%UNIX --- UPDATED ON 13/8/97  
%====================================================================%
%                  sprocl.tex     27-Feb-1995                        %
% This latex file rewritten from various sources for use in the      %
% preparation of the standard proceedings Volume, latest version     %
% by Susan Hezlet with acknowledgments to Lukas Nellen.              %
% Some changes are due to David Cassel.                              %
%====================================================================%

\documentstyle[sprocl]{article}

\font\eightrm=cmr8

%\input{psfig}

\bibliographystyle{unsrt} %for BibTeX - sorted numerical labels by
                          %order of first citation.

\arraycolsep1.5pt

% A useful Journal macro
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

% Some useful journal names
\def\NCA{\em Nuovo Cimento}
\def\NIM{\em Nucl. Instrum. Methods}
\def\NIMA{{\em Nucl. Instrum. Methods} A}
\def\NPB{{\em Nucl. Phys.} B}
\def\PLB{{\em Phys. Lett.}  B}
\def\PRL{\em Phys. Rev. Lett.}
\def\PRD{{\em Phys. Rev.} D}
\def\ZPC{{\em Z. Phys.} C}

% Some other macros used in the sample text
\def\st{\scriptstyle}
\def\sst{\scriptscriptstyle}
\def\mco{\multicolumn}
\def\epp{\epsilon^{\prime}}
\def\vep{\varepsilon}
\def\ra{\rightarrow}
\def\ppg{\pi^+\pi^-\gamma}
\def\vp{{\bf p}}
\def\ko{K^0}
\def\kb{\bar{K^0}}
\def\al{\alpha}
\def\ab{\bar{\alpha}}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\CPbar{\hbox{{\rm CP}\hskip-1.80em{/}}}%temp replacemt due to no font

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%BEGINNING OF TEXT                           
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%\title{THE PHYSICS OF FLAVOR IS THE FLAVOR OF PHYSICS}

\title{THE PHYSICS OF FLAVOR IS THE FLAVOR OF
PHYSICS\footnote{Supported in part by VW--Stiftung Hannover (I--77495)}}

\author{H. FRITZSCH}

\address{Ludwig--Maximilians--Universit\"at M\"unchen,
Sektion Physik, Theresienstra{\ss}e 37,\\ D--80333 M\"unchen,
Germany\\and\\CERN, CH-1211 Geneva 23} 

%\author{A. N. OTHER}

%\address{Department of Physics, Theoretical Physics, 1 Keble Road,\\
%Oxford OX1 3NP, England\\E-mail: other@tp.ox.uk}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% You may repeat \author \address as often as necessary      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\maketitle\abstracts{Summary Talk: International Conference of
Flavor Physics (ICFP 2001). Zhang--Jia--Jie, Hunan, China
(May / June 2001)}


%\section{Guidelines}
%\subsection{Producing the Hard Copy}\label{subsec:prod}
%The hard copy may be printed using the advice given in
%the file {\em splread.1st}, which is repeated in this
%section. You should have three files in
%total.\\


For me it is a pleasure to give the Summary Talk at this conference on
flavor physics, held near one of the most spectacular places on our planet,
the natural wonders of Wulingyan in the Hunan province of China. In the
Natural Park we have seen a very spectacular abundance of form and
structure. Wulingyuan proves once again that nature
prefers complexity instead of simplicity, once it is given the choice.
The German philosopher Leibniz suggested more than 300 years ago that we
live in the best of all possible worlds. I doubt whether this is true, but
certainly he did not mean that we live in the simplest of all possible
worlds. It is not the world of Konfuzius, following rigid laws and simple
structures, but rather the world of Laotse, full of complexity, a world in
the eternal swing between Yin and Yang.

The topic of this conference, the physics of flavor, fits very well into
these surroundings. In all of particle physics, the physics of flavor sticks
out as the field which has the highest complexity and the richest
phenomenology. Indeed, the flavor of particle physics can be seen most
clearly by looking at the physics of flavor, with all its phenomena ranging
from the spectra of heavy mesons and baryons, from particle--antiparticle
oscillations, from $CP$ violation etc. up to exotic pheonomena like
neutrino oscillations.

Flavor physics is an area which has emerged as an independent field of
high energy physics only after the Standard Model of today had come up
in its first contures, shortly after the beginning of the 70'ies. At
that time it became apparent that quarks carry color, and the color force
is of crucial importance for the understanding of the strong interaction
phenomena. Thus the need to distinguish between the color index and the
index describing the various types of quarks $u, d, s \ldots $ suddenly
 was in the air, at least at CALTECH, which was about the only place
 where such subtleties were discussed at that time. For my own private use,
 I denoted the various quarks as quark types, a notation, I still use
 today in the German language. Once Gell--Mann and I were driving to
 a lunch restaurant in Pasadena and passed by a Baskin and Robins icecream
 place,
 advertising 32 different flavors. Murray suddenly came up with the
 proposal to use the name ``flavor''. I did not like this proposal at the
 beginning, translating it into German, where it means ``Geschmack'',
 an expression one could hardly use for the description of a subatomic
 particle. Soon afterwards, however, I went along with it, expecially
 after realizing that in other languages the translation of ``flavor''
 gives quite meaningful results. For example, in Italian the word
 ``il sapore'', used e. g. in ``il sapore del vino'' could very well be
 used to distinguish the various degrees of freedom of the quarks.
 
 Compared to the present time, the flavor physics in those days was rather 
 poor. Only three flavors, i. e. $u, d, s$, were known, and the basic
 parameters of flavor physics were the three quark masses and the
 Cabibbo angle. $CP$--violation was considered to be a peculiar
 phenomenon not intrinsically related to the flavor mixing.
 
 Today we see the sharp contures of the Standard Model$^{1)}$ in front
 of us,
 like the contures of the Wulingyan mountains seen from the Golden
 Whip Stream. The physics of flavor is at the same time the physics of
 the multitude of the free parameters of the theory. Even if we disregard
 possible neutrino masses, the minimal number of parameters is 18, among
 them the six masses of the quarks, the three lepton masses, and four
 flavor mixing parameters. Especially those 13 parameters are in the focus
 of flavor physics. In the Standard Model they arise in a way, which
 can hardly be considered satisfactory, even on low standards. They appear
 as the result of a direct coupling of the fermions to the
 ``Higgs'' field, a formal device without any predictive power, as far
 as those parameters are concerned.
 
 In my view, this mechanism of fermion mass generation is the least
 attractive corner of the Standard Model, and it is quite likely that
 this is the corner where the model might deviate from reality.
 Furthermore it
 might well be that the ``Higgs'' particle responsible for the generation
 of mass for the $W$-- and the $Z$--bosons does not couple to the
 $b$--quark with a strength proportional to $m_b$ as expected in the
 Standard Model, in which case the ``Higgs'' particle would not decay
 predominantly into a $\bar b b$--system, but into other particles,
 e. g. into two gluons or into $\gamma \gamma $ (see e. g. ref. (2)).
 
 More than a year ago we have entered the new millenium with a rather
 bizarre spectrum of the lepton and quark masses, which extends (in
 the absence of neutrino masses) from about 0.5 MeV (electron mass)
 to about 175000 MeV ($t$--mass), stretching over almost six orders of
 magnitude. On a logarithmic scale, the quark masses are nearly on
 straight lines, if plotted as functions of the family index, implying
 that the mass ratios are identical:
 \begin{eqnarray}
 m_c : m_t & = & m_u : m_c \nonumber \\
 m_s : m_b & = & m_d : m_s
 \end{eqnarray}
 
 The hierarchy exhibited by the mass spectrum is impressive. Moreover,
 about 97\% of the mass is provided by the $t$--mass. The $t$--quark is
 the only fermion whose mass is comparable to the mass scale of the
 electroweak symmetry breaking, parametrized by the v.e.v. of the ``Higgs''
 field
 \begin{equation}
 v \cong 246 \, \, \, \, {\rm GeV} \, .
 \end{equation}
 
 The observed $t$--mass is very close to
 \begin{equation}
 v / \sqrt{2} \cong 174 \, \, \, \, {\rm GeV} \, \, \, i.e. \, \, \, \,
 v / m_t \cong \sqrt{2}
 \end{equation}
 
 Thus far this factor $\sqrt{2}$, which looks like a Clebsch--Gordon
 coefficient, has not been understood; it might, of course, simply be an
 accident.
 
 Nevertheless the lepton--quark mass spectrum exhibits simple features
 which ask for a deeper understanding, beyond the rather shallow
 interpretation given within the Standard Model. Slightly more than
 100 years ago the energy spectra observed e. g. for the hydrogen atom
 found their theoretical explanation within quantum theory. One can
 expect that in a similar way the fermion mass spectrum is a
 clear sign that there is physics activity beyond the frontier line
 drawn by the Standard Model, presumably not much below the presently
 explored surface.
 
 In this talk I shall not summarize the conference by going in more
 detail through the many topics discussed at the conference. Let me just
 list the main themes. After we were reminded by Wolfenstein that $CP$ violation is now with us
 for 36 years, we heard in the talks of Roos and Sagawa the news about
 the $B$--decay measurements from BaBar and Belle. The results for
 sin2$\beta$ have still large errors, but they provide clear signs that
 $CP$ is violated also in the $B$--system. For the first time 
 $CP$--violation has been observed outside the $K$--system. Further
 results for $B$--decays came in from CLEO, as reported by Gao. In the
 Standard Model $CP$ violation arises as a by-product of flavor mixing.
 But any extension of the model, e. g. towards supersymmetric theories,
 has its new sources of $CP$--violation, as discussed by Chang.
 
 Flavor physics cannot be seen disjointly from other parts of particle
 physics, in particular from QCD or from extensions of the Standard Model
 towards a deeper understanding of gravity (see the talks of Liu and Li
 on chiral symmetry and of Kim on the mysteries surrounding the
 cosmological constant).
 
 As we heard in the talks of Koepke and Hsiung, direct $CP$--violation,
 which is expected in the Standard Model, seems to be established both
 at CERN and at FNAL, although the violation in the US is larger than in
 Europe. The chairman of this conference, Yue--Liang Wu, described in
 detail the present theoretical pricture, which seems to suggest that the
 ``true'' value for direct $CP$--violation is about in the middle between
 the FNAL and CERN results. $CP$--violation has not been observed, thus far,
 for baryons. This might change in the future, as pointed out by Valencia
 in his talk on hyperon decays.
 
 Besides $CP$--violation, there are many other features to be studied in
 the physics of charmed and $b$--flavored particles (see the reviews by
 Kutschke on the FNAL results, of Lista on BaBar, and of Antilogus on
 the results from Delphi).
 
 The decays of $B$--mesons provide us with a beautiful testing ground to
 study the interplay between QCD and flavor dynamics. This interplay was
 discussed in the talks of Cheng, Chiu and Lu.
 
 Although flavor physics is the corner of the Standard Model which is
 very close to the experiments, it is not immune with respect to
 extrapolations of the Standard Model. Thus far the experiments have
 not provided a direct hint to where the exit road which takes us beyond
 the Standard Model is leaving, but this could change soon, as discussed
 by Ali and Masiero with respect to the exit towards supersymmetry and by
 Ng with respect to the path leading to the jungle of extra dimensions.
 
 New results from BES were discussed by Liu. The present situation at LEP
 concerning the still hidden ``Higgs'' particle was outlined by Jin. Yuan
 described the interplay between the physics of the $t$--quark and of
 the ``Higgs'' particle. The future of the $t$--quark physics, from the
 TEVATRON to the LHC and LC, was discussed by Yeh.
 
 During the second half of the nineties, a new field of flavor physics
 has opened its doors, the field of lepton flavor mixing, most notably seen
 in the progress which was made in the study of neutrino oscillations.
 One should be reminded that neutrino oscillations were first discussed
 by Pontecorvo about 40 years ago in connection with the
 $K^0 - \bar K^0$--oscillations. Thus they came up in close contact with
 the flavor physics of quarks. We heared from Kaneyuki about the
 impressive progress made in Kamioka. Neutrino oscillations seem to be
 firmly established, although many details, in particular the absolute
 magnitude of the neutrino masses, are still unknown. Like in the
 $K^0 - \bar K^0$--system we know the mass splittings among the mass
 eigenstates much better than the masses themselves.
 
 Presumably the only
 way to find out more about the absolute magnitude of the neutrino masses
 in the laboratory is to study the double $\beta $ decay. Unfortunately,
 we could not hear
 about the future plans in this field, since professor
 Klapdor--Kleingrothaus could not come. Of course, double $\beta $--decay
 is only feasable as a tool to investigate the neutrino mass matrix if
 neutrinos are either pure Majorana particles, or mixtures of Dirac
 and Majorana states; in case of massive Dirac states there would be
 no effect, due to the lepton number conservation.
 
 In the past neutrino physics was a part of high energy physics in which
 neutrinos were used as tools to study the structure of nuclear matter,
 like the quark structure functions of the nucleon. Now the focus has
 changed. The neutrinos themselves are the subject of investigation.
 It might well be that the new insights obtained in this field allow us
 to find important information about the dynamics inside the lepton
 sector. I do not have to stress how important it would be to confirm the
 results about neutrino mixing obtained in studying the solar and
 atmospheric neutrinos by laboratory measurements. Longley discussed this
 in connection to the Minos project in the US, Suzuki in connection to
 the Kamland projekt in Japan.
 
 Neutrinos are special elementary objects in the sense that they are
 electrically neutral. Thus the dynamics of the neutrinos can easily
 be influenced by phenomena beyond the Standard Model, e. g. by mixing
 with states which have no residence permit within the framework of the
 Standard
 Model. Unified Gauge theories, based on the gauge group $SO(10)$, are good
 examples for this phenomenon.
 
 It is well--known that within a theory based on $SO(10)^{3)}$
 one is able to describe a nontrivial mass and mixing pattern for the
 neutrinos. Moreover, the simplest schemes for the breaking of the
 symmetry suggest simple relations between the masses of the quarks and
 the leptons, in particular between the masses of the charged leptons
 and the down--type quarks$^{4)}$.
 
 Furthermore, the see--saw mechanism to generate the neutrino masses
 can easily be implemented in the $SO(10)$--framework. It connects
 the flavor mixing in the quark sector (typically described by small
 mixing angles) and in the lepton sector. However, this connection can
 only be made if something is known about the mass and mixing pattern of
 the righthanded massive Majorana partners of the observed neutrinos.
 A large mixing between the light neutrinos is possible, if the
 structure of the mass matrix for the righthanded Majorana states is
 similar to the mass--matrices of the quarks and charged leptons$^{5)}$.
 
 Recently much interest has been devoted to the study of additional large
 dimensions$^{6)}$, i. e. dimensions, which go beyond the Minkowski
 (3+1)--structure. They are possible, provided all fields of
 the Standard Model propagate only in the four--dimensional subspace,
 but fields which are allowed with respect to the S.M. gauge group, in
 particular gravitons and righthanded neutrinos, are singlets to propagate
 in a larger space--time manifold. Departures from the inverse--square
 law of Newton are expected in this case, but have not been seen down
 to the submillimeter scale. Nevertheless, if such extra dimensions are
 there, the fundamental energy scale associated with gravity will not
 be the Planck scale of about $10^{19}$ GeV, but could be much lower.
 In particular for one extra dimension the scale is expected to be
 about $10^8$ GeV, in which case neutrino physics is likely to be the
 only possibility to find out something about the extra dimension,
 as discussed by Lam. A righthanded neutrino would be derived from
 a 5--dimensional Dirac field, and it would act like a sterile
 neutrino or a tower of sterile neutrinos mixed with the ordinary
 neutrinos. The oscillation pattern of the observed neutrinos can
 be quite different to the one of the extended Standard Model with
 only three massive neutrinos. One feature of those models is that
 a considerable amount of the active neutrino flux is dispersed into
 sterile neutrinos. In the case of $\nu _e$ and $\nu_{\mu}$ neutrinos
 there are good limits for such an effect, and no dispersion is
 seen thus far. However, no limit exists for the $\tau $--neutrinos.
 In the foreseeable future the experimentalists will provide us with
 enough data to set rather stringent limits on a possible diversion
 of neutrino flux into the dark corner of extra dimensions, and this
 will at the same time provide strong constraints on the physics of
 extra dimensions. Less likely, but certainly possible, is their
 actual discovery in ``looking through the neutrino glas'' beyond
 the realm of our four--dimensional world.
 
 The phenomenon of flavor mixing is an intrinsic part of the Standard
 Model, but the part whose dynamics is not understood. The world
 would be simpler without flavor mixing, but nature seems to prefer
 to go off the simplest road. Obviously the mixing between the families
 is intrinsically related to the dynamics of quark mass generation. The
 observed fact that the flavor mixing angles in the quark sector are
 small must be related to the strong mass hierarchy observed in the
 mass spectrum.
 
 The ``standard'' parametrization of the flavor mixing matrix
 (advocated by the Particle Data Group) and the original
 Kobayashi--Maskawa parametrization$^{7)}$ were introduced without
 taking possible links between the quark masses and the flavor
 mixing parameters into account. The parametrization Xing and I
 introduced some time ago (for a review see ref. (5)) is based on
 such a connection,
 although the specific relations between flavor mixing angles and
 quark masses might be more complicated than commonly envisaged.
 It is a parametrization which allows to interpret the phenomenon
 of flavor mixing as an evolutionary or tumbling process. In the
 limit in which the masses of the light quarks $(u, d)$ and the
 medium light quarks $(c, s)$ are set to zero, while the heavy
 quarks $(t, b)$ acquire their masses, there is
 no flavor mixing. Once the masses of the $(c, s)$--quarks are
 introduced, while the $(u, d)$--quarks remain massless, the flavor
 mixing is reduced to an admixture between two families, described
 by one angle $\Theta $. As soon as the $u$-- and $d$--quark masses are
 introduced as small perturbations, the full flavor mixing matrix
 involving a complex phase parameter and two more mixing angles
 $(\Theta_u, \Theta_d)$ appears. These angles can be interpreted
 as rotations between the states $(u, c)$ and $(d, s)$, respectively.
 In either the ``standard'' parametrization or the Kobayashi--Maskawa
 representation, however, such specific limits are difficult to
 consider. The representation I prefer is given by:
 \begin{eqnarray}
 V & = & \left( \begin{array}{ccc}
                        c_u & s_u & 0 \\
                        -s_u & c_u & 0 \\
                        0 & 0 & 1 \end{array} \right)
                        \left( \begin{array}{ccc}
                        e^{-i\varphi} & 0 & 0 \\
                        0 & c & s \\
                        0 & -s & c  \end{array} \right)
                        \left( \begin{array}{ccc}
                        c_d & -s_d & 0 \\
                        s_d & c_d & 0 \\
                        0 & 0 & 1 \end{array} \right) \nonumber \\
& = & \left( \begin{array}{ccc}
                         s_u s_d c + c_u c_d e^{-i\varphi} &
                         s_u c_d c - c_u s_d c^{-i\varphi} & s_u s\\
                         c_u s_d c - s_u c_d e^{-i\varphi} & c_u c_d c +
                         s_u s_d e^{-i\varphi} & c_u s \\
                         - s_d s & - c_ds & c \end{array} \right) \, ,
\end{eqnarray}
where $s_u \equiv $ sin $\Theta_u, c_u \equiv $ cos $\Theta_u$, etc. The
three mixing angles can all be arranged to lie in the first quadrant,
i. e., all $s_u, s_d, s$ and $c_u, c_d, c$ are positive. The phase
$\varphi $ may in general take all values between $0$ and $2 \pi $.
Clearly $CP$ violation is present, if $\varphi \not= 0 $ or $\varphi \not=
\pi$.

In many models for the quark mass matrices there exist simple relations
between the mass eigenvalues and the mixing angles $\Theta_u$ and
$\Theta_d$:
\begin{eqnarray}
{\rm tan} \Theta_u & = & | V_{ub} / V_{cb} | \approx \sqrt{m_u/m_c}
\nonumber \\ \nonumber \\
{\rm tan} \Theta_d & = & | V_{td} / V_{ts} | \approx \sqrt{m_d / m_s}
\end{eqnarray} 
The typical estimates of the quark masses give $\sqrt{m_u / m_c} \sim
0.06 \ldots 0.08$, a value which is slightly lower than the observed
ratio $ |V_{ub} / V_{cb} | \approx 0.09 \pm 0.02$.

The angle $\Theta_d$ is determined rather precisely by the ratio
$\sqrt{m_d / m_s}$, if one takes the results of chiral symmetry
breaking into account. One expects $| V_{td} / V_{ts} |\cong 0.22
\ldots 0.23$.

In the representation I am advocating the mixing strength between
the first and second generation is determined by the two mixing
angles $\Theta_u$ and $\Theta_d$. Both angles vanish, if the masses
of the light quarks $m_u$ and $m_d$ are turned off. In this limit
$CP$--violation would not be present. Suppose only one of the light
quarks $u \sim d$ acquires a mass. In this case both
$\Theta_u$ and $\Theta_d$ are nonzero, but one of the angles is
extremely small. Its magnitude depends on the actual structure of
the mass matrix. In specific models$^{5)}$ $\Theta_u$ is of the order of
$\sqrt{m_d / m_s} \cdot
\left( m_s / m_b \right)$, if $m_u$ is zero, i. e. about $10^{-2}$,
an order of magnitude smaller than observed. Likewise $\Theta_d$ is of the
order of $\sqrt{m_u / m_c} \cdot \left( m_c / m_t \right)$, if
$m_d$ is zero, i. e. about two orders of magnitude smaller than the
observed value. Thus the observed value of the flavor
mixing angles $\Theta_u, \Theta_d$ give a strong hint that neither
$m_u$ nor $m_d$ are vanishing.

The mixing element $V_{us}$ is given by:
\begin{equation}
V_{us} \cong s_u - s_d e^{- i \varphi}
\end{equation}
where $s_{u, d} = sin \Theta_{u, d}$. Thus a precise determination
of $V_{us}$, along with a precise determination of $s_u$ and $s_d$,
would allow to determine the phase $\varphi$ responsible for
$CP$--violation.

In the mass matrix models mentioned above $\Theta_u$ and $\Theta_d$
are given by $\sqrt{m_u / m_c}$ and $\sqrt{m_d / m_s}$ respectively.
The relation for $V_{us}$ fixes a triangle in the complex phase, which
is congruent to the unitarity triangle\cite{Fri}.

It is well--known that the absolute value of the Cabibbo transition
$V_{us}$ is essentially identical to $\sqrt{m_d / m_s}$, which can be
determined very well from the chiral dynamics of QCD. Thus there is
little space for the contribution from $s_u$, and one concludes that
the $CP$--violating phase $\varphi$ must be close to 90$^0$, a
situation which can be described as maximal $CP$--violation\cite{Fri}.

The picture which emerges is the following:
In the absence of the $u$-- and $d$--masses only one mixing angle
$\Theta$, describing the mixing between the second and third family,
is present. At the second step the mixing angle $\Theta_d \approx
\sqrt{m_d / m_s}$ appears, while $\Theta_u \approx \sqrt{m_u / m_s}$
can be introduced in a third step. The Cabibbo transition $V_{cd}$ is
then given by
\begin{equation}
V_{cd} \approx \Theta_d - \Theta_u e^{- i \varphi} \approx
\sqrt{\frac{m_d}{m_s}} - \sqrt{\frac{m_u}{m_c}} e^{- i \varphi}
\end{equation}

$CP$ violation is clearly seen as a phenomenon related to the
generation of mass for the first family. As mentioned above, the
observed absolute magnitudes of $V_{us}$ and $V_{cd}$ agree with the
ratio $\sqrt{m_d / m_s}$, and the correction coming from
$\sqrt{m_u / m_c}$ cannot be sizeable, implying that the phase angle
$\varphi$ must be close to 90$^0$. In this case one can rewrite
$V_{cd}$ as follows:
\begin{equation}
V_{cd} \approx \sqrt{\frac{m_d}{m_s}} + \sqrt{\frac{-m_u}{m_c}}
\end{equation}
i. e. the mass of the $u$--quark enters with a negative sign, such
that a phase angle of 90$^0$ appears. This phase angle of 90$^0$ might
be a signal for a specific discrete symmetry. Since the phase
angle $\varphi$ corresponds to the angle $\alpha $ in the unitarity
triangle, it is implied that the unitarity triangle is rectangular. The
other
two angles $\beta $ and $\gamma $ are given in terms of the two mass
ratios $\sqrt{m_u / m_c}$ and $\sqrt{m_d / m_s}$. Taking the central values
of the quark masses, one finds $\beta \approx 20^{\circ} (sin 2 \beta
\approx 0.64)$ and $\gamma \approx 70^0$. More specifically, the range
of sin $2 \beta$
varies between 0.56 and 0.70, as we vary $\sqrt{m_u / m_c}$ in the most
likely range 0.06 \ldots 0.08. The experimental data from BaBar and
Belle are consistend with this range of values for sin $2 \beta$, but
differ so much from each other that a clear conclusion cannot be drawn.
Nevertheless the value sin $2 \beta \approx 0.70$ seems to me the upper
edge
of the allowed range. If the experiments eventually give a higher
value, the theoretical basis of connecting the flavor mixing angles with
quark mass ratios, as discussed becomes questionable.

But even if this would be the case, which I doubt, we must conclude
that the success of the Standard Model with respect to $CP$ violation
is impressive. Its strength both is the $K$ meson sector and in the
$B$ meson sector is predicted by parameters which on their own have
nothing to do with $CP$ violation, but rather with flavor violation,
the flavor mixing angles. The phase parameter $\varphi $ describing the
$CP$--violation is large; it might even be 90$^0$.
In view of this success it seems unlikely to me that the observed
$CP$ violation comes from something else than the mechanism offered
for free by the Standard Model, where it is linked to the flavor mixing.
 However, small deviations from the
Standard Model expectations might be there and should  be   searched for 
in the future. Whether they are there and whether they are specific enough
to point towards a specific model, like the supersymmetric extension of
the Standard Model, remains to be seen.

Closing this conference, let me thank, especially also in the name of
all foreign participants, the organizing committee and in particular its
chairman, Prof. Yue--Liang Wu, for taking on the complicated task to
organize this conference in this wonderful and remote place.

%\section*{Acknowledgments}
%This is where one places acknowledgments for funding
%bodies etc.  Note that there are no section numbers for
%the Acknowledgments, Appendix or References.

%\section*{Appendix}
%We can insert an appendix here and place equations so that they 
%are given numbers such as Eq.~(\ref{eq:app}).
%\be
%x = y.
%\label{eq:app}
%\ee

\begin{thebibliography}{99}
\bibitem{Qu }See e. g.:C Quigg in {\em Gauge Theories of the Strong, Weak and
Electromagnetic Interactions}, ed. (Perseus, 1997)

F. Halzen and A. Martin {\em Quarks and Leptons} (J. Wiley, 1984).

\bibitem{Cal}X. Calmet and H. Fritzsch, \Journal{\PLB}{496}{190}{2000}.

\bibitem{Fri}H. Fritzsch and P. Minkowski, (Annals of Physics 93, 193, 1975).
\\H. Georgi, in Particles and Fields, ed. C.E. Carlson (AIP. N.Y., 1975),
p. 575.

\bibitem{Geor}H. Georgi and C. Jarlskog, \Journal{\PLB}{86}{297}{1979}.

\bibitem{Fri} H. Fritzsch and Z. Xing, \Journal{\NPB}{45}{1-81}{2000}
See also: W. Buchm\"uller and D. Wyler,
preprint DESY 01-115 .

\bibitem{Arkan} N. Arkani--Hamed, S. Dimopoulos and G. Dvali,
\Journal{\PLB}{429}{263}{1998};
I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvoli,
\Journal{\PLB}{246}{317}{1990}
%{\it et al}

\bibitem{Kob} Kobayaschi, M. and Maskawa, T., Prog. Theor. Phys. 49, 652 (1973)
\end{thebibliography}
\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% End of sprocl.tex  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



