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%\documentclass[prb]{revtex4}% Physical Review B

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\begin{document}

%\preprint{APS/123-QED}

%%
\title{\boldmath{Indication for Large Rescatterings in Charmless Rare $B$
Decays}}
%

\author{$^{a)}$Chun-Khiang Chua}
%\altaffiliation[Also at ]{Physics Department, XYZ University.}
%Lines break automatically or can be forced with \\
\author{$^{a)}$Wei-Shu Hou}%
%\email{Second.Author@institution.edu}
\author{$^{b)}$Kwei-Chou Yang}
% \homepage{http://www.Second.institution.edu/~Charlie.Author}
\affiliation{%
$^{a)}$Department of Physics, National Taiwan University, Taipei,
Taiwan 10764, Republic of China\\
$^{b)}$Department of Physics, Chung Yuan Christian University,
Chung-Li, Taiwan 32023, Republic of China
}%
%\affiliation{$^{b)}$Department of Physics, Chung Yuan Christian
%University, Chung-Li, Taiwan 32023, Republic of China
%}%

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
%
The current wealth of charmless $B$ decay data imply 
the presence of final state rescattering. 
Good fits are found with factorized amplitudes 
plus two SU(3) rescattering phase differences 
$\delta \sim 55^\circ$ and $\sigma \sim 100^\circ$.
With $\phi_3$ free, the fit gives $\phi_3 \sim 100^\circ$,
 and $A_{\pi\pi}$, $S_{\pi\pi}$ agree with BaBar experiment. 
For $\phi_3$ fixed at $60^\circ$, the fit gives $S_{\pi\pi}\sim -0.9$,
 which agrees with Belle experiment.
With our fitted $\delta$, $\sigma$, many direct $CP$ asymmetries flip sign,
and $B^0\to \pi^0\pi^0$, $K^-K^+$ rates $\sim 10^{-6}$.
%
\end{abstract}
%

\pacs{11.30.Hv,   %Flavor symmetries
      13.25.Hw,  %Decays of bottom mesons}
      14.40.Nd}  %Bottom mesons
%\pacs{ %Valid PACS appear here
%}
% PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle



Based on data from the CLEO experiment,
it was pointed out in 1999~\cite{gamma}
that the emerging $B\to K\pi$, $\pi\pi$ rates support
factorization, {\it if} the phase angle $\gamma$
 (or $\phi_3 \equiv \arg V_{ub}^*$~\cite{PDG}) of
the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix is large.
We now have some theoretical basis for
factorization~\cite{QCD,PQCD} in charmless $B$ decays,
and it is common for $B$ physics practitioners to take
$\phi_3 \sim 80^\circ$--$90^\circ$, which is
in contrast with the $\sim 60^\circ$ value
indicated by CKM fits~\cite{CKM} to other data.

At the turn of the century, the dramatic ascent of the two B
factories brought the CLEO era to an end. As a last hurrah, CLEO
measured~\cite{CLEO_CP} direct $CP$ violating rate asymmetries
($A_{\rm CP}$) in a few modes. The central values differed from
factorization expectations, but errors were large. Some other
oddities lingered, such as the smallness of $\pi^-\pi^+$ rate
compared with $\pi^-\pi^0$ modes. It was thus
suggested~\cite{delta} that one may further need large
rescattering in these modes.
%besides large $\phi_3$.
Though speculative, it had implications beyond the shifted pattern
in $A_{\rm CP}$s: $\pi^0\pi^0$ would become prominent, and $CP$
violation in $\overline B{}^0\to \pi^-\pi^+$ could be sizable.
Now, a few years later, it is surprising that the speculated
patterns seem to be really emerging!

Let us give a snapshot of the current landscape~\cite{ichep02}.
All the $K\pi$, $\pi\pi$ modes have been measured by both Belle
and BaBar experiments to some accuracy, and with good agreement.
The $A_{\rm CP}$s in $K^-\pi^{+,0}$ modes have become significant,
and seem {\it opposite in sign} to factorization
expectations~\cite{QCD}; the previous 3$\sigma$ effect in
$\overline K{}^0\pi^-$ mode reported by Belle is no more. Both
experiments have hints for $\pi^0\pi^0\sim 10^{-6}$ level, and
both have measured the mixing dependent $CP$ asymmetries in
$\pi^-\pi^+$ mode, which, following Belle we denote as
$A_{\pi\pi}$
 (equivalent to $A_{\rm CP}$ in $\overline B{}^0\to \pi^-\pi^+$)
 and $S_{\pi\pi}$, respectively.
Here is where the two experiments diverge:
Belle finds $A_{\pi\pi}\simeq 1$ and $S_{\pi\pi}\simeq
-1$~\cite{pipi_belle}
 (hence outside of physical domain), while
BaBar finds~\cite{pipi_babar} both $\simeq 0$, although their new
$A_{\pi\pi}$ central value now agrees in sign with Belle's, hence
{\it opposite to} (QCD) factorization expectations~\cite{QCD}.

What makes rescattering in charmless rare $B$ decays even more
enticing is the unexpected emergence of the so-called
color-suppressed $\overline B{}^0\to D^{(*)0}h^0$ decay
modes~\cite{D0h0}, where $h^0$ stands for $\pi^0$, $\eta$ and
$\omega$. The experimental values are all larger than expected
from factorization, indicating the presence of rescattering in
final state. In this paper we expand on the theme of
Ref.~\cite{delta} and explore the implications of the present
wealth~\cite{ichep02} of data on both $\phi_3$ and rescattering
phases. The pattern change in rates, such as $\pi^0\pi^0$,
$K^-K^{+,0}$, the sizable ``{\it opposite sign}" $A_{\rm CP}$s in
various modes, and especially the outcome for $A_{\pi\pi}$ and
$S_{\pi\pi}$, can be tested in the very near future.

Our picture is that of factorized $B$ decay amplitudes followed by
{\it final state} rescattering (FSI), i.e.
\begin{equation}
\langle i; \mbox{\rm out}|H_{\rm W}|B\rangle
 = \sum_l {\cal S}^{1/2}_{il} {\cal A}^f_{l},
\label{eq:master}
\end{equation}
where $| i; \mbox{\rm out}\rangle$ is the measured state, ${\cal
S}$ is the strong (re)scattering matrix, and ${\cal A}^f_{l}$ is a
factorization amplitude. We expand from our work~\cite{CHY} on
$\overline B\to DP$ processes to the present case of $\overline
B\to PP$, where $D$ is the SU$(3)$ $D$-meson triplet and $P$ the
pseudoscalar octet. That is, we extend the ${\bf 3} \bigotimes
{\bf 8} \to {\bf 3} \bigotimes {\bf 8}$ final state rescattering
formalism developed for color-suppressed $D^0h^0$ modes, to ${\bf
8} \bigotimes {\bf 8} \to {\bf 8} \bigotimes {\bf 8}$ rescattering
in $\overline B\to PP$ final state. Bose symmetry then implies
that the ${\cal S}^{1/2}$ matrix in Eq.~(\ref{eq:master}) is of
the form
\begin{equation}
{\cal S}^{1/2}=
 e^{i\delta_{\bf 27}} |{\bf 27}\rangle\langle {\bf 27}|
+e^{i\delta_{\bf  8}} |{\bf  8}\rangle\langle {\bf  8}|
+e^{i\delta_{\bf  1}} |{\bf  1}\rangle\langle {\bf  1}|,
\label{eq:Smatrix}
\end{equation}
hence just two physical phase differences, which we take as
$\delta \equiv \delta_{\bf 27} - \delta_{\bf 8}$ and $\sigma
\equiv \delta_{\bf 27} - \delta_{\bf 1}$. These rescattering
phases redistribute the factorized decay amplitudes ${\cal
A}^f_{l}$ according to Eq. (\ref{eq:master}).
%
The detailed formalism would be given elsewhere.
%
%
As in~\cite{CHY}, we drop the SU$(3)$ singlet ${\bf 1}$, or
$\eta_1$, from the rescattering formulation. For the present case,
one anyway has the difficulty in explaining the huge rate
observed~\cite{PDG} for $B\to \eta^\prime K$.



\begin{table*}[t!]
\caption{ World average inputs and fitted outputs; data in
brackets are not used, while $\eta_8 K^-$, $\eta_8 \pi^-$ entries
are for $\eta K^-$, $\eta \pi^-$. Horizontal lines separate
rescattering subsets. For Fit 1 or 2, $\phi_3$ is taken free or
fixed at $60^\circ$. The results in parentheses are from 
setting $\delta = \sigma = 0$ but keeping other fitted parameters fixed;
the fitted parameters and $\chi^2_{\rm min.}$ are given in 
Table~\ref{tab:phase}. \label{tab:output} }
\begin{ruledtabular}
\begin{tabular}{lcccccc}
 Modes
 & ${\mathcal B}^{\rm expt}\times10^6$
 & ${\mathcal B}^{\rm Fit1}\times10^6$
 & ${\mathcal B}^{\rm Fit2}\times10^6$
 & $A_{\rm CP}^{\rm expt}$
 & $A_{\rm CP}^{\rm Fit1}$
 & $A_{\rm CP}^{\rm Fit2}$\\
\hline

$K^-\pi^+$
        & $18.5\pm1.0$
        & $19.2$ $(19.4)$
        & $18.2$ $(17.9)$
        & $-0.08\pm0.04$
        & $-0.04$ $(0.09)\,\,\,\,$
        & $-0.04$ $(0.07)\,\,\,\,$\\
$\overline K {}^0\pi^0$
        & $10.2\pm1.5$
        & $7.6$ $(6.9)$
        & $9.0$ $(8.6)$
        & $\,\,\,\,[0.03\pm0.37]$
        & $0.23$ $(0.00)$
        & $0.16$ $(0.00)$ \\
$\overline K {}^0 \eta_8$
        & ---
        & $3.3$ $(3.7)$
        & $3.8$ $(4.5)$
        & ---
        & $0.21$ $(0.00)$
        & $0.15$ $(0.00)$\\
\hline
$\overline K {}^0 \pi^-$
        & $18.1\pm1.7$
        & $18.6$ $(17.9)$
        & $21.2$ $(20.6)$
        & $-0.05\pm0.07$
        & $0.07$ $(0.00)$
        & $0.05$ $(0.00)$\\
$K^-\pi^0$
        & $12.7\pm1.2$
        & $11.7$ $(12.1)$
        & $10.8$ $(10.7)$
        & $-0.10\pm0.08$
        & $-0.15$ $(0.08)\,\,\,\,$
        & $-0.14$ $(0.06)\,\,\,\,$\\
$K^-\eta_8$
        & $[4.1\pm1.1]$
        & $3.4$ $(3.7)$
        & $4.4$ $(5.2)$
        & ---
        & $\,\,\,\,0.30$ $(-0.10)$
        & $\,\,\,\,0.19$ $(-0.05)$\\
\hline
$\pi^-\pi^0$
        & $5.8\pm1.0$
        & $4.9$ $(4.9)$
        & $3.1$ $(3.1)$
        & $\,\,\,\,[0.05\pm0.15]$
        & $0.00$ $(0.00)$
        & $0.00$ $(0.00)$ \\
\hline
$\pi^-\eta_8$
        & $[2.9\pm1.1]$
        & $1.3$ $(1.4)$
        & $1.5$ $(1.7)$
        & ---
        & $\,\,\,\,0.58$ $(-0.32)$
        & $\,\,\,\,0.42$ $(-0.19)$\\
$K^-K^0$
        & $< 2.0$ (90\% CL)
        & $1.6$ $(1.5)$
        & $1.2$ $(1.0)$
        & ---
        & $-0.77$ $(-0.04)$
        & $-0.79$ $(-0.03)$\\
\hline $\pi^- \pi^+$
        & $4.7\pm0.5$
        & $4.8$ $(7.9)$
        & $5.0$ $(8.5)$
        & $[A_{\pi\pi}=\,\,\,\,0.57\pm0.19]$
        & $A_{\pi\pi}=0.23$ $(-0.23)$
        & $A_{\pi\pi}=\,\,\,\,0.14$ $(-0.17)$\\
%$\pi^+\pi^-$
        &
        &
        &
        & $[S_{\pi\pi}=-0.57\pm0.25]$
        & $S_{\pi\pi}=0.06$ $(0.09)\,\,\,\,$
        & $S_{\pi\pi}=-0.90$ $(-0.88)$\\
$\pi^0\pi^0$
        & $1.9\pm 0.7$
        & $2.3$ $(0.2)$
        & $2.8$ $(0.1)$
        & ---
        & $-0.51$ $(0.01)\,\,\,\,$
        & $-0.35$ $(0.01)\,\,\,\,$\\
$K^-K^+$
        & $<0.6$ (90\% CL)
        & $0.5$ $(0.0)$
        & $0.6$ $(0.0)$
        & ---
        & $-0.24$ ( --- )$\,\,\,\,$
        & $-0.11$ ( --- )$\,\,\,\,$\\
$\overline K {}^0K^0$
        & $<4.1$ (90\% CL)
        & $1.7$ $(1.5)$
        & $1.1$ $(1.0)$
        & ---
        & $-0.84$ $(-0.04)$
        & $-0.86$ $(-0.03)$\\
$\pi^0\eta_8$
        & ---
        & $0.3$ $(0.3)$
        & $0.2$ $(0.2)$
        & ---
        & $-0.03$ $(-0.04)$
        & $-0.03$ $(-0.04)$\\
$\eta_8\eta_8$
        & ---
        & $0.3$ $(0.1)$
        & $0.2$ $(0.1)$
        & ---
        & $-0.80$ $(-0.06)$
        & $-0.92$ $(-0.06)$\\
\end{tabular}
\end{ruledtabular}
\end{table*}



For ${\cal A}^f_{l}$, the present QCD~\cite{QCD} or
PQCD~\cite{PQCD} approaches involve effects that are subleading in
$1/m_b$, such as weak annihilation contributions. To avoid double
counting hadronic effects, we shall use naive
factorization~\cite{AKL} amplitudes. In fact, for QCD
factorization, which is quite close at heart to naive
factorization, removing all subleading effects
 (but keeping the $1/m_s^{\rm eff}$ chiral enhancement) seems to give
a better fit to $K\pi$, $\pi\pi$ rates~\cite{Beneke}.


We follow the $\chi^2$ fit strategy of Ref.~\cite{HSW}, but in the
rescattering formalism of Eqs. (\ref{eq:master})
and~(\ref{eq:Smatrix}) with the phases $\delta$ and $\sigma$ as
additional parameters.
%
Some choice has to be made on input. 
We take central value of $\vert V_{cb}\vert$
 and $\vert V_{ub}\vert$ from Ref.~\cite{PDG}.
We focus on $\overline B\to PP$ modes, 
since the $VP$ (and $VV$) situation is not yet settled.
We thus take only the 7 $K\pi$ and $\pi\pi$ rates, and the better
measured $A_{\rm CP}$s
 in $K^-\pi^{+,0}$ and $K^0\pi^-$ modes.
We make the world average of current values reported in
Refs.~\cite{ichep02} and \cite{PDG} by BaBar, Belle and CLEO. We
also constrain $f_\pi F_0^{BK}/f_K F_0^{B\pi} = 0.9\pm
0.1$~\cite{QCD}. For the stringent limits on $K^-K^{+,0}$
rates~\cite{ichep02}, we choose to use the Belle upper limit of $2
\times 10^{-6}$ for $K^-K^{0}$ and the BaBar upper limit of $0.6
\times 10^{-6}$ for $K^-K^{+}$, the reasons of which will be
discussed later. The limits are implemented as boundaries. {\it We
do not use $A_{\pi\pi}$ and $S_{\pi\pi}$ as input}, since the
experimental situation is controversial. For the same reason, we
do not use $\eta\pi^-$, $\eta K^-$ rates. Actually, since we have
dropped $\eta_1$, we give $\eta_8\pi^-$, $\eta_8 K^-$ only.


\begin{table}[b!]
\caption{ The $\chi^2_{\rm min.}$ and fitted parameters for Fits
1, 2 with $f_\pi F_0^{BK}/f_K F_0^{B\pi} = 0.9\pm 0.1$ constraint,
and $1/m_s^{\rm eff}$ is the effective chiral enhancement for
$\langle O_6\rangle$~\cite{AKL}. The last column gives fit result
for $\phi_3$ free (fixed) without FSI phases. \label{tab:phase} }
\begin{ruledtabular}
\begin{tabular}{cccc}
      &Fit 1
      &Fit 2
      &No FSI
      \\
\hline
 $\chi^2_{\rm min.}/{\rm d.o.f.}$
        & $\,\,11/5$
        & $\,\,20/6$
        &$38/7$ ($48/8$)
       \\
 $\phi_3$
        & \ $99^\circ$
        & \ $[\,60^\circ]$
        &$115^\circ$ ($[\,60^\circ]$)
       \\
 $\delta$
        &  55$^\circ$
        &  63$^\circ$
        & ---
        \\
 $\sigma$
        &101$^\circ$
        &104$^\circ$
        & ---
        \\
 $F_0^{B\pi}$
        &0.30
        &0.24
        &0.27 (0.16)
    \\
 $F_0^{BK}$
        &0.34
        &0.26
        &0.29 (0.18)
    \\
 $m_s^{\rm eff}$ (MeV)
        & 90
        & 57
        & 78 (34)
    \\
\end{tabular}
\end{ruledtabular}
\end{table}


We have as fit parameters $F_0^{BK}$, the chiral enhancement
parameter $1/m_s^{\rm eff}$~\cite{gamma}, the rescattering phases
$\delta$ and $\sigma$, and possibly $\phi_3$. For the latter,
because of the $S_{\pi\pi}$ controversy, we will explore the two
cases of
\begin{eqnarray}
&& \mbox{Fit 1: \hskip1cm $\phi_3$ free in the fit}, \nonumber \\
&& \mbox{Fit 2: \hskip1cm $\phi_3 = 60^\circ$ fixed~\cite{CKM}}.
\nonumber
\end{eqnarray}
%
%
The fitted rates, $A_{\rm CP}$s, and especially
$A_{\pi\pi}$ and $S_{\pi\pi}$, together with inputs, 
are given in Table~\ref{tab:output}.
To show the cross-feed effect, we give in parentheses the results
from simply setting FSI phases to zero but keeping all
other parameters as determined by the fit. 

The $\chi^2_{\rm min.}$ and fitted parameters are given in
Table~\ref{tab:phase}. 
%
The $\chi^2_{\rm min.}/{\rm d.o.f.}$ for Fit 1 is 11/5
 (giving $\phi_3 \cong 99^\circ$),
while for Fit 2 it is 20/6. Without considering $A_{\pi\pi}$ and
$S_{\pi\pi}$, the former is better. Both fits are much worse
without FSI: $\chi^2_{\rm min.}/{\rm d.o.f.}$ become 38/7 and 48/8
 for Fits 1 and 2, as seen in the last column of Table~II.


Before we focus on the highlights of the $\pi\pi$ modes, let us
illustrate the roles played by other inputs. The $K\pi$ rates have
now been measured with relatively good precision. These, together
with the large $K^-\pi^+/\pi^-\pi^+$ ratio, have been the main
driving force behind the need for
 large $\phi_3$~\cite{gamma,HSW}.
This part has not really changed. What is new is the pull of
$A_{\rm CP}$s, which bears testimony to the bonanza from $B$
factories. With the 3$\sigma$ effect from Belle gone, $A_{\rm
CP}(K^0\pi^-)$ is consistent with zero and not very constraining.
But the $A_{\rm CP}$s in $K^-\pi^{+,0}$ modes now have some
significance, with central values {\it both opposite in
sign} with respect to factorization. As pointed out~\cite{delta},
this may call for rescattering. We illustrate in
Fig.~\ref{fig:acp} the $\delta$ dependence of $A_{\rm
CP}(K^-\pi^{+,0})$. Indeed, we see that for both fits, a finite,
positive $\sin\delta$ can turn the two $A_{\rm CP}$s negative;
though $\sin\delta<0$ is allowed by rate data,
 it is disfavored by $A_{\rm CP}$s.
Note that the two modes compete and settle on the fit output of
$\delta \sim 55^\circ$--$60^\circ$. That is, the $A_{\rm
CP}(K^-\pi^+)$ preference for larger $\delta$ is held back by
$A_{\rm CP}(K^-\pi^0)$. A similar tug of war is seen between the
rates of $\overline K{}^0\pi^0$ (and $\pi^-\pi^+$) vs.
$K^-\pi^{+,0}$,  $\overline K{}^0\pi^-$.


\begin{figure}[t!]
%\smallskip\smallskip\smallskip
\centerline{%{\hskip0.352cm
    {\epsfxsize1.75 in \epsffile{acpkpi.eps}}
\hskip-0.1cm
        {\epsfxsize1.75 in \epsffile{acpkpi0.eps}}}
%\smallskip\smallskip\smallskip\smallskip
\caption{ $A_{\rm CP}(K^-\pi^{+,0})$ vs. $\delta$. 
Solid (dashed) line is for Fit 1 (Fit 2), and
shaded bands are 1$\sigma$ experimental ranges. }
\label{fig:acp}
\end{figure}



As remarked earlier, the stringent bounds on $K^-K^{0,+}$ rates
require special care. These limits are far from Gaussian, so they
should not be included in the $\chi^2$ fit. We chose to impose
them as strict bounds. As can be seen from Table~\ref{tab:output},
while the fitted $K^-K^0$ rate is below the bound, 
for the $K^-K^+$ case the fit output in Fit 2 sits right at
the bound, while for Fit 1 it is also rather close. This implies
extreme sensitivity to these bounds and how they are actually
implemented.
%
In Fig.~\ref{fig:kk0kk} we give the $\delta$ ($\sigma$)
dependence of $K^-K^{0(+)}$, and the bounds that
we have employed. For $K^-K^0$, we took the looser Belle bound of
$2\times 10^{-6}$. The fit value of $1.6\times 10^{-6}$ for Fit 1 is 
above the BaBar bound of $1.3\times 10^{-6}$. 
Had we strictly imposed the latter, one could
smell trouble here from the
 curve in Fig.~\ref{fig:kk0kk}(a).
But data is fluctuating and the bound
 should not be over-interpreted.
For $K^-K^+$ mode, we illustrate with the more stringent
BaBar bound of $0.6\times 10^{-6}$, since zero can be reached
with finite $\sigma$. Had we used the Belle bound of $0.9\times
10^{-6}$, neither fit values would hit the boundaries. What is
clear is that $\sin\sigma < 0$ is strongly disfavored. The driving
force for large $\sigma$, however, rests in the $\pi\pi$ sector,
to which we now turn.


\begin{figure}[b!]
%\smallskip\smallskip    %\smallskip
\centerline{%{\hskip0.352cm
    {\epsfxsize1.67in \epsffile{k0k.eps}}
\hskip-0.2cm
        {\epsfxsize1.67in \epsffile{kk.eps}}}
%\smallskip\smallskip\smallskip\smallskip
\caption{ Rates ($\times 10^{6}$) for (a) $K^0K^-$ vs $\delta$ and
(b) $K^+K^-$ vs $\sigma$. Solid (dashed) line is for Fit 1
(Fit 2), with $\delta$ fixed at $55^\circ$ ($61^\circ$) in (b).
See text for choice of experimental limits. } \label{fig:kk0kk}
\end{figure}



{\it All} measurables in the $\pi\pi$ system turn out to be
 volatile --- {\it all} three rates and $A_{\pi\pi}$, $S_{\pi\pi}$.
Although the $\pi^-\pi^+$ and $\pi^0\pi^0$ rates also contribute
to $\delta$ determination, the uniqueness of these modes is that
they are sensitive to the second phase $\sigma$, which is the
phase difference between {\bf 27} and {\bf 1} rescattering
channels. To illustrate this, we fix $\delta$ to the fit values of
Table~\ref{tab:phase} and plot in Fig.~\ref{fig:pipi} the
$\sigma$ dependence of the $\pi^-\pi^+$ and $\pi^0\pi^0$ rates.
The two fits are qualitatively similar and clearly favor very
large $\sigma \sim 100^\circ$, as given in Table~\ref{tab:phase}.
Rescattering can really help reduce $\pi^-\pi^+$ rate, generating
$\pi^0\pi^0$ in return; {\it the latter is being hinted at by both
Belle and BaBar!}


\begin{figure}[t!]
%\smallskip
\centerline{
            {\epsfxsize1.6 in \epsffile{pipipi0pi0.eps}}
}
%\smallskip\smallskip\smallskip\smallskip
\caption { 
$\pi^+\pi^-$ and $\pi^0\pi^0$ rates ($\times 10^{6}$) vs. $\sigma$.
Solid (dashed) line is for Fit 1 (Fit 2) with $\delta$ fixed at
$55^\circ$ ($61^\circ$). Horizontal bands are 1$\sigma$ experimental 
ranges. } \label{fig:pipi}
\end{figure}


At this point we note that the $\pi^-\pi^0$ mode is too small in
Fit 2: being $\sim 3\sigma$ below experiment accounts for
most of the $\chi^2$ difference between Fits 1 and 2. Unless both
experiments went quite wrong, this makes Fit 2 less desirable, 
where further symptoms are the need for 
larger $1/m_s^{\rm eff}$ and lower $F_0^{B\pi(K)}$. 
The situation comes about because, with low $\phi_3$, it is hard to
get low $\pi^-\pi^+$ rate, even with rescattering. Thus, as seen
from Table~\ref{tab:phase}, the $F_0^{B\pi}$ form factor is only
80\% the value of Fit 1 case. This alone leads to a reduction of
$\pi\pi$ rates by 36\%, hence is quite drastic a measure. Since
$\pi^-\pi^0$ rate does not depend on $\phi_3$ nor receives
rescattering, the small $F_0^{B\pi}$ makes $\pi^-\pi^0$
too small compared with experiment.


What is really intriguing --- and makes considering Fit 2 worthwhile
 --- is $A_{\pi\pi}$ and $S_{\pi\pi}$,
which we plot in Figs.~\ref{fig:apipispipi}(a) and (b), respectively,
with $\delta$ fixed at fit values of Table~\ref{tab:phase}. 
We have forcefully ``combined" the present
Belle~\cite{pipi_belle} and BaBar~\cite{pipi_babar} results in the plots, 
but we are certainly aware that the two experiments are actually in
strong conflict. Belle and BaBar do, however, agree on the {\it
sign of $A_{\pi\pi}$}: Belle finds $0.94^{+0.25}_{-0.31}\pm 0.09$,
and BaBar finds $0.30 \pm 0.25 \pm 0.04$ ($\equiv -C_{\pi\pi}$).
With $\sigma \simeq 100^\circ$, Fits 1 and 2 return $A_{\pi\pi} =
0.22$ and 0.13, respectively, which agree with data in sign.
Without the FSI phases, both fits would give opposite sign.
%


It is useful to compare with the elastic (SU(2) or isospin) case
given in Ref.~\cite{delta}. For the whole range of $\delta$ and
$|\delta-\sigma|\lesssim 70^\circ$, the summed rates in
$K^-\pi^+$--$\overline K{}^0\pi^0$ ($\overline K
{}^0\pi^-$--$K^-\pi^0$) and $\pi^-\pi^+$--$\pi^0\pi^0$ systems
vary by no more than few\% and 20\%, respectively. We can
reproduce the results of Ref.~\cite{delta} for this parameter
range by taking 
$\delta_{K\pi}\equiv\delta_{3/2}-\delta_{1/2}
      \sim {\rm arg}(1+9 e^{i\delta})$
and $\delta_{\pi\pi} \equiv
     \delta_2 - \delta_0\sim {\rm arg}(1+24 e^{i \delta}+15 e^{i 
\sigma})$~\cite{erratum}.
We see from Fig.~\ref{fig:apipispipi}(a) that, to reach the Belle and
BaBar ``average" $A_{\pi\pi}$, a very large $\sigma \sim
180^\circ$ (hence large deviation from SU(2)) would be called for.
%, and would correspond to a significant
%deviation from the elastic or SU(2) case.
This situation is not supported by the absence of $K^-K^+$ and the
fact that $\pi^-\pi^+ > 4 \times 10^{-6}$.

%It is useful to compare with Ref.~\cite{delta}, where
%$A_{\rm CP}(\pi^-\pi^+) \equiv A_{\pi\pi}$ actually turns
%more negative with $\delta_{\pi\pi} \equiv \delta_2 - \delta_0$,
%the phase difference between $I = 2$ and $I = 1$ channels.
%We have checked that this behavior is reflected in our
%$\delta \equiv \delta_{\bf 27} - \delta_{\bf 8}$ dependence.
%But Fig.~\ref{fig:pipi}(b) plots
%$\sigma \equiv \sigma_{\bf 27} - \sigma_{\bf 1}$ dependence,
%which actually shows $A_{\pi\pi}$ {\it grows positive}
%for growing $\sigma$, and is positive for $\cos\delta \lesssim 0$!
%This is a new feature emerging from our extension from SU$(2)$ to SU$(3)$,
%or from rescattering between two channels to six channels
%(see the last 6 rows of Table~\ref{tab:output}).
%We also see from Fig.~\ref{fig:apipispipi}(a) that,
%if one wanted to reach the average $A_{\pi\pi}$ between Belle and BaBar,
%a very large $\sigma \sim 180^\circ$ would be called for.


\begin{figure}[t!]
%\smallskip
\centerline{
%\hskip0.4cm
            {\epsfxsize1.7 in \epsffile{apipi.eps}}
\hskip-0.25cm %\hskip0.4cm
            {\epsfxsize1.7 in \epsffile{spipi.eps}}
}
%\smallskip\smallskip\smallskip\smallskip
\caption { (a) $A_{\pi\pi}$ and (b) $S_{\pi\pi}$ vs $\sigma$,
with notation as Fig.~\ref{fig:pipi}. } \label{fig:apipispipi}
\end{figure}


As a measure of direct $CP$ violation, $A_{\pi\pi}$ is
clearly sensitive to FSI phases, but it does not distinguish
much between Fits 1 and 2. 
In fact, most results (except $\pi^-\pi^0$) are 
not qualitatively different between Fits 1 and 2. 
It is $S_{\pi\pi}$, which measures the $CP$ violating phase of
the combined mixing and decay amplitudes, 
that is sensitive to $\phi_3$, 
as is evident in Fig.~\ref{fig:apipispipi}(b).
Various $A_{\rm CP}$s and rates have constrained $\sin\delta > 0$
and $\sin\sigma>0$. 
We have checked that $S_{\pi\pi}$ is not just flat in $\sigma$ 
for $\sin\sigma > 0$ and $\delta \simeq 55^\circ$--$60^\circ$, 
as seen from Fig.~\ref{fig:apipispipi}(b), but is
relatively flat for all $\delta,\ \sigma \lesssim 180^\circ$. 
The sensitivity is with $\phi_3$, 
the source of $CP$ violation in KM model. 
The results are intriguing: 
for {\it Fit 1}, where $\phi_3 \cong 99^\circ$ from fit, 
$S_{\pi\pi} \sim 0$ is fully {\it consistent with the BaBar result}; 
for {\it Fit 2}, where one fixes $\phi_3 = 60^\circ$ to the CKM fit value,
$S_{\pi\pi} \sim -0.9$ is quite {\it consistent with the Belle result}. 
It is because of the present experimental volatility in $S_{\pi\pi}$ 
that we have included Fit~2, for otherwise it has much worse $\chi^2$, 
e.g. with low $\pi^-\pi^0$ as discussed.


Some remarks are now in order.
%
First, 
the $S_{\pi\pi}$ sensitivity to $\phi_3$ and its numerics
are similar to other discussions~\cite{QCD,CKM}, 
except that we show that 
$S_{\pi\pi}$ is {\it insensitive} to FSI 
for $\sin\delta,\ \sin\sigma >0$.
%
Second, 
with current data, not only $\pi^0\pi^0 \gtrsim 10^{-6}$, 
but also $K^-K^0$, $K^-K^+ \sim 10^{-6}$ are 
inevitable for our rescattering model. 
Besides the rates of $\pi^-\pi^+$, $\pi^0\pi^0$,
they are largely driven by $A_{\rm CP}(K^-\pi^{+,0})$
 (otherwise sign would be wrong).
It is also strongly suggested by the need to change sign for $A_{\pi\pi}$. 
We note that $K^-K^0$ is more severe in the sense that 
it has a factorization contribution at $10^{-6}$ level. 
For $K^-K^+$, it is possible to get vanishing rates at 
$\sigma\simeq\delta$, the SU(2) limit.
%
Third, 
our $\pi^-\eta_8$ rate is quite below experimental average.
This may be due to our inability to treat $\eta_1$, 
or indicate an experimental problem, or both.
%
%Fourth, 
%the most promising direct $CP$ asymmetry seems to be
%$A_{\rm CP}(K^-\pi^{0})$


Let us comment on QCD and PQCD factorization before closing. 
For the former, removing all subleading effects
 (hence $\sim$ naive factorization) 
gives better fit~\cite{Beneke} to $K\pi$, $\pi\pi$ rates. 
To account for $A_{\rm CP}$s, 
QCD factorization would have to resort to 
a sizable {\it complex} $X_{H,A}$ hadronic parameter~\cite{QCD},
which is a form of large strong phase.
By having a sizable absorptive part in some penguin annihilation diagrams,
PQCD factorization fares better with $A_{\rm CP}$s~\cite{keum}. 
Its trouble lies in 
a larger $\pi^-\pi^+$ and a smaller $\pi^-\pi^0$ rate than data,
and the inability to generate $\pi^0\pi^0$. 
$K^-K^+$ is also always very small~\cite{ChenLi}. 
We believe, therefore, 
that our results can be clearly distinguished from 
those of QCD and PQCD factorization approaches. 


In summary,
we find FSI rescattering is needed to account for
charmless $B$ decay data.
The observables to be tested in the near future are:
%
sign of $A_{\rm CP}$s, with $A_{\rm CP}(K^-\pi^{0})
\simeq -15\%$;
%
the rates of $\pi^0\pi^0 \gtrsim 10^{-6}$, $K^-K^+ \lesssim 10^{-6}$;
%
$A_{\pi\pi} > 0$, and 
$S_{\pi\pi}$ possibly agreeing with either BaBar or Belle values,
depending on $\phi_3$.
%
Last but not least, the $\pi^-\pi^0$ rate should be double
checked.
%
Quite a few $A_{\rm CP}$s are large and opposite the case without FSI, 
but these can only be checked later. 
More details of the present work would be given elsewhere.



%\vskip 0.3cm
\noindent  %{\bf Acknowledgement}.\ \
This work is supported in part by
NSC 91-2112-M-002-027, 91-2811-M-002-043, 91-2112-M-033-013,
the MOE CosPA Project,
and the BCP Topical Program of NCTS.


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\end{thebibliography}




\end{document}



