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\hyphenation{author another created financial paper re-commend-ed}

% declarations for front matter
\title{$B$ meson light-cone wavefunctions in the heavy quark limit\thanks{Talk presented by K. Tanaka
at the International Symposium Radcor 2002 and Loops and Legs 2002, Kloster Banz, September 8-13, 2002.}}

\author{H. Kawamura\address{Deutsches Elektronen-Synchrotron, DESY\\
Platanenallee 6, D 15738 Zeuthen, Germany},
%        \thanks{Footnotes should appear on the first page only to
%                indicate your present address (if different from your
%                normal address), research grant, sponsoring agency, etc.
%                These are obtained with the {\tt\ttbs thanks} command.}
%        and
        J. Kodaira\address{Department of Physics, Hiroshima University\\
Higashi-Hiroshima 739-8526, Japan}%
\thanks{Supported in part by the Monbu-kagaku-sho Grant-in-Aid
for Scientific Research No.9.},
        C.-F. Qiao$^{\rm b}$%
%\address{Department of Physics, Hiroshima University\\
%Higashi-Hiroshima 739-8526, Japan}%
\thanks{Supported by the Grant-in-Aid of JSPS committee.},
        K. Tanaka\address{Department of Physics, Juntendo University\\
Inba-gun, Chiba 270-1695, Japan}}

\begin{document}

\begin{abstract} 
We present a systematic study of the $B$ meson light-cone wavefunctions
in QCD in the heavy-quark limit.  
We construct model-independent formulae 
for the light-cone wavefunctions 
in terms of independent 
dynamical degrees of freedom,
which exactly satisfy the QCD equations of motion 
and constraints from heavy-quark symmetry.
The results demonstrate novel behaviors of longitudinal as well
as transverse momentum distribution in the $B$ mesons.
\end{abstract}

% typeset front matter (including abstract)
\maketitle

%\section{FORMAT}

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,Bu,Bauer:2001cu
%(for other approaches see \cite{kls})
(see also Ref. \cite{kls}).
%, e.g. 
%Ref.\cite{Ball:1998kk}).
Essential ingredients in this approach are
the light-cone distribution amplitudes for the participating mesons,
which express nonperturbative long-distance contribution
to the factorized amplitudes.
The light-cone distribution amplitudes
describe the probability
amplitudes 
%to find particular partons with definite light-cone momentum
%fraction
%in a meson, 
to find the meson in a state
with the constituents carrying definite 
light-cone momentum
fraction,
%longitudinal momenta
and thus are process-independent quantity.
{}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 $B$ mesons 
are 
not well-known 
%still immature
at present 
%have been unknown
and they 
provide a major source of uncertainty in the calculations of 
the decay rates.

By definition, 
the 
%light-cone 
distribution amplitudes are obtained from 
light-cone 
%(Bethe-Salpeter) 
wavefunctions at 
(almost) 
zero 
transverse separation 
of the constituents,
%\begin{equation}
$\phi(x) \sim \int_{k_{T}^{2} < \mu^{2}} d^{2}k_{T} \Phi(x, \mbox{\boldmath $k$}_{T})$.
%\label{eq:bs}
%\end{equation}
%By definition, the light-cone distribution amplitudes
%are given by the light-cone (Bethe-Salpeter) wavefunctions
%at zero transverse separation of the constituents.
%Thus the previous results of Ref.\cite{kkqt} have been obtained for
%strictly light-like separation between quark and antiquark,
%where the information on transverse momentum distribution has been
%integrated over.
%However, 
The light-cone wavefunctions with transverse momentum dependence
are also necessary for computing the power corrections to the exclusive
amplitudes,
and 
%also
%They also play key roles
for estimating the transition form factors for $B \rightarrow D$, $B
\rightarrow \pi$, etc,
which constitute another type of long-distance contributions 
appearing 
in
the factorization
approaches for the exclusive $B$ meson decays.


In this work \cite{KKQT,KKQT2}, 
we demonstrate that, in the heavy-quark limit
relevant for the factorization approaches
for the exclusive $B$ meson decays,
the $B$ meson light-cone wavefunctions obey
exact differential equations, which are based on
heavy-quark symmetry and 
%constraints from 
the QCD equations of motion.
As solution of those differential equations,
%we construct representations for the quark-antiquark
%wavefunctions in terms of independent
%dynamical degrees of freedom.
%By solving those differential equations for the case of zero transverse separation
%between quark and antiquark,
%corresponding the distribution amplitudes,
%The corresponding solution gives representation for the quark-antiquark
%wavefunctions in terms of independent
%dynamical degrees of freedom.
%In particular, 
we derive the model-independent formulae
for the light-cone wavefunctions,
%and show that the distribution amplitudes
which 
%prove to
involve not only the leading Fock-states with a minimal number of (valence) partons
but also the higher Fock-states
%with non-minimal parton configurations 
with additional dynamical gluons.
%
%we also extend the analysis to include the transverse momentum effects[2].
%Also for nonzero transverse separation,
%we give out the explicit analytic solution 
%which reveals that the transverse momentum distribution strongly depends on 
%the longitudinal momentum of the constituents.

The light-cone wavefunctions are related to the usual Bethe-Salpeter
wavefunctions at equal
light-cone time $z^{+}= (z^{0}+z^{3})/\sqrt{2}$.
% \cite{bl}.
In the heavy-quark limit,
%relevant for the factorization approaches
%for the exclusive $B$ meson decays,
%the $B$ meson matrix elements obey 
%the heavy-quark symmetry, so that 
the quark-antiquark light-cone wavefunctions $\tilde{\Phi}_{\pm}(t,z^2)$ of
the
$B$ mesons 
can be introduced 
%\cite{KKQT2} 
in terms of vacuum-to-meson matrix element of
nonlocal operators in the 
%HQET,
%Heavy Quark Effective Theory (HQET)
the heavy-quark effective theory (HQET)
%\cite{Neubert:1994mb}
\cite{sachrajda,KKQT2}:
%following Refs.\cite{sachrajda,Pirjol:2000gn}:
\bea
\lefteqn{\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}\right.}\nonumber\\
&&\!\!\!\times \left. \left\{ \tilde{\Phi}_{+}(t,z^2) - \slashs{z} \frac{\tilde{\Phi}_{+}(t,z^2)
 -\tilde{\Phi}_{-}(t,z^2)}{2t}\right\} \right].
 \label{phi}
\eea
Here 
%$z^{+}=0$, 
$z^{\mu}=(0, z^{-}, \mbox{\boldmath $z$}_{T})$, $z^{2}= - \mbox{\boldmath $z$}_{T}^{2}$,
$v^{2} = 1$, $t=v\cdot z$, 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,
$b(x) \approx \exp(-im_{b} v\cdot x)h_{v}(x)$,
and is subject to the on-shell constraint,
$\slashl{v} h_{v} = h_{v}$ \cite{Neubert:1994mb}.
$\Gamma$ 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
%
%\bea
$\langle 0 | \bar{q}(0) \gamma^{\mu}\gamma_{5} h_{v}(0) |\bar{B}(p) \rangle
   = i f_{B} M v^{\mu}$,
%\ ,
%\label{fb}
%\eea
%
so that $\tilde{\Phi}_{\pm}(t=0,z^2=0) = 1$.
Eq. (\ref{phi}) is the most general parameterization compatible with Lorentz
invariance and the heavy-quark limit.
%Note that 
%the light-cone distribution amplitudes 
%are
%given by the light-cone limit of the wavefunctions as
%$\tilde{\phi}_{\pm}(t)=\tilde{\Phi}_{\pm}(t, z^{2}=0)$
%\cite{KKQT}.

Higher Fock components in the $B$ mesons are described by
multi-particle wavefunctions. 
%In the present work,
We explicitly deal with quark-antiquark-gluon three-particle 
wavefunctions, defined as \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} )\  \right. \nonumber\\
&&\times\left( \tilde{\Psi}_A (t,u)
   - \tilde{\Psi}_V (t,u) \right)
%\qquad\qquad\qquad\qquad  
- i \, \sigma_{\mu\nu} z^{\nu}\,
           \tilde{\Psi}_V (t,u)
\nonumber\\
      & &- \left. z_{\mu} \, \tilde{X}_A (t,u)\,
+ \frac{z_{\mu}}{t} \, \slashs{z} \,\tilde{Y}_A \,(t\,,\,u)
\biggr\} \, \right] + \ldots ,
\label{3elements0}
\eea
%
where the ellipses stand for the terms which involve
one or more powers of $z^2$ and are irrelevant for the
present work.
We have the four functions
$\tilde{\Psi}_V , \tilde{\Psi}_A ,
\tilde{X}_A$ and $\tilde{Y}_A$
as the independent three-particle wavefunctions in the heavy-quark limit.


%It is well-known that 
The QCD equations of motion
impose a set of relations between the above wavefunctions \cite{KKQT,KKQT2}. 
%for the light-mesons.\cite{Braun:1990iv}
%To derive these relations, the most convenient method is to
%start with 
They can be derived most directly 
from the exact identities between the nonlocal operators:
%
\bea
 \lefteqn{\frac{\partial}{\partial x^{\mu}}
 \bar{q}(x) \gamma^{\mu} \Gamma h_{v}(0)}\nonumber\\
  &&=
%\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} \\
%\eea
%\bea
\lefteqn{ v^{\mu}\frac{\partial}{\partial x^{\mu}}
 \bar{q}(x) \Gamma h_{v}(0)}\nonumber\\
 &&= 
%- \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.
Taking the 
%vacuum-to-meson 
matrix element with $x_{\mu} \rightarrow z_{\mu}$,
%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 these identities
%Eqs. (\ref{id1}), (\ref{id2}) 
yield
$\tilde{\Phi}_{+}(t, z^2 )$, $\tilde{\Phi}_{-}(t, z^2 )$ defined in Eq. (\ref{phi}) 
and their derivatives,
$\partial \tilde{\Phi}_{\pm}(t, z^{2})/\partial t$ 
and 
$\partial \tilde{\Phi}_{\pm}(t, z^{2})/\partial z^{2}$.
%|_{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 terms in the RHS, which are given by integral of quark-antiquark-gluon operator,
are expressed by the three-particle wavefunctions of Eq. (\ref{3elements0}).
The last term of Eq. (\ref{id2}), the derivative over the total translation,
yields 
%contribution with
$\tilde{\Phi}_{\pm}(t, z^2 )$ multiplied by
\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}.

Substituting all the 
%possible 
Dirac matrices for $\Gamma$,
%each of the above identities (\ref{id1}), (\ref{id2})
%gives the two independent constraint equations between the two- and three-particle wavefunctions,
we obtain the four independent constraint equations between the two- and three-particle wavefunctions
from Eqs. (\ref{id1}) and (\ref{id2}).
%a system of four equations which are exact in QCD in the heavy-quark limit \cite{KKQT,KKQT2}.
%which are exact in QCD in the heavy-quark limit.
We solve this system of equations for the 
%following 
relevant 
two cases:
(i) In the light-cone limit $z^2 \rightarrow 0$ ($\mbox{\boldmath $z$}_{T} \rightarrow 0$), 
with fully taking into account
the contribution due to the three-particle wavefunctions.
%We show 
The solution gives exact model-independent representaions
for the light-cone distribution amplitudes in terms of independent dynamical degrees
of freedom.
(ii) For 
%the general case of 
$z^2 \neq 0$ but in the approximation neglecting the
contribution of the three-particle wavefunctions. 
%We show 
The solution gives exact analytic formulae
for the light-cone wavefunctions with transverse momentum dependence within the valence Fock-states.

Now we discuss the case (i) in detail \cite{KKQT}. In the light-cone limit,
the light-cone wavefunctions of Eq. (\ref{phi})
reduce to 
the light-cone distribution amplitudes $\tilde{\phi}_{\pm}(t)$
as \cite{Grozin:1997pq}
\begin{equation}
\tilde{\phi}_{\pm}(t)=\tilde{\Phi}_{\pm}(t, z^{2}=0)\ ,
\label{sphi}
\end{equation}
and we also introduce the
shorthand notations, 
$\tilde{\phi}_{\pm}'(t) \equiv d \tilde{\phi}_{\pm}(t)/dt$ and 
$\partial \tilde{\phi}_{\pm}(t)/\partial z^{2}\equiv
% \left. 
\partial \tilde{\Phi}_{\pm}(t, z^{2})/\partial z^{2}
%  \right
|_{z^{2} \rightarrow 0}$,
%
%\begin{equation}
%\tilde{\phi}_{\pm}'(t) \equiv \frac{d \tilde{\phi}_{\pm}(t)}{dt}
%\frac{\partial \tilde{\phi}_{\pm}(t)}{\partial z^{2}}\equiv
% \left. \frac{\partial \tilde{\Phi}_{\pm}(t, z^{2})}{\partial z^{2}}
%  \right|_{z^{2} \rightarrow 0}\ ,
%\end{equation}
%\ .
%\label{pd}
%\ee
%
which denote the derivatives with respect to the longitudinal and transverse separations,
respectively.
The first identity (\ref{id1}) yields the two 
%differential 
equations:
%connecting the two-particle distributions $\tilde{\phi}_{+}$
%and $\tilde{\phi}_{-}$ with the three-particle distribution amplitudes:
%
\bea
 \lefteqn{\tilde{\phi}_{-}'(t)- \frac{1}{t}\left(\tilde{\phi}_{+}(t)
 - \tilde{\phi}_{-}(t)\right)}\nonumber\\
 &&=
 2 t \int_0^1 du\, u\, ( \tilde{\Psi}_A (t,u) - \tilde{\Psi}_V (t,u)) \ ,
 \label{de1} \\
 \lefteqn{\tilde{\phi}_{+}'(t)
 -\tilde{\phi}_{-}'(t) - \frac{1}{t}\left(\tilde{\phi_{+}}(t)
 - \tilde{\phi}_{-}(t)\right) + 4t
 \frac{\partial \tilde{\phi}_{+}(t)}{\partial z^{2}}} \nonumber \\
%\qquad\qquad\qquad 
=&&\!\!\!\!\!\!\!\!\!\!\! 2  t \!\int_0^1 \!\!\! du\, u\, ( \tilde{\Psi}_A (t,u)
  + 2\, \tilde{\Psi}_V (t,u) + \tilde{X}_A (t,u)),
\label{de2}
\eea
%
and similarly the second identity (\ref{id2}) yields
%another set of two equations:
%Next, we proceed to the second identity (\ref{id2}).
%To use the HQET equation of motion $v \cdot D h_{v} = 0$
%\cite{Isgur:1989vq,Neubert:1994mb},
%we contract the both sides of eq.(\ref{id2}) with $v_{\mu}$.
%After similar manipulations as above, we finally obtain
%another set of two differential equations:
%
\bea
\lefteqn{\tilde{\phi}_{+}'(t)- \frac{1}{2t}\left(\tilde{\phi}_{+}(t)
 - \tilde{\phi}_{-}(t)\right)  + i \bar{\Lambda}\tilde{\phi}_{+}(t)
 + 2t \frac{\partial \tilde{\phi}_{+}(t)}{\partial z^{2}}} \nonumber \\
  &&=  t\, \int_0^1 du\,(u-1)\,
           ( \tilde{\Psi}_A (t,u) + \tilde{X}_A (t,u) )\ , \label{de3} \\
 \lefteqn{\tilde{\phi}_{+}'(t)
  -\tilde{\phi}_{-}'(t) +\left(i \bar{\Lambda} -\frac{1}{t}\right)
 \left(\tilde{\phi}_{+}(t) - \tilde{\phi}_{-}(t)\right)}\nonumber\\
&& + 2t \left(
  \frac{\partial \tilde{\phi}_{+}(t)}{\partial z^{2}}-
  \frac{\partial \tilde{\phi}_{-}(t)}{\partial z^{2}}\right) \nonumber \\
  &&=  2t \,\int_0^1 du\,(u-1)\,
           ( \tilde{\Psi}_A (t,u) + \tilde{Y}_A (t,u) ).
\label{de4}
\eea
%
These Eqs. (\ref{de1})-(\ref{de4}) are exact in QCD
in the heavy-quark limit.
%and are the new results.

%To proceed, 
Important observation is that we can eliminate the term
$\partial \tilde{\phi}_{+} (t)/\partial z^2$
by combining Eqs. (\ref{de2}) and (\ref{de3}).
%
%\be
% \tilde{\phi}_{+}' (t) + \tilde{\phi}_{-}' (t)
%  + 2 i \bar{\Lambda} \tilde{\phi}_+ (t)
%  = - 2 t \int_0^1 du\, \left(
%    \tilde{\Psi}_A (t,u) + \tilde{X}_A (t,u) + 2 u\,
%   \tilde{\Psi}_V (t,u) \right)   \ .\label{fresult5}
%\ee
%
The resulting equation, combined with Eq. (\ref{de1}), gives 
a system of two differential equations which involve the 
degrees of freedom along the light-cone only.
%can be solved for
%$\tilde{\phi}_{+}$ and $\tilde{\phi}_{-}$.
%{}For the momentum-space distribution amplitudes $\phi_{\pm}(\omega)$
%defined 
By going over to the momentum space
%In the representation via the longitudinal momentum 
by $\tilde{\phi}_{\pm}(t) = \int d\omega \ e^{-i \omega t}
  \phi_{\pm}(\omega)$, and $\tilde{F}(t, u) = \int d\omega d \xi \
  e^{-i(\omega  + \xi u)t} F(\omega, \xi)$
% \ ,
%   \qquad (
with $F=\{ \Psi_{V}, \Psi_{A}, X_{A}, Y_{A} \}$,
the corresponding differential equations read \cite{KKQT}
\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 
$I(\omega)$ and $J(\omega)$ 
%of eqs. (\ref{mde1}) and (\ref{mde2}) 
denote the 
``source'' terms
due to three-particle wavefunctions as
%
%\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 $J(\omega)$.
%%
%\begin{displaymath}
%\lefteqn{
\bea
\lefteqn{I(\omega)
 \! = 2\frac{d}{d\omega}
   \int_{0}^{\omega} \!\!\!\! d\rho \!\! \int_{\omega - \rho}^{\infty} \!\!\! \frac{d\xi}{\xi}
%}\nonumber\\
%&&\times   
\frac{\partial}{\partial \xi}\left[ \Psi_{A}(\rho, \xi)
   - \Psi_{V}(\rho, \xi)\right],}\nonumber\\
%} 
%\label{si} 
%\\
%\end{displaymath}
%\bea
\lefteqn{J(\omega) \!= \! -2\frac{d}{d\omega}
  \int_{0}^{\omega} \!\!\!\! d\rho \!\! \int_{\omega - \rho}^{\infty} \!\!\! \frac{d\xi}{\xi}
%}\nonumber\\
%&&\times  
\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}
\eea
%with the three-particle wavefunctions in the momentum space,
%$\Psi_{V}, \Psi_{A}, X_{A}$, 
%defined as $\tilde{F}(t, u) = \int d\omega d \xi \
%  e^{-i(\omega  + \xi u)t} F(\omega, \xi)$
%% \ ,
%%   \qquad (
%with $F=\{ \Psi_{V}, \Psi_{A}, X_{A}\}$.
%
Eqs. (\ref{mde1}), (\ref{mde2})
can be solved for $\phi_{+}(\omega)$ and $\phi_{-}(\omega)$.
The boundary conditions are specified as $\phi_{\pm}(\omega) = 0$
for $\omega < 0$ or $\omega \rightarrow \infty$, 
because $\omega v^{+}$ has the meaning of the light-cone projection
$k^{+}$ of the light-antiquark momentum in the $B$ meson, and 
the normalization condition is 
$\int_{0}^{\infty}d\omega \phi_{\pm}(\omega)= 
%\tilde{\phi}_{\pm}(0)
\tilde{\Phi}_{\pm}(0,0)
= 1$.
Obviously, the solution can be decomposed into two pieces as
%
\be
  \phi_{\pm}(\omega) = \phi_{\pm}^{(W)}(\omega) 
  + \phi_{\pm}^{(g)}(\omega) \ ,
\label{decomp}
\ee
%
where $\phi_{\pm}^{(W)}(\omega)$ 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 the ``Wandzura-Wilczek approximation \cite{Braun:1990iv}'' $\Psi_{V}=\Psi_{A}=X_{A}=Y_{A}=0$.
$\phi_{\pm}^{(g)}(\omega)$ denote the pieces induced by the source terms $I(\omega)$ and $J(\omega)$.

We are able to obtain 
the analytic solution for the Wandzura-Wilczek part 
%explicitly 
as 
%($\omega \ge 0$)
%
\be
 \phi_{\pm}^{(W)}(\omega) = \frac{\bar{\Lambda} \pm (\omega-\bar{\Lambda})}{2 \bar{\Lambda}^{2}} 
 \theta(\omega)\theta(2 \bar{\Lambda} - \omega)\ .
%\ , \;\;\;\;\;\;\;\;
%\label{solp} \\
% \phi_{-}^{(W)}(\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 
%$\int_{0}^{\infty}d\omega \phi_{\pm}^{(W)}(\omega) = 1$.
%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.
Moreover, 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
{\cal G}
%\mathcal{G}
(\omega)\ ,
%\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
  \label{solpg} \\
  \phi_{-}^{(g)}(\omega) &=&
  \frac{2\bar{\Lambda}-\omega}{2\bar{\Lambda}}
%\Phi
{\cal G}
(\omega)
 + \frac{J(\omega)}{\omega}\ ,\label{solmg}\\
%\ee
%
%where $\omega \ge 0$ and 
%
%\bea
% \Phi
{\cal G}(\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\} \nonumber\\
&& - \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).
%An interesting feature revealed by 
%Our results 
%is 
reveals that 
%(leading- and higher-twist) 
%the quark-antiquark distribution amplitudes 
$\phi_{\pm}$ 
contain the three-particle contributions.
This is also visualized explicitly in terms of 
%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},
Here we present some examples for a few low moments:
%\bea
$\langle \omega \rangle_{\pm} = (3\pm 1)\bar{\Lambda}/3$, and 
%$\langle \omega \rangle_{+} = 4\bar{\Lambda}/3$,  
%$\langle \omega \rangle_{-}= 2\bar{\Lambda}/3$, and  
% \label{mome1} \\
\be
\langle \omega^{2} \rangle_{\pm} = \frac{4\pm 2}{3} \bar{\Lambda}^{2}+\frac{1}{3}\lambda_{E}^{2}
 +\frac{1}{3}\lambda_{H}^{2}  \pm \frac{1}{3}\lambda_{E}^{2}\ ,
%\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  
%by different method \cite{Grozin:1997pq}.
Our solution (\ref{decomp}) allows us to further derive the analytic formulae 
for the general moment $n$ in terms of matrix element of local
two- and three-particle operators with dimension $n+3$ (see Ref. \cite{KKQT} for the detail).
%The results for $n \ge 3$ based on our solution (\ref{decomp}) 
%who analyzed
%have derived their relations by analyzing 
%matrix elements of 
%{\it local} operators corresponding to these moments.
%Our nonlocal operator approach allows 
%the generalization to $n \ge 3$.
%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 $n \ge 3$.


The behavior of 
%the RHS of 
Eq. (\ref{phi})
for a fast-moving meson, $t = v\cdot z \rightarrow \infty$, 
shows that 
$\phi_{+}$ is of
%the 
leading-twist 
%distribution amplitude,
whereas $\phi_{-}$ has subleading twist;
the three-particle contributions to 
the leading-twist 
%distribution amplitude 
$\phi_{+}$ 
%as well as
%the higher-twist $\phi_{-}$
%contains the three-particle contributions,
%which 
are 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.
%In the higher-twist distribution amplitudes of light mesons
%\cite{Braun:1990iv,Ball:1999je,Ball:1998sk,Ball:1999ff},
%the contributions of multi-particle states
%with additional gluons have been generally important and broadened the
%distributions, but they have typically
%produced corrections less than $\sim 20$\% to the main term
%given by the Wandzura-Wilczek contributions.
%Similar situation
We note that
%, in the present case, 
there exists
an 
%rough 
estimate
$\lambda_{E}^{2}/\bar{\Lambda}^{2} = 0.36 \pm 0.20$,
$\lambda_{H}^{2}/\bar{\Lambda}^{2} = 0.60 \pm 0.23$
by QCD sum rules \cite{Grozin:1997pq} (see Eq. (\ref{mome2})).
This might suggest that, in the $B$ mesons,
% distribution amplitudes, 
the three-particle contributions
could play important roles even in the leading twist level.
%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: One exapmle of such models is 
$\phi^{GN}_{+}(\omega) = (\omega/\omega_{0}^{2})
e^{- \omega/\omega_{0}}$ and 
$\phi^{GN}_{-}(\omega) = (1/\omega_{0})
e^{-\omega/\omega_{0}}$
with $\omega_{0} = 2\bar{\Lambda}/3$, inspired by the QCD sum rule estimates \cite{Grozin:1997pq}.
These have very different shape compared with 
%the Wandzura-Wilczek
%contributions 
Eq. (\ref{solm}), except
the behavior $\phi_{+}^{GN}(\omega) \sim \omega$,
$\phi_{-}^{GN}(\omega) \sim {\rm const}$,
as $\omega \rightarrow 0$.

Next we proceed to the case (ii), where we get
\bea
\label{eq:1}
\lefteqn{\omega \frac{\partial \Phi_{-}}{\partial \omega}+ \Phi_{+}
+ z^2\frac{\partial}{\partial z^2}
\left(\Phi_{+}
-\Phi_{-}\right) 
%\left(\frac{\partial \Phi_{+}}{\partial z^2}
%-\frac{\partial \Phi_{-}}{\partial z^2}\right)
= 0\ ,} \\
\label{eq:2}
\lefteqn{\left(\omega 
\frac{\partial}{\partial \omega} + 2\right)\left(\Phi_{+} -
\Phi_{-}\right)
%\left(\frac{\partial \Phi_{+}}{\partial \omega} -
%\frac{\partial \Phi_{-}}{\partial \omega}\right)
+ 4  \frac{\partial^{2}}{\partial \omega^{2}}\frac{\partial
\Phi_{+}}{\partial z^2}
%+2 \left(\Phi_{+}-\Phi_{-}\right)
= 0\ ,} \\
\label{eq:3}
\lefteqn{\left[(\omega - \bar{\Lambda}) \frac{\partial}{\partial \omega}\!+\!\frac{3}{2}\right]\Phi_{+}
%}\nonumber\\
\!-\!\frac{1}{2}\Phi_{-}
\! + 2  \frac{\partial^{2}}{\partial \omega^{2}}\frac{\partial \Phi_{+}}{\partial
z^2}
\!= \!0,} \\
%\lefteqn{(\omega - \bar{\Lambda}) \frac{\partial \Phi_{+}}{\partial \omega} +
%2  \frac{\partial^{2}}{\partial \omega^{2}}\frac{\partial \Phi_{+}}{\partial
%z^2}
%+\frac{1}{2}\left(3\Phi_{+}-\Phi_{-}\right)
%= 0\ ,}\\
\label{eq:4}
\lefteqn{\left[ (\omega - \bar{\Lambda})\frac{\partial}{\partial \omega}+ 2\right]
 \left(\Phi_{+}
- \Phi_{-}\right)}\nonumber\\ 
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+
2  \frac{\partial^{2}}{\partial \omega^{2}}
\left(\frac{\partial \Phi_{+}}{\partial z^2} -
\frac{\partial \Phi_{-}}{\partial z^2} \right)
%+ 2 \left(\Phi_{+} - \Phi_{-}\right)
= 0\ ,
%\lefteqn{(\omega - \bar{\Lambda}) \left(\frac{\partial \Phi_{+}}{\partial \omega}
%- \frac{\partial \Phi_{-}}{\partial \omega}\right)}\nonumber\\
% &&+
%2  \frac{\partial^{2}}{\partial \omega^{2}}
%\left(\frac{\partial \Phi_{+}}{\partial z^2} -
%\frac{\partial \Phi_{-}}{\partial z^2} \right)
%+ 2 \left(\Phi_{+} - \Phi_{-}\right)
%= 0\; ,
\eea
corresponding to Eqs. 
(\ref{de1}), 
(\ref{de2}), 
(\ref{de3}), 
(\ref{de4}),
respectively. 
Here 
Eqs. (\ref{eq:1})-(\ref{eq:4}) are given in the ``$\omega$-representation'' 
$\Phi_{\pm} \equiv \Phi_{\pm}(\omega, z^2)$,
instead of the ``$t$-representation'',
via $\tilde{\Phi}_{\pm}(t, z^{2}) = \int d\omega \ e^{-i \omega t}
\Phi_{\pm}(\omega, z^{2})$.
The light-cone limit is not taken
%in order to explore the tranverse momentum
%distribution, 
so that the terms proportional
to $z^{2}$  appear in Eq. (\ref{eq:1}).
Note that we 
%restrict
%our interest within the two-particle Fock states
%neglecting 
have neglected
the contribution 
from the quark-antiquark-gluon
three-particle operators.
%We refer to the latter point
%as the ``Wandzura-Wilczek approximation'' hereafter,
%similarly to the corresponding case in the light-cone limit.\cite{kkqt,ww}

By combining Eqs. (\ref{eq:2}) and (\ref{eq:3}), 
we eliminate the last term in their LHS.
%$\partial^{2}/\partial \omega^{2}(\partial \Phi_{+}/\partial z^2)$
%we obtain
%\begin{equation}
%\left(\omega - 2 \bar{\Lambda} \right)
%\frac{\partial \psi_{+}}{\partial \omega} + \omega \frac{\partial
%\psi_{-}}{\partial \omega}
%+ \psi_{+} + \psi_{-} = 0 \; .
%\label{eq:t1}
%\end{equation}
%This 
The resulting equation
can be integrated with boundary conditions
$\Phi_{\pm}(\omega, z^{2}) = 0$ for  $\omega <0$ or $\omega \rightarrow
\infty$ as
\begin{equation}
\left( \omega - 2 \bar{\Lambda} \right) \Phi_{+} + \omega \Phi_{-} = 0\; .
\label{eq:t2}
\end{equation}
{}In the limit $z^{2} \rightarrow 0$, Eqs. (\ref{eq:1}) and (\ref{eq:t2}) 
reduce to Eqs. (\ref{mde1}) and (\ref{mde2}) in the Wandzura-Wilczek approximation. 
%have been solved to give Eq. (\ref{solm}).
The corresponding solution (\ref{solm})
%for $z^{2}=0$
serves as ``boundary conditions''
to solve Eqs. (\ref{eq:1})-(\ref{eq:4}) for $z^{2} \neq 0$.
Then, from Eqs. (\ref{eq:1}) and (\ref{eq:t2}), we find,
as the solution in the Wandzura-Wilczek approximation 
for $z^{2} \neq 0$,
\begin{equation}
\Phi^{(W)}_{\pm}(\omega, z^{2}) = \phi_{\pm}^{(W)}(\omega)
\xi \left(z^{2}\omega(2\bar{\Lambda}-\omega)\right)\ .
\label{eq:chi}
\end{equation}
Here $\xi(x)$ is a function of a single variable $x$, 
%and satisfies
%$\xi(0) = 1$.
% due to Eqs. (\ref{solp}), (\ref{solm}).
%The functional form of $\xi(x)$ 
and can be 
%easily 
determined from a remaining
differential equation (\ref{eq:3}) or (\ref{eq:4}) as
%, which was useless
%in the light-cone limit\cite{kkqt} (Eq. (\ref{eq:4}) gives the
%result identical to Eq. (\ref{eq:3})). 
%We obtain
$\xi(x) = J_{0}\left(\sqrt{-x}\right)$
where $J_{0}$ is a (regular) Bessel function.
%, so that
%Substituting this,
%Eq. (\ref{eq:chi})
%the analytic solution
%for the coupled differential equations (\ref{eq:1})-(\ref{eq:4}) is given by
%\begin{equation}
%\Phi_{\pm}^{(W)}(\omega, -\mbox{\boldmath $z$}_{T}^2)
%= \phi^{(W)}_{\pm}(\omega)J_{0}\left(|\mbox{\boldmath
%$z$}_{T}|\sqrt{\omega(2\bar{\Lambda}-\omega)} \right) \ .
%\label{j0}
%\end{equation}
%These are 
This result gives analytic solution for the light-cone wavefunctions with the transverse separation
$\mbox{\boldmath $z$}_{T}^2 = -z^{2}$.
%between quark and antiquark.
The momentum-space wavefunctions 
$\Phi^{(W)}_{\pm}(\omega, \mbox{\boldmath
$k$}_{T})$,
defined by
%\begin{equation}
$\tilde{\Phi}^{(W)}_{\pm}(t, -\mbox{\boldmath $z$}_{T}^2) =
\int d\omega d^{2}k_{T}\
e^{-i\omega t + i\mbox{\boldmath $k$}_{T}\cdot\mbox{\boldmath $z$}_{T}}
\Phi^{(W)}_{\pm}(\omega, \mbox{\boldmath $k$}_{T})$,
%\ ,
%\label{momw}
%\end{equation}
%our solution gives
read
\be
\label{eq:12}
\Phi^{(W)}_{\pm}(\omega, \mbox{\boldmath $k$}_{T}) =
\frac{\phi_{\pm}^{(W)}(\omega)}{\pi}
%\nonumber\\
%\frac{\omega}
%{2 \pi \bar{\Lambda}^2} \theta(\omega)\theta(2 \bar{\Lambda} - \omega)
%&&\times
\delta \left(\mbox{\boldmath $k$}_{T}^{2} - \omega (2 \bar{\Lambda} -
\omega) \right) .
%\\
%\eea
%\bea
%\label{eq:13}
%\psi_{-}(\omega, \mbox{\boldmath $k$}_{T}) &=&
%\frac{2 \bar{\Lambda} - \omega}{2 \pi \bar{\Lambda}^2}
%\theta(\omega)\theta(2 \bar{\Lambda} - \omega)
%\delta \left(\mbox{\boldmath $k$}_{T}^{2} - \omega (2 \bar{\Lambda} -
%\omega) \right) \; .
\ee
The result (\ref{eq:12}) 
%and (\ref{eq:13}) 
gives exact description
of the valence Fock components of the $B$ meson
wavefunctions in the heavy-quark limit, and represent their transverse
momentum dependence explicitly.
These results show that the dynamics within the two-particle Fock states
is determined solely in terms of a single nonperturbative
parameter $\bar{\Lambda}$.

%Up to now, 
The transverse momentum distributions in the $B$ mesons
have been 
%completely 
unknown,
so that various models have been used in the literature.
%Sometimes the dependence of the light-cone wavefunctions on the transverse
%separation is simply neglected, such as
%$\Phi(\omega, -\mbox{\boldmath $z$}_{T}^2)=\Phi(\omega, 0)$; clearly,
%this in general contradicts our results (see Eq. (\ref{eq:chi})).
%Another 
Frequently used models assume
complete separation (factorization)
between the longitudinal and transverse momentum-dependence in the
wavefunctions (see e.g. Refs. \cite{kls,sachrajda}).
%,Bauer:fx}).
%such as $\Phi(\omega, \mbox{\boldmath $k$}_{T}) =
%\phi(\omega)\tau(\mbox{\boldmath $k$}_{T})$
%(see e.g. Refs.\cite{sachrajda,kls,Bauer:fx}).
A typical example of such models \cite{kls} is given by
%\cite{kls}
%\begin{equation}
$\Phi_{\pm}^{KLS}(\omega, \mbox{\boldmath $k$}_{T})
= N\omega^{2}(1-\omega)^{2}e^{-\omega^{2}/(2\beta^{2})}
\times e^{- \mbox{\boldmath $k$}_{T}^{2}/(2\kappa^{2})}$
with some constants $N, \beta$, and $\kappa$.
%the normalization constant $N$ and the parameters $\beta, \kappa$ \cite{kls}.
%\label{eq:model}
%\end{equation}
%where $N$ is the normalization constant and $\omega_{0} = 0.3$GeV, $K =
%0.4$GeV.
%\fnm{b}
%\fnt{b}{
%(In this model \cite{kls}, the two independent wavefunctions 
%$\psi_{\pm}$ of
%the $B$ mesons
%are set equal to each other, $\Phi^{KLS}\equiv
%\Phi_{+}^{KLS}=\Phii_{-}^{KLS}$.)
%Our result 
Eq. (\ref{eq:12}) 
%and (\ref{eq:13}) 
shows that 
%the dependence on
transverse and longitudinal momenta are strongly %coupled
correlated through the combination
$\mbox{\boldmath $k$}_{T}^{2}/[\omega(2\bar{\Lambda}-\omega)]$,
%so that 
therefore the ``factorization models''
are not justified.
%\fnm{c}
%\fnt{c}{The ``non-factorization'' in the light-cone 
%wavefunctions for the light-mesons 
%has been discussed in Ref.\cite{Zhitnitsky:1993vb}, 
%where the coupling between  
%transverse and longitudinal momenta through the variable 
%$\mbox{\boldmath $k$}_{T}^{2}/[u(1-u)]$,
%with $u$ the momentum fraction of the light quark, has been demonstrated.}
We further note that many models assume 
%\cite{kls,sachrajda}
%,Bauer:fx}
Gaussian distribution
for the $\mbox{\boldmath $k$}_{T}$-dependence as in $\Phi_{\pm}^{KLS}$.
%(\omega, \mbox{\boldmath $k$}_{T})$.
%Eq. (\ref{eq:model}).
%so that 
These models %wavefunctions damp strongly for
show strong dumping at 
large $|\mbox{\boldmath $z$}_{T}|$ as $\sim \exp\left(-
\kappa^2\mbox{\boldmath $z$}_{T}^{2}/2\right)$.
%\cite{sachrajda,kls,Bauer:fx},
In contrast,
% to this asymptotic behavior,
our 
%results
wavefunctions 
(\ref{eq:chi}) 
%(\ref{eq:12}) 
have slow-damping with oscillatory behavior
as
$\Phi^{(W)}_{\pm}(\omega, -\mbox{\boldmath $z$}_{T}^{2})
\sim \cos(|\mbox{\boldmath $z$}_{T}|\sqrt{\omega(2\bar{\Lambda}-\omega)}
-\pi/4)
/\sqrt{|\mbox{\boldmath $z$}_{T}|}$.
%(|\mbox{\boldmath $z$}_{T}|^{1/2}[\omega(2\bar{\Lambda}-\omega)]^{1/4})$.


%{}Finally a comment is in order concerning the error induced
%by the Wandzura-Wilczek approximation.
%From the study of the $B$ meson distribution amplitudes in the light-cone
%limit,
%there has been indication that, in the heavy-light quark systems,
%the higher Fock states could play important roles even in the
%leading twist level.\cite{kkqt}
%This would suggest that the shape of the wavefunctions
%as function of momenta and
%their quantitative role in the phenomenological applications would be
%modified
%when including the higher Fock states. 
Finally, we can 
%also 
%give an 
estimate the 
%multi-particle 
effects neglected in our solution (\ref{eq:12}).
%in the 
%above ``valence'' 
%Wandzura-Wilczek
%approximation.
%we derive the exact result for the first moment of 
%the transverse momentum squared $\mbox{\boldmath $k$}_{T}^{2}$
%in terms of the full light-cone wavefunctions,
%which include the higher Fock-states
%with non-minimal parton configurations
%with additional dynamical, nonperturbative gluons.
Inspecting the $t \rightarrow 0$ limit 
of Eqs. (\ref{de2}), (\ref{de4}),
one immediately obtains the exact result for 
$\partial \tilde{\phi}_{\pm}(0)/\partial z^{2}$, which gives
the first moment of $\mbox{\boldmath $k$}_{T}^{2}$
as \cite{KKQT2}
\begin{equation}
\int \!\!d\omega d^{2}k_{T}\ \mbox{\boldmath $k$}_{T}^{2}
\Phi_{\pm}(\omega, \mbox{\boldmath $k$}_{T})
%= 4 \frac{\partial \tilde{\phi}_{\pm}(0)}{\partial z^{2}}
= \frac{2}{3}\left(\bar{\Lambda}^{2}+\lambda_{E}^{2}+\lambda_{H}^{2}\right)\! ,
\label{eq:3part}
\end{equation}
with $\lambda_{E}$ and $\lambda_{H}$ of Eq. (\ref{mome2}).
Here $\Phi_{\pm}= \Phi^{(W)}_{\pm}+\Phi_{\pm}^{(g)}$ denote
the total wavefunctions which include the higher Fock contributions $\Phi_{\pm}^{(g)}$
induced by the three-particle operators.
{}From Eq. (\ref{eq:12}), we get
%in the Wandzura-Wilczek approximation 
$\int d\omega d^{2}k_{T}\ \mbox{\boldmath $k$}_{T}^{2}
\Phi_{\pm}^{(W)}(\omega, \mbox{\boldmath $k$}_{T})
= 2\bar{\Lambda}^{2}/3$, so that
%\frac{2}{3}\left(\bar{\Lambda}^{2}
%The first term $2\bar{\Lambda}^{2}/3$ 
%of Eq. (\ref{eq:3part}) 
%coincides with the moment
%of the solution (\ref{eq:12})
%in the Wandzura-Wilczek approximation, while %the following
%other terms 
the terms $2(\lambda_{E}^{2}+ \lambda_{H}^{2})/3$ of Eq. (\ref{eq:3part}) 
come from 
$\Phi_{\pm}^{(g)}$.
The result (\ref{eq:3part}), combined with a QCD sum rule estimate of $\lambda_{E}, \lambda_{H}$
mentioned below Eq. (\ref{mome2}),
%$\lambda_{E}^{2}/\bar{\Lambda}^{2} = 0.36 \pm 0.20$,
%$\lambda_{H}^{2}/\bar{\Lambda}^{2} = 0.60 \pm 0.23$
%by QCD sum rules,\cite{Grozin:1997pq} 
suggests that the higher Fock 
contributions might considerably broaden the transverse momentum distribution.
However, qualitative
features discussed above, like non-factorization of
longitudinal and transverse directions, ``slow-damping'' for transverse
directions, etc.,
will be unaltered by the effects of multi-particle states.



To summarize, 
%in this work 
we have derived a system of differential equations
for the $B$ meson light-cone wavefunctions 
%including the transverse
%degrees of freedom, 
and obtained the corresponding analytic solution. 
%for the relevant two cases, 
%the light-cone limit and the 
%within the valence Fock
%states. 
The differential equations are derived from the exact equations 
of motion of QCD in the heavy-quark limit.
The heavy-quark symmetry plays an essential role
%s in solvability within the valence Fock states.
%Similarly to 
%as in the case of the light-cone limit.\cite{kkqt,Grozin:1997pq,Beneke:2001wa}
%heavy-quark spin symmetry 
by reducing the number of 
independent wavefunctions drastically,  
so that the configuration of quark and antiquark in 
%the valence Fock-state components in 
the $B$ mesons is described by
only two light-cone wavefunctions.
As a result, a system of four differential equations from the 
equations of motion
%becomes a ``complete set'' to determine these two wavefunctions,
%and this enables 
allows us to obtain model-independent
formulae of these two wavefunctions, which reveal roles
of the leading 
Fock-states,
%but also 
as well as
the 
higher Fock-states
%with non-minimal parton configurations 
with additional dynamical gluons.
%
%in the light-cone limit,
%this enables us to obtain 
%the exact model-independent formulae 
%for the distribution amplitudes, which
%involve not only the leading Fock-states 
%but also the higher Fock-states
%%with non-minimal parton configurations 
%with additional dynamical gluons;
%this also allows us to obtain
%the explicit analytic solution of the wavefunctions for the leading Fock states
%with  
%full account of the $\mbox{\boldmath $k$}_{T}$-dependence.
Also due to
the power of heavy-quark symmetry,
our Wandzura-Wilczek parts (\ref{solm}), (\ref{eq:12}), which correspond to the leading Fock-states,
are given in simple analytic formulae involving
one single nonperturbative parameter $\bar{\Lambda}$.
Heavy-quark symmetry also guarantees that our solutions 
%in the present paper
determine 
%a complete set of the 
%provides complete description of 
the light-cone 
%valence Fock 
wavefunctions
for the $B^{*}$ mesons and also for the $D$, $D^{*}$ mesons in the
heavy-quark limit.

We emphasize that our solutions 
provide the powerful framework for building up the $B$ meson
light-cone 
wavefunctions
%distribution amplitudes 
and their phenomenological applications,
because the solutions satisfy all relevant QCD constraints.
{}Further developments like those required 
to clarify the effects of multi-particle states 
%All these further developments for going beyond the Wandzura-Wilczek
%approximation
can be exploited systematically starting from the exact results in this
work.



%This point can be
%studied in a systematic way, and %the corresponding
%more sophisticated wavefunctions $\psi_{\pm}^{(tot)}$ will 
%be discussed in detail
%in a separate publication.\cite{kkqt3}
%Still, 
%However, qualitative
%features revealed in this paper, like non-factorization of
%longitudinal and transverse directions, ``slow-damping'' for transverse
%directions, etc.,
%will be unaltered by the effects of multi-particle states,
%and helpful in elucidating QCD factorization theorems.










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\end{document}

