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Centre National de la Recherche Scientifique, UPR A0014, \\  
Ecole Polytechnique, 91128 Palaiseau Cedex, France}
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%\begin{document}

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\begin{document}
% declarations for front matter
\title{January 2003} {PC073.0103} { CP-violating asymmetry in $B^- \rarr \pi^+ \pi^- K^-$
   and  $B^- \rarr K^+ K^- K^-$    decays  
\footnotetext{${\dag}$  
Talk given at the Quark Confinement and Hadron Spectrum V , Gargnano,Italy 
10-14 September 2002}}
{\author{T. N. PHAM}
{Centre de Physique Th\'eorique, \\
Centre National de la Recherche Scientifique, UMR 7644, \\  
Ecole Polytechnique, 91128 Palaiseau Cedex, France}}        
{The recent Belle and Babar measurements of the
branching ratios $B^- \rarr \pi^+ \pi^- K^-$
and $B^- \rarr K^+ K^- K^-$ and $B^\pm \rarr \chi_{c0} K^\pm$ have renewed
interests in these decays as another mean to look for direct
CP violation in $B$ decays. In this talk, I would like to discuss 
a recent analysis of the CP violating  asymmetry in the partial widths
for the decays  $B^- \rarr \pi^+ \pi^- K^-$
and $B^- \rarr K^+ K^- K^-$, which results from the interference of the
non resonant amplitude with the resonant amplitude
$B^\pm \rarr \chi_{c0} K^\pm $ $ \rarr\pi^+ \pi^- K^\pm  $ and
$B^\pm \rarr \chi_{c0} K^\pm$ $ \rarr K^+ K^- K^\pm$.
For  $\gamma \simeq 58^o$ we predict that the 
partial width asymmetry could reach  $10 \%$ for 
the $B^- \rarr \pi^+ \pi^- K^-$ decay
and $16\%$ for the  $B^- \rarr K^+ K^- K^-$decay.}


In $B$ decays into 3 light mesons, for example, in 
$B^- \rarr \pi^+ \pi^- \pi^-$ decays, CP asymmetry could 
arise from the interference of the non resonant amplitude 
with the resonant amplitude coming from the decays 
$B^{-}\rarr \chi_{c0}\pi^{-}$ $\rarr$ $\chi_{c0}\rarr \pi^- \pi^-$ 
\cite{Eilam,Deshpande} and an  
estimate for the partial width asymmetry at the $\chi_{c0}$
resonance  was given recently\cite{Fajfer}. 
As there is no theoretical prediction
 for the $B^{-}\rarr \chi_{c0}\pi^{-}$
decay rate, no firm prediction for the asymmetry could be given. It
is now possible to make a similar
analysis for the CP asymmetries in the $B^-\rarr \pi^+ \pi^- K^- $ 
and $B^-\rarr K^+ K^- K^- $ decays using the recent 
Belle Collaboration measurements\cite{Belle1,Belle2}~:
${\rm BR}(B^-\rarr \chi_{c0} K^-) = (6.0^{+2.1}_{-1.8})\times 10^{-4} $,
${\rm BR}(B^-\rarr K^+ K^- K^-) = (37.0\pm 3.9 \pm 4.4) \times 10^{-6} $
and 
${\rm BR}(B^-\rarr \pi^+ \pi^- K^-) =(58.5\pm 7.1 \pm 8.8) \times 10^{-6} $,
and the  value 
${\rm BR}(B^-\rarr \chi_{c0} K^-) = (2.4\pm 0.7)\times 10^{-4} $
from the Babar Collaboration\cite{Babar}. 
In this talk, I would like
to discuss a recent work\cite{Fajfer1} on the CP asymmetries 
to look for direct CP violation in these decays. 
I will present only the main point as details are given in the
published work.

The effective weak Hamiltonian  for the nonleptonic Cabibbo-suppressed
$B$ meson decays is given by\cite{Ali,Deshpande1,Isola}
%\newpage
\begin{eqnarray}
{\mathcal H}_{\rm eff}&  = & \frac{G_F}{{\sqrt 2}} [V_{us}^* V_{ub}
(c_1 O_{1u} +
c_2 O_{2u} )  + V_{cs}^* V_{cb} (c_1 O_{1c} +c_2 O_{2c} ) ]\nonumber\\
& - &\sum_{i=3}^{10} ([V_{ub} V_{ud}^* c_i^u
     +  V_{cb} V_{cs}^* c_i^c + V_{tb} V_{ts}^* c_i^t) O_i ] + h.c.
\label{eq1}
\end{eqnarray}
where $O_{1}, O_{2}$ are the usual tree-level operators and $O_{3}-O_{6}$
are the penguin operators. As usual with the factorization model we use
in our analysis, the hadronic marix elements are obtained from the 
effective Hamiltonian in 
Eq.(\ref{eq1}) with $c_{i}$ replaced by $a_{i}$ where $c_{i}$
are next-to-leading Wilson coefficients. Since  $a_3$ and $a_5$
are one order of magnitude smaller than $a_4$ and $a_6$, the contributions
from $O_{3}$ and $O_{5}$ can be safely neglected. For
$N_{c}=3$, $m_b = 5 \rm \, GeV$, we use\cite{Deshpande1,Isola}~: 
$a_1= 1.05$ ,  $a_2=  0.07$ , $ a_4=-0.043- 0.016\,i$, 
$a_6=  -0.054- 0.016\,i$.
The matrix element of $O_{1}$ for the 
non resonant $B ^- \rarr \pi^+ \pi^- K^-$ amplitude is then given by
\begin{eqnarray}
&&<O_{1}>_{\rm nr} = < K^-(p_3) \pi^+(p_1) \pi^-(p_2)| O_1| B(p_B)>_{nr} = 
-[f_3 m_3^2 r^{nr} \nonumber\\
&&+ \frac{1}{2} f_3(m_B^2 - m_3^2 -s)w_+^{nr} + 
\frac{1}{2}f_3 (s + 2 t - m_B^2 - 2 m_1^2 - m_3^2)w_-^{nr}]
\label{o1mv}
\end{eqnarray}
where  $s= (p_B - p_3)^2$, $t= (p_B - p_1)^2$ and $u= (p_B-p_2)^2$ and 
the form factors $w_\pm^{nr}$  and $r^{nr}$ in 
$< \pi^-(p_1) \pi^+(p_2) | ({\bar u} b)_{V -A}| B^- (p_B)>_{nr}$ 
are computed in the Heavy Quark Effective Theory
(HQET)\cite{Fajfer,Bajc2} with the $B, B^{*}$ propagators in the
pole contributions are the full propagators.
The operator $O_4$ has the same kind of decomposition as $O_1$,
while  $O_6$ can be rewritten as the product of 
density operators. We have, in the factorization model: 
\begin{eqnarray}
&&<O_{6}>_{\rm nr} =< K^-(p_3) \pi^+(p_1) \pi^-(p_2)| O_6| B(p_B)>_{nr} 
= -(\frac{ {\mathcal B}}{m_B})[ 2 \frac{f_1 f_2}{f_3} m_3^2 r^{nr}\nonumber\\
&&+ \frac{f_1 f_2}{f_3} (m_B^2 + m_3^2 -s)w_+^{nr} 
+ \frac{f_1 f_2}{f_3} (s + 2 t - m_B^2 - 2 m_1^2- m_3^2 )w_-^{nr}].
\label{o6mv}
\end{eqnarray}
The $B^- \rarr K^- \pi^+ \pi^-$ decay amplitude is 
\be
\kern -0.5cm    {\mathcal M}_{nr} = \frac{G}{{\sqrt 2}}[ V_{ub} V_{us}^* a_1
<O_{1}>_{\rm nr} - \ V_{tb} V_{ts}^* ( a_4<O_{4}>_{\rm nr}
   + \ a_6<O_{6}>_{\rm nr} )].
\label{amp}
\ee
from which we obtain  the non resonant branching ratios:
\be
{\rm BR}(B^-\rarr K^- \pi^+ \pi^-)_{nr} = T + P + I_1 \cos \gamma
   + I_2 \sin \gamma. 
\label{kpp-b1}
\ee
with $\gamma$ the CP violating weak phase. 
$T = 7.0\times 10^{-6}$, $P= 7.5 \times 10^{-5}$,
$I_1 = -4.3 \times 10^{-5}$, $I_2 = -1.5 \times 10^{-5}$, and 
\be
{\rm BR}(B^-\rarr K^- K^+ K^-)_{nr} = T + P + I_1 \cos \gamma
   + I_2 \sin \gamma. 
\label{kkk-b1}
\ee
with $T = 3.4\times 10^{-6}$, $P= 3.7\times 10^{-5}$, 
$I_1 = -2.1 \times 10^{-5}$ ,$I_2 = -7.4 \times 10^{-6}$. 
These values should be considered as upper limit for the non resonant
branching ratios. The  CP asymmetry of the total decay rate is 
\be
A = \frac{ \sin \gamma N_1}{N_2 + \cos \gamma N_3}, 
\label{asint1}
\ee
with $N_1= -3.0 \times 10^{-5}$, $N_2= 16.4 \times 10^{-5}$ , 
$N_3= -8.6\times 10^{-5}$ for $B^- \rarr  \pi^+ \pi^- K^-$ decay
and $N_1= -1.5 \times 10^{-5}$, $N_2= 8.2 \times 10^{-5}$,
$N_3= -4.2\times 10^{-5}$ for $B^- \rarr  K^+ K^- K^-$ decay. This result is
essentially independent of the form factors since in the 
$SU(3)$ limit, $<O_{6}>= ({2\mathcal B}/m_{B})<O_{1}>$. 
The resonant contribution  is given by 
\begin{eqnarray}
&&{\mathcal M}_{r}(B^- \rarr  \chi_{c0} K^- \rarr \pi^+ \pi^- K^-)  = \nonumber\\
&&{\mathcal M}(B^{-} \rarr \chi_{c0} K^- ) \frac{1}{ s - m_{\chi_{c0}}^2 +  i
\Gamma_{\chi_{c0}} m_{\chi_{c0}}}
{\mathcal M}( \chi_{c0} \rarr \pi^+ \pi^- ).
\label{ares}
\end{eqnarray}
Using the Belle measured $B^- \rarr  \chi_{c0} K^- $ branching 
ratio\cite{Belle1}, we
computed the differential decay rates and CP asymmetries for
the two-pion and two-kaon system in the $\chi_{c0} $ mass region (see 
Figs.1 and Figs.2 of the published work\cite{Fajfer1}).
For the integrated CP asymmetry over the $\chi_{c0} $ resonance, we find
$ A_p(B^\pm\rarr K^\pm \pi^+ \pi^-) = 7.9 \sin \gamma/(73 - 1.2 \cos \gamma)$
and 
$A_p(B^\pm\rarr K^\pm K^+ K-) =  7.2 \sin \gamma/(41 - 5.6 \cos \gamma ) $ .




\bigskip 

 




 











%\section*{Acknowledgments}

\begin{thebibliography}{9}


\bibitem{Eilam} G. Eilam, M. Gronau and R. R. Mendel, 
Phys. Rev. Lett. {\bf 74} (1995) 4984.

\bibitem{Deshpande} N. G. Deshpande, G. Eilam, X. G. He and J. 
Trampeti\' c, Phys. Rev. D {\bf  52} (1995) 5354.

\bibitem{Fajfer} B. Bajc, S. Fajfer, R. J. Oakes, T. N. Pham
   and S. Prelov\v sek,  Phys. Lett. {\bf B} 447 (1999) 313.

\bibitem{Belle1} K. Abe {\em et al.}, Phys. Rev. Lett. {\bf  88} 
(2002) 031802.

\bibitem{Belle2} K. Abe {\em et al.}, .

\bibitem{Babar} B. Aubert et al., .

\bibitem{Fajfer1} S. Fajfer, R. J. Oakes and T. N. Pham,
Phys. Lett. {\bf B} 539 (2002) 67.

\bibitem{Ali} A. Ali, G. Kramer,  and Cai-Dian L\" u,
Phys. Rev. D {\bf 59} (1999) 014005.

\bibitem{Deshpande1} N. G. Deshpande and X. G. He, Phys. Lett. 
 B {\bf 336} (1994) 471; N. G. Deshpande, X. G. He, W. S. Hou, 
and S. Pakvasa, Phys. Rev. Let. {\bf 82} (1999) 2240.

\bibitem{Isola} C. Isola and T. N. Pham, Phys. Rev. D {\bf 62}  
(2000) 094002. 

\bibitem{Fajfer2} B. Bajc, S. Fajfer, R. J. Oakes,  and T. N. Pham,
Phys. Rev. D {\bf 58} (1998) 054009.

\bibitem{Bajc2} B. Bajc, S. Fajfer,  and R. J. Oakes,
Phys. Rev. D {\bf 53} (1996) 4957~; B. Bajc, S. Fajfer, R. J. Oakes,
and S. Prelov\v sek, Phys. Rev. D  {\bf 56} (1997) 7027.



\end{thebibliography}


\end{document}


