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\begin{document}

\title{
CONSTRUCTION OF
LORENTZ INVARIANT AMPLITUDES FROM 
REST FRAME WAVE FUNCTIONS  IN HQET \\
--  APPLICATION TO ISGUR-WISE FUNCTION
}
\author{
Takayuki Matsuki \\
%\email{matsuki@tokyo-kasei.ac.jp}
{\em Tokyo Kasei University,
1-18-1 Kaga, Itabashi, Tokyo 173-8602, JAPAN} \\
%
Kohichi Seo \\
%\email{seo@gifu-cwc.ac.jp}
{\em Gifu City Women's College, 7-1 Hitoichiba-Kitamachi,
Gifu 501-0192, JAPAN} \\
}
%
\maketitle
\baselineskip=11.6pt
\begin{abstract}
Succeeding in predicting/reproducing the heavy-light mesons by
our semi-relativistic quark potential model, we examine a method how to calculate the
semileptonic weak form factors out of the rest frame wave functions of heavy
mesons and numerically calculate the dynamical $1/m_Q$ corrections to those for
the process $\bar B \rightarrow D^{(*)}\ell\nu$ based on our model for heavy mesons.
It is shown that nonvanishing expressions for
$\rho_1(\omega)=\rho_2(\omega)$ and $\rho_3(\omega)=\rho_4(\omega)=0$ are obtained
in a special Lorentz frame, where $\rho_i(\omega)$ are the parameters used in the
Neubert-Rieckert decomposition of form factors.
Various values of form factors are estimated, which are compatible with recent experimental
data as well as other theoretical calculations.
\end{abstract}
%\medskip
%\pacs{11.10.St, 12.39.Hg, 12.39.Pn, 13.20.He}
%\maketitle  %% revtex4
%
\section{Introduction}
\label{sec:introduction}
Discovery of narrow meson states $D_{s0}(2317)$ by BaBar
and $D_{s1}(2460)$ by CLEO and the following confirmation by
Belle has driven many theorists to explain these states since
the former study of these states
using quark potential model seems to fail to reproduce these mass values.
More recent experiments have found many other heavy-light mesons, broad
$D_{0}^{*}(2308)$ and $D_{1}'(2427)$ mesons by
Belle collaboration, which have the quantum numbers $j^P=0^+$ and
$1^+$; narrow $B$ and $B_s$ states of $\ell=1$,
$B_1(5720)$, $B_2^*(5745)$, and $B_{s2}^*(5839)$ by CDF and D0 whose decay
widths are also narrow because of their decay through the D-waves; and
seemingly radial excitations $(n=2)$ of $0^+$ $D_{s0}(2860)$ state by BaBar
and $1^-$ $Ds^*(2715)$ state by Belle.
Furthermore $c\bar c$ quarkonium-like states have been discovered one after another,
$X(3872)$, $X(3940)$, $Y(3940)$, $Z(3930)$, and $Y(4260)$.
These mesons have been successfully predicted and/or reproduced by our formulation
\cite{Matsuki97,Matsuki06,Matsuki07}, except for the
quarkonium-like states.
What we need to do next is to show that our approach can also give a method to calculate
scattering amplitudes and decay widths using the rest frame wave functions.

Recently there appears a paper \cite{EGF06}
in which the authors use the "relativistic" formulation to calculate masses
of heavy mesons and apply it to calculate the Isgur-Wise functions for
semileptonic $B$ decays. Although this paper adopts a different approach from
ours, they present how to calculate higher order corrections in $1/m_Q$ to form
factors, which is one of the same purposes as we intend to do in this paper.
In our former paper \cite{Matsuki97}, we have developed
a semi-relativistic formulation how to calculate the meson masses and wave 
functions for heavy-light $Q\bar q$ system introducing phenomenological dynamics.
Meson wave functions obtained thereby and expanded in $1/m_Q$ can be used in principle
to calculate ordinary form factors or Isgur-Wise functions and their higher order
corrections in $1/m_Q$ for semileptonic weak or other decay processes.
However what we have obtained are wave functions in the rest frame so that we need
to develop a method to obtain Lorentz invariant amplitudes or Lorentz-boosted wave
functions.

The problem how to construct a Lorentz-boosted wave function is in that there is
ambiguity to determine space-time coordinates of two composite particles from
information of one bound state, in this case a heavy meson. We study four
cases of reference frames for composite particles and then give a prescription how
to calculate matrix elements of currents using the rest frame wave functions.
We give form factors in the zeroth order for four
reference frames, show that they agree with each other in the HQET limit, and give
those in the first order for just one special reference frame. The lowest order
calculation in $1/m_Q$ gives the numerical value of the slope for 
the Isgur-Wise function at the origin and the semi-leptonic weak form factors
are calculated up to the first order in $1/m_Q$.
It is derived that {\it there are no dynamical contributions to the form factors},
i.e., the first order corrections to the wave functions do not contribute to
the form factors.
Studying $\bar B \rightarrow D\ell\nu$ and
$\bar B \rightarrow D^{*}\ell\nu$ processes, physical quantities related to the
CKM matrix elements are obtained.
The summary and discussions are given for this paper.
\footnote{This paper is based on the recent paper \cite{Matsuki071}.}

\section{Results}
\label{sec:results}
As an example of this formulation, we have calculated the semi-leptonic weak form
factors up to the first order in $1/m_Q$. What we have found in
this paper is
%
\begin{enumerate}
\item The Isgur-Wise function has the following form up to the first order in
$1/m_Q$ and in $(\omega-1)$.
\[
  \xi(\omega)=1-\left({1\over 2}+{1\over 3}{\bar \Lambda}^2\left<r^2\right>
  \right)\left(\omega-1\right),\qquad 
  {\bar \Lambda}={\bar \Lambda_u}=\lim_{m_Q\rightarrow \infty}\tilde E_D
  =\lim_{m_Q\rightarrow \infty}\tilde E_B,
\]
and hence
\[
  \xi(1)=1,\qquad \xi'(1)=-{1\over 2}-{1\over 3}{\bar \Lambda}^2
  \left<r^2\right>.
\]
Here since $\bar\Lambda$ depends only on light quark mass and we treat only
heavy mesons $D$, $D^*$, and $B$ which include only $u$ and $d$ light quarks
with $m_u=m_d$, the subscript of $\bar\Lambda_u$ expresses this fact.
%
\item We find that there is no contribution from correction terms in the first order
of $m_Q$ for the rest frame wave functions to the six form factors. That is, some terms
do not contribute to the physical quantities.
%
\item The first order corrections are derived from phase factors of the wave
functions and also given by kinetic terms and there are no contributions from
the first order corrections to the wave functions. That is, in the terminology
of Neubert and Rieckert in \cite{POL},
%
%\begin{subequations}
\begin{eqnarray*}
  \rho_1(\omega) &=& \rho_2(\omega) = -\frac{1}{3}{C^1 \bar \Lambda}
  \left<r^2\right> \left(\omega-1\right), \quad
  \rho_3(\omega) = \rho_4(\omega) = 0.
\end{eqnarray*}
%\end{subequations}
%
\item We have calculated the values for the form factor ${\cal F}(\omega)$ at
the zero recoil and/or their first derivatives up to the first order in $1/m_Q$ as
%
\begin{eqnarray*}
%
  &&{\cal F}_D(1)=1.07, \qquad {{\cal F}_D}'(1)=-0.875, \\
%
  &&{\cal F}_{D^*}(1)=\xi(1)=1, \qquad  {{\cal F}_{D^*}}'(1)=-1.09, \\
  && R_1(1)=1.45, \quad R_1(1)=-0.222, \quad R_2(1)=0.942, \quad R_2'(1)=0.0286,
\end{eqnarray*}
%
the first equations are obtained by analyzing $\bar B \rightarrow D\ell\nu$ process
and the second by $\bar B \rightarrow D^{*}\ell\nu$. These values are consistent
with experimental data as well as other theoretical estimates listed in Tables I and
II of \cite{EGF06}.
%
\item The above values can be used to estimate the CKM matrix element
$\left|V_{cb}\right|$ which we have obtained as
%
\[
  \left| V_{cb}\right|= 0.0387\pm 0.0060,
\]
%
for the $\bar B \rightarrow D\ell\nu$ process and
%
\[
  \left| V_{cb}\right|=0.0380\pm 0.0021,
\]
%
for the $\bar B \rightarrow D^*\ell\nu$ process. These values are consistent with the value
in PDG \cite{PDG}
%
\[
  \left| V_{cb}\right|=0.0409\pm 0.0018 \quad({\rm exclusive}).
\]
%
Here to give theoretical predictions for $\left| V_{cb}\right|$ we have neglected
theoretical undertainties although experimental errors are taken into account.
\end{enumerate}
%

We have developed a method to obtain the relativistically invariant results using
the rest frame wave functions,
and to do this the four different Lorentz-boosted frames are adopted to check the
validity of our results. The same form is obtained for the Isgur-Wise function in
all four cases up to the zeroth order in $1/m_Q$ and the first order in $(\omega-1)$, 
however their first order corrections in $1/m_Q$ are not the same for all four
cases. Only the case, written as 2-ii) in \cite{Matsuki071}, i.e., in the Breit 
frame with $t'=x'^0=y'^0$ both for $\bar B$ and $D^{(*)}$, gives consistent
results with the relativistic ones given by \cite{POL}.
%

%

%%%%%%%%%%%%%%%%%% reference %%%%%%%%%%%%%%%%%%%
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}
\def\NIM{Nucl. Instrum. Methods}
\def\NIMA{Nucl. Instrum. Methods A}
\def\NPB{Nucl. Phys. B}
\def\NPA{Nucl. Phys. A}
\def\PLB{Phys. Lett. B}
\def\PRL{Phys. Rev. Lett.}
\def\PTP{Prog. Theor. Phys.}
\def\PRD{Phys. Rev. D}
\def\ZPC{Z. Phys. C}
\def\EPJ{Eur. Phys. J. C}
\def\EPJA{Eur. Phys. J. C}
\def\PR{Phys. Rept.}
\def\IJM{Int. J. Mod. Phys. A}
\def\AP{Ann. Phys.}
\def\RMPA{Rev. Mod. Phys.}

\begin{thebibliography}{00}
\newcommand{\etal}{{\it et al.}}
%\begin{references}
\bibitem{Matsuki97} T. Matsuki \etal, 
     \Journal{\PRD}{56}{5646}{1997};
     \Journal{\PLB}{606}{329}{2005}.
\bibitem{Matsuki06} 
     T. Matsuki \etal, \Journal{\EPJA}{31}{701}{2007}, hep-ph/0610186.
\bibitem{Matsuki07}
     T. Matsuki \etal, \Journal{\PTP}{117}{1077}{2007},
     hep-ph/0605019 (2006).
\bibitem{REVIEW} See for review,
    M. Neubert, Phys. Rep. {\bf 245}, 259 (1994).
\bibitem{EGF06} D. Ebert \etal, hep-ph/061130.
\bibitem{Matsuki071}
    T. Matsuki \etal, hep-ph/0703158, to be published in Prog. Theor. Phys.
    Vol. 118, No. 6.
\bibitem{POL}
    M. E. Luke, \Journal{\NPB}{252}{447}{1990};
    C.G. Boyd and D.E. Brahm, \Journal{\NPB}{257}{393}{1991};
    M. Neubert and V. Rieckert, \Journal{\NPB}{382}{97}{1992}.
\bibitem{PDG}
  Particle Data Group, W.-M. Yao et al., J. Phys. {\bf G 33}, 1 (2006).
%\bibitem{Seo}
%    T. Matsuki and K. Seo, Bulletin of Gifu City Women's College, {\bf 46}, 7 
%    (1997).
%\end{references}
\end{thebibliography}
%

\end{document}
