%Title: Exploring New Physics in the $B\to \phi K$ System
%Author: Robert Fleischer, Thomas Mannel
%Published: Phys. Lett. B511 (2001) 240-250.
%hep-ph/0103121
\documentclass[12pt]{article}
\usepackage{epsf}
\setlength{\oddsidemargin}{-0.2cm}
\setlength{\textwidth}{16.8cm}
\setlength{\topmargin}{-1cm}
\setlength{\textheight}{23cm}
\addtolength{\jot}{10pt}
\addtolength{\arraycolsep}{-3pt}
\renewcommand{\textfraction}{0}
%\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\begin{document}
%%%%%%%%%%% Titlepage %%%%%%%%%%%
\begin{titlepage}
\begin{flushright}
\begin{tabular}{l}
DESY 01--030\\
CERN--TH/2001--072\\
TTP01--07\\
hep--ph/0103121\\
March 2001
\end{tabular}
\end{flushright}
\vspace*{1.8truecm}
\begin{center}
\boldmath
{\Large \bf Exploring New Physics in the $B\to \phi K$ System}
\unboldmath
\vspace*{2.1cm}
\smallskip
\begin{center}
{\sc {\large Robert Fleischer}}\footnote{E-mail: {\tt
Robert.Fleischer@desy.de}} \\
\vspace*{2mm}
{\sl Deutsches Elektronen-Synchrotron DESY, Notkestra\ss e 85,
D--22607 Hamburg, Germany}
\vspace*{1truecm}\\
{\sc{\large Thomas Mannel}}\footnote{E-mail: {\tt
Thomas.Mannel@cern.ch}}\\
\vspace*{2mm}
{\sl CERN Theory Division, CH--1211 Geneva 23, Switzerland}
\\ and \\
{\sl Institut f\"{u}r Theoretische Teilchenphysik,
Universit\"{a}t Karlsruhe, \\ D--76128 Karlsruhe, Germany}
\end{center}
\vspace{2.0truecm}
{\large\bf Abstract\\[10pt]} \parbox[t]{\textwidth}{
Employing the $SU(2)$ isospin symmetry of strong interactions and
estimates borrowed from effective field theory, we explore the
impact of new physics on the decays $B^\pm\to \phi K^\pm$ and
$B_d\to \phi K_{\rm S}$ in a model-independent manner. To this end,
we introduce -- in addition to the usual mixing-induced CP asymmetry
in $B_d\to \phi K_{\rm S}$ -- a set of three observables, which may
not only provide ``smoking-gun'' signals for new-physics contributions
to different isospin channels, but also valuable insights into hadron
dynamics. Imposing dynamical hierarchies of amplitudes, we discuss
various patterns of these observables, including also scenarios with
small and large rescattering processes. Whereas the $B\to\phi K$
system provides, in general, a powerful tool to search for indications
of new physics, there is also an unfortunate case, where such effects
cannot be distinguished from those of the Standard Model.
}
\vskip1.5cm
\end{center}
\end{titlepage}
\thispagestyle{empty}
\vbox{}
\newpage
\setcounter{page}{1}
\setcounter{footnote}{0}
\renewcommand{\thefootnote}{\arabic{footnote}}
%
%
%
\section{Introduction}\label{sec:intro}
%
%
%
The experimental data collected at the $B$ factories will allow stringent
tests of the Kobayashi--Maskawa picture of CP violation \cite{KM}.
Among the various $B$ decays that can be used to achieve this goal
\cite{revs}, $B\to \phi K$ decays play an important role. In these
modes, which are governed by QCD penguin processes \cite{BphiK-old}, also
electroweak (EW) penguins are sizeable \cite{BphiK-EW}, and physics beyond
the Standard Model may have an important impact \cite{BphiK-NP}. In the
summer of 2000, the observation of the $B^\pm\to \phi K^\pm$ channel was
announced by the Belle and CLEO collaborations. The present results for
the CP-averaged branching ratio are given as follows:
\begin{equation}\label{EXP}
\mbox{BR}(B^\pm\to \phi K^\pm)=\left\{\begin{array}{ll}
\left(1.39^{+0.37+0.14}_{-0.33-0.24}\right)
\times 10^{-5} & \mbox{(Belle \cite{belle})}\\
\left(5.5^{+2.1}_{-1.8}\pm0.6\right)
\times 10^{-6} & \mbox{(CLEO \cite{cleo}).}
\end{array}
\right.
\end{equation}
The Belle and CLEO results are only marginally compatible with each other.
Evidence for the neutral mode $B^0_d\to \phi K^0$ at the $2.9\sigma$ level,
corresponding to a branching ratio of
$\left(5.4^{+3.7}_{-2.7}\pm0.7\right)\times 10^{-6}$, was also reported by
CLEO, whereas a significant signal for this decay has not yet been observed
by the Belle collaboration.
In our discussion of the $B\to \phi K$ system, we follow closely our
recent $B\to J/\psi K$ analysis \cite{FM}, and make use of the $SU(2)$
isospin symmetry of strong interactions to derive a model-independent
parametrization of the $B^+\to \phi K^+$, $B^0_d\to \phi K^0$ decay
amplitudes. After recapitulating the structure of the Standard-Model
amplitudes, we include new-physics effects in a general manner, and
estimate their generic size with the help of arguments borrowed from the
picture of effective field theory. In order to deal with hadronic matrix
elements, we impose certain dynamical hierarchies of decay amplitudes,
where we distinguish between small and large rescattering effects. Although
we do not consider the latter case, which is also not favoured by the
``QCD factorization'' approach \cite{PQCD} and the present experimental
upper bounds on $B\to K K$ branching ratios \cite{BF}, as a very
likely scenario,\footnote{Arguments against this possibility, i.e.\
large rescattering effects, were also given in \cite{GIW}.} it deserves
careful attention to separate possible new-physics effects from those of
the Standard Model. Moreover, following the strategies proposed below,
we may not only obtain insights into new physics, but also into hadron
dynamics. To this end, we introduce -- in addition to the usual
mixing-induced CP asymmetry in $B_d\to \phi K_{\rm S}$ -- a set of three
observables, providing ``smoking-gun'' signals for new-physics contributions
to different isospin channels. Two of these new-physics observables may be
significantly enhanced by large rescattering processes. In general, the
$B\to\phi K$ system offers powerful tools to search for new physics.
However, there is also an unfortunate case, where such effects cannot
be disentangled from those of the Standard Model.
The outline of this paper is as follows: in Section~\ref{sec:pheno},
we employ a low-energy effective Hamiltonian and the isospin
symmetry of strong interactions to parametrize the $B^\pm\to \phi K^\pm$,
$B_d\to \phi K_{\rm S}$ decay amplitudes arising within the Standard
Model. The impact of new physics on these amplitudes is explored in a
model-independent way in Section~\ref{sec:NP}, where we make use of
dimensional estimates following from effective field theory, and
introduce plausible dynamical hierarchies of amplitudes. The set of
observables to search for ``smoking-gun'' signals of new-physics
contributions to different isospin channels of the $B\to \phi K$
decay amplitudes is introduced in Section~\ref{sec:obs}, and is discussed
in further detail in Section~\ref{sec:disc}. In Section~\ref{sec:concl},
our conclusions are summarized.
%
%
%
\boldmath
\section{Phenomenology of $B\to \phi K$ Decays}\label{sec:pheno}
\unboldmath
%
%
%
The $B\to \phi K$ system is described by the following low-energy effective
Hamiltonian:
\begin{equation}\label{Heff}
{\cal H}_{\rm eff}=\frac{G_{\rm F}}{\sqrt{2}}\left[V_{cs}V_{cb}^\ast
\left({\cal Q}_{\rm CC}^c-{\cal Q}_{\rm QCD}^{\rm pen}-
{\cal Q}_{\rm EW}^{\rm pen}\right)+V_{us}V_{ub}^\ast
\left({\cal Q}_{\rm CC}^u-{\cal Q}_{\rm QCD}^{\rm pen}-
{\cal Q}_{\rm EW}^{\rm pen}\right)\right],
\end{equation}
where the ${\cal Q}$ are linear combinations of perturbative Wilson
coefficient functions and four-quark operators,
consisting of current--current
(CC), QCD penguin and EW penguin operators. As discussed in \cite{FM},
this Hamiltonian is a combination of isospin $I=0$ and $I=1$
pieces:
\begin{equation}\label{ham-decom}
{\cal H}_{\rm eff} = {\cal H}_{\rm eff}^{I=0} + {\cal H}_{\rm eff}^{I=1},
\end{equation}
where ${\cal H}_{\rm eff}^{I=0}$ receives contributions from all of
the operators appearing in (\ref{Heff}), whereas ${\cal H}_{\rm eff}^{I=1}$
is due to only ${\cal Q}_{\rm CC}^u$ and ${\cal Q}_{\rm EW}^{\rm pen}$.
If we employ the $SU(2)$ isospin flavour symmetry of strong interactions,
we obtain
\begin{eqnarray}
\langle \phi K^+|{\cal H}_{\rm eff}^{I=0}|B^+\rangle&=&
+\langle \phi K^0|{\cal H}_{\rm eff}^{I=0}|B^0_d\rangle\label{iso1}\\
\langle \phi K^+|{\cal H}_{\rm eff}^{I=1}|B^+\rangle&=&
-\langle \phi K^0|{\cal H}_{\rm eff}^{I=1}|B^0_d\rangle,\label{iso2}
\end{eqnarray}
yielding
\begin{equation}\label{AMPLp1}
A(B^+\to \phi K^+)=\frac{G_{\rm
F}}{\sqrt{2}}\left[V_{cs}V_{cb}^\ast\left\{
{\cal A}_c^{(0)}-{\cal A}_c^{(1)}\right\}+V_{us}V_{ub}^\ast\left\{{\cal
A}_u^{(0)}-{\cal A}_u^{(1)}\right\}\right]
\end{equation}
\begin{equation}\label{AMPLd1}
A(B^0_d\to \phi K^0)=\frac{G_{\rm F}}{\sqrt{2}}\left[V_{cs}V_{cb}^\ast
\left\{{\cal A}_c^{(0)}+{\cal A}_c^{(1)}\right\}+V_{us}V_{ub}^\ast
\left\{{\cal A}_u^{(0)}+{\cal A}_u^{(1)}\right\}\right],
\end{equation}
where the CP-conserving strong amplitudes\footnote{The labels of
${\cal A}^{(0)}$ and ${\cal A}^{(1)}$ refer to the isospin channels
$I=0$ and $I=1$, respectively.}
\begin{equation}\label{ampl-c}
{\cal A}_c^{(0)}={\cal A}_{\rm CC}^c-{\cal A}_{\rm QCD}^{\rm pen}-
{\cal A}_{\rm EW}^{(0)},\quad
{\cal A}_c^{(1)}=-{\cal A}_{\rm EW}^{(1)}
\end{equation}
\begin{equation}\label{ampl-u}
{\cal A}_u^{(0)}={\cal A}_{\rm CC}^{u (0)}-{\cal A}_{\rm QCD}^{\rm pen}-
{\cal A}_{\rm EW}^{(0)},\quad
{\cal A}_u^{(1)}={\cal A}_{\rm CC}^{u (1)}-{\cal A}_{\rm EW}^{(1)}
\end{equation}
can be expressed in terms of hadronic matrix elements
$\langle \phi K|{\cal Q}|B\rangle$. Taking into account that
\begin{equation}
V_{cs}V_{cb}^\ast=\left(1-\frac{\lambda^2}{2}\right)\lambda^2A,\quad
V_{us}V_{ub}^\ast=\lambda^4 A\, R_b\, e^{i\gamma},
\end{equation}
where $\gamma$ is the usual angle of the unitarity triangle of the CKM
matrix \cite{revs}, and
\begin{equation}
\lambda\equiv|V_{us}|=0.22, \quad A\equiv|V_{cb}|/\lambda^2=0.81\pm0.06,
\quad R_b\equiv|V_{ub}/(\lambda V_{cb})|=0.41\pm0.07,
\end{equation}
we finally arrive at
\begin{equation}\label{AMPLp2}
A(B^+\to \phi K^+)=\frac{G_{\rm F}}{\sqrt{2}}
\left(1-\frac{\lambda^2}{2}\right)\lambda^2A\left\{{\cal A}_c^{(0)}-
{\cal A}_c^{(1)}\right\}\left[1+\frac{\lambda^2 R_b}{1-\lambda^2/2}
\left\{\frac{{\cal A}_u^{(0)}-{\cal A}_u^{(1)}}{{\cal A}_c^{(0)}-
{\cal A}_c^{(1)}}\right\}e^{i\gamma}\right]
\end{equation}
\begin{equation}\label{AMPLd2}
A(B^0_d\to \phi K^0)=\frac{G_{\rm F}}{\sqrt{2}}
\left(1-\frac{\lambda^2}{2}\right)\lambda^2A\left\{{\cal A}_c^{(0)}+
{\cal A}_c^{(1)}\right\}\left[1+\frac{\lambda^2 R_b}{1-\lambda^2/2}
\left\{\frac{{\cal A}_u^{(0)}+{\cal A}_u^{(1)}}{{\cal A}_c^{(0)}+
{\cal A}_c^{(1)}}\right\}e^{i\gamma}\right].
\end{equation}
At first sight, expressions (\ref{AMPLp2}) and (\ref{AMPLd2}) are
completely analogous to the ones for the $B^+\to J/\psi K^+$ and
$B^0_d\to J/\psi K^0$ amplitudes given in \cite{FM}. However, the
dynamics, which is encoded in the strong amplitudes ${\cal A}$, is
very different. In particular, the current--current operators
${\cal Q}_{\rm CC}^c$ cannot contribute to $B\to \phi K$ decays, i.e.\
to ${\cal A}_{\rm CC}^c$, through tree-diagram-like topologies; they may
only do so through penguin topologies with internal charm-quark
exchanges, which include also
\begin{equation}\label{rescatter-c}
B^+\to\{D_s^+\overline{D^0},...\}\to\phi K^+,\quad
B^0_d\to\{D_s^+D^-,...\}\to\phi K^0
\end{equation}
rescattering processes \cite{BFM}, and may actually play an important
role \cite{pens}. On the other hand, the ${\cal A}_{\rm CC}^{u (0,1)}$
amplitudes receive contributions from penguin processes with internal
up- and down-quark exchanges, as well as from annihilation
topologies.\footnote{Note that the isospin projection operators
${\cal Q}\sim(\overline{u}u\pm\overline{d}d)(\overline{b}s)$ involve
also $\overline{d}d$ quark currents.} Such penguins may also be
important, in particular in the presence of large rescattering processes
\cite{BFM,FSI}; a similar comment applies to annihilation topologies. In
the $B\to\phi K$ system, the relevant rescattering processes are
\begin{equation}\label{rescatter-u}
B^+\to\{K^+\pi^0,...\}\to\phi K^+,\quad
B^0_d\to\{K^+\pi^-,...\}\to\phi K^0,
\end{equation}
containing -- in addition to long-distance penguins -- also annihilation
processes (see Figs.~\ref{fig:rescatter1} and \ref{fig:rescatter2}). In
contrast to (\ref{rescatter-c}), large rescattering effects of the kind
described by (\ref{rescatter-u}) may affect the search for new physics
with $B\to \phi K$ decays, since these processes are associated with the
weak phase factor $e^{i\gamma}$. Moreover, they involve ``light''
intermediate states, and are hence expected to be enhanced more easily,
dynamically, through long-distance effects than (\ref{rescatter-c}),
which involve ``heavy'' intermediate states.
As is well known, the $\phi$-meson is an almost pure $\overline{s}s$ state;
the mixing angle with its isoscalar partner $\omega\sim (\overline{u}u+
\overline{d}d)/\sqrt{2}$ is small, i.e.\ at the few per cent level. Whereas
$\omega$--$\phi$ mixing does not at all affect the isospin relations
(\ref{iso1}) and (\ref{iso2}), which rely on the fact that the $\phi$ is
an isospin singlet, it has an impact on the size of the amplitudes
${\cal A}_{\rm CC}^{u (0,1)}$, since an $\omega$ component of the $\phi$
state permits current--current operator contributions through
tree-diagram-like topologies. However, the arguments given below are not
modified by the small $\omega$--$\phi$ mixing.
Let us now have a closer look at the structure of the $B\to \phi K$
decay amplitudes, focusing first on the case corresponding to small
rescattering effects. Looking at (\ref{ampl-c}) and (\ref{ampl-u}),
we expect
\begin{equation}\label{hier1}
\left|{\cal A}_u^{(0,1)}/{\cal A}_c^{(0)}\right|={\cal O}(1).
\end{equation}
In the case of the amplitude ${\cal A}_c^{(1)}$, the situation is different.
Here we have to deal with an amplitude that is essentially due to
EW penguins. Moreover, the $B\to \phi K$ matrix elements of $I=1$ operators,
having the general flavour structure
\begin{equation}\label{I1-ops}
{\cal Q}_{I=1}\sim (\overline{u}u-\overline{d}d)(\overline{b}s),
\end{equation}
are expected to suffer from a dynamical suppression. In order to keep track
of these features, we introduce, as in \cite{FM}, a ``generic'' expansion
parameter $\overline{\lambda}={\cal O}(0.2)$ \cite{hierarchy}, which is of
the same order as the Wolfenstein parameter $\lambda=0.22$, and suggests
\begin{equation}\label{hier2}
\left|{\cal A}_c^{(1)}/{\cal A}_c^{(0)}\right|=
\underbrace{{\cal O}(\overline{\lambda})}_{{\rm EW\, penguins}}
\times\underbrace{{\cal O}(\overline{\lambda})}_{{\rm Dynamics}}
={\cal O}(\overline{\lambda}^2).
\end{equation}
Consequently, we obtain
\begin{equation}\label{SM-ampl}
A(B^+\to \phi K^+)={\cal A}_{\rm SM}^{(0)}
\left[1+{\cal O}(\overline{\lambda}^2)\right]=A(B^0_d\to \phi K^0),
\end{equation}
with
\begin{equation}\label{ASM0}
{\cal A}_{\rm SM}^{(0)}\equiv
\frac{G_{\rm F}}{\sqrt{2}}\lambda^2A\,{\cal A}_c^{(0)}.
\end{equation}
The terms entering (\ref{SM-ampl}) at the $\overline{\lambda}^2$ level
contain also pieces that are proportional to the weak-phase factor
$e^{i\gamma}$, thereby leading to direct CP violation in the
$B\to \phi K$ system.
\begin{figure}
\begin{center}
\leavevmode
\epsfysize=4.8truecm
\epsffile{rescatter1.eps}
\end{center}
\vspace*{-0.4truecm}
\caption{Rescattering processes contributing to $B\to\phi K$ through
penguin-like topologies with internal $q$-quark exchanges ($q\in\{u,d\}$).
The shaded circle represents insertions of the corresponding current--current
operators.}\label{fig:rescatter1}
\end{figure}
\begin{figure}
\vspace*{0.6truecm}
\begin{center}
\leavevmode
\epsfysize=4.4truecm
\epsffile{rescatter2.eps}
\end{center}
\vspace*{-0.4truecm}
\caption{Rescattering processes contributing to $B\to\phi K$
through annihilation topologies. The shaded circle represents
insertions of current--current operators
($q\in\{u,d\}$).}\label{fig:rescatter2}
\end{figure}
Let us now consider large rescattering effects of the kind given in
(\ref{rescatter-u}). In the worst case, (\ref{hier1}) would be dynamically
enhanced as
\begin{equation}\label{hier1-res}
\left|{\cal A}_u^{(0,1)}/{\cal A}_c^{(0)}\right|=
{\cal O}(1/\overline{\lambda}),
\end{equation}
and the dynamical suppression in (\ref{hier2}) would no longer be
effective, i.e.
\begin{equation}\label{hier2-res}
\left|{\cal A}_c^{(1)}/{\cal A}_c^{(0)}\right|={\cal O}(\overline{\lambda}).
\end{equation}
In such a scenario, (\ref{SM-ampl}) would receive corrections of
${\cal O}(\overline{\lambda})$, involving also $e^{i\gamma}$. This
feature may complicate the search for new physics with the help of
CP-violating effects in $B\to\phi K$ decays. On the other hand, the
rescattering processes described by (\ref{rescatter-c}) may only affect
the amplitude ${\cal A}_{\rm SM}^{(0)}$ sizeably through its
${\cal A}_{\rm CC}^c$ piece, and are not related to a CP-violating
weak-phase factor within the Standard Model.
Before turning to new physics, we would like to point out an interesting
difference between the $B\to \phi K$ system and the $B\to J/\psi K$ decays
discussed in \cite{FM}. In the $B\to J/\psi K$ case, the Standard-Model
amplitudes corresponding to (\ref{SM-ampl}) receive corrections at the
$\overline{\lambda}^3$ level, which may be enhanced to
${\cal O}(\overline{\lambda}^2)$ in the presence of large
rescattering effects. Consequently, within the Standard Model, there may
be direct CP-violating effects in $B\to J/\psi K$ transitions of at most
${\cal O}(\overline{\lambda}^2)$, whereas such asymmetries may already
arise at the $\overline{\lambda}$ level in the $B\to \phi K$ system.
On the other hand, as $B\to \phi K$ modes are governed by penguin processes,
their decay amplitudes are more sensitive to new physics. In the next
section, we have a closer look at the generic size of such effects, using
dimensional arguments inspired by effective field theory.
%
%
%
\section{Effects of Physics Beyond the Standard Model}\label{sec:NP}
%
%
%
%
The most general way of introducing new physics is to employ the picture
of effective field theory, where we write down all possible dim-6
operators and construct the generalization of the Standard-Model effective
Hamiltonian (\ref{Heff}) at the scale of the $b$ quark. The relevant
operators are again dim-6 operators, where the Wilson coefficients of
those already present in the Standard Model now contain also a possible
piece of new physics. The problem with this generic point of view is that
the list of possible dim-6 operators contains close to one hundred entries
\cite{dim-6}, not yet taking into account the flavour structure, making
this general approach almost useless. However, as we are dealing with
non-leptonic decays in which, because of our ignorance of the hadronic
matrix elements, there is no senistivity neither to the helicity structure
of the operators nor to their colour structure, only the flavour structure
of the operators is relevant for us.
A discussion of the $\Delta B = \pm 2$ operators mediating
$B^0_d$--$\overline{B^0_d}$ mixing within this framework was given
in \cite{FM}. For a characteristic new-physics scale $\Lambda$ in the
TeV regime, new-physics contributions could in principle be as large as
those of the Standard Model. This well-known feature was also found in
several model-dependent studies of physics beyond the Standard Model
\cite{NP-revs}. As far as CP violation is concerned, new-physics
contributions involving also new CP-violating phases are of particular
interest. In this case, not only the ``strength'' of
$B^0_d$--$\overline{B^0_d}$ mixing is affected, which would manifest
itself as an inconsistency in the usual ``standard analysis'' of the
unitarity triangle \cite{AL}, but also ``mixing-induced'' CP-violating
asymmetries \cite{revs}, arising, for instance, in $B_d\to J/\psi K_{\rm S}$
or $B_d\to \phi K_{\rm S}$ decays.
Let us now turn to the $B\to \phi K$ amplitudes. As in (\ref{ham-decom}),
also in the presence of new physics, the corresponding low-energy effective
Hamiltonian can be decomposed into $I=0$ and $I=1$ pieces, where the new
physics may affect the Wilson coefficients and may introduce new
dim-6 operators, thereby modifying (\ref{SM-ampl}) as follows:
\begin{eqnarray}
A(B^+\to \phi K^+)&=&{\cal A}_{\rm SM}^{(0)}
\left[1+\sum_k v_0^{(k)}e^{i\Delta_0^{(k)}}e^{i\Phi_0^{(k)}}-
\sum_j v_1^{(j)}e^{i\Delta_1^{(j)}}e^{i\Phi_1^{(j)}}
\right]\label{ampl-NPp}\\
A(B^0_d\to \phi K^0)&=&{\cal A}_{\rm SM}^{(0)}
\left[1+\sum_k v_0^{(k)}e^{i\Delta_0^{(k)}}e^{i\Phi_0^{(k)}}+
\sum_j v_1^{(j)}e^{i\Delta_1^{(j)}}e^{i\Phi_1^{(j)}}\right].\label{ampl-NPd}
\end{eqnarray}
Here $v_0^{(k)}$ and $v_1^{(j)}$ correspond to the $I=0$ and $I=1$ pieces,
respectively, $\Delta_{0}^{(k)}$ and $\Delta_{1}^{(j)}$ are CP-conserving
strong phases, and $\Phi_{0}^{(k)}$ and $\Phi_{1}^{(j)}$ the corresponding
CP-violating weak phases. The amplitudes for the CP-conjugate processes
can be obtained straightforwardly from (\ref{ampl-NPp}) and
(\ref{ampl-NPd}) by reversing the signs of the weak phases. The labels
$k$ and $j$ distinguish between different new-physics contributions to the
$I=0$ and $I=1$ sectors.
As we have already noted, within the Standard Model, $B\to \phi K$
decays are governed by QCD penguins. Neglecting, for simplicity,
EW penguins and the proper renormalization-group evolution, we may
write
\begin{equation}\label{SM-pen-expr}
{\cal A}^{(0)}_{\rm SM}\sim\frac{G_{\rm F}}{\sqrt{2}}\lambda^2 A
\left[\frac{\alpha_s}{4\pi}\, {\cal C}\right] \langle P_{\rm QCD}\rangle,
\end{equation}
where ${\cal C}={\cal O}(1)$ is a perturbative short-distance coefficient,
which is multiplied by the characteristic loop factor $\alpha_s/(4\pi)$,
and $P_{\rm QCD}$ denotes an appropriate linear combination of QCD penguin
operators \cite{BF-rev}. Since (\ref{SM-pen-expr}) is a doubly
Cabibbo-suppressed loop amplitude, new physics could well be of the same
order of magnitude. If we assume that the physics beyond the Standard Model
is associated with a scale $\Lambda$ and impose that it yields contributions
to the $B\to \phi K$ amplitudes of the same size as the Standard Model, we
obtain
\begin{equation}\label{generic}
\frac{G_{\rm F}}{\sqrt{2}}\frac{M_{W}^2}{\Lambda^2}\sim
\frac{G_{\rm F}}{\sqrt{2}}\lambda^2 A \left[\frac{\alpha_s}{4\pi}\, {\cal C}
\right],
\end{equation}
corresponding to $\Lambda\sim 3$\,TeV.\footnote{In our numerical
estimate, we have assumed $A \times {\cal C}\sim1$ and
$\alpha_s=\alpha_s(m_b)\sim 0.2$.}\,
Consequently, for a generic new-physics scale in the TeV regime, which
was also considered in our $B\to J/\psi K$ analysis \cite{FM}, we may have
\begin{equation}
v_0^{(k)}={\cal O}(1).
\end{equation}
In the case where the new-physics effects are less pronounced, it may
be difficult to disentangle them from the Standard-Model contributions.
We shall come back to this issue in Section~\ref{sec:disc}, where we
shall discuss various scenarios.
Concerning possible new-physics contributions to the $I=1$ sector, we
assume a ``generic strength'' of the corresponding operators similar
to (\ref{generic}). However, since the $I=1$ operators have the general
flavour structure given in (\ref{I1-ops}), they are expected to suffer from
a dynamical suppression in $B\to\phi K$ decays. As in (\ref{hier2}), we
assume that this brings a factor of $\overline{\lambda}$ into the game,
yielding
\begin{equation}\label{gen-strength1}
v_1^{(j)}={\cal O}(\overline{\lambda}).
\end{equation}
If we impose such a hierarchy of amplitudes, the new-physics contributions
to the $I=1$ sector would be enhanced by a factor of
${\cal O}(\overline{\lambda})$ with respect to the $I=1$ Standard-Model
pieces. This may actually be the case if new physics shows up, for example,
in EW penguin processes.
Consequently, we finally arrive at
\begin{equation}\label{ampl-hier1}
A(B\to \phi K)={\cal A}_{\rm SM}^{(0)}\biggl[1+
\underbrace{{\cal O}(1)}_{{\rm NP}_{I=0}}+
\underbrace{{\cal O}(\overline{\lambda})}_{{\rm NP}_{I=1}}+
\underbrace{{\cal O}(\overline{\lambda}^2)}_{{\rm SM}}\biggr].
\end{equation}
In deriving this expression, we have assumed that the $B\to \phi K$ decays
are not affected by rescattering effects. On the other hand, in the presence
of large rescattering processes of the kind described by (\ref{rescatter-u}),
the dynamical suppression assumed in (\ref{gen-strength1}) would no longer
be effective, thereby yielding $v_1^{(j)}={\cal O}(1)$. Analogously, the
$B\to \phi K$ matrix elements of $I=0$ operators with flavour structure
\begin{equation}\label{I0-ops1}
{\cal Q}_{I=0}^{\overline{u}u,\overline{d}d}
\sim (\overline{u}u+\overline{d}d)(\overline{b}s)
\end{equation}
would no longer be suppressed with respect to those of the dynamically
favoured $I=0$ operators
\begin{equation}\label{I0-ops2}
{\cal Q}_{I=0}^{\overline{s}s}\sim (\overline{s}s)(\overline{b}s),
\end{equation}
and would also contribute to $v_0^{(k)}$ at ${\cal O}(1)$. A similar
comment applies to the matrix elements of the $I=0$ operators with
the following flavour content:
\begin{equation}\label{I0-ops3}
{\cal Q}_{I=0}^{\overline{c}c}\sim (\overline{c}c)(\overline{b}s),
\end{equation}
whose dynamical suppression in $B\to\phi K$ decays may be reduced through
rescattering effects of the kind given in (\ref{rescatter-c}), which may
also affect the ${\cal A}_{\rm SM}^{(0)}$ amplitude, as we have noted
above. Consequently, in the presence of large rescattering effects,
the decay amplitude (\ref{ampl-hier1}) is modified as follows:
\begin{equation}\label{ampl-hier2}
\left.A(B\to \phi K)\right|_{\rm res.}=
\left.{\cal A}_{\rm SM}^{(0)}\right|_{\rm res.}\times
\biggl[1+\underbrace{{\cal O}(1)}_{{\rm NP}_{I=0}}+
\underbrace{{\cal O}(1)}_{{\rm NP}_{I=1}}+
\underbrace{{\cal O}(\overline{\lambda})}_{{\rm SM}}\biggr].
\end{equation}
Let us emphasize once again that the rescattering contributions to the
prefactor on the right-hand side of this equation are due to
(\ref{rescatter-c}), whereas the hierarchy in square brackets is
governed by large rescattering processes of the type described
by (\ref{rescatter-u}).
In the following section, we introduce a set of observables, allowing
us to separate the $I=0$ contributions from the $I=1$ sector. These
observables play a key role to search for new physics and to obtain,
moreover, valuable insights into the $B\to \phi K$ hadron dynamics.
%
%
%
\boldmath
\section{Observables of $B\to \phi K$ Decays}\label{sec:obs}
\unboldmath
%
%
%
The decays $B^+\to \phi K^+$, $B^0_d\to \phi K^0$ and their
charge conjugates are described by four decay amplitudes $A_i$.
If the corresponding rates are measured, the $|A_i|^2$ can be
extracted. Since we are not interested in the overall normalization
${\cal A}_{\rm SM}^{(0)}$ of these amplitudes, we may construct the
following three independent observables with the help of the $|A_i|^2$:
\begin{equation}\label{def-AP}
{\cal A}_{\rm CP}^{(+)}\equiv
\frac{|A(B^+\to \phi K^+)|^2-|A(B^-\to \phi K^-)|^2}{|A(B^+\to
\phi K^+)|^2+|A(B^-\to \phi K^-)|^2}
\end{equation}
\begin{equation}\label{def-A0}
{\cal A}_{\rm CP}^{{\rm dir}}\equiv
\frac{|A(B^0_d\to \phi K^0)|^2-|A(\overline{B^0_d}\to \phi
\overline{K^0})|^2}{|A(B^0_d\to \phi K^0)|^2+
|A(\overline{B^0_d}\to \phi\overline{K^0})|^2}
\end{equation}
\begin{equation}\label{def-A}
{\cal B} \equiv\frac{\langle|A(B_d\to \phi K)|^2\rangle-
\langle|A(B^\pm\to \phi K^\pm)|^2\rangle}{\langle|A(B_d\to
\phi K)|^2\rangle+\langle|A(B^\pm\to \phi K^\pm)|^2\rangle},
\end{equation}
where the ``CP-averaged'' amplitudes are defined in the usual way:
\begin{eqnarray}
\langle|A(B_d\to \phi K)|^2\rangle&\equiv&
\frac{1}{2}\left[|A(B^0_d\to \phi K^0)|^2
+|A(\overline{B^0_d}\to \phi \overline{K^0})|^2\right]\\
\langle|A(B^\pm\to \phi K^\pm)|^2\rangle&\equiv&
\frac{1}{2}\left[|A(B^+\to \phi K^+)|^2+|A(B^-\to \phi K^-)|^2\right].
\end{eqnarray}
It should be noted that ${\cal B}$ is a CP-conserving quantity. In the
case of the neutral decay $B_d\to \phi K_{\rm S}$, interference between
$B^0_d$--$\overline{B^0_d}$ mixing and decay processes yields an additional
observable \cite{revs}:
\begin{equation}\label{time-dep}
\frac{\Gamma(B^0_d(t)\to \phi K_{\rm S})-
\Gamma(\overline{B^0_d}(t)\to \phi K_{\rm S})}{\Gamma(B^0_d(t)\to
\phi K_{\rm S})+\Gamma(\overline{B^0_d}(t)\to \phi K_{\rm S})}
={\cal A}_{\rm CP}^{{\rm dir}}\,\cos(\Delta M_d t)+
{\cal A}_{\rm CP}^{\rm mix}\,\sin(\Delta M_d t),
\end{equation}
where the ``direct'' CP-violating contribution
${\cal A}_{\rm CP}^{{\rm dir}}$ was already introduced in (\ref{def-A0}),
and the ``mixing-induced'' CP asymmetry is given by
\begin{equation}\label{def-mix}
{\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}
=\frac{2\,\mbox{Im}\,\xi}{1+|\xi|^2}\,,
\end{equation}
with
\begin{equation}\label{xi-def}
\xi=e^{-i\phi}\left[\frac{1+\sum_k v_0^{(k)}e^{i\Delta_0^{(k)}}
e^{-i\Phi_0^{(k)}}+\sum_j v_1^{(j)}e^{i\Delta_1^{(j)}}
e^{-i\Phi_1^{(j)}}}{1+\sum_k v_0^{(k)}e^{i\Delta_0^{(k)}}
e^{+i\Phi_0^{(k)}}+\sum_j v_1^{(j)}e^{i\Delta_1^{(j)}}
e^{+i\Phi_1^{(j)}}}\right].
\end{equation}
Here (\ref{ampl-NPd}) was used to parametrize the decay amplitudes.
The CP-violating phase $\phi$ is given by $\phi=\phi_{\rm M}+\phi_{K}$,
where $\phi_{\rm M}$ and $\phi_K$ are the weak $B^0_d$--$\overline{B^0_d}$
and $K^0$--$\overline{K^0}$ mixing phases, respectively.
As in our analysis of the $B\to J/\psi K$ system \cite{FM}, it is useful
to introduce the following combinations of the observables
(\ref{def-AP}) and (\ref{def-A0}):
\begin{equation}
{\cal S}\equiv\frac{1}{2}\left[{\cal A}_{\rm CP}^{{\rm dir}}+{\cal A}_{\rm
CP}^{(+)}
\right],\quad
{\cal D}\equiv\frac{1}{2}\left[{\cal A}_{\rm CP}^{{\rm dir}}-{\cal A}_{\rm
CP}^{(+)}
\right].
\end{equation}
The interesting feature of these combinations is that ${\cal S}$ is
governed by the $I=0$ pieces of the $B\to \phi K$ amplitudes, whereas
${\cal D}$ and ${\cal B}$ are proportional to the $I=1$ amplitudes.
As the general expressions are very complicated and not very instructive,
let us focus on the simplified case where the new-physics contributions to
the $I=0$ and $I=1$ sectors involve either the same weak or strong phases:
\begin{eqnarray}
A(B^+\to \phi K^+)&=&{\cal A}_{\rm SM}^{(0)}
\left[1+v_0 e^{i\Delta_0} e^{i\Phi_0}-v_1 e^{i\Delta_1}e^{i\Phi_1}
\right]\label{ampl-NPp-simple}\\
A(B^0_d\to \phi K^0)&=&{\cal A}_{\rm SM}^{(0)}
\left[1+v_0 e^{i\Delta_0} e^{i\Phi_0}+v_1 e^{i\Delta_1} e^{i\Phi_1}
\right].\label{ampl-NPd-simple}
\end{eqnarray}
Then we obtain
\begin{eqnarray}
{\cal S}&=&-2\left[\frac{a c-b d\,v_1^2}{c^2-d^2\,v_1^2}\right]=
-2\,v_0\left[\frac{\sin\Delta_0\sin\Phi_0}{1+
2\,v_0\cos\Delta_0\cos\Phi_0+v_0^2}\right]+{\cal O}(v_1^2)\label{S-expr}\\
{\cal D}&=&
-2\,v_1\left[\frac{b c - a d}{c^2-d^2\,v_1^2}\right]\label{D-expr}\\
{\cal B}&=&v_1\,\frac{d}{c}=2\,v_1\left[\frac{\cos\Delta_1\cos\Phi_1+
v_0\cos(\Delta_0-\Delta_1)\cos(\Phi_0-\Phi_1)}{1+
2\,v_0\cos\Delta_0\cos\Phi_0+v_0^2+v_1^2}\right],\label{B-expr}
\end{eqnarray}
with
\begin{eqnarray}
a&=&v_0\sin\Delta_0\sin\Phi_0\\
b&=&\sin\Delta_1\sin\Phi_1+v_0\sin(\Delta_0-\Delta_1)\sin(\Phi_0-\Phi_1)\\
c&=&1+2\,v_0\cos\Delta_0\cos\Phi_0+v_0^2+v_1^2\\
d&=&2\left[\cos\Delta_1\cos\Phi_1+v_0\cos(\Delta_0-\Delta_1)
\cos(\Phi_0-\Phi_1)\right].
\end{eqnarray}
Let us finally give the expression for the mixing-induced CP asymmetry
(\ref{def-mix}):
\begin{equation}\label{Amix-calc}
{\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}=-\sin\phi-
2\,\frac{z}{n},
\end{equation}
where
\begin{eqnarray}
z&=&\left[v_0\cos\Delta_0\sin\Phi_0+v_1\cos\Delta_1\sin\Phi_1\right]\cos\phi
\nonumber\\
&&+v_0v_1\left[\cos(\phi+\Phi_1)\sin\Phi_0+\cos(\phi+\Phi_0)\sin\Phi_1\right]
\cos(\Delta_0-\Delta_1)\nonumber\\
&&+v_0^2\cos(\phi+\Phi_0)\sin\Phi_0+v_1^2\cos(\phi+\Phi_1)\sin\Phi_1,
\end{eqnarray}
and
\begin{eqnarray}
n&=&1+2\,v_0\cos\Delta_0\cos\Phi_0+2\,v_1\cos\Delta_1\cos\Phi_1\nonumber\\
&&+2\,v_0v_1\cos(\Delta_0-\Delta_1)\cos(\Phi_0-\Phi_1)+v_0^2+v_1^2.
\end{eqnarray}
%
%
%
\section{Discussion}\label{sec:disc}
%
%
%
Because of large uncertainties, the data on $B\to \phi K$ decays that
are available at present from the Belle and CLEO collaborations (see
(\ref{EXP})) do not allow us to speculate on new physics. However, the
experimental situation should significantly improve in the next couple of
years. In this section, we discuss various patterns of the observables
${\cal S}$, ${\cal D}$ and ${\cal B}$ introduced above that may shed
light both on new physics and on the $B\to \phi K$ hadron dynamics.
An interesting probe for new physics is of course also provided by the
mixing-induced CP asymmetry of the $B_d\to \phi K_{\rm S}$ channel, which
can be compared with the one of the ``gold-plated'' mode
$B_d\to J/\psi K_{\rm S}$ \cite{BphiK-NP}. Using the expression for
${\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}
(B_d\to J/\psi K_{\rm S})$ given in \cite{FM}, the parametrization for
${\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}
(B_d\to \phi K_{\rm S})$ given in (\ref{Amix-calc}), and the hierarchy
arising in (\ref{ampl-hier1}), we obtain
\begin{equation}\label{mix-diff1}
{\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}
(B_d\to \phi K_{\rm S})-
{\cal A}^{\mbox{{\scriptsize mix}}}_{\mbox{{\scriptsize CP}}}
(B_d\to J/\psi K_{\rm S})=\underbrace{{\cal O}(1)}_{{\rm NP}_{I=0}} +
\underbrace{{\cal O}(\overline{\lambda})}_{{\rm NP}_{I=1}}
+\underbrace{{\cal O}(\overline{\lambda}^2)}_{\rm SM}\, ,
\end{equation}
where the $\sin\phi$ terms, which may deviate from the Standard-Model
expectation \cite{AL}, cancel. The contributions entering at the
$\overline{\lambda}$ and $\overline{\lambda}^2$ levels may also contain
new-physics effects from $B_d\to J/\psi K_{\rm S}$, whereas the
${\cal O}(1)$ term would essentially be due to new physics in
$B_d\to \phi K_{\rm S}$.
In order to disentangle the $I=0$ and $I=1$ isospin sectors, the
observables ${\cal S}$, ${\cal D}$ and ${\cal B}$ play a key role.
As can be seen in (\ref{S-expr})--(\ref{B-expr}), ${\cal S}$ provides a
``smoking-gun'' signal for new-physics contributions to the $I=0$ amplitude,
whereas ${\cal D}$ and ${\cal B}$ probe new-physics effects in the $I=1$
sector. Using the hierarchy arising in (\ref{ampl-hier1}), we obtain
\begin{equation}\label{obs-hier1}
{\cal S}=\underbrace{{\cal O}(1)}_{{\rm NP}_{I=0}}+
\underbrace{{\cal O}(\overline{\lambda}^2)}_{\rm SM},\quad
{\cal D}=\underbrace{{\cal O}(\overline{\lambda})}_{{\rm NP}_{I=1}}
+\underbrace{{\cal O}(\overline{\lambda}^2)}_{\rm SM},\quad
{\cal B}=\underbrace{{\cal O}(\overline{\lambda})}_{{\rm NP}_{I=1}}
+\underbrace{{\cal O}(\overline{\lambda}^2)}_{\rm SM},
\end{equation}
where the Standard-Model contributions are not under theoretical control.
If the dynamical suppression of the $I=1$ contributions were larger
than ${\cal O}(\overline{\lambda})$, ${\cal D}$ and ${\cal B}$ would be
further suppressed with respect to ${\cal S}$. On the other hand, if
the rescattering effects described by (\ref{rescatter-u}) were very
large -- and not small, as assumed in (\ref{obs-hier1}) -- {\it all}
observables would be of ${\cal O}(1)$. In such a situation, we would not
only have signals for physics beyond the Standard Model, but also for
large rescattering processes.
The discussion given above corresponds to the most optimistic scenario
concerning the generic strength of the new-physics effects in the
$B\to \phi K$ system. Let us now consider a more pessimistic case,
where the new-physics contributions are smaller by a factor of
${\cal O}(\overline{\lambda})$:
\begin{equation}\label{ampl-hier3}
A(B\to \phi K)={\cal A}_{\rm SM}^{(0)}\biggl[1+
\underbrace{{\cal O}(\overline{\lambda})}_{{\rm NP}_{I=0}}+
\underbrace{{\cal O}(\overline{\lambda}^2)}_{{\rm NP}_{I=1}}+
\underbrace{{\cal O}(\overline{\lambda}^2)}_{{\rm SM}}\biggr].
\end{equation}
Now the new-physics contributions to the $I=1$ sector can no longer be
separated from the Standard-Model contributions. However, we
would still get an interesting pattern for the $B\to \phi K$ observables,
providing evidence for new physics: whereas (\ref{mix-diff1}) and
${\cal S}$ would both be sizeable, i.e.\ of ${\cal O}(10\%)$ and within
reach of the $B$-factories, ${\cal D}$ and ${\cal B}$ would be strongly
suppressed. However, if these two observables, in addition to
(\ref{mix-diff1}) and ${\cal S}$, are found to be also at the $10\%$
level, new physics cannot be distinguished from Standard-Model
contributions, which could also be enhanced to the $\overline{\lambda}$
level by large rescattering effects. This would be the most unfortunate
case for the search of new-physics contributions to the $B\to \phi K$
decay amplitudes. An analogous discussion of the $B\to J/\psi K$ system
was given in \cite{FM}.
%
%
%
\section{Summary}\label{sec:concl}
%
%
%
The decays $B^\pm\to \phi K^\pm$ and $B_d\to \phi K_{\rm S}$ offer
an interesting probe to search for physics beyond the Standard Model.
Using estimates borrowed from effective field theory, we have
explored the impact of new physics on the $B\to \phi K$ system in a
model-independent manner, and have derived parametrizations of the
corresponding decay amplitudes, which rely only on the $SU(2)$ isospin
symmetry of strong interactions. We have introduced -- in addition to
the mixing-induced CP asymmetry of the $B_d\to \phi K_{\rm S}$ channel --
a set of three observables, providing the full picture of possible
new-physics effects in $B\to\phi K$ decays. In particular, these observables
allow us to separate the new-physics contributions to the $I=0$ and $I=1$
isospin sectors, and offer, moreover, valuable insights into hadron dynamics.
Imposing dynamical hierarchies of amplitudes, we have discussed various
patterns of these observables, including also scenarios corresponding to
small and large rescattering effects. We find that $B\to\phi K$ decays
may provide, in general, powerful ``smoking-gun'' signals for new physics.
However, there is also an unfortunate case, where such effects cannot be
distinguished from those of the Standard Model. We strongly encourage our
experimental colleagues to focus not only on the measurement of the
mixing-induced CP asymmetries in $B_d\to \phi K_{\rm S}$ and
$B_d\to J/\psi K_{\rm S}$, but also on the observables ${\cal S}$,
${\cal D}$ and ${\cal B}$. Hopefully, these will yield evidence for physics
beyond the Standard Model.
\section*{Acknowledgements}
The work of T.M. is supported by the DFG Graduiertenkolleg
``Elementarteilchenphysik an Beschleunigern'', by the
DFG Forschergruppe ``Quantenfeldtheorie, Computeralgebra und Monte
Carlo Simulationen'', and by the
``Ministerium f\"ur Bildung und Forschung'' (bmb+f).
\begin{thebibliography}{99}
\bibitem{KM}
M. Kobayashi and T. Maskawa,
{\it Prog.\ Theor.\ Phys.}~{\bf 49} (1973) 652.
%%CITATION = PTPKA,49,652;%%
\bibitem{revs}For reviews, see
M. Gronau, TECHNION-PH-2000-30
[hep-ph/0011392];\\
%%CITATION = HEP-PH 0011392;%%
J.L. Rosner, EFI-2000-47 [hep-ph/0011355];\\
%%CITATION = HEP-PH 0011355;%%
R. Fleischer, DESY 00-170 [hep-ph/0011323];\\
%%CITATION = HEP-PH 0011323;%%
Y. Nir, IASSNS-HEP-99-96 [hep-ph/9911321].
%%CITATION = HEP-PH 9911321;%%
\bibitem{BphiK-old}N.G. Deshpande and J. Trampetic,
{\it Phys.\ Rev.}~{\bf D41} (1990) 895 and 2926;\\
%%CITATION = PHRVA,D41,895;%%
%%CITATION = PHRVA,D41,2926;%%
J.-M. G\'erard and W.-S. Hou, {\it Phys.\ Rev.}~{\bf D43} (1991) 2909;
%%CITATION = PHRVA,D43,2909;%%
{\it Phys.\ Lett.}~{\bf B253} (1991) 478.
%%CITATION = PHLTA,B253,478;%%
\bibitem{BphiK-EW}R. Fleischer, {\it Z.\ Phys.}~{\bf C62} (1994) 81;\\
%%CITATION = ZEPYA,C62,81;%%
N.G. Deshpande and X.-G. He,
{\it Phys.\ Lett.}~{\bf B336} (1994) 471.
%%CITATION = HEP-PH 9403266;%%
\bibitem{BphiK-NP}Y. Grossman and M.P.~Worah,
{\it Phys.\ Lett.}~{\bf B395} (1997) 241;\\
%%CITATION = HEP-PH 9612269;%%
R. Fleischer, {\it Int.\ J.\ Mod.\ Phys.}~{\bf A12} (1997) 2459;\\
%%CITATION = HEP-PH 9612446;%%
D. London and A. Soni, {\it Phys.\ Lett.}~{\bf B407} (1997) 61.
%%CITATION = HEP-PH 9704277;%%
\bibitem{belle}A. Bozek (Belle Collaboration), talk at BCP4,
19--23 February 2001, Ise-Shima, Japan;\\
see also Belle Collaboration (A. Abashian {\it et al.}),
BELLE-CONF-0007.
\bibitem{cleo}CLEO Collaboration (R.A. Briere {\it et al.}),
CLEO 01-03 [hep-ex/0101032].
%%CITATION = HEP-EX 0101032;%%
\bibitem{FM}R. Fleischer and T. Mannel, DESY 01-006 [hep-ph/0101276],
to appear in {\it Phys.\ Lett.}~{\bf B}.
%%CITATION = HEP-PH 0101276;%%
\bibitem{PQCD}M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda,
{\it Phys.\ Rev.\ Lett.}~{\bf 83} (1999) 1914;
%%CITATION = HEP-PH 9905312;%%
{\it Nucl.\ Phys.}~{\bf B591} (2000) 313;\\
%%CITATION = HEP-PH 0006124;%%
see also Y.Y. Keum, H.-n. Li and A.I. Sanda, KEK-TH-642 [hep-ph/0004004].
%%CITATION = HEP-PH 0004004;%%
\bibitem{BF}A.J. Buras and R. Fleischer, {\it Eur.\ Phys.\ J.}~{\bf C16}
(2000) 97.
%%CITATION = HEP-PH 0003323;%%
\bibitem{GIW}Y. Grossman, G. Isidori and M.P. Worah, {\it Phys.\
Rev.}~{\bf D58} (1998) 057504.
%%CITATION = HEP-PH 9708305;%%
\bibitem{BFM}A.J. Buras, R. Fleischer and T. Mannel, {\it Nucl.\
Phys.}~{\bf B533} (1998) 3.
%%CITATION = HEP-PH 9711262;%%
\bibitem{pens}A.J. Buras and R. Fleischer,
{\it Phys.\ Lett.}~{\bf B341} (1995) 379;\\
%%CITATION = HEP-PH 9409244;%%
M. Ciuchini, E. Franco, G. Martinelli and L. Silvestrini,
{\it Nucl.\ Phys.}~{\bf B501} (1997) 271.
%%CITATION = HEP-PH 9703353;%%
\bibitem{FSI}L. Wolfenstein, {\it Phys.\ Rev.}~{\bf D52} (1995) 537;\\
%%CITATION = PHRVA,D52,537;%%
J.F. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares,
{\it Phys.\ Rev.\ Lett.}~{\bf 77} (1996) 2178;\\
%%CITATION = HEP-PH 9604283;%%
M. Neubert, {\it Phys.\ Lett.}~{\bf B424} (1998) 152;\\
%%CITATION = HEP-PH 9712224;%%
J.-M. G\'erard and J. Weyers, {\it Eur.\ Phys.\ J.}~{\bf C7} (1999) 1;\\
%%CITATION = HEP-PH 9711469;%%
A. Falk, A. Kagan, Y. Nir and A. Petrov,
{\it Phys.\ Rev.}~{\bf D57} (1998) 4290;\\
%%CITATION = HEP-PH 9712225;%%
D. Atwood and A. Soni, {\it Phys.\ Rev.}~{\bf D58} (1998) 036005.
%%CITATION = HEP-PH 9712287;%%
\bibitem{hierarchy}
A.J. Buras and R. Fleischer,
%``Towards the control over electroweak penguins in nonleptonic B decays,''
{\it Phys.\ Lett.}~{\bf B365} (1996) 390;\\
%%CITATION = HEP-PH 9507303;%%
M. Gronau, O.F. Hern\'andez, D. London and J.L. Rosner,
%``Broken SU(3) symmetry in two body B decays,''
{\it Phys.\ Rev.}~{\bf D52} (1995) 6356 and 6374.
%%CITATION = HEP-PH 9504326;%%
%%CITATION = HEP-PH 9504327;%%
\bibitem{dim-6}W. Buchm\"uller and D. Wyler, {\it Nucl.\ Phys.}~{\bf B268}
(1986) 621.
%%CITATION = NUPHA,B268,621;%%
\bibitem{NP-revs}For reviews, see Y. Grossman, Y. Nir and R. Rattazzi,
in {\it Heavy Flavours II}, eds.\ A.J.~Buras and M. Lindner,
World Scientific, Singapore (1998), p.\ 755 [hep-ph/9701231];\\
%%CITATION = HEP-PH 9701231;%%
Y. Nir and H.R. Quinn, {\it Annu.\ Rev.\ Nucl.\ Part.\
Sci.}~{\bf 42} (1992) 211;\\
%%CITATION = ARNUA,42,211;%%
M. Gronau and D. London, {\it Phys.\ Rev.}~{\bf D55} (1997) 2845.
%%CITATION = HEP-PH 9608430;%%
\bibitem{AL}A. Ali and D. London, DESY 00-182 [hep-ph/0012155].
%%CITATION = HEP-PH 0012155;%%
\bibitem{BF-rev}For a detailed discussion, see A.J. Buras and R. Fleischer,
in {\it Heavy Flavours II}, eds.\ A.J.~Buras and M. Lindner,
World Scientific, Singapore (1998), p.\ 65 [hep-ph/9704376].
%%CITATION = HEP-PH 9704376;%%
\end{thebibliography}
\end{document}