td ts A .
1 24 10
at the leading order in . The s quark mass has to be evaluated in the  m 1.3GeV
c

scale. If the whole spectrum of the available theoretical predictions for m were taken
s

into account (the range 60 MeV m 170 MeV, for  2GeV , is quoted by the
s

PDG12), a rather indefinite estimate of would be obtained. On the other hand, if
K K

an `average' ( 120 MeV ) value is assumed for m , Eq. (2.7-44) and the measurement
s

of cannot be made compatible. The lattice QCD results show a much better
K K

agreement with the experimental data. The range of the quenched results is136


-4 K -4
- 2 1
. 10 13.310
[137]
K
K = (4.63.00 4.) -4
10
[138]
K
K = ( 14+ -
17 10
10
- ) 4
[139]
K
.
5 3 .
3 8 10 4 using ( ) 150 20
K (  ) - m m
s c = (  )
= MeV [140]
4
K (8 5. .59) -
10 using m (m ) 125 20
s c = (  )MeV
K = ( -
.
15 0  .
4 ) 4
8 10
[141]
K


Table XII.  Expected values of the parameter according to some Standard
K K

Model calculations.



42



100 MeV m GeV
2 150 MeV; (2. 7-48)
s ( )quenched
if the quenching effects are taken into account, there should be a 40% reduction:142
60 MeV m 2 90 MeV; (2. 7-49)
s ( GeV )

the corresponding values in the  = m scale are
c
70 MeV m m 100 MeV. (2.7-50)
s ( c )

The following range of allowed values is obtained from Eqs. (2. 7-44, 45, 46, 47):
0.7 10-3< K
Re < 4.3 10-3, (2.7-51)
K

which is fully compatible with the average experimental value in Eq. (2. 7-43).
is directly proportional to the imaginary part of the CKM matrix (Eq. 2.7-47);
K K

however, due to the large uncertainty in its expected value, the use of this measurement
as a constraint would have a completely negligible effect.



2.7.5 The constraint
K


The PDG12 averages for the CP violation parameters of the kaon system are
+- = (2 284
.  0 018
. ) -3
10 ,
(2. 7-52)

00 +- = 0 9930
.  0 020
. ;
(2. 7-53)

their product is
00 = (2 268
.  0.018) -3
10 .
(2. 7-54)

Inverting the relations (2. 7-40) and (2. 7-41) to extract the modulus of
, one gets
K
+-
= +- ,
(2. 7-55)
K
K
K
1+ 1+ Re
K
K
00
= 00 (2. 7-56)
K
K
K 1- 2
1- 2 Re
K
K

and using Eq. (2. 7-43) together with either (2. 7-52) or (2. 7-54), from the preceding
expressions the measurement
= (2.
(2. 7-57)
K 279  0 018
. ) -3
10
is obtained. The contribution of the c quark loop cannot be neglected in the kaon
system:96
2 2
i 4 G m m
F W K 2
Im Re (2. 7-58)
K = e f B
K K ( M + 2
12 M12 )
2
12 2 mK
2 2

with M
12 = S(x ) V V S (x ) V V S (x , x V
) V V V .
cc c ( cs cd ) + tt t ( ts td )
+ 2 ct c t cs cd ts td

(2. 7-59)
M12 is proportional to the non-diagonal element of the neutral kaon mass matrix, which
0 0 0 0
represents K - K mixing: M K + ( ) ; is the ratio
eff S = 2
12 K




43


Im [
A K ( )I =0 ]
= ,
(2. 7-60)
Re [
A K ( )I =0 ]
m
the mass difference between the K and K autostates; the QCD corrections to
K L S

the Inami-Lim functions are the factors
=1 38
. , = 0 574
. , = 0. ,
ct 47  0 04
.
tt  0 004
.
cc  0.53
(2. 7-61)
which have been calculated up to the next-to-leading order95,96. The decay constant f
K

and the bag parameter B determine the magnitude of the hadronic matrix element
K
2
0 5 0 2
K [s (1- )d] K m f B . A precise value of f can be deduced from the
K K K K

measurement of the leptonic decay rate of kaons12:
f = 159.8 1.5 . (2. 7-62)
K ( )MeV
On the contrary, the parameter B cannot be measured. The two most recent lattice
K

QCD computations give perfectly compatible results:
Kilcup et al. 143 B ( GeV
2 ) = 62
.
0  02
.
0  .
0 02 ,
(2. 7-63)
K stat syst

JLQCD144 B ( .
(2. 7-64)
K 2GeV ) = 0 628
.  0.042
JLQCD's datum is the result of an extended simulation, which included a detailed
analysis of the systematic effects of discretization and the renormalization scheme
conversion. The scale-independent constant B entering into the expression 2.7-58 has
K

to be determined by applying a suitable transformation to the value computed at
 = 2GeV . This can be done either with reference to the physical situation in which
three light dynamical quarks ( n ) exist, or within the quenched approximation
f = 3

( n ) already used to deduce the results 2. 7-63 and 2. 7-64. Allowing for both
f = 0

possibilities133, which lead to slightly different results, the relation
B 1.34
K = =1.36  0 02 (2.7-65)
B (
K 2 .
GeV ) 1.38 (quenched)
will be used. If the contribution of the dynamic quarks is included (unquenching) and the
deviation from the SU(3)-symmetry assumed for light quarks ( m = m = m ) is taken
u d s

into account, the value of B should increase by a further 10% amount:145
K

BK =1.05  0 02
. , (2. 7-66)
(B )
K quenched

BK =1.04 -1 05
. . (2. 7-67)
(B )
K SU (3)

When Eq. (2. 7-64) is multiplied by Eq. (2. 7-65) and the corrective factor
B (B ) is applied (with a conservative increase in the margin
K K quenched+SU (3) =1.10  0 05
.

of uncertainty in order to allow for a possible systematic error in the evaluation of the
unquenching correction145) the result
B = 0.

(2. 7-68)
K 94  0.08
is obtained.
The term proportional to the real part of the box diagram will be neglected in the
expression of , since it is numerically insignificant (it has been estimated96 as a
K

correction not exceeding 2% compared to the size of the term proportional to the




44


imaginary part). This approximation is justified on account of the considerable
theoretical uncertainty in B , but the whole expression will have to be considered (and
K

the parameter will also have to be evaluated) as soon as the error in the determination
of B is reduced. For the present, an additional 2% error is attributed to the
K

measurement of .
K

The following constraint on the CKM matrix elements is provided by the present
0 0
experimental and theoretical information about CP violation in the K - K system:
2 2 2
G m m f
B S(x )Im V V S(x ) Im V V
K = F W K K K { cc c ([ cs cd)2 + tt t ts td +
2 ] ([ )2]
12 2 m (2. 7-69)
K
+ 2 S(x , x )Im V V V V
ct c t (
cs cd ts td )}
with
= (2. . (2. 7-70)
K 28  0 05
. ) -3
10


2.7.6 b s penguin decays

The first observation of the bs electromagnetic penguin decays (see Figure 6),
performed by CLEO146, consisted in the identification of the exclusive channel
B
K (892
) . The measured decay rate of this process cannot be used to extract
accurate information about the weak coupling of the quarks involved in the interaction,
because of major theoretical uncertainties in the description of the hadronization process.
Recently, CLEO and ALEPH have carried out independent measurements of the
inclusive branching ratio 7 (b X , providing quite a reliable way of constraining
s )

the CKM matrix elements V and V . The Standard Model prediction, normalized to
ts tb

the background processes b
c , has been calculated up to the next-to-leading
order147, assuming V V V and 7 (b X
c ) = (10.4  .
0 4)%
ts tb = 0.
cb 976  0 010
.

(the values are taken from [67]):
7 (b
s ) = (3.28  0. ) -4
33 10 .
(2. 7-71)






b s
t

W Vts

Figure 6.  Example of electromagnetic penguin diagram describing the process b s.


Since
7 (b s ) 2 2
V V
ts tb , (2. 7-72)
7 (b c) 2
Vcb



45


the following expression relates the CKM matrix elements to the measured quantities:

7 (b X 7 b X V V
s ) ( c ) 2 2
ts tb
( =
- ; (2. 7-73)
28
.
3  .
0 30) 4
10 .
0 104 ( 976
.
0 )2 2
Vcb

the errors coming from 7 (b X and V V V have been subtracted from
c ) ts tb cb

the total error. CLEO and ALEPH have reported the following results:
CLEO148 7 (B X , (2. 7-74)
s )= ( -
15
.
3  .
0 35stat  41
.
0 syst ) 4
10

ALEPH149 7 (b X . (2. 7-75)
s )= ( -
.
3 11 80
.
0 stat  72
.
0 syst ) 4
10

The different way in which b quarks are produced in these experiments (at the (4#)
resonance and from Z decay) should be taken into account choosing the corresponding
measured value for 7 (b X . However, since the CLEO measurement is the
c )

main contribution to the weighted average
7 (b X , (2. 7-76)
s ) = (3.14  0. ) -4
48 10
it will be assumed that
7 (b X 7 12
) = (B X ) = 
,
(2. 7-77)
(4S ) ( .
10 45 )
21
.
0 %

therefore, after subtracting the b X contribution (Eq. 2. 4-2),
u
7 (b X .
(2. 7-78)
c ) = ( 25
.
10  .
0 )
23 %

By substituting the values 2. 7-76 and 2. 7-78 into Eq. (2. 7-73), the result
2 2
V V
ts tb = 93
.
0  14
.
0  08
.
0 = 93
.
0  .
0 17 , (2. 7-79)
2
Vcb
is obtained. The first error is the uncertainty in the measurement of the branching ratio;
the second error is associated with its theoretical computation. This corresponds, with

V 1 and V given by Eq. (2.5-10), to the following determination of V :
tb cb ts

V = (38.1  3.8)10-3. (2. 7-80)
ts
























46


2.8 Summary


The best values obtained so far in the present work for the CKM matrix elements
are listed in Table XIII, where they are compared to those quoted by the Particle Data
Group (updated in January, 1998).
In most cases, a reduction in the experimental uncertainties has been obtained by
taking into account the most recent measurements. For example, data from deep inelastic
scattering of neutrinos on nucleons and from hadronic 
W decays have been included in
the V average, leading to a reduction by a factor of 7 in the final uncertainty. A
cs

significant improvement has also been achieved in the determination of V V , since
ub cb

new, independent measurements of V and V V have made it possible to
ub ub cb

discriminate between the disagreeing theoretical predictions which were used in the end-
point inclusive analyses of non-charmed B decays.
In other cases, the determinations of the matrix elements have not changed
considerably; however, all the values obtained are the result of a detailed re-analysis of
the available information, in which the relevant experimental parameters have been
updated to the most recent world averages.


PDG'98 this
analysis

V 0.9740  0.0010 0.9743  0.0008
ud

V 0.2196  0.0023 0.2200  0.0025
us

V (3.3  0.8)10-3 (3.6  0.5)10-3
ub

V V 0.080  0.020 0.090  0.008
ub cb

V 0.224  0.016 0
+ .013
cd 0 225
. - .
0 011

V 1.04  0.16 0.996  0.024
cs

V (39.5  1.7)10-3 (39.5  1.7)10-3
cb

V 0.99  0.15 + 16
.
0
tb 0.96- 12
.
0

V V
tb td (8.4  1.8)10-3 (7.9  1.4)10-3
(from B - B oscillation)
d d


V V
ts td > 3.7 95% C.L. > 4.5 95% C.L.
(from B - B oscillation)
s s


V V V
ts tb cb 1.1  0.43 0.96  0.09
(from b s)

Table XIII.  Comparison between the values of the CKM matrix elements obtained in
the present analysis and those from the Review of Particle Physics (January `98).









47


3 An up-to-date profile of the unitarity triangle

As far as possible, all the experimental information collected in the previous Section
has been treated in such a way that any correlation among the single determinations has
been avoided or reduced to a negligible level. As a result, a set of independent
constraints on the CKM matrix elements has been outlined. In this Section, we refine the
determination of the CKM matrix and obtain estimates for other parameters of physical
interest by imposing all these constraints simultaneously and requiring explicitly that the
three-family unitarity condition of the matrix be satisfied. This further constraint is
imposed (together with the removal of the non-physical complex phases) by expressing
the matrix elements in terms of a four-variable parametrization (either Wolfenstein's or
the canonical parametrization). The number of independent constraints which is needed
to make the problem completely determined is then exceeded by a wide margin.
Therefore, the precision in the values of the CKM matrix elements can be improved
considerably with respect to those obtained from direct measurements. At the same time,
the unitarity triangle, which represents one of the unitarity relations on the complex
plane, can be determined, and the magnitudes of the CP asymmetries in B decays can be
evaluated.
Two different methods are followed independently for the overall determination of the
CKM matrix and the unitarity triangle. The first procedure, a 2-minimization, is
discussed in Sect. 3. 1 and the results are reported in Sect. 3. 3. A number of additional
trials is also performed, in which the principal constraints are removed in turn to show
how they affect the results. In Sect. 3. 2 the information which is effective in
constraining the position of the vertex of the unitarity triangle is represented
geometrically on the (, ) plane using Wolfenstein's parametrization. An account of
the second method, based on Bayesian statistics, and the results obtained from its
application are presented in Sect. 3. 4. Finally, in Sect. 3. 5 the impact of the
forthcoming experiments at the B factories is evaluated, and the importance they have as
essential instruments for the verification of the Standard Model is emphasized.




3.1 Description of the fit procedure


The information used in the present analysis to constrain the CKM matrix is
summarized in Table XIV. The matrix elements have been expressed in terms of
Wolfenstein's unitary parametrization. Each term has been calculated up to the fourth
order in . For this purpose, the expansion of V has been extended further than in
CKM

Eq. (1. 1-7), considering the real and the imaginary parts separately. While it is true that
a fourth-order computation is not justified by the present degree of precision of the data,
future improvements in the theoretical and experimental uncertainties will increase the
sensitivity of the results to the accuracy of these computations. For example, the fourth-
order correction is already comparable to the magnitude of the error in the case of V
ud

and it would not even be possible to use the direct measurement of V as a constraint if
tb

the expansion were truncated at the second order.



48


The maximum-likelihood estimates of the parameters , A, and result from the
minimization of the function
2
2
( , ,
,
, A,
, ) (O A O
i ( )- i )
= 2
.
(3. 1-1)
i Oi

In each term, the unknown member O of a constraint equation is compared
i (, ,
A , )
with the measured value O within the error corresponding to one standard
i Oi

deviation (all the input data of the fit are assumed to be normally distributed).


Term Value Expression Eq./Ref.
2 4

V 0.9743  0.0008 1- - (2.1-25)
ud 2 8
V 0.2200  0.0025 (2.2-16)
us

V ( -
6
.
3  0 5
. ) 3
3 2 2
A + (2.4-4)
ub 10

V V 0.090  0.008 2 2
+ (2.6-4)
ub cb

4
2
V + .
0 013
225
.
0 1- A (1- 2 ) (2.3-32)
cd -
.
0 011 2
2
1 2
4 A
V 0.996  0.024 1- - + (2.3-33)
cs
2 8 2
V (39.5 1.7) -3
10 2
A (2.5-10)
cb

4
V + .
0 16
96
.
0 1- 2 (2.7-1)
tb - A
0 12
. 2
2 2
V V
ts tb 1- 2 1- 2 - 4[ 2
A -2 + 1- ]
2 0.93  0.17 ( ) ( ) (2.7-79)
Vcb
2
2 2
G fB m
F 2
m m m
= d f B S t A
Bd 2 W Bd ( Ds )2 2 6
B B 2 ( 2 2 2 2 2
{1- ) + - ( + - )+
6 f m
m ( - Ds W
0.473  0.016) 1
B ps (2.7-7,28)
d
4 1 2
+ - A ( 2
+ 2
)+ 2
A (2 - )
1
4
2
2
G f
B 2
m
F 2 s 2
m -1
> m = m m f B S t A A (2.7-8,29)
Bs 2 W Bs ( Ds ) 24 2 4 2 2
B B 2 {1- (1-2)- [ - + (1- )]}
B 14.3 ps 95% C. .
L
s
6 fDs mW
2 2 2
G m m f
F W K K 2 2
6 m 2 4
=
B A S c A
K 2 K - 2
cc 2 1- - (4 (3-2 )+ )
1 +

6 2 m
K mW 2 8
2
m
t 2 4 2 2 2 1 4 1 2
2 1 2
(2 28
.  0. ) 3
-
05 10 + S (2.7-69,70)
tt A A
2 1- - + - 2 + + - + + (4 - )
1 +
K mW 2 2 2 8
2 2
m m 2

c t 4 2 5 1
+ S ,
ct A
2 2 1- - - 2 +
m m
W W 2 2 8
2 (1- )
+0 41
sin2 0.79 .
-0 44 [13]
. (1- )2 2
+

Table XIV.  Constraints on the CKM matrix in Wolfenstein's parametrization. Each
single term in the expressions has been calculated up to ( 4
) corrections.


A number of theoretical or experimental quantities, such as B , B , f and m ,
B K D t
s
on which the expressions of m
, m
and are dependent, are not known
B B K
d s

precisely enough to be regarded as constants. Therefore, they are handled as variables of



49


the minimization together with , A, and . Their measurements or theoretical
estimates are used as additional constraints, which are imposed by adding further terms
to the total 2, such as (f f
D - D
s s )2
2
. (3. 1-2)
f = 2
s
D f sD
The constrained variables of the fit and the parameters which, because of the
comparatively small errors (< 1%), are chosen to be fixed at their central values are
listed in Table XV.
By allowing some `known' parameters to vary, a strong coupling between the
equations for m
and m
is induced. Therefore, B , f , f f and f f ,
B B D B D B D
d s B s d s s s

instead of f B and f B , will be used as parameters of the fit when the
B B B B
d d s s

constraints m
and m
(2. 7-7 and 2. 7-8) are applied simultaneously: in this way,
B B
d s

the correlation arising from the common use of the measurement of f is taken into
Ds

account. f f and f f will be further constrained by means of Eq. (2. 7-20) (the
B D B D
d s s s

determination of f f is regarded as an independent piece of information):
B B
s d

f f
B D
s s = 1.15  0 04
. . (3.1-3)
f f
B D
d s


Variables Eq./Ref. Constants Ref.
f f = 0 574
.
tt  0.
B D = .
0 76  04
.
0 (2.7-18) 004 [95,96]
d s

f f 5
- -
G [12]
F = 
B D = 87
.
0  .
0 04 (2.7-19) (1.16639 0 ) 2
00001
. 10 GeV
s s
f = 254
.
0  025
.
0 m = .
80 41  0.10
W ( )GeV
D ( )GeV (2.7-17) [12]
s
B B B m = 2792
.
5  0018
.
0
B ( )GeV
B (= B = B (2.7-25) [12]
d s ) = .
1 30  .
0 15
d
= 0 55
. [95,96] m = .
5 3692  .
0 0020
B ( )GeV [12]
B  0.01
s

m = 166  5 15
-
= 
t ( )GeV [12] m 3.489 0.009 10 [12]
K ( ) GeV

m = 1 25
.  0.10 m = .
0 497672  .
0 000031
c ( )GeV [12] [12]
K ( )GeV
=1. [95,96] f = 0.1598  0.0015 [12]
K ( )GeV
cc 38  0 53
.
= 0 47
.
ct  0.04 [95,96]


B = 0. (2.7-68)
K 94  0.08

Table XV.  Variable and constant parameters of the fit.


The m
constraint is imposed following the same pattern of the procedure adopted
Bs
in the search for 0 0
B B oscillations (see Sect 2. 7. 3). The variable is expected to be
s - s

distributed normally with mean value equal to 1 in the presence of oscillations. For each
value of m
, the value assumed by is determined using the data represented in
Bs

Figure 5 and the term
2
(m )
B -1
s
(3. 1-4)
(m )
Bs


50


is added to the total 2. m
is in turn determined by the current values of the
Bs

parameters of the minimization using equation (2. 7-8). Therefore, the maximum
2
probability ( - 2
P e ) is attributed to the values of the parameters that yield
measurements of compatible with 1. Following this method, the information contained
in the whole amplitude spectrum become effective, while the single measurement
( 1
-
m was used to deduce the lower limit (2. 7-29).
B = 3
.
14 ps
s )
The complete expression of the total 2 is given below. The central values of the
experimental or theoretical determinations are indicated with a superscript bar. Gaussian
errors corresponding to one standard deviation are assigned to them. The remaining
symbols represent the unknown terms of the constraint equations calculated as functions
of the parameters.
(V V V V V V V V V V
ud - ud )2
2 ( us - us )2 ( ub - ub )2 ( ub cb - ub cb )2
= + + + +
2 2 2 2

V V V V V
ud us ub ub cb
(V V V V V V V V
cd - cd )2 ( cs - cs )2 ( cb - cb )2 ( tb - tb )2
+ + + + +
2 2 2 2

V V V V
cd cs cb tb

2
2 2 2
V V V V V V
ts tb cb - 2 2 2
ts tb cb
2
(m m
B - B
d d )
+ + +
2 2

2 2 2 m
V V V d
B
ts tb cb

(m f f f f f f f f
B K K B D B D B D B D
s ) 2 2 2 2
+ -1 ( - ) ( -
d s d s ) ( -
s s s s )

2 2 2
(m
B f f f f
s ) + + + +
K B
s
D
d
B s
D
s
2
f f
B D
s s - f f
B B
s d
f f 2 2 2 2
f f
B D B B m m
d s ( D - D
s s ) ( B - B) ( B - B) ( t - t)
+ + + + + +
2 2 2 2 2

f f f B m
B B B t
d
B
s s
D
(m m B B
c - c )2 (cc -cc)2 (ct -ct)2 ( K - K)2 2
sin2 - sin2
+ + + + +
2 2 2 2
m B
c
cc
ct K (sin2 )
(3. 1-5)
The asymmetric errors in the measurements of V , V and sin2 have been handled
cd tb

as already described in the footnote on page 24.
24 constraints (i. e. the number of 2 terms) and 14 parameters (, A, , and those
listed in the first column of Table XV) correspond to 2414= 10 degrees of freedom in
the fit.
The minimization has been performed using the numerical libraries of MINUIT150.
The errors quoted in the results are the ones, asymmetric, computed by the MINOS
subprogram taking into account the non-linearity of the problem.









51


3.2 Constraints on the unitarity triangle in Wolfenstein's
parametrization.


One of the advantages in the use of Wolfenstein's parametrization is that the
effectiveness of a constraint can be estimated at first sight. In fact, the present status of
the uncertainties is such that some constraints are almost ineffective in comparison with
the much stronger requirement of unitarity. For example, the ( 5
) term of
4
2
V = 1- A (1- 2) ,
(3. 2-1)
cd
2
i. e. the fourth-order correction in the magnitude of the matrix element, is completely
negligible compared to the experimental error. Therefore, the measurement of V and
cd

that of 7
V = + ( ) impose essentially the same constraint on the parameter , but
us

the former is six times less precise and thus insignificant. Analogously, V , V and
cs tb
2 2 2 2
V V V differ from unity by ( ) terms (see Table XIV), which can be
ts tb cb

neglected given the comparatively low precision of the measurements. The error in V
ud

is probably just below the level of precision which would make the constraint useful.
2 2 2
The expressions of m
and V V V have the same functional dependence on
B ts tb cb
s

the parameters, which occurs only at the second order in :
2 2 2 2
m V V V . (3. 2-2)
B ts tb cb 1- (1- 2 )+
s ( 4)
However, the `measurement' of m
does have a non-negligible effect as a constraint
Bs

on the vertex of the unitarity triangle. The equations for m
and m
are in fact
B B
s d

strongly coupled together by the common factor 2
f B S m m and the additional
D B B
s ( 2 2
t W )

constraint (3. 1-3). The m
constraint can thus be replaced by the equivalent equation
Bs
2
m m f V m f
B B B B B
s s s ts s s 1 1
= = +
2 2 ( 2 , (3.2-3)
2 )

m m f V m f
B B B td B B
d d d d d (1- )
+
which is significant at the lowest order in .
The way in which the constraints applied to the CKM matrix define an allowed region
for the vertex ( 2
, ) = (1- 2)(
, ) of the unitarity triangle is shown in Figure 7. The
measurement of the ratio V V defines an annulus centred in the origin on the ( , )
ub cb

plane:
2 2 2 2
2
V V . (3. 2-4)
ub cb = + = + 1-
2
Since V = A2 V V has the same dependence on and , the graph represents
ub ub cb

the combined measurements of V V , V and V (with average V V =
ub cb ub cb ub cb
0.090  0.007). At the lowest order in , an annular region with centre in ( , ) =
(1,0) is favoured by the measurement of m
(see Table XIV):
Bd
(1- )2 2
+ (1- )2 2 3
+ m . (3. 2-5)
B
A
d





52


The dotted arc shown in the figure circumscribes the area defined by the constraint
-1
m , which has been expressed as a lower bound of m
m (3.2-3),
B > 3
.
14 ps B B
s s d

with f f fixed at its central value. The dependence of on and is, at the
B B K
s d

lowest order in ,
B A2
6 - S x + S x , x + A24 1- S x , (3. 2-6)
K K { cc ( c) ct ( c t) ( ) tt ( t )}
where (see Eqs. 2. 7-3, 2. 7-4 and the mass values in Table XV)
S (xc )= (2 42
.  0.39) -4
10
S (x

(3. 2-7)
t ) = 2.38  0 11
.

S (x , x
c t ) = (2.15  0. ) -3
31 10
The foregoing expressions define the region included between the two lines shaped like
hyperbolas. The direct measurement of the angle is represented by a cone with the
vertex in (1, 0). Clearly, this measurement is not precise enough to constitute an effective
constraint.



10.9 m
m Bd
0.8 Bs
(95% C.L.)
0.7

0.6 | / |
V V
ub cb
0.5
| |K
0.4


0.3

0.2 sin2

0.1


0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
Figure 7.  Graphic representation of the experimental constraints on the vertex of the
unitarity triangle. The continuous lines mark the boundary of the regions favoured by the
constraints , m
, V V and sin2 at the 68% confidence level. They have
K B ub cb
d

been calculated by allowing all the parameters to vary inside their respective 1
intervals. The limit on m
(95% C. L. ) is represented by the dotted line. The graph
Bs
also indicates the most probable shape of the unitarity triangle as determined by the
simultaneous application of all constraints.









53





3.3 Results


The complete solution of the fit is given in Table XVI (CKM parameters and matrix
elements) and Table XVII (output values of the additional parameters). The most
probable shape of the unitarity triangle and the contours of the 68 and 95% confidence
regions for the vertex (, ) are shown in Figure 8.
The maximum-likelihood estimates of , , sin2, sin2, = , ,
13 12 23 and 13
have been obtained by repeating the minimization with the use of different
parametrizations. Wolfenstein's parameters and can be replaced by the coordinates

, of the vertex of the unitarity triangle using the one-to-one relations (1. 2-6). A

choice between several non-equivalent ways of inverting the relations (1. 2-8) has to be
made in order to express the parametrization in terms of sin2 and sin2. In fact, while
and can be placed into one-to-one correspondence with tan and tan
tan(-- ),
tan
= tan =
1+ tan tan

(3. 3-1)
=
tan = tan tan
1-
1+ tan tan
sin2 and sin2 cannot be inverted unless the domains of and are already known.
Assuming that and belong to the intervals

0 < 2 <
3
and <
2 < , (3. 3-2)
2 2 2
which are favoured by the previously obtained results for and , the relations

1
= - 1
arcsin sin
(
2 ) , = 2 (3. 3-3)
2 2 arcsin sin
( )
2
hold. This choice is also supported a posteriori, for the minimum 2 and the output
values of all other parameters are the same as those yielded by the primary fit. When the
constraint equations are expressed in terms of the canonical parametrization (1-3), the
values of the three Euler angles , , and the phase = are obtained as a
12 23 13 13
result of the fit. An alternative possibility of estimating the angle is given by Eqs. (3.3-
1) and (3. 3-3), from which and can be extracted as functions of sin2, or
sin2, . The results are displayed in Figure 8 in terms of three different couples of
parameters.

The constraints have in turn been released by subtracting the corresponding terms
from the total 2. In this way, the influence of each single constraint on the
determination of the unitarity triangle has been ascertained. The most significant results
are listed in Table XVIII. The way in which the confidence regions for (
, ) vary in

the four most interesting cases can be seen in Figure 9.






54




68% C.L. 95% C.L.
+
0020
.
0
2219
.
0 - 0.2179  0.2258
0 0021
.

A 0.798  0.029 0.743  0.868
+0 046
.
175
.
0 - 0.103  0.288
0 034
.

+0 031
.
354
.
0 - 0.275  0.415
0 032
.

+
sin2 0 20
.
- .
0 11- -0.73  0.26
0 22
.
+
sin2 0 044
.
725
.
0 - 0.632  0.809
0 046
.
= ( +
.
63 7- 45.4  74.4
7 0
. )
3
.
5
13

(12.82  0.12) 12.58  13.05
12
( + .
0
.
2 250- 2.12  2.43
0 071
. )
074
23
( +0
.
0 202- 0.176  0.230
0 013
. )
014
.
13

97508
.
0 (+45 25
- 0 2218
. 20 0 00353
. .
0 97417  97597
.
0 0 2179
.  0.2258 0 00309
.  00402
.
0
46 ) ( ) (+-24)

V 0 2217
. (20) 0 97432
. (46) 0 0393
. (+12- 0 2178
.  .
0 2257 0 97341
.  .
0 97522 0369
.
0  0424
.
0
13 )
ij +
32 +13 +
00782
.
0 ( 48 00701
.
0  0.00846 0.0363  .
0 0418 0 999118
. 
- 0386
.
0 999223
.
0 0 999316
.
33 ) (-12) (-51)
2 = 2.6

Table XVI.  Results of the fit.



68% C.L. 95% C.L.

f f 0
+ 29
. 0.705  0.817
B D 760
.
0 -
d s 0 28
.

f f 0.872  0.031 0.811  0.933
B D
s s
f +16 228  289
D 257 MeV
- MeV
s 15
B = B = B 1.31  0.13 1.06  1.58
B ( B B
d s )
0.550  0.010 0.531  0.570
B
m 165.9  4.9
t GeV 156.3  175.5 GeV
m 1.244  0.098
c GeV 1.05

1.44
GeV
1.42  0.51 0.42  2.42
cc
0.468  0.039 0.391  0.545
ct
+
B .
0 073
930
.
0 0.790  1.074
K -0 072
.


Table XVII.  Output values of the additional parameters entering into the expressions

of , m and m
.
K B B
d s








55


1 1
2
0.9 0.9


0.8 0.8
sin
0.7 0.7


0.6 0.6


0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
sin 2

1 1
2 2
0.9 0.8


0.8
sin 0.6
sin
0.7 0.4


0.6 0.2


0.5 0


0.4 -0.2


0.3 -0.4


0.2 -0.6


0.1 -0.8


0 -1
30 40 50 60 70 80 90 100 30 40 50 60 70 80 90 100
(degrees) (degrees)

Figure 8.  Contours of the 68 and 95% two-dimensional confidence regions for the vertex (, ) of the unitarity triangle and the
observables sin2, sin2 , .




56


sin2 sin2 V (68%)
CKM

97508
.
0 (+45 25
- 2218
.
0 20 0 00353
.
46 ) +
+ ( ) (-24)
0 046
. + + +
.
0 175 0 031
. 0 20
. 0 044
. ( +
.
63 7
- 0
.
7 )3.
5
- .
0 354- - .
0 11- .
0 725- +
all the constraints 0.034 0 032
. 0 22
. 0.046 0 2217
. (20) 97432
.
0 (46) .
0 0393 ( 12
-13 )
0.103  0.288 0.275  0.415 -0.73  0.26 0.632  0.809 45.4  74.4 +32 +13 +
00782
.
0 ( 48
- .
0 0386 999223
.
0
33 ) (-12) (-51)
+28
+ .
0 97508 (46) 0 2218
. (20) .
0 00355 (-27 )
064
.
0 + + +
0 186
. 042
.
0
0 350
. .
0 47
- 0 17
. .
0 044
0 725
. (62+
10
- )
14
without m
- .
0 090 - .
0 044 - 36
.
0 - 048
.
0 0.2217 (20) .
0 97431 (+47 17
- .
0 0394
46 ) (+-16)
s
B -0.041  0.299 0.266  0.429 -0.83  0.82 0.624  0.812 43  96 +62 +
.
0 00775 ( 62
- 0 0388
. 17 0 999216
.
49 ) ( ) (-66)
.
0 97508 (46) 0 2218
. (20) 0 00357
. (+29-28)
+
068
.
0
0 196
.  + 53
.
0
- 0 22
. + 045
.
0
0 726
. (60+
10
- )
16
without 0 346
. 0 045
.
m
and m
- .
0 104 - .
0 37 - .
0 049 .
0 2217 (20) 0 97431
. (+46- .
0 0395 16
47 ) ( )
Bd s
B -0.386  0.314 0.261  0.430 > -0.82 0.625  0.811 42  154 +73 +
.
0 00767 ( 64
- 0 0389
. 17 0 999213
.
53 ) ( ) (-66)
+29
+ .
0 97508 (46) 0 2218
. (20) 0.00357 (-28)
046
.
0 + + +
0 172
. .
0 032
0 358
. .
0 20
- 0 08
. .
0 045
0 728
. ( + .5
64 3
.
- .
7 6 )
3
without - .
0 035 - .
0 035 - .
0 24 - .
0 048 0.2217 (20) 0 97431
. (+46 16
- 0 0395
.
47 ) (+-15)
K 0.099  0.312 0.232  0.421 -0.91  0.30 0.629  0.812 38  75 0 00791
. (45) +16 +
0 0389
. ( 61
- 0 999211
.
15 ) (-64)
0 97508
. (46) 0 2218
. (20) .
0 00339 (+74
-54 )
without V V + 065
.
0 + 081
.
0 + .
0 23
- .
0 10 ( + .5
64 0
.
- .
7 2 )
6
ub cb 0 166
. - 0 340
. 0 13
. 70
.
0 +
.
0 048 - .
0 060 -0 25
. - .
0 11 .
0 2217 (20) 0 97432
. (+46 16
- 0 0394
.
47 ) (+-15)
and V -0.074  0.363 0.234  0.544 -0.74  0.33 0.495  0.990 +44 +
ub 44  76 .
0 00788 ( 60
- 0 0388
. 15 0 999216
.
43 ) ( ) (-63)
.
0 97508 (46) 2218
.
0 (20) 0 00357
. (29)
only
V
ij -0.427  0.427 -0.427  0.427 -0.773  0.773
undetermined undetermined .
0 2217 (+21 50 17
- 97431
.
0 .
0 0395
20 ) (+-46) (+-16)
and V V -0.457  0.457 -0.457  0.457 -0.813  0.813 +50 +21 +
ub cb .
0 00782 ( 64
- 0389
.
0 999212
.
0
29 ) (-27) (-67)

Table XVIII.  Results (68 and 95% C. L. ) obtained after releasing in turn the main constraints.





57


0.6 0.6
0.5 0.5





0.4 0.4





0.3 0.3





0.2 0.2
witout m
without constraint
m Bs
Bs m
and constraints
B
0.1 0.1 d





0 0
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0.6 0.6
0.5 0.5





0.4 0.4





0.3 0.3

without | |
V ub

0.2 without constraint

| | 0.2 and | / |
V V
K ub cb

constraints
0.1 0.1





0 0
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4


Figure 9.  Regions of maximum probability (68% and 95%) for the vertex (, ) of the unitarity triangle, as determined by four
different sets of constraints. Each graph has been superimposed over the contours obtained in the complete fit (see Figure 8).



58


It should be noted that, even when the only constraint which is based on the present
experimental evidence for CP violation, namely the measurement of , is released, it
K

is unequivocally predicted that the observables parametrizing the magnitude of CP
violation ( , , sin2 ) assume non-zero values. (The measurement of sin2 is not
incompatible with the absence of CP-violating effects and, given its precision, has no
influence on the results). The current precision in the determination of the phase is
almost entirely due to the much more effective V V and V constraints: when
ub cb ub

these are removed, the error in sin2, for example, is nearly tripled. However, the
imaginary part of the CKM matrix becomes totally undetermined after the removal of all
constraints the theoretical interpretation of which assumes the dominant contribution of
the top quark in the virtual intermediate state of the S= 2 and/or B= 2 processes. On
the other hand, it is evident that the remaining constraints (the direct measurements of

the moduli V except V ) cannot be used to confine the domain of the vertex of the
ij tb

unitarity triangle to any definite region, except for a spherically symmetrical area around
the origin ( V V and V ).
ub cb ub

As expected, the results have proved to be nearly or completely insensitive to the
2 2 2
removal of the constraints V , V , V , V and V V V .
ud cd cs tb ts tb cb


It is not clear to what extent the minimum 2 value can be interpreted as an estimate
of the goodness-of-fit. The contribution of non-Gaussian uncertainties should be taken
into account. Moreover, the ineffective constraints should reasonably not be numbered
among the degrees of freedom of the fit. Whatever its exact statistical interpretation, the
small 2 value (=2.6 unnormalized) obtained in the fit with all the constraints, is the
clear sign of a high degree of consistency in the experimental data and between the data
and the model.


The value of m
has been computed using the relation linking m
to m
:
B B B
s s d

2
m f
B B V
s s ts
m
= m .

(3. 3-4)
B B
s d m f V
B B td
d d

The results are
+3 0
. -1
m (3. 3-5)
B = .
15 4- ps
0 7
.
s
with -1 -1
.
14 1ps < m 95% C. L. ,
B < .
21 5 ps
s

when the term depending on the measured amplitude ( m
is included in the 2 and
Bs )
+3 2
. -1
m (3. 3-6)
B = .
15 9- ps
3 1
.
s
with -1 -1
9
.
9 ps < m 95% C. L. ,
B < .
22 4 ps
s

after releasing the direct experimental constraint on m
. When the other constraints in
Bs
turn are removed, the results listed in Table XIX are obtained.






59


68% C. L. m (ps -1)
B 95% C. L.
s

+ 0
.
3
complete fit .
15 4- 14.1  21.5
0 7
.

+
without m
2
.
3
B .
15 9
s - 9.9  22.4
3 1
.

+
without m
and m
4 4
.
B B .
15 2
d s - 8.9  24.8
3 5
.

+ 0
.
3
without K .
15 4- 14.0  23.5
0 7
.

+
without V V and V 1
.
3
.
15 4
ub cb ub -0 7
. 14.0  23.6
Table XIX.  m
predictions.
Bs


The combination of CKM matrix element which determines the magnitude of the
direct-CP-violation parameter (Eq. 2.7-44) has also been calculated:
K K

ImV V = (1.24  0.14) 10-4 (3. 3-7)
td ts
1.0 10-4 <
ImV V < 1.5 10-4 95% C.L.
td ts


The major theoretical uncertainty in the results of the fit is the one affecting the
knowledge of B and B (the value of is even more uncertain, but a more precise
K B cc

determination would be of little importance, since in the term multiplied by is
K cc

roughly ten times smaller than those containing and ). The corresponding
ct tt
2 2
constraints (i. e. the terms (B B and (B B of the 2) have been
B - B ) 2
K - K ) 2
B B
K B

removed in order to establish their influence on the results. Moreover, as B and B
K B

are included among the `free' parameters of the fit, their values are determined, together
with those of the CKM parameters, on the basis of the remaining data, which are mostly
experimental in origin (see Table XX).


free parameters B B
B K
+ .
0 046 +
0.175 .
0 031
0.354
-0 034
. -0 032
.
0.103  0.288 0.275  0.415
+ .
0 071 +0 032
. +0 36
.
B 176
.
0 354
.
0 .
1 34
- .
0 036 - .
0 031 -0 26
.
B
0.103  0.303 0.270  0.415 0.89  2.19
+0 047
. +0 033
. + .
0 17
B 172
.
0 358
.
0 89
.
0
-0 035
. -0 035
. - .
0 14
K
0.099  0.311 0.236  0.421 0.64  1.27
+0 122
. +0 033
. +0 37
. + .
0 19
B 172
.
0 358
.
0 .
1 31 89
.
0
and B -0 035
. -0 035
. - 27
.
0 -0 15
.
K K 0.098  0.426 0.104  0.421 0.85  3.90 > 0.63

Table XX.  Simultaneous determination of the vertex of the unitarity triangle and the
bag parameters (68 and 95% C. L. ), performed by removing the theoretical constraints
on B and B . The results are perfectly compatible with the (more precise) theoretical
B K

determinations B and B used in the complete fit.
K = 0.94  0.08
B = 30
.
1  15
.
0




60


The purely theoretical determinations of the bag parameters clearly have considerable
weight, although this is not evident when the corresponding constraints are released one
at a time (the cause may be found in the strong correlation between the constraints
m , m and ). In particular, both B and the -coordinate of the vertex (the
B B K B
d s

two parameters are highly correlated) are determined with much less precision when
only the `experimental' constraints are used.
On the other hand, a significant increase in the precision of the results will not be
achieved by reducing the theoretical errors in B and B , unless a general
B K

improvement in the level of the uncertainties is obtained (particularly in the quark
masses, which occur in the factor S (x , x , and in the decay constants). The values of
c t )

B and B are in fact already known at the highest useful level of precision: if the
K B

errors in B and B B ,
B = .
1 300  .
0
B K are simultaneously reduced by a factor of 10 ( 015
B ), the results
K = .
0 940  0 008
.
+ .
0 044
= 176
.
0 =
- , 353
.
0  031
.
0 (3. 3-8)
.
0 036

104
.
0 < < 282
.
0 .
0 277 < < .
0 413 %
95 C.L
are obtained, which in practice coincide with the ones reported in Table XVI.

Our results for the CKM parameters and for the vertex of the unitarity triangle are
compatible with those obtained in previous analyses151,152 which made use of similar
techniques but different sets of experimental data.




3.4 Bayesian determination of the CKM matrix

To have an estimate of the goodness of the values obtained with the 2 minimization
method and obtain the probability distribution functions (p. d. f. ) of the CKM elements
and of all the estimated quantities, a complementary method153, based on Bayesian
statistics154, has been applied to the data set of Table XIV. This method has the
considerable advantage that it keeps track of all the assumptions made from beginning to
end, whereas other methods usually do not. For example, the standard 2-minimization
method starts from the assumption that all the input values are Gaussian-distributed and
provide estimates that are assumed to be Gaussian-distributed. This is reliable only at a
first approximation, since the asymmetric errors and the shapes of the maximum
probability regions clearly indicate that at least some quantities depart from Gaussianity.
The Bayesian approach for the statistic estimations starts from the definition of a set of
parameters ( x ) with a prior probability P(x,i) and a set of experimental data e with a
probability distribution function P(e, x | i), which depends on the x parameters. If x is
a set of continuous variables, their final (`posterior') probability distribution function is
given by
, |  ,
P(x,i | e) P(e x i) P(x i)
= . (3. 4-1)
P(e, x | i) P(x,i) x
d




61


Hence, if an initial probability distribution for certain variables ( x ) is used and new
experimental data are expressed by P(e, x | i), from Eq. (3.4-1) we get a new
distribution of probability for the initial variables (the posterior p. d. f. ). From the shape
of this distribution, the best estimations or the allowed ranges of the starting variables
can be extracted. The prior p. d. f. P(x,i) and the allowed range for x are generally a
matter of assumption. As long as the P(e, x | i) function contains more information than
the prior p. d. f. , these assumptions do not play any role in the final results155.
Eq. (3. 4-1) can be used to estimate the p. d. f. of the CKM elements. To allow the
comparison with the 2 minimization method, we consider exactly the same experimental
data set (Table XIV). Therefore, in e we consider the quantities V , V , V , V ,
ud us cd cs

2 2 2
V , V V , V , V V V and sin2, which depend on the four CKM
cb ub cb ub ts tb cb

parameters only, and the quantities , m
, ( m
which depend also on other
Bs )
K Bd

experimental or theoretical parameters. These parameter dependencies fix the set of the
x quantities. Concerning the CKM parameters, we take the four angles that define
completely a unitarized three-family CKM matrix (Eq. 1. 1-3): = ( , , , .
12 13 23 13 )

The parameter space is given by all the allowed space, that is [ ,
0 [23 [ ,0
2 [.
For the prior distribution P( ,
i) of , since we have no information to use, we assume
a uniform distribution in the allowed space. As is customary in the Bayesian method,
this general and weak assumption has no (or weak) effect on the final result154. As far as
, m
, and m
are concerned, we use the following expressions:
K B B
d s

2 2 2
G f m m
F K K W B = (3. 4-2)
K Im M K Im M
K = 12 12
2
12 2 mK
2 2 2
G m m f B
F B W B B B
m d d
= S x V V
= S x V V (3.4-3)
B ( t ) 2
d 2 td tb B ( t ) 2
td tb
6
2 2 2
G m m f B
F B W B B B
m s s
= S x V V = S x V V (3.4-4)
B ( t ) 2
s 2 ts tb B ds ( t ) 2
ts tb
6

where M has the same expression as in Eq. (2. 7-57), while , and are
12 K B ds

given by
2 2 2 2
G2 f 2m m2 G m m f B m f
F B W B
F K K W
= B , B B B B
d d
= , s s
= . (3. 4-5)
K K
2 B ds
12 2 m 2
6 2
m f
K B B
d d

The factors , which have been isolated in these expressions, will be considered as
i

three other initial parameters in the set x , which is finally composed by
(,)= ( , , , , , , . The distinction between and in m

12 13 23 13 K B ds ) B ds Bs

takes correctly into account that a strong correlation of the theoretical errors between
m
and m
exists. For each of them, we assume a Gaussian prior p. d. f. defined on
B B
d s

the positive side only with the mean values and standard deviations given in table XVI.







62


parameter value
(18.0  1.6)103
K
(3.10  0.69)103
B GeV
1.34  0.09
ds


Table XXI.  The factors, part of the x parameter set, which enter into the
expressions of , m and m . Values and errors listed have been used to define
K B B
d s

the prior p. d. f. .


For the expression of the probability distribution function P(e, x | i) which enters in Eq.
(3. 4-1), it is possible to take advantage of the fact that all the measurements in the set e
are independent. Therefore, we use:

P(e, x | i)= P  , x | i
(3. 4-6)
i ( X i )
i

where the product runs over all the measurements considered and P are their
i

probabilities. These are given by the Gaussian distribution functions, associated to the
X measured values, times an interval value X
:
P  , | = ,  ,
(3. 4-7)
i ( x i
X ) g(X(x) X X ) X
with
(X (x)-X )2
-
g(X (x )
,  , = e (3. 4-8)
X X ) 2
1 2 X

2 X
The P  , | written in this way represents the probability to measure X in an interval
i ( x i
X )
[ , + X
when the mean value and the standard deviation of the Gaussian
X X ]
distribution are given by X (x) and . The actual choice of the interval width X is
X

not relevant since, when P(e, x | i) is built and substituted into Eq. (3.4-1), all the X
i
cancel out in the ratio. Apart from these X
terms, P(e, x | i) has the structure of a
i

likelihood.
For the posterior probability function P(x,i | e), we can calculate the probability
distribution of any variable z which is a function of x , z = f (x), by means of the
following relation:
P(z,i | ) d
e = (z - f (x))P(x,i | e) d
x
d = P(x,i | e)
(3. 4-9)
z > f (x ) x
d
dz dz
where the integral must be evaluated in all the allowed space and (x) is the usual
Heavyside function, that is 1 for x > 0 and 0 for x < 0.
In order to obtain the p. d. f. for any quantity z which we are interested in using Eq.
(3. 4-9), we applied extensively the Monte Carlo method for the integral evaluation,
generating a sample of 9
10 points in the x space with a probability distribution given
by the P(x,i).
The p. d. f. for the most important quantities, which are evaluated simultaneously,
are shown in Figures 10-13. As can be seen in Figure 10, the three angles can vary
ij

in a rather small range and have an approximate Gaussian distribution, whereas the




63


phase can vary in a much wider range and it is clearly not Gaussian-distributed. The
13

p. d. fs. for sin2, sin2, m
and the vertex of the unitarity triangle are shown in
Bs

Figure 11 and Figure 13. They have a marked non-Gaussian behaviour, which is
particularly evident in the shape of sin2. The p.d.f. for the moduli of the CKM
elements are shown in Figure 12. As it can be seen, the elements V , V , V and
cb td ts

V are clearly not Gaussian-distributed.
tb













Figure 10.  Posterior probability distribution functions for and . Points with
ij 13

error bars refer to the solution obtained with the 2-minimization method.










64















Figure 11.  Posterior probability distribution functions for sin2, sin2 and m .
Bs












65


















Figure 12.  Posterior probability distribution functions for V .
ij








66




Figure 13.  Posterior probability distribution functions for and , 68% and 95%
probability contours for the vertex of the unitarity triangle.





In Table XXII, we present the complete result of this method for comparison with
the one obtained with the 2-minimization method (Table XVI). As Bayesian estimators
of the p. d. f. parameters, we take the mean and root mean square of the relative p. d. f. .
The comparison with the results given in Table XVI shows a good agreement on all data
within one sigma. As expected, the biggest differences can be observed on the variables
having the clearest non-Gaussian behaviour, for which the Bayesian estimation of the
root mean square gives always a bigger value. As regards the 95% limits, since these
are mainly determined by the properties of the p. d. f. tails (which are assumed to be
Gaussian in the 2 method) the Bayesian estimation always favour wider intervals. The
95% regions quoted are central-interval estimates, computed as in the previous analysis
of Ref. 153.






67


parameter mean
 r.m.s. 95% C.L.
0.2218  0.0020 0.2178  0.2257
0.183  0.063 0.045  0.293
0.343  0.032 0.283  0.412
= (64  13) 47  96
13
(12.81  0.12) 12.6  13.0
12
(2.27  0.06) 2.16  2.42
13
(0.203  0.015) 0.175  0.234
23
sin2 -0.15  0.33 -0.67  0.74
sin2 0.71  0.06 0.60  0.80
V 0.9751  0.0005 0.9742  0.9760
ud

V 0.2218  0.0020 0.2178  0.2258
us

V 0.0036  0.0003 0.0030  0.0041
ub

V 0.2217  0.0020 0.2177  0.2256
cd

V 0.9743  0.0003 0.9734  0.9752
cs

V 0.0398  0.0011 0.0379  0.0424
cb

V 0.0078  0.0006 0.0071  0.0093
td

V 0.0392  0.0012 0.0371  0.0420
ts

V 0.99921  0.00004 0.99910  0.99928
tb

V V 0.0894  0.0066 0.0765  0.1025
ub cb

m
(15.5  0.9) -1 13.8  17.5 -1
B ps ps
s



Table XXII.  Results of the Bayesian estimation method. The results can be directly
compared with those obtained with the 2-minimization method (see Table XVI), with
which they are fully compatible.





3.5 New prospects

According to the experimental results concerning kaon decays, a single complex
phase in the CKM matrix seems a suitable explanation for CP violation. On the other
hand, insofar as the baryon number of the universe has to be considered as a
dynamically generated quantity rather than an initial condition, the Standard Model fails
to account for the present abundance of matter by several orders of magnitude156. The
measurement of the CP-violating phenomena which are expected to occur in heavier
quark sectors will provide an independent way of testing the present theoretical
knowledge. The study of B mesons, which are ten times heavier than kaons and thus
undergo a wide range of (mostly rare) decays, seems to be an almost ideal field of
research. Large CP asymmetries are predicted to occur in the decays of B mesons and




68


the experiments will have many possibilities of revealing the presence of new physics.
The measurement of CP violation in the b quark sector will be the subject of an intense
activity in the so-called B factories after 2000.
A quite clear strategy can be adopted in the measurement of sin2. The already
mentioned CP asymmetry (1-17) which is expected in the rates of 0 0
B B J K
d d S

decays will be measured either as an oscillating function of time, whose amplitude is
equal to sin2, or by integration over a definite interval of time.
The extraction of sin2 from the measurement of the asymmetry
CP ( 0 + -
B B
d )-( 0 + -
d )
a + - = (2. 4-1)
( 0 + -
B B
d )+( 0 + -
d )
is less straightforward, since `tree' and `penguin' amplitudes interfere with different
weak and strong phases in 0 0 + -
B B decays, which therefore violate CP symmetry
d d
directly. Additional measurements of the isospin-related processes 0 + -
B ,
d
+ + 0 0 0
B , 0
B and their CP-conjugates will be required in order to estimate
d
the relative phase of the amplitudes ( 0 + - 0 + -
B and (B .
d )
d )
An even greater uncertainty affects the prospects for the determination of the angle
. A measurement of can in principle be obtained from the observation of the time-
dependent CP asymmetries involved in 0 0
B B oscillations, such as those between the
s - s
rates of 0 0 - 0 +
B t K and 0
B t K , or B and B . The
s (t ) -
D K
s (t ) +
D K
s ( ) 0
s
s ( ) 0
s s s

major experimental difficulty is due to the rapidity of the oscillations, which have not
yet been resolved. The most promising method in the short run is probably the one
consisting in the measurement of the direct CP asymmetry between the decays
0 + - 0 -
B and +
B ; the extraction of requires some additional branching
d K
d K

ratio measurements in order to eliminate the hadronic uncertainties which are introduced
by a phase difference between tree and penguin amplitudes.

A distinction can be made between the experiments at the hadronic machines
(HERA-B, CDF, LHCb) and those exploiting the B production induced by
electromagnetic interactions between colliding +
e and -
e (BaBar and Belle). While the
experiments in the first class can detect the production of b
b couples with a high cross-
section, but with a low signal-to-background ratio, a low cross-section is expected
BB

at the electromagnetic B factories, which, on the other hand, will be operating in a frame
of quite a few background events.
The measurement of the 0 0
B B oscillation frequency m
is within reach of the
s - s Bs
hadronic B factories only. The chances of success of each experiment are restricted by
the maximum time resolution it can achieve, which imposes an upper limit on the
measurable value of m
.
Bs
0 
HERA-B will study the reactions pN 0
b
b B , B , B , in fixed-target mode
d s b

( s GeV
42 ). The expected cross-section is high ( nb
12 ), but, because of a very
bb
slow signal-to-background ratio ( -6
), an extremely sensitive trigger is
bb tot 10

needed.



69


At BaBar and Belle, the B mesons will be produced by electromagnetic interaction
in collider mode ( + - 0 
e e (4S) B , B ). The favourable signal-to-background ratio
d
( ) is counterbalanced by a very low production cross-section
B
B tot 28
.
0
( nb
1
.
1 ), so that a quite high luminosity is required: integrated luminosities of
BB

respectively 30 and 100 fb-1 can be reached after one year by the two experiments. For
both BaBar and Belle the B physics is inhibited.
s

By the end of 2000, CDF Run II will become operative. The reactions
0 
p
p 0
b
b B , B , B , will be studied in collider, with s TeV
0
.
2 . The expected
d s b
cross-section is 50b , but a large amount of background events will have to be
bb
rejected ( -3
).
bb tot 10

Finally, LHCb is expected to start in 2005. b-flavoured hadrons will be produced in

pp collisions at s TeV
14 . With such a high energy in the centre of mass, the b
b
cross section will reach the value 500b . At the same time, the background
bb
sensitivity will be confined to .
bb tot 1 160


The predictions obtained in the present analysis,

sin2 = 0
+ 20
.
- + + + -
0 11
. 0 .
5 3 0
. 1
- , sin2 = 044
.
0 725
. .
63 7 m , (2. 4-2)
B = .
15 4 ps
22
.
0 - , = ( - ,
0
.
7 )3
.
0 046 -0 7
.
s

already provide an estimate of the limits beyond which the new measurements will have
to be definitely interpreted as signals of new physics. For example, the prediction for
sin2 is precise enough to define a wide range of values which the Standard Model
would be unable to account for, even within the `physical' range [0,1].
The most general supersymmetric (SUSY) extensions of the Standard Model would
be able to change the expected shape of the unitarity triangle (the existence of only three
generations of particles is not called into question) in quite an unpredictable way, due to
the contribution of new, presently unconstrained phases. Some restrictive assumptions
have to be made in order to give a quantitative account of the possible modifications of
the present scenario, as allowed by the currently available experimental data. In the
Minimal Supersymmetric Standard Model (MSSM) and its several variants (see Ref. 152
for a recent review), the flavourchanging processes are controlled by essentially the
same mixing matrix (V ) as in the Standard Model. The direct measurements of the
CKM

CKM matrix elements are not affected by these theories, while the contribution of the
supersymmetric particles to the effective FCNC processes (B-meson and K-meson
mixing) can be parametrized by adding a `supersymmetric' term to the top-quark loop

function in the expressions of m
, m and . Equations (1. 7-7), (1. 7-8) and
B B K
d s

(1. 7-65) have thus to be modified by making the substitutions

m
: S (x d
1+ S x
t ) ( BSUSY ) ( t)
Bd
m : S (x s
1+ S x
(2. 4-3)
t ) ( BSUSY ) ( t)
Bs
: S (x 1+ S x
t ) ( KSUSY) ( t)
K





70


The quantities B B
d
, s
and K
are positive definite functions of the masses of
SUSY SUSY SUSY

the supersymmetric particles involved (top squark, chargino and charged Higgs) and of
other SUSY parameters. To an excellent approximation, they are equal to each other152:
B B K
d s
= = = .

(2. 4-4)
SUSY SUSY SUSY SUSY

Therefore, as far as the determination of the vertex of the unitarity triangle is concerned,
only one additional degree of freedom ( ) has to be considered, so that a suitable
SUSY

level of predictivity can be reached in the computation of the relevant parameters. The
experimental constraints on the masses of the supersymmetric particles, on the electric
dipole moments of electron and neutron and on the branching ratio of B X decays
s

can be used to set an upper bound on : the values allowed in different MSSM
SUSY

models range157 from (Standard Model) to .
SUSY = .
0 6
SUSY = 0


-
m
B ( 1
ps
SUSY sin2 sin2 )
s


0
+ 39
.
- + .
0 087 + .
0 032 + .
0 23
- 0
+ 045
. ( 8
.
6
.
64 0+ 4
.
15 +
- .
15 1 )
4
.
5
EE 03
.
0 - 0 174
. 0.355 .
0 10 0.726
.
0 19 - .
0 036 - .
0 033 - .
0 56 -0 048
. -0 7
.
FR < 1.9 95% 0.099  0.385 0.171  0.417 -1  0.34 0.629  0.811 24.8  75.0 14.0 32.3
+ .
0 046 + 031
.
0 + 20
.
0 + .
0 044 ( + .5 3
+ 0
.
63 7
. 4
.
15
- .
7 0 )
3
0.0 0 175
. 0 354
. - .
0 11 0 725
.
(SM) - .
0 034 - .
0 032 - 22
.
0 - .
0 046 - 7
.
0
0.103  0.288 0.275  0.415 -0.73  0.26 0.632  0.809 45.4  74.4 14.1 21.5
+ .
0 103 + .
0 033 0
+ 24
. 0
+ 047
. ( + .5 + .
6 2
.
61 5 7
.
15
- .
17 2 )
9
0.2 0.187 -
- 0 345
. 0 19
. 0.720
0 038
. -0 071
. -0 59
. - .
0 049 - .
0 8
D 0.113  0.333 0.236  0.409 -0.98  0.26 0.620  0.809 37.4  72.8 14.2 25.3
IXE 0
+ 078
. 0
+ 034
. 0
+ 21
. + .
0 053
F ( .
5 + .
5 8
0
.
52 + 18 5
.
- .
13 3 )
1
0.4 0.242 -
- 0 311
. 0 52
. 0 702
.
.
0 031 - 049
.
0 -0 40
. - .
0 061 - .
1 4
0.140  0.356 0.213  0.387 -1  0.102 0.569  0.799 33.0  68.6 14.7 27.7
0
+ 040
. + .
0 042 0
+ 25
. + .
0 063 ( + .
3 5
.
41 7+ 22 6
.
- 6
.
6 )1.
9
0.6 0.299 -
- 0.267 0 82
. 0.665
.
0 052 -0 039
. -0 17
. - .
0 068 - .
3 6
0.192  0.373 0.197  0.358 -1  - 0.216 0.530  0.783 29.8  60.2 15.9 29.7

Table XXIII.  MSSM predictions.


The data displayed in the first row of Table XXIII are the result of a new fit1 performed

using the modified formulas for m
, m , and treating as an additional
B B K SUSY
d s

free parameter. Although values of greater than 0. 36 are disfavoured at the 84%
SUSY

C. L. , SUSY effects as large as 190% of the Standard Model expectation are possible
according to the 95% result. The prediction for the observable sin2 is completely
insensitive to the new physics at the present level of precision of the data, while the
lowest values of sin2 and and the highest ones of m allowed at the 95% C.L.
Bs

( < 45, sin2 < -0.73, m > 22ps-1) fall outside the corresponding ranges of the
Bs

Standard Model predictions. A lower value of the imaginary part and, in general, less
marked CP-violating effects are predicted in the MSSM than in the Standard Model
itself.
Figure 14 shows the 95% contours obtained after fixing the parameter at four
different values belonging to the allowed range ( = 0.0, 0.2, 0.4, 0.6). The variation
SUSY



1 2 2 2
The constraints sin2 and 7 (B X ( V V V ) have been removed.
s ) ts tb cb





71


of the minimum 2 value with respect to the Standard Model ( ) result ( 2
) is
SUSY = 0 0

also indicated. The numerical results are given in Table XXIII. Even though the
Standard Model scenario is slightly favoured by the present experimental constraints, the
two extreme regions = 0.0 and = 0.6 of the (, ) plane are not mutually
SUSY SUSY

exclusive. It will not be possible to test the reliability of the Standard Model with respect
of its minimal extensions until more precise (and independent) experimental data are
made available by the experiments at the B factories. Figure 15 shows how the improved
determination of the unitarity triangle may be able to exhibit significant divergences
between the models already in the first years of running of the experiments.


0.6
1999
0.5




0.4




0.3


/
2 2
0.2 SUSY 0
(SM) 0.0 1
0.2 1.3
0.1 0.4 1.4
0.6 1.5

0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7



Figure 14.  95% C. L. allowed regions for the vertex of the unitarity triangle in four
different SUSY scenarios.


The approximate error magnitudes expected in each experiment for the measurements of
sin2 and sin2 are reported in Table XXIV as functions of the year. The error values
in sin
2 are the results of very unequal estimations, in which different weights have
been attributed to the unknown contribution of the penguin amplitude. The measurement
of has not been taken into account, since most of the experiments refrain from making
predictions on the error. The largest measurable values of m
are also indicated.
Bs
Using these data, the projections represented in Figure 15 have been calculated. They
describe the probable way in which the results of the experiments running at the B
factories will progressively increase the accuracy in the determination of the vertex of
the unitarity triangle, according to different Minimal Supersymmetric models (including
the Standard Model itself). It has been assumed that the new measurements are
compatible with the values predicted inside each respective model (Table XXIII) and that
the first measurement of the 0 0
B B oscillation frequency is provided by LHCb in
s - s

2006 with the error (m . The results expected from single experiments
Bs ) 1
-
= .
0 009 ps

have been combined. These hypothetical measurements have been used as the additional
constraints of a new fit procedure, as a result of which the contours shown in Figure 15
have been calculated.



72




2000 2002 2004 2006
m

Bs (sin 2 ) (sin
2 )

HERA-B pN b
b B0, B , B0, 1
d s b 22 -
ps 0.17 / 0.35 0.12 / 0.25 0.09 / 0.18 0.07 / 0.14
BaBar + - 0 
e e (4S) B B - 0.08 / 0.31 0.05 / 0.18 0.04 / 0.14 0.03 / 0.12
d ,

Belle + - 0 
e e (4S) B B - 0.10 / 0.15 0.06 / 0.09 0.04 / 0.07 0.04 / 0.06
d ,


CDF p
p b
b B0, B, B0, 1
d s b 35 -
ps - 0.07 / 0.09 0.05 / 0.06 0.04 / 0.05
LHCb pp b
b B0, B , B0, 1
d s b 50 -
ps - - - 0.023 / 0.06
Average 0. 06 / 0. 13 0.03 / 0.06 0.02 / 0.04 0.01 / 0.03

Table XXIV.  Expected uncertainties in the measurements of sin2 and sin2 and
largest measurable values of m
in the experiments at the B factories.
Bs





0.6
0.6

2000 2002
0.5 0.5




0.4 0.4




0.3 0.3




0.2 0.2




0.1 0.1




0 0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


0.6
0.6

2004 2006
0.5 0.5




0.4 0.4




0.3 0.3




0.2 0.2




0.1 0.1




0 0
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7


Figure 15.  Expected evolution in the determination of the vertex of the unitarity
triangle resulting from forthcoming measurements of sin2, sin2 and (in 2006) m
at
Bs
the B factories. Each graph represents the 95% C. L. regions corresponding respectively
to (Standard Model scenario, the upper left plot in all four cases) and
SUSY = 0
= 0.2, 0.4, 0.6. The present (1999) scenarios were shown in Figure 14.
SUSY








73


Conclusions

An intensive program for the study of CP violation in weak decays involving the b-
quark is about to be carried on by dedicated experiments. Only indirect estimates can be
given at present for the magnitude of these phenomena; moreover, they are obtained in
the framework of a critical sector of the Standard Model, the one describing the flavour-
changing processes and parametrized by the CKM matrix, the verification of which
offers the most favourable prospects for the discovery of new physics. With the aim of
setting the status of the present knowledge about the CKM matrix before the start of the
new experimental venture, a detailed and comprehensive review of the theoretical and
experimental information until now available in this field has been presented.
A great amount of precise information, obtained in some cases by exploiting new
experimental sources, has been made available by the exceptionally thriving activity of
the experiments in the very last years, leading in some cases to a noticeable
improvement in the direct determination of the CKM matrix elements. For example, the
precise measurement of W hadronic decays can be translated into a determination of the
element V being a factor of 6 more precise than the former best value derived from
cs

the analysis of D semileptonic decays. Moreover, thanks to a new measurement of the
size of the nucleon strange-quark sea, it has been possible to extract a value for V
cs

from the neutrino-nucleon scattering data. Especially valuable information comes from
the recent progress in the observation of B decays. More stringent constraints on V
td

and (in the form of a lower limit) V are now provided by the study of 0 0
B - B and
ts d d

0 0
B - B oscillations, while the first determination of V has been made possible by
s s ts

new measurements and accurate theoretical calculations of the b s decay rate.
Furthermore, new or recently improved measurements of B semileptonic decays have
led to better determinations of V , V and V V .
ub cb ub cb

Both the latest measurements and earlier (but not always unproblematic) results have
been critically reviewed in the present work, calling attention to the least clearly defined
experimental situations and pointing out the main theoretical uncertainties and questions.
Among the results obtained, the theoretical uncertainty on the earliest experimental
determinations of V V has been reduced by exploiting the results of the recent
ub cb

measurements of V , V and V V to discriminate between the disagreeing
ub cb ub cb

predictions of different models. The resulting information provides at present the most
effective constraint on the values of the imaginary part of the CKM matrix and of the
CP-violating parameters.
A set of independent constraints has then been defined on the basis of the collected
information. The unitarity condition of the 33 CKM matrix has been imposed as a
further constraint by expressing each quantity in terms of either Wolfenstein's or the
canonical parametrization. With the explicit assumption of unitarity, an over-determined
problem is obtained, the solution of which gives much more precise values for the
matrix elements; at the same time, other quantities of physical interest, such as the
angles of the unitarity triangle and the B -oscillation parameter, can be determined.
s





74


Two independent procedures have been followed, consisting respectively in a 2-
minimization and in a more refined approach based on Bayesian statistics. The results
obtained from the application of the two methods are in good agreement and, in some
cases, constitute a great improvement with respect to the direct experimental
determinations of the CKM elements. For example, the errors in V and V are
cd cs

reduced respectively by factors of 6 and 8, while the precision on V is increased by 4
tb

orders of magnitude. The probability distribution functions of the CKM elements and of
all other relevant parameters are obtained as a result of the Bayesian determination.
Highly asymmetric distributions are found for some quantities, such as the angles of the
unitarity triangle, indicating that a complete description of the final knowledge of these
parameters cannot be given if a simple confidence interval is quoted, as in the results of
the 2 minimization. On the other hand, the 2 method has the advantage of providing a
measure of the compatibility between the data and the model. The low 2 value at the
minimum denotes a high degree of consistency between the experimental data and the
description of the CP violating phenomena considered inside the Standard Model.
The following predictions concerning the physics of B-meson decays have been
obtained:

sin2 = 0
+ 20
.
- + + + -
0 11
. 0 .
5 3 0
.
m .
B =
- , sin2 = 044
.
0 725
. .
63 7 .
15 4 ps
22
.
0 - , = ( - , 1
0
.
7 )3
.
0 046 -0 7
.
s

They indicate with absolute certainty the presence of CP violation in B decays according
to the Standard Model. Moreover, this conclusion is independent of the experimental
evidence for the parallel phenomenon already observed in the kaon system, since the
removal of the constraint does not alter significantly the results.
K

The above-mentioned results constitute the reference values to which the direct CP-
violation and 0
B -mixing measurements will have to be compared in the search for new
s

physics. On the other hand, their constraining effect on at least one class of minimal
extensions of the Standard Model is still rather weak. In fact, the predictions of the
Standard Model and those of its Minimal Supersymmetric extensions are at present
nearly indistinguishable. However, as it has been shown in the last Section of this work,
the direct measurements of CP violation expected in the first years of running of the B
factories should already be able to bring the knowledge of the unitarity triangle to a
substantially higher level of precision, offering the chance of detecting even the smallest
non-Standard-Model effects.





Acknowledgments

We would like to thank F. Buccella, N. Cabibbo, E. A. Paschos, P. Santorelli and H.
Schrder for stimulating discussions and many helpful suggestions.






75


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