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\title{Stability of two-fermion bound states in the explicitly
covariant Light-Front
Dynamics}



\author{M. Mangin-Brinet,\address{Institut des Sciences
Nucl\'eaires, \\ 
        53, avenue des Martyrs, 38 026 Grenoble, France}%
J. Carbonell$^{\mbox{{\small a}}}$        
and
V. A. Karmanov\address{Lebedev Physical Institute,\\ 
Leninsky Pr. 53, 119991 Moscow, Russia}}

       

\begin{document}

\maketitle

\section{Introduction}
The system of two bound fermions covers a huge number of
interesting
problems from atomic, nuclear and subnuclear physics. It is one
of the
most difficult problems in field theory due to the fact that
bound states
necessarily involve an infinite number of diagrams. We studied
this
problem in the framework of the explicitly covariant light-front
dynamics \cite{karm76}
(CLFD). In this approach, the state vector is defined on an
hyperplane given 
by the invariant equation $\omega\cdot x=0$ with
$\omega^2=0$. The standard light-front, 
reviewed in \cite{BPP_PR_98}, is recovered for
$\omega=(1,0,0,-1)$. 
The CLFD equations have been solved exactly for a two fermion 
system with different boson exchange ladder kernels
\cite{These_MMB,fermions}.
We have considered separately the usual couplings between two
fermions 
(scalar, pseudo-scalar, pseudo-vector, and vector)
and we were interested in states
with given angular momentum and parity $J^\pi=0^{\pm},1^{\pm}$. 
Each coupling leads to a system of integral equations, which in
practice are solved 
on a finite momentum domain $[0; k_{max}]$. If the solutions
necessarily exist when 
the integration domain is finite -- for the kernels are compact,
it is not a
priori obvious that the equations admit stable solutions when
$k_{max}$ goes to infinity. Particular attention must therefore 
be paid to the stability of 
the equations relative to the cutoff $k_{max}$. We develop
hereafter an analytical 
method to study the cutoff dependence of the
equations and to determine whether they need to be regularized
or not. 
 
The method  will here be detailed for a $J=0^+$ state 
in the Yukawa model but it can be applied to any coupling. 
Results will be presented for scalar and pseudo-scalar 
exchange. This latter furthermore exhibits some strange
particularities which will be discussed.


\section{Scalar exchange}

Let us consider a system of two fermions in a $J^{\pi}=0^+$
state, bound by a
scalar exchange, whose Lagrangian density
is given by ${\cal{L}}=g_s\bar{\Psi}\Phi\Psi$. 
Its wave function, constructed using all possible 
spin structures, is determined in the $0^+$ case by two
components \cite{ckj0},
$f_1$ and $f_2$, which depend on the two scalar variables $k$
and
$\theta$:
\begin{eqnarray*}
\psi={1 \over \sqrt{2}}w^{\dag}_{\sigma_2}
\left( f_1+i{\vec{\sigma} \cdot [\vec{k}\times \hat{n} ] \over 
k\sin\theta} f_2\right) \sigma_y
w^{\dag}_{\sigma_1}
\end{eqnarray*}
$\vec{k}$ is the momentum of one particle in the system of
reference where $\vec{k}_1+\vec{k}_2=0$, $\hat{n}$ is the
spatial part
of the normal $\omega$ to the light-front plane, $\theta$ is the
angle 
between $\vec{k}$ and $\hat{n}$, and $w$ is the two component
spinor. The
appearence of a second component compared to the non
relativistic
case is due to vector $\hat{n}$, which induces additional 
spin structures. 

$f_1$ and $f_2$ satisfy the system of coupled equations :
\begin{eqnarray}\label{eqfiJ0}  
&&\left[M^2-4(k^2 +m^2)\right]f_1(k,\theta)\nonumber \\ 
&=&\frac{m^2}{2\pi^3} \int
\left[K_{11}
f_1(k',\theta')+K_{12}
f_2(k',\theta')\right]\frac{d^3k'}{\varepsilon_{k'}}
\nonumber\\
&&\left[M^2-4(k^2 + m^2)\right] f_2(k,\theta) 
\nonumber\\
&=&\hspace{-0.3cm}\frac{m^2}{2\pi^3} \int \left[K_{21}
f_1(k',\theta')+K_{22}
f_2(k',\theta')\right]\frac{d^3k'}{\varepsilon_{k'}}
\end{eqnarray}
$M^2$ si the total mass squared of the system, $m$ is the
constituent 
mass and $\varepsilon_k=\sqrt{\vec{k}^2+m^2}$. 
The kernels $K_{ij}$ result from a first integration of more
elementary 
quantities:
\begin{eqnarray*} 
K_{ij}(k,\theta;k',\theta')&=&\int_0^{2\pi}{\kappa_{ij} \over
(Q^2+\mu^2)m^2\varepsilon_k \varepsilon_{k'}}{d\varphi'\over
2\pi}, 
\end{eqnarray*} 
where $\kappa_{ij}$ depend on the type of coupling.
The analytical expressions of $\kappa_{ij}$ for the scalar
coupling, read
\begin{eqnarray}\label{eqap1}
\kappa^{S}_{11}\hspace{-0.2cm}&=&\hspace{-0.2cm}-\alpha\pi
\left[2 k^2 k'^2+3k^2 m^2+3k'^2 m^2+4 m^4 \right. \nonumber \\
\hspace{-0.2cm}&&\hspace{-0.2cm}-2 k k'\varepsilon_k \varepsilon_{k'} \cos\theta \cos\theta'
\nonumber\\
\hspace{-0.2cm}&&\hspace{-0.2cm}\left.- k k' (k^2 + k'^2 + 2 m^2)
\sin\theta \sin\theta' \cos\varphi'\right]\nonumber\\
\kappa^{S}_{12}\hspace{-0.2cm}&=&\hspace{-0.2cm}-\alpha\pi m
(k^2 - k'^2) \left(k'\sin\theta' + k\sin\theta\cos\varphi'
\right)
\nonumber\\
\kappa^{S}_{21}\hspace{-0.2cm}&=&\hspace{-0.2cm}-\alpha\pi m
(k'^2 - k^2) \left(k\sin\theta + k'\sin\theta'\cos\varphi'
\right)
\nonumber\\
\kappa^{S}_{22}\hspace{-0.2cm}&=&\hspace{-0.2cm}-\alpha\pi
\left[\left(2 k^2 k'^2+3k^2 m^2+3k'^2 m^2+4 m^4 \right.
\right.\nonumber \\
\hspace{-0.2cm}&&\hspace{-0.2cm}\left. - 2 k k' \varepsilon_k\varepsilon_{k'}
\cos\theta \cos\theta'\right)\cos\varphi' 
\nonumber\\
\hspace{-0.2cm}&&\hspace{-0.2cm}\left.-k k'(k^2 + k'^2 + 2 m^2) \sin\theta\sin\theta'\right]
\end{eqnarray}
In practice, the integration region over the momenta is reduced
to a finite
domain $[0,k_{max}]$. The kinematical term $[M^2-4(k^2+m^2)]$
on l.h.s. of equation (\ref{eqfiJ0}) does not
generate any singularity and the kernels $K_{ij}$ 
are smooth functions
of the $\theta$ variable. Thus, the stability of the solution 
depends only on
the asymptotical behavior of the kernels in the $(k,k')$ plane.

Variables $(k,k')$ can tend to infinity following different
directions:  for a 
fixed value of $k$, $K_{11}$ decreases as $1/k'$, and vice
versa. As the
integration volume contains the factor $\varepsilon_{k'}$, this
means 
that the total kernel decreases as $1/k'^2$, that is like a
Yukawa 
potential. In contrast, $K_{22}$ does not decrease in any
direction of
the $(k,k')$ plane, but tends to a positive constant 
with respect to $k$ and $k'$. $K_{22}$ 
is thus asymptotically repulsive and does not generate any
unstability. In the domain
where both $k,k'$ tend to infinity with a fixed ratio ${k'\over
k}=\gamma$, it is useful to introduce the functions $A_{ij}$
defined by
\begin{eqnarray*}
K_{ij}=-\frac{\pi\alpha}{m^2}\left\{
\begin{array}{ll}
\sqrt{\gamma}      \;A_{ij}(\theta,\theta',\gamma)   & \mbox{if
$\gamma\leq 1$}\\
{1\over\sqrt\gamma}\;A_{ij}(\theta,\theta',1/\gamma) & \mbox{if
$\gamma\geq 1$}
\end{array}\right.
\end{eqnarray*}
Since $K_{22}$ is repulsive and does not generate any collapse,
we
consider only the first channel. We have 
\begin{eqnarray*}
&&\hspace{-0.5cm}A_{11}(\theta,\theta',\gamma)=
\frac{1}{\sqrt{\gamma}}\int_0^{2\pi}\frac{d\varphi'}{2\pi}
{1\over D} \times\\ 
&&\hspace{-0.5cm}\left\{2\gamma(1-\cos\theta\cos\theta')-   
(1+\gamma^2)\sin\theta\sin\theta'\cos\varphi'\right\}  
\end{eqnarray*}
where  
\begin{eqnarray*}
D&=&(1+\gamma^2)(1+|\cos\theta-\cos\theta'|-\cos\theta\cos\theta')\\
&-&2\gamma\sin\theta\sin\theta'\cos\varphi'
\end{eqnarray*}
Let us now majorate the function $A_{11}$. For fixed $\gamma$, 
the maximum of $A_{11}$ is achieved at $\theta=\theta'$ 
and for any $\theta=\theta'$ it reads: 
$A_{11}(\theta=\theta',\gamma)=\alpha'\sqrt{\gamma}$. The 
maximum value of kernel $K_{11}$ is thus reached for $\gamma=1$.
The majorated kernel obtained this way coincides with the
non-relativistic potential $U(r)=-\alpha'/r^2$ in the momentum
space  
with $\alpha'=\alpha/(2m\pi)$.
As well known \cite{ll}, for this potential,
the binding energy does not depend on cutoff if
$\alpha'<\alpha_{cr}=1/(4m)$
what restricts the coupling constant to:
$\alpha<\pi/2$. If $\alpha'>1/(4m)$, the binding 
energy is cutoff dependent and tends to $-\infty$ when
$k_{max}\to \infty$. 
A finer majoration of $A_{11}$ was done by  
taking into account its dependence on $\gamma$ \cite{mck_prd1}. 
In this way we have found  $\alpha_{cr}=\pi$, instead of
$\pi/2$. 
As the kernel was majorated, the critical coupling constant is
expected to
be larger than $\pi$. 

It can be determined, together with the asymptotical
behavior of the wave functions, by considering  the limit
$k\to\infty$ of
equation (\ref{eqfiJ0}) for $f_1$
\begin{eqnarray*}
&&-4 f(k,z)=  \\
&& \frac{m^2}{\pi^2} 
\int_0^{\infty} \gamma d\gamma \int_{-1}^{+1}dz' 
K(k,z;\gamma k,z')f(\gamma k,z') 
\end{eqnarray*}
where we have neglected the binding energy, supposing that it is
finite, and omitted the indices
for $f_1$ and $K_{11}$. This can also be written
\begin{eqnarray}\label{as2_1}  
4f(k,z)&=&{\alpha\over\pi} \int_{-1}^1 dz' 
 \int_0^1  d\gamma\; A(\theta,\theta',\gamma) \nonumber\\
&\times&\hspace{-0.3cm}\left\{\gamma^{3/2} f(\gamma
k,z')+\gamma^{-5/2} f({k\over\gamma},z')\right\} 
\end{eqnarray}
Looking for a solution which behaves as 
\begin{equation}\label{pl}
f(k,z) \sim {h(z)\over k^{2+\beta}},  \qquad 0\le\beta< 1. 
\end{equation}
we are led for $h(z)$ to the eigenvalue equation
\begin{eqnarray*} 
h(z)=\alpha\int_{-1}^{+1} dz' H_{\beta}(z,z')\;h(z')  
\end{eqnarray*}
with
\begin{eqnarray*} 
H_{\beta}(z,z')=\int_0^1{d\gamma\over2\pi\sqrt\gamma} 
A(z,z',\gamma)\,\cosh{(\beta\log\gamma)} 
\end{eqnarray*} 
The relation 
between the coupling constant $\alpha$ and the coefficient
$\beta$, 
determining the power law  of the asymptotic wave function, can
be found in 
practice by solving the eigenvalue equation (\ref{Sta_5}) for a
fixed value 
of $\beta$
\begin{equation}\label{Sta_5}
\lambda_{\beta} \, h(z)=\int_{-1}^{+1} dz'
H_{\beta}(z,z')\;h(z')  
\end{equation}
and taking $\alpha(\beta) =1/\lambda_{\beta}$
The relation $\alpha(\beta)$ obtained that way is represented in
Figure
\ref{alpha_beta}. The value $\beta=0$ corresponds to the maximal
-- that is the critical -- value of $\alpha$: 
$\alpha_c=\alpha(\beta=0)=3.72$, in agreement with the previous 
analytical estimations. It is independent of the exchanged mass
$\mu$.

\begin{figure}[htbp]
\vspace{-1.5cm}
\begin{center}
\epsfxsize=7.5cm\epsfysize=7.5cm\mbox{\epsffile{alpha_beta_Y.eps}}
\end{center}
\vspace{-1.2cm}
\caption{Function $\alpha(\beta)$ for LFD Yukawa model with
$K_{11}$ channel only.}
\label{alpha_beta} 
\vspace{-0.3cm}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\vspace{-1.4cm}
\epsfxsize=7.5cm\epsfysize=7.2cm\mbox{\epsffile{M2_kmax_alpha=3-4_mu=0.25_B22=0.eps}}
\end{center}
\vspace{-0.8cm}
\caption{Cutoff dependence of the binding energy in the $J=0^+$ 
state ($\mu=0.25$), in the one-channel problem ($f_1$), for two fixed values
of the coupling 
constant below and above the critical value.}\label{B_kmax}
\vspace{-0.3cm}
\end{figure}
Figure \ref{B_kmax} shows the two different regimes, whether the
coupling 
constant is
below $(\alpha=3)$ or above $(\alpha=4)$ the critical value
$\alpha_c$.
\begin{figure}[htbp]
\vspace{-1.9cm}
\begin{center}
\epsfxsize=7.5cm\epsfysize=7.5cm\mbox{\epsffile{f1f2_map_B=50MeV_mu=0.25_2.eps}}
\end{center}
\vspace{-1.0cm}
\caption{Asymptotical behavior of the $J=0^+$ wave function 
components $f_i$ for $B$=0.05, $\alpha$=1.096, $\mu$=0.25. The
slope coefficient are $\beta_1=0.82$ and  $\beta_2\approx0$.}
\label{wf_as_50} 
\vspace{-0.6cm}
\end{figure}
As it can be seen in Figure \ref{wf_as_50}, the wave functions
accurately follow the power law
asymptotical behavior $1/k^{2+\beta}$ with a coefficient 
$\beta(\alpha)$ given in Figure \ref{alpha_beta}. 
It is worth noticing that -- at least in the framework of this
model --
one could measure the coupling constant
from the asymptotic behavior of the bound state wave function.

A similar study has been done for the $J=1^+$ state, which is
shown to be unstable
without regularization \cite{mck_prd1,mck_prd2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Pseudo-scalar coupling}

The stability of the pseudo-scalar (PS) coupling is analyzed
similarly to
the scalar one. The same method leads to the conclusion that 
the equations for the PS coupling are quite surprisingly 
stable without any regularization.

However, the results show a "quasi-degeneracy" of the coupling
constant,
for a wide range of binding energies. One has for instance 
(see Figure \ref{PS_total1_500}) $\alpha=49.5$ 
for a system with $B=0.001$, and $\alpha=48.6$ for a system five
hundred 
times more deeply bound ($B=0.5$), that is only a 2\%
difference. 
\begin{figure}[htbp]
\begin{center}
\vspace{-1.5cm}
\mbox{\epsfxsize=7.5cm\epsfysize=7.5cm\epsffile{PS_B=500MeV_g2_kmax.eps}}
\vspace{-0.9cm}
\caption{Convergence of coupling constants as function of the
cutoff $k_{max}$ for $B=0.001$ and $B=0.5$. The exchange mass is $\mu=0.15$.}
\label{PS_total1_500}
\end{center}
\vspace{-0.7cm}
\end{figure}

This peculiar behavior can be shown to come from the second
channel:
\begin{eqnarray}\label{lfd_K22}
&&[M^2-4(k^2+m^2)] f_2(k,\theta)= \nonumber \\
&&{m^2\over 2\pi^3} \int 
K^{PS}_{22}(k,k',\theta,\theta',M^2)f_{2}(k',\theta'){d^3k'\over
\varepsilon_{k'}}
\end{eqnarray}
\par

The kernel $K_{22}(k,k',\theta,\theta',M^2)$, whose expression 
is explicitly given in \cite{These_MMB}, is represented in 
Figure \ref{K22PSplot} for fixed values of $\theta,\theta'$. It 
vanishes for 
$k=0$ or $k'=0$ and tends towards a positive constant in all the
$(k,k')$ plane. 
\begin{figure}[htbp]
\vspace{-1.5cm}
\begin{center}
\begin{minipage}{40mm}
\vspace{0.5cm}
\hspace{-2.0cm}
\mbox{\epsfxsize=4.0cm\epsfysize=4.0cm\epsffile{K22_1.eps}}
\end{minipage}\hspace{1.5cm}
\begin{minipage}{50mm}
\vspace{-4.0cm}
\hspace{2.3cm}
\mbox{\epsfxsize=4.0cm\epsfysize=3.5cm\epsffile{K22_2.eps}}
\end{minipage}
\vspace{-0.5cm}
\caption{$K_{22}$ kernel in $(k,k')$ plane.}\label{K22PSplot}
\end{center}
\vspace{-1.0cm}
\end{figure}

Let us modelize this kernel by a kind of "potential barrier" in
the momentum space ($k,k'$), displayed in Figure
\ref{pot_modele_PS}, whose advantage is to be analytically
solvable. 
\begin{figure}[htbp]
\vspace{-1.5cm}
\begin{center}
\begin{minipage}{40mm}
\vspace{0.5cm}
\hspace{-2.0cm}
\mbox{\epsfxsize=4.0cm\epsfysize=4.0cm\epsffile{K22model_1.eps}}
\end{minipage}\hspace{1.5cm}
\begin{minipage}{50mm}
\vspace{-4.0cm}
\hspace{2.3cm}
\mbox{\epsfxsize=4.0cm\epsfysize=3.5cm\epsffile{K22model_2.eps}}
\end{minipage}
\vspace{-1.0cm}
\caption{Modelization of $K_{22}$ by a simpler kernel in the
$(k,k')$
plane.}\label{pot_modele_PS}
\end{center}
\vspace{-1.0cm}
\end{figure}
\begin{eqnarray*}
&\hspace{-0.3cm}&K(k,k')= {\alpha U_1\over
m^2}\Theta(k'-k_1)\Theta(k_2-k') \\
&\hspace{-0.3cm}\times&\hspace{-0.3cm} 
\left[\Theta(k-k_1)\Theta(k_2-k)+\Theta(k-k_2)\Theta(k_{max}-k)\right]\\
&\hspace{-0.3cm}+&\hspace{-0.3cm}
\Theta(k'-k_2)\Theta(k_{max}-k')\left[{\alpha U_1\over m^2}
\Theta(k-k_1)\right.\\
&\hspace{-0.3cm}\times&\hspace{-0.3cm} \left.\Theta(k_2-k)+
{\alpha U_2\over m^2}
\Theta(k-k_2)\Theta(k_{max}-k)\right]
\end{eqnarray*}
with $\Theta(x)=1,\,x>0$ and $\Theta(x)=0, \; x\le0$.
This kernel has the same characteristics than $K^{PS}_{22}$ 
since it is zero when $k,k'\to 0$,and tends towards a constant
when
$(k,k')$ go to infinity with a fixed ratio $\gamma=k'/k$. 

$f_2$ satisfies the Schr\"odinger type equation
\begin{eqnarray}\label{modele_PS}
[k^2+\kappa^2] f_2(k)=\hspace{-0.1cm}-{m^2\over (2\pi)^3} \int 
K(k,k')f_2(k'){d^3k'\over k'}\hspace{-0.17cm}
\end{eqnarray}
with $\kappa^2=m^2-{M^2\over 4}$.
We assume that $k_1<k_2<k_{max}$ et $U_2<U_1$.
The term $\varepsilon_{k'}$ in the volume element of
(\ref{lfd_K22}) 
was replaced by its large momentum behavior, that is by $k'$. 
We define $\Gamma(k)=[k^2+\kappa^2] f_2(k)$. The equation for 
$\Gamma(k)$, which is analytically solvable, reads:
\begin{eqnarray*} 
\Gamma(k)=-{m^2\over 2\pi^2} \int_{-\infty}^{+\infty} 
K(k,k'){\Gamma(k')\over k'^2+\kappa^2} k'dk'
\end{eqnarray*}
The solution $\Gamma(k)$ is constant for $k_1<k<k_2$ and
$k_2<k<k_{max}$ :
\begin{eqnarray*}
\Gamma(k)&=&\Gamma_1 \Theta(k-k_1) \Theta(k_2-k) \\
&+&\Gamma_2 \Theta(k-k_2) \Theta(k_{max}-k)
\end{eqnarray*}
The $\Gamma_i$ satisfy the coupled equations
\begin{eqnarray*}\label{syst_G1G2}
\left\{
\begin{array}{ccc}
(1+\alpha u_1 a)\;\Gamma_1&=& -\alpha u_1 b\Gamma_2     \\
(1+\alpha u_2 b)\;\Gamma_2&=& -\alpha u_1 a\Gamma_1     
\end{array}
\right.
\end{eqnarray*}
where we have defined $u_i={m^2\over 2\pi^2}U_i$ and
\begin{equation}\label{ab}
a=\log\left(k_2^2+\kappa^2 \over k_1^2+\kappa^2\right), \quad
b=\log\left(k_{max}^2+\kappa^2 \over k_2^2+\kappa^2\right)  
\end{equation}
Replacing $\Gamma(k)$ by its definition in terms of $f_2(k)$,
we finally get the solution of equation (\ref{modele_PS}) on the
form:
\begin{eqnarray*}
f_2(k)&=& {N\over
k^2+\kappa^2}\left[\Theta(k-k_1)\Theta(k_2-k)\right. \\
&-&\left.{\alpha a u_1 \over (1+\alpha b u_2)} 
\Theta(k-k_2)\Theta(k_{max}-k)\right]
\end{eqnarray*}
where $N$ is a normalisation constant.
For a given $\kappa$ the coupling constant is
\begin{eqnarray*}
\alpha(\kappa)={(au_1+bu_2)+\sqrt{(au_1-bu_2)^2+4u_1^2ab}\over 
2abu_1(u_1-u_2)}
\end{eqnarray*}
and the results provided by this simple kernel 
are summarized in Table \ref{alphamodel2}.
\begin{table}[htbp]\caption{Coupling constant as a function of
the binding energy 
for the analytical model of the PS coupling second
channel.}\label{alphamodel2}
\[
\begin{array}{||c|c||}\hline
B              & \alpha \\\hline
  0.001        & 22.5327        \\\hline
  0.010    & 22.5334    \\\hline
  0.100    & 22.5396    \\\hline
  0.500    & 22.5639    \\\hline
  1,00     & 22.5862    \\\hline
  1.50     & 22.5995    \\\hline
  2.00     & 22.6040    \\\hline
\end{array}
\]
\vspace{-1.0cm}
\end{table}
$\alpha(\kappa)$ depends on $\kappa$ through logarithms in 
$a,b$, eq. (\ref{ab}). Besides, 
the value of $\kappa$ is much smaller than $k_1,k_2,k_{max}$.
This explains the 
very weak dependence of $\alpha(\kappa)$ v.s. $\kappa$.


We conclude from the above discussion that,  
even if the PS coupling does not formally need any
regularization to insure its
stability, calculations without form factors -- though
analytically understood 
-- lead to results which are hardly interpretable on the
physical point of vue.

\vspace{0.2cm}
{\bf Acknowledgements:}
The numerical calculations were performed
at CGCV (CEA Grenoble) and  IDRIS (CNRS).
We are grateful to the staff members
of these two organizations for their constant support.

\begin{thebibliography}{9}
\bibitem{karm76}  V.A. Karmanov, ZhETF 71 (1976) 399 
(transl.: JETP  44 (1976) 210).
\bibitem{BPP_PR_98}   S.J. Brodsky, H.-C. Pauli and S.S. Pinsky,
Phys. Rep., 
{\bf 301} (1998) 299. 
\bibitem{These_MMB} M. Mangin-Brinet, {\it Th\`ese Universit\'e
de Paris} (2001).
\bibitem{fermions} M. Mangin-Brinet, J. Carbonell and V.A.
Karmanov, 
submitted for publication.
\bibitem{cdkm} J. Carbonell, B. Desplanques, V.A. Karmanov and 
J.-F. Mathiot, Phys. Reports 300 (1998) 215.
\bibitem{ckj0}    J. Carbonell and V.A. Karmanov, 
Nucl. Phys.  A589 (1995) 713.
\bibitem{ll} L.D. Landau, E.M. Lifshits, {\it Quantum
mechanics}, \S 35, Pergamon press, 1965. 
\bibitem{mck_prd1} M. Mangin-Brinet, J. Carbonell and V.A.
Karmanov, 
                   Phys Rev D  64 (2001) 027701.
\bibitem{mck_prd2} M. Mangin-Brinet, J. Carbonell and V.A.
Karmanov, 
                   accepted in Phys Rev D (2001).
\end{thebibliography}


\end{document}
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                These are obtained with the {\tt\ttbs thanks} command.},
        R. de Maas\addressmark\thanks{For following authors with the same
                address use the {\tt\ttbs addressmark} command.},
        X.-Y. Wang\address{Economics Department, University of Winchester, \\
        2 Finch Road, Winchester, Hampshire P3L T19, United Kingdom}
        and
        A. Sheffield\addressmark[MCSD]\thanks{To reuse an addressmark
                later on, label the address with an optional argument to the
                {\tt \ttbs address} command, e.g. {\tt\ttbs
                address[MCSD]}, and repeat the label
                as the optional argument to the {\tt\ttbs addressmark}
                command, e.g. {\tt\ttbs addressmark[MCSD]}.}}
       
\begin{document}

\begin{abstract}
These pages provide you with an example of the layout and style for
100\% reproduction which we wish you to adopt during the preparation of
your paper. This is the output from the \LaTeX{} document class you
requested.
\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle

\section{FORMAT}

Text should be produced within the dimensions shown on these pages:
each column 7.5 cm wide with 1 cm middle margin, total width of 16 cm
and a maximum length of 19.5 cm on first pages and 21 cm on second and
following pages. The \LaTeX{} document class uses the maximum stipulated
length apart from the following two exceptions (i) \LaTeX{} does not
begin a new section directly at the bottom of a page, but transfers the
heading to the top of the next page; (ii) \LaTeX{} never (well, hardly
ever) exceeds the length of the text area in order to complete a
section of text or a paragraph. Here are some references:
\cite{Scho70,Mazu84}.

\subsection{Spacing}

We normally recommend the use of 1.0 (single) line spacing. However,
when typing complicated mathematical text \LaTeX{} automatically
increases the space between text lines in order to prevent sub- and
superscript fonts overlapping one another and making your printed
matter illegible.

\subsection{Fonts}

These instructions have been produced using a 10 point Computer Modern
Roman. Other recommended fonts are 10 point Times Roman, New Century
Schoolbook, Bookman Light and Palatino.

\section{PRINTOUT}

The most suitable printer is a laser or an inkjet printer. A dot
matrix printer should only be used if it possesses an 18 or 24 pin
printhead (``letter-quality'').

The printout submitted should be an original; a photocopy is not
acceptable. Please make use of good quality plain white A4 (or US
Letter) paper size. {\em The dimensions shown here should be strictly
adhered to: do not make changes to these dimensions, which are
determined by the document class}. The document class leaves at least
3~cm at the top of the page before the head, which contains the page
number.

Printers sometimes produce text which contains light and dark streaks,
or has considerable lighting variation either between left-hand and
right-hand margins or between text heads and bottoms. To achieve
optimal reproduction quality, the contrast of text lettering must be
uniform, sharp and dark over the whole page and throughout the article.

If corrections are made to the text, print completely new replacement
pages. The contrast on these pages should be consistent with the rest
of the paper as should text dimensions and font sizes.

\section{TABLES AND ILLUSTRATIONS}

Tables should be made with \LaTeX; illustrations should be originals or
sharp prints. They should be arranged throughout the text and
preferably be included {\em on the same page as they are first
discussed}. They should have a self-contained caption and be positioned
in flush-left alignment with the text margin within the column. If they
do not fit into one column they may be placed across both columns
(using \verb-\begin{table*}- or \verb-\begin{figure*}- so that they
appear at the top of a page).

\subsection{Tables}

Tables should be presented in the form shown in
Table~\ref{table:1}.  Their layout should be consistent
throughout.

\begin{table*}[htb]
\caption{The next-to-leading order (NLO) results
{\em without} the pion field.}
\label{table:1}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\tabcolsep}{2pc} % enlarge column spacing
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular}{@{}lllll}
\hline
$\Lambda$ (MeV)           & \cc{$140$} & \cc{$150$} & \cc{$175$} & \cc{$200$} \\
\hline
$r_d$ (fm)                & \m1.973 & \m1.972 & \m1.974 & \m1.978 \\
$Q_d$ ($\mbox{fm}^2$)     & \m0.259 & \m0.268 & \m0.287 & \m0.302 \\
$P_D$ (\%)                & \m2.32  & \m2.83  & \m4.34  & \m6.14  \\
$\mu_d$                   & \m0.867 & \m0.864 & \m0.855 & \m0.845 \\
$\mathcal{M}_{\mathrm{M1}}$ (fm)   & \m3.995 & \m3.989 & \m3.973 & \m3.955 \\
$\mathcal{M}_{\mathrm{GT}}$ (fm)   & \m4.887 & \m4.881 & \m4.864 & \m4.846 \\
$\delta_{\mathrm{1B}}^{\mathrm{VP}}$ (\%)   
                          & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ \\
$\delta_{\mathrm{1B}}^{\mathrm{C2:C}}$ (\%) 
                          & \m0.03  & \m0.03  & \m0.03  & \m0.03  \\
$\delta_{\mathrm{1B}}^{\mathrm{C2:N}}$ (\%) 
                          & $-0.19$ & $-0.19$ & $-0.18$ & $-0.15$ \\
\hline
\end{tabular}\\[2pt]
The experimental values are given in ref. \cite{Eato75}.
\end{table*}

\begin{sidewaystable}
\caption{The next-to-leading order (NLO) results
{\em without} the pion field.}
\label{table:2}
\newcommand{\m}{\hphantom{$-$}}
\newcommand{\cc}[1]{\multicolumn{1}{c}{#1}}
\renewcommand{\arraystretch}{1.2} % enlarge line spacing
\begin{tabular*}{\textheight}{@{\extracolsep{\fill}}lllllllllllll}
\hline
& $\Lambda$ (MeV) & \cc{$140$} & \cc{$150$} & \cc{$175$} & \cc{$200$} & \cc{$225$} & \cc{$250$} &
\cc{Exp.} & \cc{$v_{18}$~\cite{v18}} &  \\
\hline
%b
 & $r_d$ (fm)                        & \m1.973 & \m1.972 & \m1.974 & \m1.978 & \m1.983 & \m1.987 & 1.966(7) & \m1.967 & \\[2pt]
 & $Q_d$ ($\mbox{fm}^2$)             & \m0.259 & \m0.268 & \m0.287 & \m0.302 & \m0.312 & \m0.319 & 0.286    & \m0.270 & \\[2pt]
 & $P_D$ (\%)                        & \m2.32  & \m2.83  & \m4.34  & \m6.14  & \m8.09  & \m9.90  & $-$      & \m5.76  & \\[2pt]
 & $\mu_d$                           & \m0.867 & \m0.864 & \m0.855 & \m0.845 & \m0.834 & \m0.823 & 0.8574   & \m0.847 & \\[5pt]
 & $\mathcal{M}_{\mathrm{M1}}$ (fm)             & \m3.995 & \m3.989 & \m3.973 & \m3.955 & \m3.936 & \m3.918 & $-$      & \m3.979 & \\[5pt]
 & $\mathcal{M}_{\mathrm{GT}}$ (fm)             & \m4.887 & \m4.881 & \m4.864 & \m4.846 & \m4.827 & \m4.810 & $-$      & \m4.859 & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{VP}}$ (\%)   & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ & $-0.45$ & $-0.44$ & $-$      & $-0.45$ & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{C2:C}}$ (\%) & \m0.03  & \m0.03  & \m0.03  & \m0.03  & \m0.03  & \m0.03  & $-$      & \m0.03  & \\[2pt]
 & $\delta_{\mathrm{1B}}^{\mathrm{C2:N}}$ (\%) & $-0.19$ & $-0.19$ & $-0.18$ & $-0.15$ & $-0.12$ & $-0.10$ & $-$      & $-0.21$ & \\
\hline
\end{tabular*}\\[2pt]
The experimental values are given in ref. \cite{Eato75}.
\end{sidewaystable}

Horizontal lines should be placed above and below table headings, above
the subheadings and at the end of the table above any notes. Vertical
lines should be avoided.

If a table is too long to fit onto one page, the table number and
headings should be repeated above the continuation of the table. For
this you have to reset the table counter with
\verb|\addtocounter{table}{-1}|. Alternatively, the table can be turned
by $90^\circ$ (`landscape mode') and spread over two consecutive pages
(first an even-numbered, then an odd-numbered one) created by means of
\verb|\begin{table}[h]| without a caption. To do this, you prepare the
table as a separate \LaTeX{} document and attach the tables to the
empty pages with a few spots of suitable glue.

\subsection{Useful table packages}

Modern \LaTeX{} comes with several packages for tables that
provide additional functionality. Below we mention a few. See
the documentation of the individual packages for more details. The
packages can be found in \LaTeX's \texttt{tools} directory.

\begin{description}
  
\item[\texttt{array}] Various extensions to \LaTeX's \texttt{array}
  and \texttt{tabular} environments.
  
\item[\texttt{longtable}] Automatically break tables over several
  pages. Put the table in the \texttt{longtable} environment instead
  of the \texttt{table} environment.
  
\item [\texttt{dcolumn}] Define your own type of column. Among others,
  this is one way to obtain alignment on the decimal point.

\item[\texttt{tabularx}] Smart column width calculation within a
  specified table width.
  
\item[\texttt{rotating}] Print a page with a wide table or figure in
  landscape orientation using the \texttt{sidewaystable} or
  \texttt{sidewaysfigure} environments, and many other rotating
  tricks. Use the package with the \texttt{figuresright} option to
  make all tables and figures rotate in clockwise. Use the starred
  form of the \texttt{sideways} environments to obtain full-width
  tables or figures in a two-column article.

\end{description}

\subsection{Line drawings}

Line drawings may consist of laser-printed graphics or professionally
drawn figures attached to the manuscript page. All figures should be
clearly displayed by leaving at least one line of spacing above and
below them. When placing a figure at the top of a page, the top of the
figure should align with the bottom of the first text line of the other
column.

Do not use too light or too dark shading in your figures; too dark a
shading may become too dense while a very light shading made of tiny
points may fade away during reproduction.

All notations and lettering should be no less than 2\,mm high. The use
of heavy black, bold lettering should be avoided as this will look
unpleasantly dark when printed.

\subsection{PostScript figures}

Instead of providing separate drawings or prints of the figures you
may also use PostScript files which are included into your \LaTeX{}
file and printed together with the text. Use one of the packages from
\LaTeX's \texttt{graphics} directory: \texttt{graphics},
\texttt{graphicx} or \texttt{epsfig}, with the \verb|\usepackage|
command, and then use the appropriate commands
(\verb|\includegraphics| or \verb|\epsfig|) to include your PostScript
file.

The simplest command is: \newline
\verb|\includegraphics{file}|, which inserts the
PostScript file \texttt{file} at its own size. The starred version of
this command: \newline
\verb|\includegraphics*{file}|, does the same, but clips
the figure to its bounding box.

With the \texttt{graphicx} package one may specify a series of options
as a key--value list, e.g.:
\begin{tabular}{@{}l}
\verb|\includegraphics[width=15pc]{file}|\\
\verb|\includegraphics[height=5pc]{file}|\\
\verb|\includegraphics[scale=0.6]{file}|\\
\verb|\includegraphics[angle=90,width=20pc]{file}|
\end{tabular}

See the file \texttt{grfguide}, section ``Including Graphics Files'',
of the \texttt{graphics} distribution for all options and a detailed
description.

The \texttt{epsfig} package mimicks the commands familiar from the
package with the same name in \LaTeX2.09. A PostScript file
\texttt{file} is included with the command
\verb|\psfig{file=file}|.

Grey-scale and colour photographs cannot be included in this way,
since reproduction from the printed CRC article would give
insufficient typographical quality. See the following subsections.

\begin{figure}[htb]
\vspace{9pt}
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Good sharp prints should be used and not (distorted) photocopies.}
\label{fig:largenenough}
\end{figure}
%
\begin{figure}[htb]
\framebox[55mm]{\rule[-21mm]{0mm}{43mm}}
\caption{Remember to keep details clear and large enough.}
\label{fig:toosmall}
\end{figure}

\subsection{Black and white photographs}

Photographs must always be sharp originals ({\em not screened
versions\/}) and rich in contrast. They will undergo the same reduction
as the text and should be pasted on your page in the same way as line
drawings.

\subsection{Colour photographs}

Sharp originals ({\em not transparencies or slides\/}) should be
submitted close to the size expected in publication. Charges for the
processing and printing of colour will be passed on to the author(s) of
the paper. As costs involved are per page, care should be taken in the
selection of size and shape so that two or more illustrations may be
fitted together on one page. Please contact the Author Support
Department at Elsevier (E-mail: \texttt{authorsupport@elsevier.nl})
for a price quotation and layout instructions before producing your
paper in its final form.

\section{EQUATIONS}

Equations should be flush-left with the text margin; \LaTeX{} ensures
that the equation is preceded and followed by one line of white space.
\LaTeX{} provides the document class option {\tt fleqn} to get the
flush-left effect.

\begin{equation}
H_{\alpha\beta}(\omega) = E_\alpha^{(0)}(\omega) \delta_{\alpha\beta} +
                          \langle \alpha | W_\pi | \beta \rangle 
\end{equation}

You need not put in equation numbers, since this is taken care of
automatically. The equation numbers are always consecutive and are
printed in parentheses flush with the right-hand margin of the text and
level with the last line of the equation. For multi-line equations, use
the {\tt eqnarray} environment.

For complex mathematics, use the \AmS math package. This package
sets the math indentation to a positive value. To keep the equations
flush left, either load the \texttt{espcrc} package \emph{after} the
\AmS math package or set the command \verb|\mathindent=0pt| in the
preamble of your article.

\begin{thebibliography}{9}
\bibitem{Scho70} S. Scholes, Discuss. Faraday Soc. No. 50 (1970) 222.
\bibitem{Mazu84} O.V. Mazurin and E.A. Porai-Koshits (eds.),
                 Phase Separation in Glass, North-Holland, Amsterdam, 1984.
\bibitem{Dimi75} Y. Dimitriev and E. Kashchieva, 
                 J. Mater. Sci. 10 (1975) 1419.
\bibitem{Eato75} D.L. Eaton, Porous Glass Support Material,
                 US Patent No. 3 904 422 (1975).
\end{thebibliography}

References should be collected at the end of your paper. Do not begin
them on a new page unless this is absolutely necessary. They should be
prepared according to the sequential numeric system making sure that
all material mentioned is generally available to the reader. Use
\verb+\cite+ to refer to the entries in the bibliography so that your
accumulated list corresponds to the citations made in the text body. 

Above we have listed some references according to the
sequential numeric system \cite{Scho70,Mazu84,Dimi75,Eato75}.
\end{document}

