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%{\bf HEAVY QUARK PHYSICS\\}

%\vspace*{1cm}
%AUTHOR'S NAME\\
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%\vspace*{0.5cm}
%and
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%\vspace*{1.5cm}
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%2. M. L. Cohen and P. W. Anderson, in {\it Sperconductivity in d- and
%f-Band Metals}, ed.

%\hspace*{3.5mm} D. H. Douglas (AIP, New York, 1972).

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%\hspace*{3.5mm} 1968), p. 160.

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\newcommand{\beq}{\begin{eqnarray}}
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\newcommand{\no}{\nonumber}
\newcommand{\D}{{\mathcal D}}

\newcommand{\vk}{v\!\!\:\cdot\!\!\: k}
\newcommand{\vsk}{v'\!\!\:\cdot\!\!\: k}
\newcommand{\vx}{v\!\!\:\cdot\!\!\: x}
\newcommand{\vq}{v\!\!\:\cdot\!\!\: q}
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%find some improved version of the Feynman slash
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\def\slash#1{#1 \hskip -0.5em / }

\def\Pp{\frac{1 + \slash{v}}{2}}
\def\Pm{\frac{1 - \slash{v}}{2}}

\def\gl#1{Eq.~(\ref{#1})}

\def\L#1{{\cal L}_{\mbox{\scriptsize #1}}}

\def\tr#1{{\rm tr}\left[#1\right]}
\def\trF#1{{\rm tr}_F\left[#1\right]}
\def\Tr#1{{\rm Tr}\left[#1\right]}
\def\RL{{\mbox{\tiny R/L}}}
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\begin{document}
\begin{flushright}

\end{flushright}
\vspace{2em}

\begin{center}
{\bf RECENT RESULTS IN THE NJL MODEL WITH HEAVY QUARKS\footnote
{Talk presented at the III.\ German-Russian Workshop on
 {\it Theoretical Progress in \sc Heavy Quark Physics},
  Dubna, 20-22 May 1996. To appear in the conference proceedings.}}

\vspace*{1cm}
THORSTEN FELDMANN\footnote{Supported by
{\it Deutsche Forschungsgemeinschaft} under contract Eb 139/1--2.}
\\
{\it Institute of Physics, Humboldt University,
Invalidenstra\ss{}e~110,\\ 10115~Berlin, Germany\\}
\end{center}

\vspace*{1.5cm}
\begin{abstracts}
{\small We investigate the interplay of chiral and heavy quark
symmetries by using the NJL quark model.
Heavy quarks with finite masses $m_Q$ as
well as the limit $m_Q \to \infty$ are studied. We found 
large corrections to the heavy mass scaling law for
the 
%heavy 
pseudoscalar decay constant.
%Special
%emphasis is put on the treatment of the IR region of
%the quark determinant and the influence of external momenta
%on heavy meson observables. 
The influence of external momenta
on
the shape parameters of the Isgur-Wise form factor is discussed.}
\vspace*{1.5cm}
\end{abstracts}

\section{Introduction}


%In the framework of the Standard Model,
%the theory of strong interaction is given by
%Quantum Chromodynamics (QCD), which is
%formulated as a renormalizable
%non-abelian gauge theory in 
%terms of quark and gluon fields.
%Due to asymptotic freedom of the theory, the
%short distance part of the QCD dynamics can
%be treated perturbatively.
%However from the phenomenological point of view,
%in the low-energy region
%one would rather like to formulate an effective theory
%in terms of hadronic degrees of freedom.
%The long-distance QCD effects leading to the confinement
%of quark and gluons and their binding to hadrons,
%however, can only be understood non-perturbatively, 
%e.g.\ by studying QCD on the 
%lattice~\cite{wittig}. 


%In this talk we like to approximate QCD at low
%energies by an effective quark
%lagrangian which has to be restricted by the 
%symmetries of QCD. From this we derive an effective
%theory in terms of meson degrees of freedom.
%This idea should be viewed complementary to direct
%non-perturbative methods like QCD on the lattice\cite{wittig}
%as well as to approaches based on phenomenological
%effective meson lagrangians\cite{DonoghueWise,Casalbuoni}.
 
As is well known 
for heavy quark masses
$m_Q \gg \Lambda_{QCD}$ one can expand the QCD lagrangian
connected with heavy quark dynamics in terms of $1/m_Q$,
leading to
\beq
\L{HQL} &=&
\bar Q_v \, (i \vD ) \, Q_v
+  {\cal K}_v + {\cal M}_v + O(1/m_Q^2) \ ,
\eeq
with $D_\mu$ being the QCD covariant derivative
and ${\mathcal K}_v$ and ${\mathcal M}_v$
%\beq
%{\mathcal K}_v &=& \frac{1}{2m_Q} \,
%	\bar Q_v \, \vec {D}_\perp^2 \, Q_v
%\ , \qquad
%{\mathcal M}_v =\frac{1}{2m_Q} \,
%	\bar Q_v \, \sigma^{\mu\nu} \, {G_\mu\nu} \, Q_v
%\ ,
%\eeq
being the kinetic and chromomagnetic energy
of the heavy quark, respectively\cite{MRR}.
In the heavy quark limit $m_Q \to \infty$ (HQL)
 the contribution of the latter vanishes, and the remaining
lagrangian is independent of the heavy quark flavor and
spin.
Consequently, %the heavy quark spin is separately conserved, and 
heavy mesons are organized in spin symmetry doublets
with $J= j_l \pm 1/2$ where $j_l$ is the spin
of the light degrees of freedom\cite{Falk}.
%\beq
%L^{P_L} = 0^+   &\Rightarrow&  j_l^{P_l} = 
%  1/2^- \Rightarrow  J^P = (0^-,1^-) \ , 
%\no \\[0.2em]
%L^{P_L} = 1^-   &\Rightarrow&  j_l^{P_l} = 
%\left\{ 
%\begin{array}{l} 
%1/2^+  \Rightarrow  J^P = (0^+,1^+) \\
%3/2^+  \Rightarrow  J^P = (1^+,2^+) % \mbox{\ (siehe unten)}
%\end{array} 
%\\right. \ ,\no \\ 
%\ldots
%\label{schema}
%\eeq
%

In the sector of light quark flavors
$q=(u,d,s)$, QCD possesses an approximate $SU(3)_L \times
SU(3)_R$ chiral symmetry which is spontaneously broken
to $SU(3)_V$, leading to the emergence of (pseu\-do)\-Gold\-stone
bosons $\pi,K,\eta$.
We use
the common non-linear representation, where the Goldstone bosons
(denoted by $\pi$) and their transformations under
chiral symmetry $SU(3)_L \times SU(3)_R$ are given by
\beq
&&
 \xi = e^{i\pi/F} \to L \, \xi \, U^\dagger = U \, \xi \, R^\dagger \ ,
\eeq
which defines the matrix $U(\pi,L,R)$ as a non-linear function of
its arguments.
Then heavy meson fields $\Phi$ transform
under chiral symmetry as\cite{DonoghueWise,Casalbuoni}
$\Phi \to \Phi \, U^\dagger $.


\section{The NJL model with heavy quarks}

%In the following 
We will use the above symmetry considerations
in order to construct the NJL quark model as a low-energy
approximation to QCD.
%, which will then be converted into an
%effective theory for heavy mesons.
%Assuming
%that gluon effects have already 
%been integrated out, one may chose suitable local
%quark operators 
%of lowest dimension (6) 
%which are %still 
%consistent with the above mentioned (approximate) QCD symmetries,
One choses suitable 4-quark operators
 representing the scalar, pseudoscalar,
vector and axial-vector channel, respectively,
\beq
\L{}^{\bar q Q} &=&
 2 \, G_3 \, \left( 
 	(\bar Q \, q) \, (\bar q \, Q)
	+ ( \bar Q \, i \gamma_5 \, q )\, (\bar q \, i \gamma_5 \, Q)
  \right. \no 
\\
&& \qquad \left. 
	- \frac{1}{2} \, (\bar Q \, \gamma_\mu \, q) \,
			(\bar q \, \gamma^\mu \, Q)
        - \frac{1}{2} \, (\bar Q \, i \gamma_\mu\gamma_5 \, q) \,
			(\bar q \, i\gamma_5\gamma^\mu \, Q)	
       \right) 
\no \\
\stackrel{m_Q \to \infty} \longrightarrow
\L{}^{\bar q Q_v} &=&
G_3 \, \left( ( \bar Q_v \, i \gamma_5 \, q )\, (\bar q \, i \gamma_5 \, Q_v)
 	     - (\bar Q_v \, \gamma_\mu^\perp \, q) \,
			(\bar q \, \gamma^\mu_\perp \, Q_v)
	  \right. \no 
\\
&& \qquad \left. 
	+ ( \bar Q_v \, q )\, (\bar q \, Q_v)
 	     - (\bar Q_v \, i\gamma_\mu^\perp\gamma_5 \, q) \,
			(\bar q \, i\gamma_5\gamma^\mu_\perp \, Q_v)
       \right) 
\ .
\label{LqQ}
\eeq
Here the coupling constant $G_3$ has dimension~(-2) and is
related to an UV cut-off $\Lambda$, reflecting
%which is assumed to be
%connected with 
the scale of chiral symmetry breaking of the order of 1~GeV.
Note that 
%although the heavy quark masses are larger
%than $\Lambda$ 
it is 
%rather 
the heavy quark residual
momentum $(P_Q - m_Q \, v^\mu)^2 \sim O(p_q^2) < \Lambda^2$
which is regularized, and in this way
the NJL model may indeed be applied.
Note also, that all reference to gluons is gone, and 
consequently the model has no idea about confinement.

The above lagrangian can be transformed {\em exactly}\/
into a quark-meson interaction by integrating in
auxiliary fields, which in the HQL leads to 
\beq
\tilde \L{}^{\bar \chi Q} 
%&=&
%- \bar Q \, \left[ \varphi + i\gamma_5 \varphi_5
%+ \slash \varphi +  i \slash \varphi_5 \gamma_5 \right] \, \chi
%- \bar \chi \, \left[ \varphi^\dagger + i\gamma_5 \varphi_5^\dagger
%+ \slash \varphi^\dagger +  i \gamma_5 \slash \varphi_5^\dagger  \right] \, Q
%\no \\
%&& - \frac{1}{2 G_3} \,
%\left[  \varphi \varphi^\dagger +  \varphi_5 \varphi_5^\dagger
%  - 2 \varphi^\mu  \varphi^\dagger_\mu
%  - 2  \varphi_5^\mu \varphi^\dagger_{5\,\mu} \right]
%\\
& \stackrel{m_Q \to \infty} \longrightarrow &
- \bar Q_v \, \left[ H + K \right] \, \chi
- \bar \chi \, \left[ \bar H + \bar K \right] \, Q_v
%\no \\
%&& 
+ \frac{1}{2 G_3} \,
\Tr{\bar H  H - \bar K K}
\label{eq5}
\ .
\eeq
Here we introduced the above mentioned non-linear representation
by rotating the light quark fields
$\chi_R = \xi \, q_R $ , \ 
	$\chi_L = \xi^\dagger \, q_L $.
The Dirac matrices for the spin-symmetry doublets with
$J^P = (0^-,1^-)$ and $J^P = (0^+,1^+)$, respectively,
are given by
\beq
 H &\equiv& 
\Pp 
%1/2 \, (1 + \slash v) 
\, ( i \Phi_5 \gamma_5 + \slash\Phi ) \ , \qquad
 K \equiv 
\Pp 
%1/2 \, (1 + \slash v) 
\, ( \Phi + i \slash\Phi'_{5}\gamma_5 )  \ ,
\eeq
with $v_\mu \, \Phi^\mu = v_\mu \, \Phi_5^\mu = 0$.
%Note that the interaction is now quadratic in the quark fields, which
The quark fields are now integrated out explicitly by a functional Gaussian 
integration. The real part of the
resulting quark determinant %[det~$iM$]
after Wick-rotation ($ D \to D_E$)
is regularized by
a proper-time integral 
\beq
\ln \left| \det D \right|
&\to&
- \frac{N_c}{2} \, \int_{1/\Lambda^2}^{1/\mu^2} \,
\frac{ds}{s} \, \int \Tr{ e^{-s \, D_E^\dagger  D_E}}
\ .
\label{reg}
\eeq
Here, $\mu$ serves as an adjustable parameter and may be
interpreted as the scale 
up to which the quarks have been integrated out.
Especially, at $\mu = \Lambda$ the contribution of the 
quark determinant vanishes by construction, and we recover
the original interaction of quarks with static meson fields
in \gl{eq5}. 
%\section{Higher Resonances}

It is a straight forward task to generalize the presented
ideas to heavy mesons of higher excitations like the
$J^P=(1^+,2^+)$ multiplet.
It can be represented as\cite{ChoKoerner}
\beq
T^\mu &=& \frac{1 + \slash{v}}{2} \,
      \left\{ {\Phi}^{\mu\nu} \gamma_\nu +
      \sqrt{\frac{3}{2}} {\Phi_5}^\nu \, i\gamma_5\,
   \left(g_{\mu\nu} - \frac{1}{3} \gamma_\nu (\gamma_\mu -
v_\mu)\right)
  \right\}
\eeq
with $v_\nu \Phi^{\mu\nu} = v_\nu {\Phi_5}^\nu = 0$,
$\Phi^{\mu\nu}= \Phi^{\nu\mu}$ and
$\Phi^\mu_\mu = 0 $.
For the light degrees of freedom with $j_l = 3/2$ we
construct a Rarita-Schwinger representation
 from the light quark fields, introducing a covariant
derivative 
$i {\cal D}_\nu \equiv i \partial_\nu + i/2 \,
(\xi^\dagger \partial_\nu \xi + \xi \partial_\nu \xi^\dagger)$
of the chiral $SU(3)_V$, 
%Consequently, the resulting quark-meson lagrangian 
%includes derivative couplings
\beq
\tilde \L{}^{\chi Q_v} &\to&
- \bar Q_v \, 
T^\nu \, i \!\stackrel{\rightarrow} {\cal D}_\nu  \chi
+ \bar \chi \, 
 i \!\stackrel{\leftarrow} {\cal D}_\mu
 \bar T^\mu \, Q_v
- \frac{1}{2 G_4} \, \Tr{\bar T^\mu T_\mu}
\ ,
\eeq
introducing a new coupling constant $G_4 \sim O$(1~GeV)$^{-4}$.
For further details we refer to our original
works\cite{we,next}.


%\section{Dynamical Breaking of Chiral Symmetry}
\section{Results}
\begin{figure}[htb]
\begin{center}
\psfig{file = Mudep.chiralgap.ps, bb = 50 70 610 720, width=6cm, angle = -90}
\end{center}
\caption{\small The solutions  $m_q$ of \gl{gap} as
an implicit function of the scale $\mu$.} 
\label{gapfig}
\end{figure}

For low $\mu$ the contribution of the quark determinant 
leads to dynamical breaking of chiral symmetry. In the
NJL model this 
is governed by the gap-equation for the light quark
constituent mass\cite{EbRe}
\beq
m_q = m_q^{(0)} + 8 \, G_1 \, m_q \, I_1
\ , \qquad
I_1 = \frac{N_c}{16\,\pi^2} \,\left(
		\Lambda^2 - \mu^2 + O(m_q^2) \right) 
\label{gap}\ .
\eeq
Fig.~\ref{gapfig} shows the behaviour of the solution
for $m_q$ as a function of the scale $\mu$
in the chiral limit, i.e.\ for a vanishing current quark mass
$m_q^{(0)} = 0$.
We may use $\mu$ for separating the IR region of 
the proper-time integral in \gl{reg} -- which
is believed to be governed by confinement and can not be
well described in our model -- from the region where 
the model is assumed to lead to reasonable
results, namely to a consistent description
of dynamical chiral symmetry breaking.
The dashed vertical line indicates the convenient
choice $\mu = 300$~MeV.
%which seems to be convenient to us, since the values
%of $m_q$ remain rather constant for values of $\mu<300$~MeV.
Note that in the chiral limit the critical scale for
the phase transition can be read off analytically from
\gl{gap}
\beq
\mu_c^2 &=& \Lambda^2 - \frac{2 \, \pi^2}{3 \, G_1}
\approx (550~\mbox{MeV})^2 \ , \qquad 
(G_1 = 5.25~\mbox{GeV}^{-2} , \
 \Lambda = 1.25~\mbox{GeV})\mbox{\cite{we}}
\ .
\eeq

%\section{Masses and Weak Decay constants}

With the self-energy contributions of the quark-determinant 
for scales $\mu < \Lambda$ the several meson fields in the
NJL model become dynamical degrees of 
freedom. 
%The crucial difference between heavy and light mesons
%-- although they have been
% originally introduced on the same footing -- 
%stems from the different
%treatment of the external momenta, which are
%small for light mesons but scale with the heavy quark mass
%for heavy mesons. 
%For example,
%performing the HQL for
%the self-energy in the pseudoscalar channel results in
%\beq
%\Pi_5(P^2) & 
%%\stackrel{P^2 \ll \Lambda^2,\ m_Q \to m_q} \longrightarrow &
%%	4 \, I_1 + 2 \, P^2 \, I_2 \qquad \quad \mbox{(light)}
%%\\[0.5em]
%%&
%\stackrel{P^2=(m_Q+\vp)^2,\ m_Q \to \infty}
%\longrightarrow &
%	I_1 + (\vp + m_q) \, I_3(\vp) 
%\ ,
%\eeq
%where $I_3$ is a proper-time integral diverging
%linearly with the cut-off\cite{we}. 
We stress, that the limit $\mu \to 0$ in the proper-time
integrals is in general 
not suitable if the external momenta exceed
the sum of the internal quark masses since by analytic
continuation the self-energy then receives an imaginary part,
which by the optical theorem is
connected with unphysical quark-antiquark thresholds.

The mass-spectrum of the heavy mesons in the NJL model is
then determined by the value of the 4-quark coupling constant
$G_3$ (with all other parameters fixed in the light meson
sector). 
The NJL model further gives 
simple predictions for the weak decay constants of the
heavy pseudoscalar ($P$) and vector
mesons ($V$) in terms of the wave function renormalization
factors\footnote{Note that in the HQL a
factor of $M_H$ is absorbed into the normalization
of the heavy meson fields.} $Z$ and the coupling constant $G_3$,
\beq
\left.
\begin{array}{l}
\sqrt{M_P} \, f_P = (1+ \delta) \, 
 \frac{\sqrt{Z_P/M_P}}{G_3}
\\
\sqrt{M_{V}} \, f_{V} = \frac{\sqrt{Z_{V}/M_{V}}}{G_3}
\end{array}   
\right\} 
&\stackrel{\rm HQL} \longrightarrow 
& \sqrt{M_H} \, f_H = \frac{ \sqrt{Z_H} }{ G_3}
\label{fH}
\eeq
%The $1/m_Q$ corrections arise from the $Z$-factors,
%the meson masses and in case of the pseudoscalars
%due to an additional correction $\delta$ coming from
%the mixing with the heavy axial vector meson\cite{next}.
Here, $\delta$ denotes the contribution from the
mixing of the pseudoscalar and axial-vector states
beyond the HQL.
The perturbative corrections to \gl{fH} are known;
in the leading-log approximation one has\cite{LLog}
%, and
%in our case can be put in by hand by allowing the value
%of $G_3$ to be $m_Q$-dependent. In the leading-log approximation
%this is equivalent to 
$f_D = f_B \, \left(\alpha_s(m_c)/\alpha_s(m_b)\right)^{6/25}$.


%For concreteness let us chose a value of $G_3 = 6.2~$GeV$^{-2}$
%which corresponds to a value of $\bar\Lambda \approx$~300~MeV.
%From an $1/m_Q$ analysis of the experimental data, one would
%then estimate heavy quark masses $m_b \approx 5~$GeV and
%$m_c \approx 1.6$~GeV. Table~\ref{tab1} shows the NJL result
%for finite and infinite heavy quark masses. The
%NJL model seems to overestimate the fine-splitting
%between pseudoscalar and vector mesons while it
%underestimates the $SU(3)_F$ splitting
%which is induced via different light quark masses $m_u = 300$~MeV,
%$m_s = 510$~MeV.
The results are collected in Table~\ref{table}.
Compared to experiment 
the $SU(3)_F$-mass splitting
due to the different light quark masses $m_u = 300$~MeV
and $m_s = 510$~MeV is somewhat underestimated. 
The experimental fine-splitting between the spin-symmetry
partners is recovered for rather small values of $m_b$
and $m_c$. We further observe large
corrections to the HQL scaling law
for the pseudoscalar decay constant,
whereas the deviations for the vector decay constant
are within the usual expectation. This is mostly
due to the large value of the mixing contribution
$\delta$ which obtains values up to 50\% in
the case of $D$-mesons.  
The values for the weak decay constants are on the
lower side of the range obtained from lattice results\cite{wittig}.

\begin{table}[bth]
\caption{\small Heavy Meson masses and decay constants 
	in the NJL model for a) the HQL and b) finite values
	of $m_Q$. Here $\bar\Lambda$ is the mass-difference
	between heavy meson and heavy quark in the HQL. The
	heavy quark masses are fitted to the averaged heavy
	meson masses $\bar M_H = 1/4 \, (3 \, M_{V} + M_P)$.
	These are then used to estimate the mass-splittings
	$\delta_H = M_{V} - M_P$ and the decay constants.}
\label{table}
\begin{center}

a)
\begin{tabular}[t]{l|ccc|cc}
\hline
$G_3$ 
& $\bar\Lambda_u$ & $\bar\Lambda_s - \bar\Lambda_u$ & $M_K - M_H$
& $f_B$ & $f_{B_s}/f_B$ 
\\
{}[GeV]$^{-2}$ & \multicolumn{3}{c|}{[MeV]} & [MeV] & \\
\hline \hline
%8.2  & 200 & 55 & 300 & 135 & 1.04\\
6.3  & 300 & 60 & 245 & 140 & 1.04 \\ 
4.5  & 400 & 75 & 190 & 150 & 1.04 \\ 
2.9  & 500 & 80 & 140 & 160 & 1.03 
\\
\hline
Exp.\cite{PD} & & 100 &  &  &
\\ \hline
\end{tabular}\\
\vspace{1em}
b)
\begin{tabular}[t]{l|cccc|cccc}
\hline 
$G_3$ 
& $m_b$ & $m_c$ & $\delta_B$ & $\delta_D$ 
& $f_B$ & $f_{B^*}$ & $f_D$ & $f_{D^*}$ 
\\
{}[GeV]$^{-2}$ & \multicolumn{2}{c}{[GeV]} &
\multicolumn{2}{c|}{[MeV]} &
\multicolumn{4}{c}{[MeV]} \\
\hline \hline
%8.2 & 5.06 & 1.61 & 99 & 282 & 120 & 125 & 140 & 185  \\
6.3 & 4.97 & 1.53 & 81 & 239 & 130 & 135 & 160 & 200  \\
4.5 & 4.88 & 1.43 & 63 & 192 & 140 & 145 & 170 & 225  \\
2.9 & 4.78 & 1.33 & 46 & 146 & 150 & 155 & 190 & 260
\\
\hline 
Exp.\cite{PD} & & & 46 & 141 & 
\\ \hline
\end{tabular}
\end{center}
\end{table}

%\section{Isgur-Wise function}

The NJL result for the Isgur-Wise function is obtained
from the quark determinant by 
insertion of a heavy quark current.
We present the NJL results for 
%the shape parameters, i.e.\
the slope parameter $\rho = \sqrt{-\xi'(1)}$ and the
curvature $c_0 = \xi''(1)/2$,
\beq
\vp = 300~\mbox{MeV} :& \quad \rho = 0.84 \ , \ \ & c_0 = 0.49 \no \ ; \\
\vp = 400~\mbox{MeV} :& \quad \rho = 0.94 \ , \ \ & c_0 = 0.72 \no \ ; \\
\vp = 500~\mbox{MeV} :& \quad \rho = 1.07 \ , \ \ & c_0 = 1.18
\ .
\eeq
Note that here the values of $\vp = \bar\Lambda$ and $\rho$
and $c_0$
are positively correlated. This should be compared
with the estimated theoretical relation\cite{neubert} 
$c_0 \simeq 0.72 \, \rho^2 - 0.09$ together
with the phenomenological bounds 
$0.84 \leq \rho \leq 1.00$.


\section{Conclusions}

We have presented the 
NJL quark model as a low-energy approximation to
QCD including the approximate symmetries for light
and heavy flavors. 
It has been shown that this model is equivalent to a relativistic
quark-meson model for the several heavy meson states.
%of spin/parity $J^P=(0^-,1^-,0^+,1^+)$.
We have introduced an adjustable scale parameter $\mu$
in the proper-time regularized quark determinant, such that
at $\mu=\Lambda$ the model matches to
the case of static meson fields which become
then dynamical for $\mu < \Lambda$. 
The dynamical
breaking of chiral symmetry occurs around $\mu \approx 550$~MeV. 
We then chose
a finite value of $\mu \approx 300$~MeV in order to separate
the IR region of the proper-time integration.
%%The heavy quark mass limit is related to a proper treatment
%%of the external meson momenta which do scale with the heavy
%%quark mass. 
With this one obtains reasonable results for the
heavy meson mass spectrum both, with finite and infinite heavy
quark masses, for the decay constants and for the Isgur-Wise
form factors. 
%The pseudoscalar decay constant is shown to
%have large $1/m_Q$-corrections in accordance with other
%approaches. The NJL predictions for the shape parameters
%of the Isgur-Wise function are updated with the inclusion of
%finite (residual) momenta confirming the phenomenological
%results. 
%It is a straight forward task to generalize the presented
%ideas to heavy mesons of higher excitations like the
%$J^P=(1^+,2^+)$ multiplet\cite{ChoKoerner}. A detailed
%discussion will be given elsewhere\cite{next}.
%%Finally, we have shown how to include even
%%higher heavy meson resonances with $J^P=(1^+,2^+)$.


\vspace{2em}

\noindent{\bf Acknowledgements}
\vspace{1em}

{\it 
I would like to express my gratitude to the Organizing Committee
of the workshop and the people at the JINR Dubna for their
warm hospitality as well as to the Heisenberg-Landau Program
for financial support.
It is also a pleasure to thank D.~Ebert,
H.~Reinhardt and R.~Friedrich for their
contributions to
the results presented here. 
}


\vspace{1em}

\noindent{\bf References}
\begin{thebibliography}{44}
\bibitem{MRR}{see e.g.\ T.\  Mannel, W.\  Roberts and Z.\ Ryzak,
	{\it Nucl.\  Phys.\ }{\bf B368} (1992) 204.}
\bibitem{Falk}{A.\  F.\  Falk, {\it Nucl.\  Phys.\ }{\bf B378} (1992) 79.}
\bibitem{DonoghueWise}{G.\  Burdman and J.\  F.\  Donoghue,
		{\it Phys.\  Lett.\ }{\bf B280} (1992) 287;\\
		M.\  B.\  Wise, {\it Phys.\  Rev.\ }{\bf D45} (1992) R2188.}
\bibitem{Casalbuoni}{R.\  Casalbuoni et al.,
		 (1996).}
\bibitem{ChoKoerner}{J.\  G.\  K\"orner, D.\  Pirjol and K.\  Schilcher,
		{\it Phys.\  Rev.\ }{\bf D47} (1993) 3955;\\
	      %M.\  A.\  Nowak and I.\  Zahed,
	      %	{\it Phys.\  Rev.\  }{\bf D48} (1993) 356;
	      P.\  Cho and S.\  P.\  Trivedi,
		{\it Phys.\  Rev.\  }{\bf D50} (1994) 381.}
\bibitem{we}{D.\ Ebert, T.\ Feldmann, R.\ Friedrich and H.\ Reinhardt,
		{\it Nucl.\  Phys.\ }{\bf B434} (1995) 619.}
%	     see also: M.\  A.\  Nowak, M.\  Rho and I.\  Zahed,
%		{\it Phys.\  Rev.\ }{\bf D48} (1993) 4370;
%		W.\  A.\  Bardeen and C.\  T.\  Hill,
%		{\it Phys.\ Rev.\ }{\bf D49} (1993) 409.}

\bibitem{next}{D.\  Ebert, T.\  Feldmann and H.\  Reinhardt, in
		preparation. }

\bibitem{EbRe}{D.\  Ebert and H.\  Reinhardt, 
		{\it Nucl.\  Phys.\ }{\bf B271} (1986) 188.}
\bibitem{LLog}{see e.g.\ H.\  Politzer and M.\  B.\  Wise,
		{\it Phys.\  Lett.\ }{\bf B206} (1988) 681.}
\bibitem{wittig}{H.\  Wittig, contribution to this workshop,
 and refs.\ therein.}

\bibitem{neubert}{M.\  Neubert, ;
	 I.\  Caprini and M.\  Neubert, .}

\bibitem{PD}{L.\ Montanet et al.\ (Particle Data Group),
             {\it Phys.\  Rev.\ }{\bf D50} (1994), 1173.}
\end{thebibliography}




%\ \ \ \
%\parbox[t]{7cm}{
%\begin{center}
%\psfig{file = fout.ps, bb = 50 70 610 720, width=5.7cm, angle = -90}
%\end{center}
%{\small Figure 2: Deviations of the heavy meson
%decay constants from the HQL scaling law.}}
%\end{center}



\end{document}






