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%%International Journal of Modern Physics A --- IJMPA %%%%%



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\markboth{H. Kawamura, J. Kodaira, C.F. Qiao and K. Tanaka}
{$B$ Meson Light-cone Distribution Amplitudes and Heavy-quark Symmetry}

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\title{$B$ MESON LIGHT-CONE DISTRIBUTION AMPLITUDES\\
AND HEAVY-QUARK SYMMETRY\footnote{Talk presented by K. Tanaka.}
}

\author{\footnotesize HIROYUKI KAWAMURA%\footnote{
%Typeset names in
%10 pt roman, uppercase. Use the footnote to indicate the
%present or permanent address of the author.}
}

\address{Deutsches Elektronen-Synchrotron, DESY\\
Platanenallee 6, D 15738 Zeuthen, GERMANY
%\footnote{State completely without abbreviations, the
%affiliation and mailing address, including country. Typeset in 8 pt
%italic.}
}

\author{JIRO KODAIRA\, and CONG-FENG QIAO\footnote{JSPS Research Fellow.}}

\address{Dept. of Physics, Hiroshima University\\
Higashi-Hiroshima 739-8526, JAPAN
}

\author{KAZUHIRO TANAKA}

\address{Dept. of Physics, Juntendo University\\
Inba-gun, Chiba 270-1695, JAPAN
}


\maketitle

%\pub{Received (received date)}{Revised (revised date)}

\begin{abstract}
%We investigate 
We present a systematic study of the $B$ meson light-cone distribution amplitudes
%in the heavy-quark limit 
which are relevant for the QCD factorization
approach for the exclusive $B$ meson decays. 
%We take into account derive exact relations between two- and three-particle
%distribution amplitudes from the QCD equations of motion
%and heavy-quark symmetry constraint.
%As solution of these relations, 
We construct representations for the
quark-antiquark distribution amplitudes in terms of independent 
dynamical degrees of freedom,
which exactly satisfy the QCD equations of motion 
and constraints from heavy-quark symmetry.
% in the heavy-quark limit.
%In particular, we find that the Wandzura-Wilczek-type contributions
%are determined uniquely in analytic form in terms of $\bar{\Lambda}$,
%a fundamental mass parameter of heavy-quark effective theory,
%and that both leading- and higher-twist distribution amplitudes
%receive the contributions of multi-particle states with additional gluons.
\end{abstract}

%\section{General Appearance}	%) A SECTION HEADING
\vspace{0.7cm}
\noindent
Recently systematic methods based on the QCD factorization
have been developed for the exclusive $B$ meson decays
into light mesons\cite{Beneke:2000ry,Beneke:2001wa,Beneke:2001at,Bauer:2001cu}
(for other approaches see\cite{Ball:1998kk}).
%, e.g. 
%Ref.\cite{Ball:1998kk}).
Essential ingredients in this approach are
the light-cone distribution amplitudes for the participating mesons,
which constitute nonperturbative long-distance contribution
to the factorized amplitudes.
{}For the light mesons ($\pi$, $K$, 
%$\eta$, 
$\rho$, 
%$\omega$, 
$K^{*}$, etc.)
%$\phi$)
appearing in the final state,
systematic model-independent study of the light-cone distributions 
exists for both leading 
and higher twists.\cite{Braun:1990iv}
On the other hand, 
%unfortunately,
the light-cone distribution amplitudes for
the $B$ meson are not well-known at present and 
provide a major source of uncertainty in the calculations of 
the decay rates.
In this work,\cite{KKQT} 
we demonstrate that 
heavy-quark symmetry and constraints from the equations of motion
determine a unique analytic solution for the $B$ meson light-cone
distribution amplitudes within the two-particle Fock states.
We also derive the exact integral representations 
for the effects of higher Fock states with additional gluons.
%A complete set of the $B$ meson distribution amplitudes
%are constructed in terms of independent dynamical degrees of freedom,
%which satisfies all relevant QCD constraints.

In the heavy-quark limit, the $B$ meson matrix elements obey 
the heavy-quark symmetry,
and is described by the heavy-quark
effective theory (HQET).\cite{Neubert:1994mb}
Following Refs.\cite{Grozin:1997pq,Beneke:2001wa},
we introduce the quark-antiquark light-cone distribution 
amplitudes $\tilde{\phi}_{\pm}(t)$ of the $B$ meson in terms of matrix element
%in terms of vacuum-to-meson matrix element of nonlocal 
%light-cone operators 
in the HQET:
%
\be
\langle 0 | \bar{q}(z) \Gamma h_{v}(0) |\bar{B}(p) \rangle
 = - \frac{i f_{B} M}{2} {\rm Tr}
 \left[ \gamma_{5}\Gamma \frac{1 + \slashs{v}}{2}
 \left\{ \tilde{\phi}_{+}(t) - \slashs{z} \frac{\tilde{\phi}_{+}(t)
 -\tilde{\phi}_{-}(t)}{2t}\right\} \right]\ . 
 \label{phi}
\ee
%
where $z_{\mu}$ is a light-like vector ($z^{2}=0$), $v^{2} = 1$,
$t=v\cdot z$,
and $p^{\mu} = Mv^{\mu}$
is the 4-momentum of the $B$ meson with mass $M$.
$h_{v}(x)$ denotes the effective $b$-quark 
field,\cite{Neubert:1994mb}
$b(x) \approx \exp(-im_{b} v\cdot x)h_{v}(x)$, 
and is subject to the on-shell 
constraint, $\slashl{v} h_{v} = h_{v}$.
$\Gamma$ is a generic Dirac matrix and,
here and in the following, the path-ordered gauge
factors are implied in between the constituent fields.
$f_{B}$ is the decay constant defined as usually as
%
%\be 
$\langle 0 | \bar{q}(0) \gamma^{\mu}\gamma_{5} h_{v}(0) |\bar{B}(p) \rangle
   = i f_{B} M v^{\mu}$
%\ , 
%\label{fb}
%\ee
%
so that $\tilde{\phi}_{\pm}(t=0) = 1$.
%The behavior of the RHS of eq.(\ref{phi}) 
%for a fast-moving meson, $t = v\cdot z \rightarrow \infty$, shows that 
$\tilde{\phi}_{+}$ is of leading-twist,
% distribution amplitude,
whereas $\tilde{\phi}_{-}$ has subleading twist.\cite{Grozin:1997pq} 

It is well-known that the QCD equations of motion
impose a set of relations between distribution 
amplitudes for the light-mesons.\cite{Braun:1990iv}
%To derive these relations, the most convenient method is to
%start with 
The corresponding relations can be derived
from the identities between the nonlocal operators:
%
\bea
 \frac{\partial}{\partial x^{\mu}}
 \bar{q}(x) \gamma^{\mu} \Gamma h_{v}(0)
  &=& 
%\bar{q}(x) \stackrel{\leftarrow}{\slashl{D}} \Gamma h_{v}(0)
%  +
i \int_{0}^{1}duu \ \bar{q}(x) gG_{\mu \nu}(ux) x^{\nu}
  \gamma^{\mu}\Gamma h_{v}(0) \ ,
 \label{id1} \\
 v^{\mu}\frac{\partial}{\partial x^{\mu}}
 \bar{q}(x) \Gamma h_{v}(0)
 &=& 
%- \bar{q}(x) \Gamma D_{\mu} h_{v}(0)
% +
i \int_{0}^{1}du (u-1)\ \bar{q}(x) gG_{\mu \nu}(ux) 
 v^{\mu}x^{\nu}\Gamma h_{v}(0) \nonumber \\
 &+& 
 v^{\mu}\left.
   \frac{\partial}{\partial y_{\mu}}\
  \bar{q}(x+y) \Gamma h_{v}(y)\right|_{y \rightarrow 0}\ ,
%\partial^{\mu}\left\{
% \bar{q}(x) \Gamma h_{v}(0) \right\} 
%\ ,
\label{id2}
\eea
%
%where 
%$x^{\mu}$ is not restricted on the light-cone,
%and 
%$G_{\mu \nu}$
%= (i/g)[D_{\mu}, D_{\nu}]$
%is the gluon field strength tensor.
%These are exact up to the operators containing the equations of motion
where $G_{\mu \nu}$ is the gluon field strength tensor, and
we have used the equations of motion
%$\bar{q}\stackrel{\leftarrow}{\slashl{D}}=0$ 
$\slashl{D}q=0$ 
and $v \cdot D h_{v} = 0$
with $D_{\mu}= \partial_{\mu} - igA_{\mu}$ the covariant derivative.
%for the light-quark
%and effective heavy-quark fields.
%, and 
%we have used the equations of motion
%$\bar{q}\stackrel{\leftarrow}{\slashl{D}}=0$ 
%and $v \cdot D h_{v} = 0$ for the light-quark
%and effective heavy-quark fields.
%, respectively.
%$D_{\mu}= \partial_{\mu} - igA_{\mu}$, 
%$\stackrel{\leftarrow}{D}_{\mu} =
%\stackrel{\leftarrow}{\partial}_{\mu} + ig A_{\mu}$
%are the covariant derivatives, 
%$G_{\mu \nu}$
%= (i/g)[D_{\mu}, D_{\nu}]$
%is the gluon field strength tensor, and
% 
%\be
%  \partial^{\mu}\left\{
%    \bar{q}(x) \Gamma h_{v}(0) \right\} \equiv \left.
%   \frac{\partial}{\partial y_{\mu}}\
%  \bar{q}(x+y) \Gamma h_{v}(y)\right|_{y \rightarrow 0} 
%\label{trans}
%\ee
%
%stands for the derivative over the total translation.
%These identities simply describe the response of the nonlocal operators
%to the change of the interquark separation and/or total translation.
We take the vacuum-to-meson matrix element of these identities and
%the operators involving the equations of motion vanish.
go over to the light-cone limit $x_{\mu} \rightarrow z_{\mu}$.
%the vacuum-to-meson matrix elements of the identities (\ref{id1}) and (\ref{id2})
%yield respectively the two exact relations between the distribution amplitudes.
%we obtain exact relations between the distribution amplitudes in QCD in the heavy-quark limit.
The terms in the LHS of eqs. (\ref{id1}) and (\ref{id2}) yield  
%the 
%appropriate
%distribution amplitudes 
$\tilde{\phi}_{+}(t)$, $\tilde{\phi}_{-}(t)$ defined above and their derivatives:
$d \tilde{\phi}_{\pm}(t)/dt$ 
and 
%\be
% \frac{\partial \tilde{\phi}_{+}(t)}{\partial z^{2}} \equiv 
$\partial \tilde{\phi}_{\pm}(t, x^{2})/\partial x^{2}|_{x^{2} \rightarrow 0}$,
%\label{pd}
%\ee
%
where, via  $\tilde{\phi}_{\pm}(t) \rightarrow \tilde{\phi}_{\pm}(t, x^{2})$,
we extend the definitions in eq.(\ref{phi}) to the case $z \rightarrow x$
($x^{2} \neq 0$).
%, since the derivative in the LHS of
%eq.(\ref{id1}) has to be taken before going to the light-cone limit.
The last term of (\ref{id2}), the derivative over total translation,
yields contribution with
\be 
  \bar{\Lambda} = M - m_{b} =
 \frac{iv\cdot \partial \langle 0| \bar{q} \Gamma h_{v} |\bar{B}(p) \rangle}
  {\langle 0| \bar{q} \Gamma h_{v} |\bar{B}(p) \rangle}\ .
\label{lambda}
\ee
%
This is the usual ``effective mass'' of meson states in the 
HQET.\cite{Neubert:1994mb}


The terms given by an integral of quark-antiquark-gluon operator
are expressed by the three-particle distribution amplitudes
corresponding to the higher-Fock components of the meson wave function.
Through the Lorentz decomposition of the three-particle light-cone 
matrix element, we define the three functions 
$\tilde{\Psi}_V \,(t\,,\,u)$, $\tilde{\Psi}_A \,(t\,,\,u)$,
$\tilde{X}_A \,(t\,,\,u)$
as the independent three-particle distributions:\cite{KKQT}
%
\bea
 \lefteqn{\langle 0 | \bar{q} (z) \, g G_{\mu\nu} (uz)\, z^{\nu}
      \, \Gamma \, h_{v} (0) | \bar{B}(p) \rangle}\nonumber \\  
  &=& \frac{1}{2}\, f_B M \, {\rm Tr}\, \left[ \, \gamma_5\,
      \Gamma \, 
        \frac{1 + \slashs{v}}{2}\, \biggl\{ ( v_{\mu}\slashs{z}
         - t \, \gamma_{\mu} )\  \left( \tilde{\Psi}_A (t,u) 
   - \tilde{\Psi}_V (t,u) \right)
      \right. \nonumber\\
  & & \qquad\qquad\qquad\qquad  - i \, \sigma_{\mu\nu} z^{\nu}\,
           \tilde{\Psi}_V (t,u)
       + \left. \frac{z_{\mu}}{t} \, ( \slashs{z} - t ) \, 
   \tilde{X}_A (t,u)\, \biggr\} \, \right]\ . \label{3elements}
\eea
%
%This is the most general parameterization
%compatible with Lorentz invariance and the heavy-quark limit.


{}From the two identities (\ref{id1}) and (\ref{id2}), we eventually obtain 
the four independent 
differential equations for the distribution amplitudes,\cite{KKQT}
which are exact in QCD in the heavy-quark limit.
% and are the new results.
This system of four differential equations can be organized
into the two sets, so that the first set of two equations  
does not involve the derivatives with respect to the transverse separation,
$\partial \tilde{\phi}_{\pm}(t, x^{2})/\partial x^{2}|_{x^{2} \rightarrow 0}$, 
while the second set of two equations involves them;
the second set is uninteresting for our purpose.
The first set of equations is given by\cite{KKQT}
%By going over to the momentum space, we obtain from
%eqs.(\ref{de1}) and (\ref{fresult5}),
%
\bea
 \omega \frac{d \phi_{-}(\omega)}{d \omega}
  &+& \phi_{+}(\omega) = I(\omega)\ ,
  \label{mde1} \\
  \left(\omega - 2 \bar{\Lambda}\right)\phi_{+}(\omega)
 &+& \omega \phi_{-}(\omega) = J(\omega) \ , \label{mde2}
\eea
where we have introduced the 
momentum-space 
distribution amplitudes $\phi_{\pm}(\omega)$ as
%
%\be
$\tilde{\phi}_{\pm}(t) = \int d\omega \ e^{-i \omega t}
  \phi_{\pm}(\omega)$
%\ , 
%\label{mom}
%\ee
%
with  $\omega v^{+}$ the light-cone projection
of the light-antiquark momentum in the $B$ meson.
$I(\omega)$ and $J(\omega)$ of eqs. (\ref{mde1}) and (\ref{mde2}) denote the 
``source'' terms due to three-particle amplitudes as\cite{KKQT}
%
\be
 I(\omega) = 2\frac{d}{d\omega}
   \int_{0}^{\omega}d\rho \int_{\omega - \rho}^{\infty}\frac{d\xi}{\xi}
  \frac{\partial}{\partial \xi}\left[ \Psi_{A}(\rho, \xi) 
   - \Psi_{V}(\rho, \xi)\right] \ , \label{si} 
%\\
% J(\omega) &=& -2\frac{d}{d\omega}
%  \int_{0}^{\omega}d\rho \int_{\omega - \rho}^{\infty}\frac{d\xi}{\xi}
%  \left[ \Psi_{A}(\rho, \xi) + X_{A}(\rho, \xi)\right]
%  \nonumber \\
%  && -4 \int_{0}^{\omega}d\rho \int_{\omega - \rho}^{\infty}\frac{d\xi}{\xi}
%  \frac{\partial \Psi_{V}(\rho, \xi)}{\partial \xi} \ , \label{sj}
\ee
and similarly for $J(\omega)$.
%
Here the three-particle amplitudes in the momentum space are defined as
%
%\be
$\tilde{F}(t, u) = \int d\omega d \xi \
  e^{-i(\omega  + \xi u)t} F(\omega, \xi)$
% \ ,
%   \qquad (
with $F=\{ \Psi_{V}, \Psi_{A}, X_{A}\}$.
%)
%\ . \label{ff3}
%\ee
%
%Here $\omega v^{+}$ and $\xi v^{+}$ denote 
%the light-cone projection
%of the momentum carried by the light antiquark and the gluon, 
%respectively.
% and $F(\omega, \xi)$ vanishes unless
%$\omega \ge 0$ and $\xi \ge 0$. 

%
A system of equations (\ref{mde1}), (\ref{mde2})
can be solved for $\phi_{+}(\omega)$ and $\phi_{-}(\omega)$
with boundary conditions $\phi_{\pm}(\omega) = 0$
for $\omega < 0$ or $\omega \rightarrow \infty$, and with
normalization condition 
$\int_{0}^{\infty}d\omega \phi_{\pm}(\omega)= \tilde{\phi}_{\pm}(0)= 1$.
The solution can be decomposed into two pieces as
%
\be
  \phi_{\pm}(\omega) = \phi_{\pm}^{(WW)}(\omega) 
  + \phi_{\pm}^{(g)}(\omega) \ ,
\label{decomp}
\ee
%
where $\phi_{\pm}^{(WW)}(\omega)$ are the solution of
eqs. (\ref{mde1}) and (\ref{mde2}) with 
%the source terms set to zero,
$I(\omega)=J(\omega)=0$,
which corresponds to $\Psi_{V}=\Psi_{A}=X_{A}=0$ (``Wandzura-Wilczek approximation'').
$\phi_{\pm}^{(g)}(\omega)$ denote the pieces induced by the source terms.

%First, let us discuss $\phi_{\pm}^{(WW)}$.
Eq.(\ref{mde1}) alone, with $I(\omega)=0$, is 
equivalent to a usual Wandzura-Wilczek type relation derived
in Ref.\cite{Beneke:2001wa}:
%
%\be
$\phi_{-}^{(WW)}(\omega) = \int_{\omega}^{\infty} 
  d\rho \ \phi_{+}^{(WW)}(\rho)/\rho$.
% \ .
%\label{ww}
%\ee
%
Combining eqs.(\ref{mde1}) and (\ref{mde2}), we are able to obtain 
the analytic solution explicitly as ($\omega \ge 0$)
%
\be
 \phi_{+}^{(WW)}(\omega) = \frac{\omega}{2 \bar{\Lambda}^{2}} 
 \theta(2 \bar{\Lambda} - \omega) \ , \;\;\;\;\;\;\;\;
%\label{solp} \\
 \phi_{-}^{(WW)}(\omega) = 
  \frac{2 \bar{\Lambda} - \omega}{2 \bar{\Lambda}^{2}} 
  \theta(2 \bar{\Lambda} - \omega) \ . \label{solm}
\ee
%
%for $\omega \ge 0$.
%These are normalized as 
%$\int_{0}^{\infty}d\omega \phi_{\pm}^{(WW)}(\omega) = 1$.
%They vanish for $\omega > 2\bar{\Lambda}$, and we note that
%$2\bar{\Lambda}$ is actually the kinematical upper bound of $\omega$
%allowed for the two-particle Fock states of the $B$ meson 
%in the heavy quark limit.
The solution for $\phi_{\pm}^{(g)}$ can be obtained
straightforwardly, and reads ($\omega \ge 0$):
%
\bea
 \phi_{+}^{(g)}(\omega) &=& \frac{\omega}{2\bar{\Lambda}}\Phi(\omega)\ ,
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
%  \label{solpg} \\
  \phi_{-}^{(g)}(\omega) =
  \frac{2\bar{\Lambda}-\omega}{2\bar{\Lambda}}\Phi(\omega)
 + \frac{J(\omega)}{\omega}\ ,\label{solmg}\\
%\ee
%
%where $\omega \ge 0$ and 
%
%\bea
 \Phi(\omega) &=& \theta(2\bar{\Lambda}-\omega)
 \left\{\int_{0}^{\omega}d\rho \frac{K(\rho)}{2\bar{\Lambda} - \rho}
 -\frac{J(0)}{2\bar{\Lambda}}\right\}
 - \theta(\omega - 2\bar{\Lambda}) \int_{\omega}^{\infty}
  d\rho \frac{K(\rho)}{2\bar{\Lambda} - \rho} \nonumber \\
   &-& \int_{\omega}^{\infty}d\rho
  \left( \frac{K(\rho)}{\rho} + \frac{J(\rho)}{\rho^{2}} \right) \ ,
\label{Phi}
\eea
%
with
%
%\be
$K(\rho) = I(\rho) + \left[ 1/(2\bar{\Lambda}) -
    d/d\rho\right]J(\rho)$.
%\ .
%\label{K}
%\ee
%
%These functions obey
%$\int_{0}^{\infty}d\omega \phi_{\pm}^{(g)}(\omega) = 0$, so that
%The total amplitudes are normalized as 
%$\int_{0}^{\infty}d\omega \phi_{\pm}(\omega)= \tilde{\phi}_{\pm}(0)= 1$.
The solution (\ref{decomp}) with eqs.(\ref{solm})-(\ref{Phi})
%for the distribution amplitudes 
is exact and 
presents our principal result.
%Eq.(\ref{decomp}) might exhibit discontinuity at $\omega = 2\bar{\Lambda}$,
%but this does not constitute a problem because, as is well-known,
%the distribution amplitudes should generally be understood
%as distributions (in the mathematical sense).

It is also straightforward to derive the Mellin moments
$\langle \omega^{n} \rangle_{\pm} \equiv \int d\omega \ \omega^{n} \phi_{\pm}(\omega)$ 
($n= 0, 1, 2, \cdots$)
of our solution. 
Because the analytic expression for general moment $n$ is somewhat complicated,\cite{KKQT}
we present some examples for a few low moments:
%\bea
$\langle \omega \rangle_{+} = 4\bar{\Lambda}/3$, 
% \qquad \qquad \qquad \qquad 
$\langle \omega \rangle_{-}= 2\bar{\Lambda}/3$, and  
% \label{mome1} \\
\be
\langle \omega^{2} \rangle_{+} = 2 \bar{\Lambda}^{2}+\frac{2}{3}\lambda_{E}^{2}
 +\frac{1}{3}\lambda_{H}^{2} \ ,\quad\ \;\;\;\; 
 \langle \omega^{2} \rangle_{-} = \frac{2}{3} \bar{\Lambda}^{2}
 +\frac{1}{3}\lambda_{H}^{2} \ ,
\label{mome2}
\ee
where $\lambda_{E}$ and $\lambda_{H}$ 
%parameterize the two independent reduced
%matrix elements of local quark-antiquark-gluon operators of dimension 5,
%and 
are due to $\phi_{\pm}^{(g)}$, and 
are related to the chromoelectric and chromomagnetic fields
in the $B$ meson rest frame as
%
%\bea
$\langle 0 |\bar{q} g \mbox{\boldmath $E$}\cdot\mbox{\boldmath $\alpha$}
 \gamma_{5}h_{v} |\bar{B}(\mbox{\boldmath $p$}=0)\rangle
 = f_{B}M \lambda_{E}^{2}$,
% \ ,
% \label{lambdae}\\
$\langle 0 |\bar{q} g \mbox{\boldmath $H$}\cdot\mbox{\boldmath $\sigma$}
 \gamma_{5}h_{v} |\bar{B}(\mbox{\boldmath $p$}=0)\rangle
 = if_{B}M \lambda_{H}^{2}$.
%\ ,
% \label{lambdah}
%\eea
%
%with $E^{i}=G^{0i}$, $H^{i}=-\frac{1}{2}\epsilon^{ijk}G^{jk}$, and 
%$\mbox{\boldmath $\alpha$}= \gamma^{0}\mbox{\boldmath $\gamma$}$.
These results for $n=1, 2$ coincide with the relations
obtained by Grozin and Neubert,\cite{Grozin:1997pq}
who have derived their relations by analyzing matrix elements of 
{\it local} operators corresponding to these moments.
Our solution (\ref{decomp}) from nonlocal operators
gives generalization of theirs to $n \ge 3$.

%For $n=1, 2$, the results
%agree with those 
%exactly coincide with the relations
%obtained by Grozin and Neubert\cite{Grozin:1997pq},
%who have derived their relations by analyzing matrix elements of some
%{\it local} operators.
%Our results (\ref{mel})-(\ref{dm}) from nonlocal operators
%give generalization of theirs to $n \ge 3$.


%An interesting feature revealed by 
Our results 
%is 
reveal that
the leading-twist distribution amplitude $\phi_{+}$ as well as
the higher-twist $\phi_{-}$
contains the three-particle contributions,
which is in contrast with the case of the light mesons.\cite{Braun:1990iv}
%where the leading-twist amplitudes correspond to
%the ``valence'' Fock component of the wave function,
%while the higher-twist amplitudes involve contributions of  
%multi-particle states.
We note that
%, in the present case, 
there exists
an estimate of $\lambda_{E}$ and $\lambda_{H}$
%$\lambda_{E}^{2}/\bar{\Lambda}^{2} = 0.36 \pm 0.20$,  
%$\lambda_{H}^{2}/\bar{\Lambda}^{2} = 0.60 \pm 0.23$
by QCD sum rules,\cite{Grozin:1997pq} 
%(see eq.(\ref{mome2})),
but any estimate of the higher moments is not known.
%It is obvious that 
{}Further investigations are required 
to clarify the effects of multi-particle states.
%In the applications to the physical amplitudes,
%the evolution effects including the three-body operators also enter the game.
%All these further developments for going beyond the Wandzura-Wilczek approximation
%can be exploited systematically starting from the exact results in this paper,
%as it has been done for light mesons. 
%so that the role of the three-body operators.
%in the evolution should be 
%Therefore, it is expected that
%the dominant contribution to the $B$ meson light-cone distribution amplitudes
%will be given by our solutions (\ref{solp}) and (\ref{solm}) in the
%Wandzura-Wilczek approximation.
In this connection, we note that 
the shape of our Wandzura-Wilczek
contributions (\ref{solm}),
which are determined uniquely in analytic form in terms of $\bar{\Lambda}$, 
%as function of $\omega$
is rather different from various ``model'' distribution amplitudes that have been used
in the existing literature (see Ref.\cite{Grozin:1997pq} and references therein).

%Inspired by the QCD sum rule estimates, Grozin and Neubert\cite{Grozin:1997pq}
%have proposed model distribution amplitudes
%$\phi^{GN}_{+}(\omega) = \left(\frac{\omega}{\omega_{0}^{2}}\right)
%\exp \left(- \frac{\omega}{\omega_{0}}\right)$,
%$\phi^{GN}_{-}(\omega) = \left(\frac{1}{\omega_{0}}\right)
%\exp \left(-\frac{\omega}{\omega_{0}}\right)$,
%where $\omega_{0} = 2\bar{\Lambda}/3$.
%The shape of their model distributions is rather different from that
%of the Wandzura-Wilczek
%contributions (\ref{solp}) and (\ref{solm}), except
%the behavior $\phi_{+}^{GN}(\omega) \sim \omega$, 
%$\phi_{-}^{GN}(\omega) \sim {\rm const}$,
%as $\omega \rightarrow 0$.
%Actually, such behavior when the light-antiquark becomes ``soft''
%is suggested by the corresponding behavior  
%of the light-mesons\cite{Braun:1990iv,Ball:1999je,Ball:1998sk,Ball:1999ff}.
%But we note that the gluon correction (\ref{solpg}) to $\phi_{+}(\omega)$
%would modify such behavior if 
%$J(0) = -2 \int_{0}^{\infty}(d\xi/\xi)\left(\Psi_{A}(0, \xi)
% + X_{A}(0, \xi)\right) \neq 0$.
%\footnote{$\phi_{-}^{(g)}(0)$ of eq.(\ref{solmg})
%is finite even if $J(0) \neq 0$.}

To conclude, our solution 
provides the powerful framework for building up the $B$ meson
light-cone distribution amplitudes and their phenomenological applications,
because the solution is exact and satisfies all relevant QCD constraints.

%Our results represent
%the quark-antiquark distribution amplitudes
%in terms of independent dynamical degrees of freedom,
%and satisfy the constraints from the equations of motion exactly.
%The other essential constraints are imposed by
%heavy-quark symmetry, which lead to the reduced set of the 
%distribution amplitudes and thus allow us to determine the Wandzura-Wilczek 
%contributions explicitly in analytic form. 
%The Wandzura-Wilczek contributions give the  
%effects corresponding to the valence distributions, while the corrections due to  
%the additional gluons are also obtained as exact integral representations
%involving the three-particle distribution amplitudes.
%A detailed study of the gluon corrections requires systematic treatment of the
%three-particle distributions, and will be presented elsewhere.



%\begin{equation}
%\mu(n, t) = {\sum^\infty_{i=1} 1(d_i < t, N(d_i) = n) \over
%\int^t_{\sigma=0} 1(N(\sigma) = n)d\sigma}\,. \label{this}
%\end{equation}
%
%Equations should be referred to in abbreviated form,
%e.g.~``Eq.~(\ref{this})'' or ``(2)''. In multiple-line
%equations, the number should be given on the last line.




%\section{Illustrations and Photographs}
%
%Figures are to be inserted in the text nearest their first
%reference.  Original india ink drawings of glossy prints are
%preferred. Please send one set of originals with copies. If the
%author requires the publisher to reduce the figures, ensure that
%the figures (including letterings and numbers) are large enough
%to be clearly seen after reduction. If photographs are to be
%used, only black and white ones are acceptable.
%
%\begin{figure}[htbp] %ORIGINAL SIZE: width=1.4TRUEIN; height=1.5TRUEIN
%\figurebox{}{}{ijmpdf1} %100 percent
%\caption{Labeled tree {\it T}.}
%\end{figure}
%
%Figures are to be sequentially numbered in Arabic numerals. The
%caption must be placed below the figure. Typeset in 8 pt
%roman with baselineskip of 10~pt. Use double spacing between a
%caption and the text that follows immediately.
%
%Previously published material must be accompanied by written
%permission from the author and publisher.


%\section*{Acknowledgments}

%The authors would like to thank T. Onogi for fruitful discussions.
\bigskip
We 
%wish to 
thank the organizers and all others who helped make
the symposium successful.
We thank T. Onogi for fruitful discussions.
The work of J.K. was supported in part by the Monbu-kagaku-sho Grant-in-Aid
for Scientific Research No.9.
The work of C-F.Q. was supported by the Grant-in-Aid of JSPS committee.

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\end{document}

