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\begin{document}
\hbadness=10000 \pagenumbering{arabic}

\preprint{{\vbox{\hbox{
IPAS-HEP-01-k003, NCKU-HEP-01-06}}}} \vspace{1.5cm}

\title{\Large \bf
Three-body nonleptonic $B$ decays in perturbative QCD
}
\date{\today}
\author{\large \bf  Chuan-Hung Chen$^{a}$\footnote{Email:
chchen@phys.sinica.edu.tw} and Hsiang-nan
Li$^{a,b}$\footnote{Email: hnli@phys.sinica.edu.tw}}

\vskip1.0cm

\address{ $^{a}$ Institute of Physics, Academia Sinica,
Taipei, Taiwan 115, Republic of China}
\address{ $^{b}$ Department of Physics, National Cheng-Kung University,
Tainan, Taiwan 701, Republic of China}
\maketitle

\begin{abstract}

We develop perturbative QCD formalism for three-body nonleptonic $B$
meson decays. Leading contributions are identified by defining the power
counting rules for various topologies of amplitudes. The analysis is simplified
into the one for two-body decays by introducing two-pion distribution
amplitudes. This formalism predicts both nonresonant and resonant
contributions, and can be generalized to baryonic decays.

\end{abstract}

%\newpage
\vskip 1.0cm

The perturbative QCD (PQCD) formalism for charmful and charmless
two-body $B$ meson decays has been developed for some time
\cite{YL,KLS,LUY}. This formalism is based on factorization theorem
\cite{BL,L4}, in which nonperturbative dynamics of the above
processes is factorized into meson distribution amplitudes and the
remaining contribution, being calculable in perturbation theory, is grouped
into hard amplitudes. A meson distribution amplitude, though not calculable,
is universal, and can be determined by nonperturbative methods. Hence, PQCD
has a predictive power, whose predictions for $B\to PP$, $VP$, and $VV$
modes are all in agreement with data \cite{Keum}. Besides, PQCD is
self-consistent in the sense that the involved {\em hard} amplitudes are free of
end-point singularities due to the inclusion of Sudakov effects \cite{TLS}.

Three-body nonleptonic $B$ meson decays have been observed recently
\cite{Belle,Bar}. Viewing the experimental progress, it is urgent to construct a
reliable framework for these modes. Motivated by its theoretical
self-consistency and phenomenological success, we shall generalize PQCD
to three-body nonleptonic $B$ meson decays. A direct evaluation of the
hard amplitudes, which contain two virtual gluons at lowest order, is, on one
hand, not practical due to the enormous number of diagrams. On the other
hand, the region with the two gluons being hard simultaneously is
power-suppressed and not important. Therefore, a new input is necessary in
order to catch dominant contributions to three-body decays in a simple manner.
The idea is to introduce two-meson distribution amplitudes \cite{MP}, by
means of which the analysis is simplified into the
one for two-body decays (number of diagrams is greatly
reduced). Both nonresonant contributions and resonant contributions through
two-body channels can be included. Moreover, the application
of this formalism to three-body baryonic decays is straightforward.

Three-body decay amplitudes are classified into four topologies,
depending on number of light mesons emitted from the four-fermion
vertices. Topologies I and III are associated with one light meson emission and
three light meson emission, respectively. The former involves transition of the
$B$ meson into two light mesons. In the latter case a $B$ meson annihilates
completely. For two light meson emission, we assign IIs to the special
amplitude corresponding to the scalar vertex, and II to the rest of the
amplitudes. Both topologies II and IIs are expressed as a product of a
heavy-to-light form factor and a time-like light-light form factor in the
heavy-quark limit.

The dominant kinematic region for three-body $B$ meson decays is the one,
where at least one pair of light mesons has the invariant mass of
$O(\bar\Lambda M_B)$, $\bar\Lambda=M_B-m_b$ being the $B$ meson
and $b$ quark mass difference. This region includes the configurations,
where one of the light mesons carries momentum of $O(\bar\Lambda)$
(strongly overlaps with the $B$ meson), and the other two carry momenta of
$O(M_B)$, and where all three mesons carry momenta of $O(M_B)$, but two
of them move almost parallelly. The contribution from the region, where all
three pairs have the invariant mass of $O(M_B^2)$ is power-suppressed.

We define the power counting
rules for the various topologies in the dominant kinematic region, and
identify the leading ones. Topology I behaves like $(\bar\Lambda M_B)^{-2}$,
where one power of $(\bar\Lambda M_B)^{-1}$ comes from the hard gluon,
which kicks the soft spectator in the $B$ meson into a fast one in a light
meson, and another power is attributed to the invariant mass of the two light
mesons from the $B$ meson transition. The overall product of the meson
decay constants is not shown explicitly. Topology II exhibits the
same power behavior as topology I.
The scalar vertex introduces an extra power $m_0/M_B$, $m_0$ being the
chiral symmetry breaking scale, to topology IIs.  Topology III must involve
large energy release for producing at least a pair of fast mesons with the
invariant mass of $O(M_B^2)$. That is, it behaves like
$(\bar\Lambda M_B)^{-1}M_B^{-2}$. Hence, we have the relative importance
of the decay amplitudes,
\begin{eqnarray}
{\cal M}_{\rm I}:{\cal M}_{\rm II}:{\cal M}_{\rm IIs}:{\cal M}_{\rm III}
=1:1:\frac{m_0}{M_B}:\frac{\bar\Lambda}{M_B}\;,
\end{eqnarray}
indicating that topology III is negligible.

Take the $B^+\to K^+\pi^+\pi^-$ mode as an example. The $\pi^+$ and
$\pi^-$ mesons carry the momenta $P_1$ and $P_2$, respectively.
The $B$ meson momentum $P_B$, the total momentum of the two pions,
$P=P_1+P_2$, and the kaon momentum $P_3$ are chosen as
$P_B=M_B/\sqrt{2}(1,1,{\bf 0}_T)$, $P=M_B/\sqrt{2}(1,\eta,{\bf 0}_T)$, and
$P_3=M_B/\sqrt{2}(0,1-\eta,{\bf 0}_T)$,
with the variable $\eta=w^2/M_B^2$, $w^2=P^2$ being the invariant mass
of the two-pion system. The light-cone coordinates have been adopted here.
Define $\zeta=P_1^+/P^+$ as the $\pi^+$ meson momentum fraction, in
terms of which, the other kinematic variables are expressed as
$P_2^+=(1-\zeta)P^+$, $P_1^-=(1-\zeta)\eta P^+$, $P_2^-=\zeta\eta P^+$,
and $P_{1T}^2=P_{2T}^2=\zeta(1-\zeta)w^2$.

The two pions from the $B$ meson transition possess the invariant mass
$w^2\sim O(\bar\Lambda M_B)$, implying the orders of magnitude
$P^+\sim O(M_B)$, $P^-\sim O(\bar\Lambda)$ and
$P_T\sim O(\sqrt{\bar\Lambda M_B})$. In the heavy-quark limit, the
hierachy $P^+\gg P_T\gg P^-$ corresponds to a collinear configuration.
Therefore, we introduce the two-pion distribution amplitudes \cite{MP},
\begin{eqnarray}
\Phi_i(z,\zeta,w^2)&=&\frac{N_i}{2\sqrt{2N_c}}\int \frac{dy^-}{2\pi}
e^{-izP^+y^-}\langle\pi^+(P_1)\pi^-(P_2)|\bar\psi(y^-)\Gamma_iT
\psi(0)|0\rangle\;,\;\;\;\;i=v,s,t\;,
\label{pa}
\end{eqnarray}
with $N_v=1$ and $\Gamma_v=\not n_-$ for the twist-2 component, and with
$N_s=1$ and $\Gamma_s=P\cdot n_-/w$ and $N_t=1/[(2\zeta-1)w]$ and
$\Gamma_t=i\sigma_{\mu\nu}n_-^\mu P^\nu$ for the twist-3 components.
The structures with $\mu$ or $\nu$ in $\sigma_{\mu\nu}$
being a transverse index do not contribute. 
$T=\tau^3/2$ is for the isovector $I=1$ state, $\psi$ the $u$-$d$ doublet,
$z$ the momentum fraction carried by the valence $u$ quark,
and $n_-=(0,1,{\bf 0}_T)$ a dimensionless vector on the light cone. The matrix
element with the structure $\gamma_5\not n_-$ vanishes for topologies I
and IIs, and contributes to topology II at twist 4. The one with the structure
$\gamma_5$ simply vanishes.
For other two-pion systems, the distribution amplitudes can be defined with
the appropriate choice of the matrix $T$. For instance, $T=1/2$ is for the
$\pi^0\pi^0$ isoscalar ($I=0$) state. Obviously, the two infrared
configurations mentioned before have been included in the two-pion
distribution amplitudes.

A two-pion distribution amplitude can be related to the pion distribution 
amplitude through the calculation of the process 
$\gamma\gamma^*\to \pi^+\pi^-$ at large invariant mass $w^2$ \cite{DFK}.
The extraction of the two-pion distribution amplitudes from the
$B\to \pi\pi l\bar\nu$ decay has been discussed in \cite{M}. Here
we pick up the leading term in the complete Gegenbauer expansion of
$\Phi_i(z,\zeta,w^2)$ \cite{MP}:
\begin{eqnarray}
\Phi_{v}(z,\zeta,w^2)=\frac{3F_{\pi}(w^2)}{\sqrt{2N_c}}z(1-z)(2\zeta-1)\;,
\;\;\;\;
\Phi_{s,t}(z,\zeta,w^2)=\frac{3F_{s,t}(w^2)}{\sqrt{2N_c}}z(1-z)\;,
\label{2pi}
\end{eqnarray}
where $F_{\pi,s,t}(w^2)$ are the time-like pion electromagnetic, scalar and
tensor form factors with $F_{\pi,s,t}(0)=1$. That is, the two-pion distribution
amplitudes are normalized to the time-like form factors. For the $B$ meson
distribution amplitude, we adopt the model
$\Phi_{B}(x)=N_{B}x^{2}(1-x)^{2}\exp [-(xM_{B}/\omega _{B})^{2}/2]$
\cite{KLS}
with the shape parameter $\omega_{B}=0.4$ GeV, and the normalization
constant $N_{B}$ related to the decay constant $f_{B}=190$
MeV (in the convention $f_{\pi}=130$ MeV)
via $\int_{0}^{1}\Phi_{B}(x)dx=f_{B}/(2\sqrt{2N_c})$.

The total decay rate is written as
\begin{eqnarray}
\Gamma=\frac{G_F^2M_B^5}{512\pi^4}\int_0^1 d\eta (1-\eta)\int_0^1 d\zeta
|{\cal M}|^2\;,\;\;\;\;{\cal M}={\cal M}_{\rm I}+{\cal M}_{\rm II}+
{\cal M}_{\rm IIs}\;,
\end{eqnarray}
with the amplitudes,
\begin{eqnarray}
{\cal M}_{\rm I}&=&f_{K}\Big(V^{*}_{t}\sum_{i=4,6}{\cal F}^{P(u)}_{ei}
- V^{*}_{u}{\cal F}_{e2}\Big)\;, \;\;\;\;
{\cal M}_{\rm IIs}=V^{*}_{t}F_{s}(\omega^2)F^{P(d)}_{e6} \;,
\nonumber \\
{\cal M}_{\rm II}&=&
(2\zeta-1)F_{\pi}(\omega^2)\Big[V^{*}_{t}\Big(\sum_{i=3}^5 F^{P(d)}_{ei}
+\sum_{i=3,5}F^{P(u)}_{ei}\Big)-V^{*}_{u} F_{e1}\Big]\;.
\end{eqnarray}
For a simpler presentation, we have assumed that the form factors
involving a kaon are equal to those involving a pion multiplied by the
ratio of the decay constants $f_K/f_\pi$.
We calculate the hard amplitudes by contractng the structures,
\begin{equation}
\frac{(\not{P}_B+M_B)\gamma_5}{\sqrt{2N_c}}\Phi_B(x)\;,\;\;\;\;
\frac{1}{\sqrt{2N_c}}\left[\not P\Phi_v(z,\zeta,w^2)
+w\Phi_s(z,\zeta,w^2)
+\frac{\not P_1\not P_2-\not P_2\not P_1}{w}
\Phi_t(z,\zeta,w^2)\right]\;,
\end{equation}
to the lowest-order diagrams.

The factorization formulas for the $B\to 2\pi$ transition amplitudes
are given by
\begin{eqnarray}
{\cal F}^{P(u)}_{e4}&=& 8\pi C_{F} M^2_{B} (1-\eta) \int^{1}_{0}
dx_{1} dz\frac{\Phi_{B}(x_{1})}{x_{1}zM^{2}_{B}+P_{T}^2}
\nonumber \\
&& \times \Big\{\Big[
(1+z)\Phi_{v}(z,\zeta,w^2)-\sqrt{\eta}(1-2z)(1-2\zeta)\Phi_{t}(z,\zeta,w^2)
+\sqrt{\eta}(1-2z)\Phi_{s}(z,\zeta,w^2)\Big]
\frac{\alpha _{s}( t^{(1)}_{e})a_{4}^{(u)}(t^{(1)}_{e})}
{zM^2_{B}+P_{T}^2}
\nonumber \\
&&-\Big[ \eta \Phi_{v}(z,\zeta,w^2)
-2\sqrt{\eta}\Phi_{s}(z,\zeta,w^2)\Big]
\frac{\alpha _{s}( t^{(2)}_{e})a_{4}^{(u)}(t^{(2)}_{e})}
{x_{1}M^2_{B}}\Big\}\;,
\nonumber\\
{\cal F}^{P(u)}_{e6}&=& -16\pi C_{F} M^2_{B} r_{0} \int^{1}_{0}
dx_{1} dz\frac{\Phi_{B}(x_{1})}{x_{1}zM^{2}_{B}+P_{T}^2}
\nonumber \\
&& \times \Big\{\Big[ (1+\eta-2z\eta)\Phi_{v}(z,\zeta,w^2)
+\sqrt{\eta} z(1-2\zeta)\Phi_{t}(z,\zeta,w^2)
+\sqrt{\eta}(2+z)\Phi_{s}(z,\zeta,w^2)\Big]
\frac{\alpha _{s}( t^{(1)}_{e})a_{6}^{(u)}(t^{(1)}_{e})}
{zM^2_{B}+P_{T}^2}
\nonumber\\
&&-\Big[ \eta\Phi_{v}(z,\zeta,w^2)-2\sqrt{\eta}
 \Phi_{s}(z,\zeta,w^2) \Big]
\frac{\alpha _{s}( t^{(2)}_{e})a_{6}^{(u)}(t^{(2)}_{e})}
{x_{1}M^2_{B}}\Big\}\;,
\label{cf}
\end{eqnarray}
with $r_0=m_0/M_B$.
${\cal F}_{e2}$ is the same as ${\cal F}^{P(u)}_{e4}$ but with $a^{(u)}_{4}$
replaced by $a_{2}$ (here $a_2$ is close to unity). The definitions of
the Wilson coefficients $a^{(q)}(t)$ are referred to \cite{CKL1}, where the
superscript $q$ stands for the light quark pair emitted from the penguin
operators. The hard scales are defined by
$t_e^{(1)}=\max[\sqrt{z}M_B, P_T]$ and $t_e^{(2)}=\max[\sqrt{x_1}M_B, P_T]$.
Note that in three-body decays the invariant mass of the two-pion
system, proportional to $P_T$, provides natural smearing of the
end-point singularities. In two-body decays the parton transverse
momenta $k_T$ and the corresponding Sudakov factor, which
describes the $k_T$ distribution, must be taken into account to
achieve the same effect \cite{KLS,TLS}.

In the dominant kinematic region the $B$
to light meson transition occurs at fast recoil, and is calculable in PQCD.
The form factors for topologies II and IIs are
\begin{eqnarray}
F^{P(d)}_{e4}(\zeta,\eta) &=&8\pi C_{F}M_{B}^{2}
\int_{0}^{1}dx_1dx_3\int_{0}^{\infty}b_{1}db_{1}b_{3}db_{3}
\phi_{B}( x_{1},b_{1})
\nonumber \\
&&\times \Big\{ \Big[ (1-\eta)(1+(1-\eta)x_{3})\phi_{K}(x_{3})
+r_{0}(1+\eta-2(1-\eta)x_{3})\phi_{K}^{p}(x_{3})
\nonumber \\
&&+r_{0}(1-\eta)(1-2x_{3})\phi_{K}^{\sigma}(x_{3})\Big]
E_4^{(d)}( t_{e}^{( 1) }) h_e(x_{1},(1-\eta)x_{3},b_{1},b_{3})
\nonumber \\
&& +2r_{0}(1-\eta)\phi _{K}^{p}(x_{3}) E_4^{(d)}( t_{e}^{( 2) })
h_e((1-\eta)x_{3},x_{1},b_{3},b_{1}) \Big\} \;,
\end{eqnarray}
\begin{eqnarray}
F^{P(d)}_{e6}(\zeta,\eta) &=&16\pi
C_{F}M_{B}^{2}\sqrt{\eta}\int_{0}^{1}dx_1dx_3\int_{0}^{\infty}
b_{1}db_{1}b_{3}db_{3}\ \phi_{B}( x_{1},b_{1})
\nonumber \\
&&\times \Big\{ \Big[ (1-\eta)\phi _{K}(x_{3})
+2r_{0}\phi_{K}^{p}(x_{3})+r_{0}(1-\eta)x_{3}\Big(\phi_{K}^{p}(x_{3})
-\phi_{K}^{\sigma}(x_{3})\Big)\Big]
\nonumber \\
&&\times E_6^{(d)}( t_{e}^{( 1) }) h_e(x_{1},(1-\eta)x_{3},b_{1},b_{3})
\nonumber \\
&& +2r_{0}(1-\eta)\phi _{K}^{p}(x_{3}) E_6^{(d)}( t_{e}^{( 2) })
h_e((1-\eta)x_{3},x_{1},b_{3},b_{1}) \Big\} \;.
\end{eqnarray}
The definitions of the evolution factors $E_i^{(q)}(t)$, which contain the
Wilson coefficients $a^{(q)}_{i}(t)$, of the hard functions $h_{e}$, and
of the kaon distribution amplitudes $\phi_{K}$, $\phi_{K}^{p}$ and
$\phi_{K}^{\sigma}$ are referred to \cite{CKL1}.
$F^{P(q)}_{e3}$, $F^{P(q)}_{e5}$  and $F_{e1}$ are obtained from
$F^{P(d)}_{e4}$ by substituting $a^{(q)}_{3}$, $a^{(q)}_{5}$ and $a_1$
for $a^{(d)}_{4}$, respectively.

The PQCD evaluation of the form factors indicates the power behavior
 in the asymptotic region, $F_\pi(w^2)\sim 1/w^2$, and their relative
importance: $F_{s,t}(w^2)/F_\pi(w^2)\sim m_0/w$. Therefore, the twist-3
contributions in Eqs.~(\ref{cf}) are down by a power of $m_0/M_B$
compared to the twist-2 ones, which is the accuracy considered here.
To calculate the nonresonant contribution, we propose the parametrization,
\begin{eqnarray}
F^{(nr)}_\pi(w^2)=\frac{m^2}{w^2+m^2}\;,\;\;\;\;
F^{(nr)}_{s,t}(w^2)=\frac{m_0 m^2}{w^3+m_0 m^2}\;,
\label{non}
\end{eqnarray}
where the the parameter $m=1$ GeV is determined by the fit to
the  experimental data $M_{J/\psi}^2|F_\pi(M_{J/\psi}^2)|^2\sim 0.9$
GeV$^2$ \cite{PDG}, $M_{J/\psi}$ being the $J/\psi$ meson mass. These
form factors can carry strong phases, which are
assumed to be not very different, {\it i.e.}, overall and negligible. This
assumption is supported by their PQCD analysis at large $w^2$
\cite{KLS,CKL1}.

To calculate the resonant contribution, we parametrize
it into the time-like form factors,
\begin{eqnarray}
F^{(r)}_{\pi,s,t}(w^2)=\frac{M_V^2}{\sqrt{(w^2-M_V^2)^2+\Gamma_V^2w^2}}
-\frac{M_V^2}{w^2+M_V^2}\;,
\label{res}
\end{eqnarray}
with $\Gamma_V$ being the width of
the meson $V$.  The subtraction term renders Eq.~(\ref{res}) exhibit
the features of resonant contributions: the normalization
$F^{(r)}_\pi(0)=0$ and the asymptotic behavior
$F^{(r)}_\pi(w^2)\sim 1/w^4$, which decreases at large $w$ faster than
the nonresonant parametrization in Eq.~(\ref{non}). Equation~(\ref{res})
is motivated by the pion time-like form factor measured at the $\rho$
resonance \cite{RR}. It is likely that all
$F^{(r)}_{\pi,s,t}$ contain the similar resonant contributions.
The relative phases among different resonances will be
discussed elsewhere by employing the more sophisticated
parametrization \cite{GS}. Here we assume the absence of the inteference
effect.

We adopt $m_0=$ 1.4 (1.7) GeV for the pion (kaon) and the unitarity angle
$\phi_3=90^o$ \cite{KLS}. For the $B^+\to\rho^0(770) K^+$
and $B^+\to f_0(980) K^+$ channels, we choose $\Gamma_\rho=150$ MeV
and $\Gamma_{f_0}=50$ MeV \cite{KH}.
%, which is determined by the fit to the
%$e^+e^-\to \pi^+\pi^-$ cross section around the $\rho$ resonance \cite{RR}.
%For the $B^+\to f_0(980) K^+$ channel, we choose $\Gamma_{f_0}=0.045$,
%such that the width of the $f_0$ meson is about 44 MeV \cite{EA}.
The nonresonant contribution $0.61\times 10^{-6}$ to the $B^+\to
K^+\pi^+\pi^-$ branching ratio is obtained. Our results $1.8\times
10^{-6}$ and $13.2\times 10^{-6}$ are consistent with the measured
three-body decay branching ratios through the $B^+\to \rho(770)
K^+$  and $B^+\to f_0(980) K^+$ channels, $< 12\times 10^{-6}$ and
$(9.6^{+2.5+1.5+3.4}_{-2.3-1.5-0.8})\times 10^{-6}$ \cite{Belle},
respectively. Since the $f_0$ width has a large uncertainty, we
also consider $\Gamma_{f_0}=60$ MeV, and the branching ratio
reduces to $10.5\times 10^{-6}$. The resonant contributions from
the other channels can be analyzed in a similar way. For example,
the $K^*(892)$ resonance can be included into the $K$-$\pi$ form
factors by choosing the width $\Gamma_{K^*}=50$ MeV. The
nonresonant and resonant contributions to the $B^+\to
K^+\pi^+\pi^-$ decay spectrum are displayed in Fig.~1, which
confirms that the region with the two meson invariant mass of
$O(\bar\Lambda M_B)$ dominates.

The factorization formulas for the decay $B^+\to K^+\pi^+\pi^-$ hold for
$B^+\to K^+K^+K^-$ with the appropriate replacement of $m_0$, the meson
distribution amplitudes, and the form factors.
The nonfactorizable contributions and topology III, being of
$O(\bar\Lambda/M_B)$, can also be evaluated systematically by means
of the two-pion distribution amplitudes, from which nontrivial strong phases
are generated. Therefore, CP asymmetries in three-body nonleptonic $B$
meson decays are expected to be smaller than in two-body decays, in which
strong phases are produced at $O(m_0/M_B)$ \cite{KLS}.
The framework presented here is not only applicable to the study
of both the nonresonant and resonant contributions to three-body
mesonic $B$ meson decays, but also to baryonic decays, such as
$B\to p\bar p K$. We simply introduce two-proton distribution amplitudes,
and the calculation of the corresponding hard amplitudes is the same. That
is, our formalism is general enough and provides a simple tool for the
analysis of three-body $B$ meson decays.

We thank S. Brodsky, H. Cheng, M. Diehl and A. Garmash for useful
discussions. The work was supported in part by the National Science Council
of R.O.C. under Grant No. NSC-90-2112-M-001-077, by the National Center for
Theoretical Sciences of R.O.C., and by Theory Group of KEK, Japan.


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\begin{figure}[tbp]
\vspace{2cm} \centerline{ \psfig{figure=figab.eps,height=3.in } }
\caption{(a) [(b)] Nonresonant (resonant) contribution to the
$B^+\to K^+\pi^+\pi^-$ decay spectrum with respect to the two-pion
invariant mass $M(\pi^+\pi^-)$. The sharp peak corresponds to the
$f_0$ resonance with the width 50 MeV.}
\end{figure}



\end{document}

