Equivalence of model space techniques and the renormalization group for a separable model problem
S.K. Bogner and T.T.S. Kuo
Lee-Suzuki similarity transformations and Krenciglowa-Kuo folded diagrams are two common methods used to derive energy independent model space effective interactions for nuclear many-body systems. We demonstrate that these two methods are equivalent to a Renormalization Group (RG) analysis of a well-studied problem in quantum mechanics. The effective low-momentum potentials $V_{eff}$ obtained from model space methods are shown to obey the same scaling equation for $V_{eff}$ that RG arguments predict. This indicates that model space methods might be of interest to those studying low-energy nuclear physics using Effective Field Theories (EFT). We find the new result that all of the different energy independent model space techniques yield a unique low-momentum $V_{eff}$ when applied to the toy model under consideration.
nucl-th0009077.tex
%12.15.00 shorten and make changes to satisy the PLB referee
%8.12.00 rough draft for separable model veff 
\documentstyle[preprint,aps,epsfig]{revtex}
\tightenlines
\begin{document}
\title{Equivalence of model space techniques    
 and the renormalization group
 for a separable model problem}
\author{S.K. Bogner and T.T.S. Kuo }
\address{ Department of Physics, SUNY, Stony Brook, New York 11794, USA\\}
\date{\today}
\maketitle
\begin{abstract}
Lee-Suzuki similarity transformations
and Krenciglowa-Kuo folded diagrams are two common methods used 
to derive energy independent model space effective interactions for nuclear many-body systems. 
We demonstrate that these two methods 
are equivalent to a Renormalization Group (RG) analysis 
of a well-studied problem in quantum mechanics.
The effective low-momentum potentials $V_{eff}$ obtained from model space methods are  
shown to obey the same scaling equation
for $V_{eff}$ that RG arguments predict.    
This  
indicates that model space methods might be 
of interest to those studying low-energy nuclear 
physics using Effective Field Theories (EFT).
We find the new result that all of the different energy independent model space techniques yield a unique low-momentum $V_{eff}$ when applied to 
the toy model under consideration. 
  
 
 
 
\end{abstract}
\draft
\pacs{21.60.Cs; 21.30.Fe; 27.80.+j}
\newpage
\section{Introduction}
There has been much effort over the past decade to describe 
low-energy nuclear phenomena  
using the techniques of Effective Field Theory (EFT) \cite
{Wein,ksw,orv,epel98,fsteele}. 
The main goal of an EFT description of these phenomena is not to out-perform
the traditional methods at fitting the experimental data.  Rather it is to 
provide a {\itshape model independent\/} effective low energy theory 
whose form is constrained by the 
symmetries of QCD, as well as to eliminate the uncontrolled
and unjustified approximations that are made in the traditional approaches.  
One of the more appealing features of an EFT treatment is the ability to 
reliably estimate the errors in calculations.  This allows   
one to consistently calculate observables to any chosen level of accuracy 
in principle. 
The success of 
various low energy effective theories (Landau Fermi Liquid theory,
the Fermi theory of Beta decay, the Standard Model, etc) results from the 
fundamental tenet of EFT:  {\itshape low energy observables are insensitive to 
the details 
of the high energy dynamics\/}.  Even if the 
high energy dynamics are unknown, one can still mimic their effects on the
low energy physics with local, model-independent operators that are 
consistent with the low-energy symmetries.  The corresponding 
coupling constants then implicitly contain the effects of the integrated out 
high energy degrees so that the low-energy physics of the full theory
is preserved.  EFT's are founded upon the ideas first
put forward by Wilson in his formulation of the "modern" 
renormalization group in the study of phase transitions.  
The early review article of Wilson and Kogut \cite{Wilson} and the more recent
papers of Lepage and Polchinski \cite{lepage,polch} illustrate these ideas 
with pedagological examples and a minimum of formalism. 

Shell-model theorists  
have long employed the notion of effective interactions that are 
defined only within a truncated model space, but that implicitly contain
the effects of the states that are being thrown away (i.e.- integrated
out) so that certain low energy observables 
are preserved from the full-theory 
\cite{ko90,krmku74,suzuki80,suzuki94,andre96,jensen}.
The qualitative similarities between the modern RG techniques 
and model space methods for deriving effective
interactions within a truncated model space are
obvious.
It would be interesting to study the {\itshape 
quantitative\/} similarities between the two approaches along the lines of the 
recent work of Haxton and Song \cite{haxton}, who seek to employ RG concepts
to formulate a "rigorous" shell model free of ad-hoc assumptions 
with the ability to reliably estimate errors. 
The scope of the current paper is a bit more modest and is similar 
in spirit to the 
earlier work of Fields et. al. \cite{fields}, where they  studied 
the effective interaction for a simple model problem    
by writing down a scaling equation for the 
"generalized G-matrix" as the UV regulator is 
varied and eventually sent to infinity. Our approach is 
more in line with the Wilsonian view of renormalization, 
as we keep the UV regulator 
finite throughout and instead study how $V_{eff}$ scales as we vary the 
boundary of the model space.    

The main
purpose of this paper is to study a separable model 
problem that allows closed form solutions of  
the Lee-Suzuki (LS) 
and the Krenciglowa-Kuo (KK) iteration schemes.  We find that
$V_{eff}$ derived from both schemes obeys the same scaling equation 
that one would obtain from a RG
analysis of the model problem \cite{Birse}.  Moreover,
we find the new and interesting result that {\itshape all\/} 
of the various 
model space techniques corresponding to different methods of solving
the decoupling equation 
(LS and KK are two specific schemes) give
the same low-momentum $V_{eff}$.  
In other words, the low-momentum $V_{eff}$ is unique and 
independent of the particular model space scheme one employs for the toy
model we consider. 
This is a suprising result as it is known that  
when one considers bound state (e.g- shell
model) problems the different model space techniques often converge to
different (i.e.- non unique) $V_{eff}$'s 
\cite{ko90,krmku74,suzuki80,suzuki94,andre96,jensen}.

\section{Model Space Techniques}  
  We turn now to a quick review of the methods one uses to derive energy 
independent
model space effective interactions.  Andreozzi has shown that both the Lee-Suzuki (LS)
and Kuo-Krenciglowa (KK) methods 
can be recast in the form of a general similarity transformation \cite
{andre96}.
In this formalism we  
perform a similarity transformation to obtain an effective 
Hamiltonian that acts only within the model space, but that 
preserves low-energy physics (spectra, scattering amplitudes, etc)
from the full problem.   
If we denote the 
projection operator onto the model space as $P$ and its complement as 
$Q$ (using the eigenkets of the unperturbed Hamiltonian as our basis) 
we have  

\begin {equation}
 S^{-1}HSS^{-1}\mid \Psi \rangle =ES^{-1} \mid \Psi \rangle 
\end {equation}

  We can write the above equation in an obvious way
by defining $H_{eff}\equiv S^{-1}HS$ and $\mid \chi \rangle  = S
^{-1}\mid \Psi \rangle $.  A convenient form of the similarity 
transformation is $S= 1 + \omega$, where $\omega$ is called the wave
operator and is defined to satisfy $\omega= Q\omega P$.  
It is easy to see that $S^{-1}= 1 - \omega$  and that the P-space
projections of the transformed and original states are the same, 
$P\mid \chi \rangle= P\mid \Psi \rangle$.  We want to exploit the
invariance of eigenvalues under similarity transformations to 
preserve a subset of the exact eigenvalues, but with  $H_{eff}$ {\itshape
acting only within the model space\/}.  In other words we want
$H_{eff}= PH_{eff}P$ and $\mid \chi\rangle = P\mid \Psi\rangle$ so that
the excluded Q-space completely decouples from the problem.  
A necessary and sufficient
condition for this is $QH_{eff}P= 0$.  Explicitly writing this out
gives the so-called decoupling equation for $\omega$

\begin {equation}
\omega PHQ \omega + \omega PHP - QHQ\omega - QHP = 0
\end {equation}
   
  The decoupling equation is a non-linear operator equation and different
methods of solution can give different answers.  Andreozzi has recently
shown that two iteration methods of solving it are equivalent to the Lee-Suzuki 
method and the Krenciglowa-Kuo folded diagram theory \cite{andre96}.  The
differences between the LS and the KK methods are most apparent when
one is dealing with bound state problems and the basis states are discrete
(e.g.- harmonic oscillator orbits).  If the model space is d-
dimensional, then the LS $H_{eff}$ reproduces the lowest d eigenvalues of
the full problem and the corresponding P-space projections of the exact
eigenstates.  Conversely, the KK $H_{eff}$ reproduces the d eigenvalues of
the full problem whose eigenstates have the largest P-space component (i.e.-
maximum overlap).  
The great benefit of Andreozzi's work is that it contains solutions of the
decoupling equation that are formally equivalent to the LS and KK schemes, 
but his methods are non-perturbative in nature and hence eliminate the need
to calculate irreducible vertex functions (i.e.-$\hat{Q}$-boxes)  and perform G-matrix resummations.   
Andreozzi's methods can furnish analytic results for simple toy models, and yet
they are extremely robust and easy to implement numerically for realistic bare
potentials (Paris, Bonn-A, V-18, etc...) \cite{Bogner}.
In deference to Andreozzi's simplification of KK and LS calculations, we 
hereafter refer to them as the Andreozzi-Krenciglowa-Kuo (AKK) and the 
Andreozzi-Lee-Suzuki (ALS) schemes.  We simply quote the relevant equations
and refer the interested reader to Andreozzi's paper for details.
\section *{The ALS Method}  
\noindent
Writing $\omega=\sum_{n=0}^{\infty}X_{n}$ and $\sigma_{n}=\sum_{m=0}^{n}X_{m}$, the ALS equations are

\begin {equation}
X_{0}=\frac{-1}{QHQ}QHP
\end {equation}

\begin {equation}
X_{n}=\frac{1}{q(\sigma_{n-1})}X_{n-1}p(\sigma_{n-1})\quad n=1,2,...
\end {equation}

\begin {equation}
p(\sigma_{n-1})=PHP + PHQ\sigma_{n-1}
\end {equation}

\begin {equation}
q(\sigma_{n-1})=QHQ - \sigma_{n-1}PHQ
\end {equation}

\noindent
Note that $p(\sigma_{n-1})$ and $q(\sigma_{n-1})$ are the P-space
and Q-space effective Hamiltonians at each step in the iteration, and
the iteration converges when $\sigma_{n}\approx  \sigma_{n-1}$. 

\section *{The AKK method} 
\noindent
Using the same notation as the ALS scheme,
the $X_{n}$ in the AKK scheme are given by the following equations,

\begin {equation}
X_{0}PHP-QHQX_{0} - QHP=0
\end {equation}

\begin {equation}
X_{n}p(\sigma_{n-1})-QHQX_{n}+\sigma_{n-1}PHQX_{n-1}=0 \quad n=1,2,...
\end {equation}

\section{Schematic Model with Separable Potential}
Both the ALS and AKK methods are easy to implement numerically for realistic
models of the bare $V_{NN}$ such as the Paris and Bonn-A potentials.  However, 
it would be nice to construct a toy model for which the ALS and AKK 
equations yield analytic solutions.  For this purpose, we consider the 
following schematic underlying (or full space) Hamiltonian  

\begin {equation}
H=H_{0} + g \mid \eta\rangle\langle\eta\mid
\end {equation}

\noindent
where

\begin {equation}
\langle i\mid H\mid j \rangle
= \epsilon_{i}\delta_{ij}
+ g\eta_{i}\eta_{j}
\end {equation}

\noindent
We will use the following formula many times in the analysis
that follows.
\begin {equation}
\langle i\mid H^{-1}\mid j \rangle   
= \frac{\delta_{ij}}{\epsilon_{i}}
- \frac{g\eta_{i}\eta_{j}}{\epsilon_{i}\epsilon_{j}}
\frac{1}{1 + gF}
\end {equation} 

\noindent
where 

\begin {equation}
F=\sum_{l}\frac{\eta^{2}_{l}}{\epsilon_{l}}
\end {equation}

\noindent
Note that we don't have to worry about the above expressions becoming 
singular ($\epsilon_{i}=0$), as we are always inverting 
in Q-space (where the $\epsilon_{i}\neq 0$).  

\section{ALS and AKK  Analysis of the Schematic Model}
For the remainder of this paper, all state 
labels refer to relative momenta, and it is assumed we are working with a 
particular partial wave (labels are suppressed).  
If we assume that the underlying potential
$V(k',k)\approx\ 0$ for $k,k' >\Lambda$ (either by a form factor or a sharp 
cut-off), then the truncated model space will consist of all states with 
$k\leq\frac{\Lambda}{s}$, where $s > 1$.  The excluded Q-space will then consist
of all states with momenta lying in the shell $\frac{\Lambda}{s}< k \leq \Lambda$. 
For a finite shrinking or decimation (e.g. $s=2$), we can use the formulas of the 
previous two sections to calculate the $X_{n}$ of both the ALS and AKK schemes analytically.
Unfortunately the expressions quickly become unwieldy for $n>1$, and it is not
at all obvious how one can sum all of the $X_{n}$ to obtain an analytic expression for the wave operator (and 
hence for $V_{eff}$).  The trick that gets us off the hook is to consider infinitesimal
decimations where $\frac{\Lambda}{s}=\Lambda - \delta\Lambda$.  In this way we are able
to calculate the wave operator analytically and obtain a scaling equation for
$V_{eff}$.  
Using the definition $V_{eff}=H_{eff}-H_{0}$, cavalierly ignoring factors of $2\pi$, and working in units where $\epsilon_{p}=p^{2}$, we have

\begin {equation}
V_{eff}(k',k)=V(k',k) + \int\limits_{\Lambda -\delta\Lambda}^{\Lambda}
V(k',q)\omega(q,k)
q^{2}dq
\qquad  k,k'
\leq \Lambda - \delta\Lambda
\end {equation} 

\noindent
For $\delta\Lambda\rightarrow 0$, this becomes a flow equation for $V_{eff}$

\begin {equation}
\frac{\partial}{\partial \Lambda}V_{eff}(k',k)=
-\Lambda^{2}V_{eff}(k',\Lambda)\omega(\Lambda,k)
\end {equation}

\noindent
Since we take the limit of $\delta\Lambda\rightarrow 0$, we can safely ignore
contributions to $\omega$ that are of $O(\delta\Lambda)$
and higher.
With this in mind, both the ALS and the AKK equations simplify considerably.
\section*{ALS for Infinitesimal Decimations}
Recalling that
each intermediate projection operator $Q=\int\limits_{\Lambda-\delta\Lambda}^{\Lambda}
q^{2}\mid q\rangle\langle q\mid dq\quad \sim O(\delta\Lambda)$, we can replace all
of the $p(\sigma_{n-1})=PHP + PHQ\sigma_{n-1}$ by $PHP$ in the ALS equations.  
It can be shown that at each step in the ALS iteration scheme we can write 

\begin {equation}
\langle q'\mid q(\sigma_{n-1}) \mid q\rangle=
\epsilon_{q}\delta_{q'q} + g\bar{\eta_{q'}}\eta_{q}
\end {equation}

\noindent
Where $\bar{\eta_{q'}}=\eta_{q'}(1+constants)$.
Consequently, we can use equations 11 and 12 to write 

\begin {equation}
\langle q'\mid q^{-1}(\sigma_{n-1}) \mid q\rangle=
\frac{\delta_{q'q}}{\epsilon_{q}} - \frac{g\bar{\eta_{q'}}\eta_{q}}{\epsilon_q'
\epsilon_{q}} + O(\delta\Lambda)
\end {equation}

\noindent
Naively, one might think that the $X_{n}$ are all infinitesimals of $O(\delta\Lambda)$
since the ALS equations all have an intermediate $Q$ projection operator. {\itshape This
is wrong\/}.  One must remember that by equation 16 , the operators 
$(QHQ)^{-1}$ and $q^{-1}(\sigma_{n-1})$ consist of a diagonal part proportional
to a $\delta$-function plus an off-diagonal part.  It is clear that when an 
intermediate infinitesimal projection operator $Q$ acts on the off-diagonal part it
results in a term of $O(\delta\Lambda)$ that can be dropped.  However, the $\delta$-function
portion survives the infinitesimal $Q$ projection and gives a finite contribution.
Therefore, we can make a second major simplification to the ALS equations by
replacing $(QHQ)^{-1}$ and $q^{-1}(\sigma_{n-1})$ by $(QH_{0}Q)^{-1}$.
The simplified ALS equations can be solved by addition, resulting in the following
linear integral equation for $\omega$

\begin {equation}
\omega=X_{0} + \frac{1}{QH_{0}Q}\omega PHP 
\end {equation}

\noindent
Rearranging terms, explicitly displaying indices, and abbreviating integrals
as sums,  we obtain

\begin {equation}
\sum_{p'}^{\Lambda-\delta\Lambda}\omega(\Lambda,p')A(p',p)=-\epsilon_{\Lambda}X_{0}(\Lambda,p)
\end {equation}

\noindent
Where we have defined the matrix $A(p',p)=(\epsilon_{p'}-\epsilon_{\Lambda})\delta_{p'p}
+ g\eta_{p'}\eta_{p}$.  Multiplying from the right by $A^{-1}$ we obtain

\begin {equation}
\omega(\Lambda,p)=-\epsilon_{\Lambda}\sum_{p'}X_{0}(\Lambda,p')A^{-1}(p',p)
\end {equation}

\noindent
Once again, we make use of equation 11 to obtain an explicit expression
for $A^{-1}$, viz,

\begin {equation}
A^{-1}(p',p)= \frac{\delta_{p'p}}{\epsilon_{p}-\epsilon_{\Lambda}}
-\frac{g\eta_{p'}\eta_{p}}{(\epsilon_{p}-\epsilon_{\Lambda})(\epsilon_{p'}
- \epsilon_{\Lambda})}\frac{1}{1+g\bar{F}}
\end {equation}
    
\noindent
where we have defined $\bar{F}$ by the following equation

\begin {equation}
\bar{F}=\int\limits_{0}^{\Lambda-\delta\Lambda}\frac{p^{2}\eta_{p}^{2}}
{p^{2}-\Lambda^{2}}dp
\end {equation}

\noindent
Since we eventually take $\delta\Lambda\rightarrow 0$, it is
clear that $\bar{F}\rightarrow \infty$ allowing us to ignore the second term in 
the above expression for $A^{-1}$. Noting that $X_{0}(\Lambda,p)=\frac{-g\eta_{\Lambda}
\eta_{p}}{\epsilon_{\Lambda}}$ and utilizing the simplified expression for $A^{-1}$
, we obtain our final expressions for the ALS wave operator $\omega(\Lambda,p)$
and the ALS flow equation 

\begin {equation}
\omega(\Lambda,p)=\frac{-g\eta_{\Lambda}\eta_{p}}{\Lambda^{2}-p^{2}} =
\frac{- V_{eff}(\Lambda,p)}{\Lambda^{2}-p^{2}}
\end {equation}

\noindent
and     

\begin {equation}
\frac{\partial}{\partial\Lambda}V_{eff}(p',p)=\frac{\Lambda^{2}V_{eff}(p',\Lambda)
V_{eff}(\Lambda,p)}{\Lambda^{2}-p^{2}}
\end {equation}

\section*{AKK for Infinitesimal Decimations}
For infinitesimal
model space reductions $\Lambda\rightarrow\Lambda-\delta\Lambda$, we simply
state the simplified AKK equations as we use the same arguments we used to 
simplify the ALS equations in the previous section.  Equation 7 simplifies to

\begin {equation}
X_{0}PHP - QH_{0}QX_{0}
=QHP
\end {equation}

\noindent
while equation 8 simplifies to

\begin {equation}
X_{n}PHP - QH_{0}QX_{n}=0
\quad n=1,2,...
\end {equation}

\noindent
It is easily shown 
that all of the $X_{n}$ are zero except for $X_{0}$.  {\itshape Hence, the AKK
wave operator is given by $X_{0}$ and it is found to be the same as the ALS
wave operator\/}.  
\indent
We can generalize the above results to cover {\itshape any\/} method of
solving the decoupling equation, by once again considering an 
infinitesimal $\Lambda\rightarrow\Lambda-\delta\Lambda$.  In this
case the non-linear decoupling equation simplifies to 

\begin {equation}
\omega PHP - QH_{0}Q \omega - QHP=0
\end {equation}  

\noindent
It is easily shown that the solution to this equation is the same as the ALS 
and 
AKK solutions. Hence, when  
we shrink the boundary of P-space by an infinitesimal amount, all of the 
various methods of solving the decoupling equation 
give the same equation for $\partial_{\Lambda}V_{eff}$.   
We shall find that we obtain the same scaling equation if we cut off all loop 
integrals in the Lippman-Schwinger 
equation and demand the observable scattering amplitude be independent of 
$\Lambda$; this is the RG method described in the next section.    
We can now invoke the semi-group  property of the RG 
and make a much stronger statement about our model space techniques.   
Even for {\itshape finite
\/} $\Lambda\rightarrow\frac{\Lambda}{s}$, the
different methods of solving the decoupling equation 
give a unique low-momentum
$V_{eff}$ for the toy problem under consideration.   
To the best of our knowledge,
this is a new and interesting result.  This result brings out yet another 
similarity
between model space techniques and the Wilsonian RG. In the application of
the Wilsonian RG to critical phenomena, there are many different methods of
integrating out the short distance physics to obtain an effective long-wavelength theory.  For example, there are infinitely many prescriptions for performing block
spin transformations and assigning values to the block variables for the Ising
model.  However, the differences between the various coarse graining procedures
are eventually washed out and one always converges to the same long-wavelength
effective theory.
\section{renormalization group analysis}
Since we claim that the model space techniques of nuclear many-body physics 
give the same scaling equation for $V_{eff}$ that a renormalization group 
analysis predicts, it is convenient to quickly review exactly what we mean by 
a "renormalization group analysis". In relativistic quantum field theory 
when one 
calculates physical observables perturbatively as a 
power series in the coupling $g$, one finds that the coefficients of the higher
order powers of $g$ are divergent.  These infinities are the result of loop 
integrals that diverge at high momentum, or equivalently at short distances.  
One can isolate these infinities by regulating the divergent integrals with 
a UV cut-off or any other convenient method (dimensional regularization, Pauli-
Villars, etc...), but then one is faced with the unpleasant result that  
calculated observables depend on the arbitrary regulator mass 
$\Lambda$ that has been introduced into the problem, as well as the particular
method of regularization.  As is well known, the way out of this dilemna is to
notice that for certain field theories (the so-called "renormalizable" theories)
we can add counterterms to the theory that have the same operator structure as
the terms present in the initial Lagrangian.  If we are sufficiently clever, we
can hide the regulated infinities in the coefficients of the 
counterterms so that when we calculate graphs using the new Feynman rules 
(that are induced by the counterterms), everything is finite.  Hence, one can absorb
the divergences by redefining the couplings and making them $\Lambda$-dependent
in such a way that the {\itshape physical observables\/} are $\Lambda$-independent and finite as $\Lambda\rightarrow\infty$.  Thus, a change in the UV cut-off
scale is compensated by a simultaneous change in the couplings and masses 
leaving the physics invariant.  This is the essence, albeit a grotesquely 
simplified description, of the renormalization group in QFT designed to
hide the infinities at short distances.  But by allowing $\Lambda\rightarrow
\infty$ we are assuming that the dynamics encoded in the original Lagrangian
holds true at arbitrarily large energies.  In the language of old fashioned 
perturbation theory, allowing loop momenta to go up to infinity corresponds
to summing over intermediate states of arbitrarily large energy and momentum.  But this presupposes that the intermediate states we are summing are devoid of
any "new" physics at arbitrarily small distances, which is clearly a bold 
assumption (supersymmetric partners?  quark and lepton substructure?  strings?)
since $1$ TeV $\neq\infty$!  About the only thing we can say for certain about
the high energy intermediate states is that they are highly virtual and thus 
localized in space-time.

   The modern approach to renormalization views $\Lambda$ as a 
physical scale at which unknown physics comes into play, and not as an 
artifact that is to be taken to infinity at the end of the calculation.  In
this way we don't sum over intermediate states we don't fully understand.  Yet
we can model the effects of these excluded states on the low energy physics by
recalling that their virtual nature means that they are propagated over very 
small distances and times (i.e.- they are localized).  We can include 
their effects on the low energy physics by writing down every possible local 
interaction that is consistent with low energy symmetries (model independence!), and we can parameterize our ignorance of the the excluded high energy states 
inside of the tunable couplings that multiply each local operator. 
Although there are an infinite number of local operators consistent with the
symmetry requirements, they all scale as inverse powers of $\Lambda$ 
allowing one to truncate the effective Lagrangian to any chosen level of 
accuracy.  The corresponding couplings can then be obtained from a finite 
number of independent experimental measurements, giving the theory predictive 
power to the chosen level of accuracy.  In this way the effective theory is
consistent with the underlying symmetries, the errors are reliably estimated, 
different quantities can be calculated consistently to the same level of 
accuracy, and one can infer the presence of "new physics" when the effective
theory fails.  

  We can employ similar ideas in analysing our toy potential model.
Here, we can impose a UV cut-off $\Lambda$ in the loop integrals of the 
Lippmann-Schwinger equation to reflect our ignorance of what is really going
on at short distances.  We then demand that $V_{eff}$ depend on $\Lambda$
in such a way that the low energy observables (i.e.- the scattering amplitudes)
are independent of $\Lambda$. 
For our separable toy model we can write the half-onshell (HOS) T-matrix as  

\begin {equation}
\frac{1}{\langle k'\mid T(\omega=k^{2})\mid k \rangle}
= \frac{1}{g\eta_{k'}\eta_{k}} 
-\frac{1}{\eta_{k'}\eta_{k}}\int\limits_{0}^{\Lambda}\frac{q^{2}\eta_{q}^{2}}
{k^{2}-q^{2}}dq
\end {equation}

\noindent
Since we want the physical quantities to be independent of this UV cut-off
$\Lambda$, we allow the coupling to "run" with $\Lambda$.  By setting
$\partial_{\Lambda}T^{-1}=0$ (and noting that all the $\Lambda$-dependence is 
inside $g$), we obtain a 
scaling equation for $V_{eff}$

\begin {equation}
\frac{\partial}{\partial \Lambda}V_{eff}(k',k)=
\frac{\Lambda^{2}V_{eff}(k',\Lambda) V_{eff}(\Lambda,k)}{\Lambda^{2}
- k^{2}}
\end {equation}

\noindent
This is same scaling equation for $V_{eff}$ that we derived using
model space methods.  We also note that this is the same equation obtained
by Birse et. al. \cite{Birse}, except that their $V_{eff}$ depends on the
energy as well (i.e. $V_{eff}(k',k)\rightarrow V_{eff}(k',k;E)$).  
We therefore conclude that model space techniques preserve the low energy HOS
T-matrix and  are equivalent to renormalization group methods when applied
to the separable model. 
\section{conclusions}
We have found that model space methods and renormalization group techniques 
give the same scaling equation for the low-momentum $V_{eff}$.  
This was accomplished by considering a separable potential model 
in which the model space techniques yield analytic expressions.
Moreover, we 
have found that all methods of solving the decoupling equation give the same
$V_{eff}$ for the toy model under consideration.  
To
the best of our knowledge, these are new and interesting results.  The
generalization of these results to realistic $V_{NN}$'s will appear in a 
forthcoming paper. 
\section*{acknowledgements}
We thank the INT at the University of
Washington for its hospitality and the Department of Energy for
partial support during the completion of this work.  
\begin{references}
\bibitem{Wein} S. Weinberg, Phys. Lett. {\bf B251}, 288 (1990).
\bibitem{ksw} D.B. Kaplan, M.J. Savage and M.B. Wise,
Phys. Lett. {\bf B424} 390 (1998).
\bibitem{orv} C. Ordonez, L. Ray, U. van Kolck, Phys. Rev. Lett. {\bf
72}, 1982 (1995).
%\bibitem{lynn} B. Lynn, Nucl. Phys.  {\bf B402}, 281 (1993).
\bibitem{fsteele} J.V. Steele and R.J. Furnstahl, Nucl. Phys. {\bf A663},
999c (2000).
\bibitem{polch} J. Polchinski, "Effective Field Theory and the Fermi Surface," Proceedings of 1992 Theoretical Advanced Studies Institute in Elementary Particle Physics, edited by J. Harvey and J. Polchinski (World Scientific, Singapore, 1993).
\bibitem{lepage} G.P. Lepage, "How to Renormalize the Schr$\ddot o$dinger Equation", [nucl-th/9706029].
\bibitem{epel98} E. Epelbaum, W. Gl$\ddot o$ckle, and Ulf-G Meiszner,
Phys. Lett. {\bf B439} 1 (1998); Nucl. Phys. {\bf A637} 107 (1998).
\bibitem{Wilson} K. Wilson and J. Kogut, Phys. Rep. {\bf 12}, 75
(1974).
\bibitem{Birse} M. Birse, J. McGovern, and K. Richardson, Phys. Lett.
{\bf B464} 169 (1999).
\bibitem{Bogner} S. Bogner and T.T.S. Kuo, nucl-th/9912056.
\bibitem{ko90} T.T.S. Kuo and E. Osnes, Springer Lecture Notes of Physics,
{\bf Vol. 364}, p.1 (1990).
\bibitem{krmku74} E.M. Krenciglowa and T.T.S. Kuo, Nucl. Phys. {\bf A235} 171
(1974).
\bibitem{suzuki80} K. Suzuki and S. Y. Lee, Prog. Theor. Phys. {\bf 64}, 2091 
(1980).
\bibitem{suzuki94}K. Suzuki, R. Okamoto, P.J. Ellis and T.T.S. Kuo,
Nucl. Phys. {\bf A567} 576 (1994).
\bibitem{andre96} F. Andreozzi, Phys. Rev. {\bf C54}, 684 (1996).
\bibitem{jensen}M. Jorth-Jensen, T.T.S. Kuo and E. Osnes, Physis Report
{\bf 261} 126 (1995).
\bibitem{fields} T.J. Fields, J.P. Vary, and K.S. Gupta, Mod. Phys. Lett. 
{\bf A11}, 2233 (1996). 
\bibitem{haxton} W. Haxton and C.-L. Song, Phys. Rev. Lett. {\bf 84}, 5484 (2000
).  
\end{references}
\end{document}




such as the Bonn \cite{bonn} and Paris \cite{paris} potentials,
which all describe the observed
deuteron and NN scattering data very well. Because of the strong short range
repulsion contained in these potentials, their momentum space matrix elements $V(k,k')$ are
still significant at large momentum. A main purpose of the present
work is to study if we can reduce such realistic potentials,
in a physically equivalent way, to certain effective low-momentum NN
potentials which only have momentum components below a chosen cut-off
momentum $k_{cut}$. We shall also discuss the possibility of using
this momentum-reduced potential directly in 
nuclear many-body calculations without first calculating the Brueckner 
G-matrix.
  Recently there has been much interest in studying nuclear physics problems
using the low-momentum effective field
theory (EFT) \cite{kaplan,epel98}, and this development has been a
main motivation of our present work.  It would be nice if one could calculate
the NN potentials directly from QCD. As it is well known, this is still
not feasible at the present time, owing to the nonperturbative nature of QCD
at energies characteristic of low-energy nuclear phenomena. However, the methods
of the low-momentum EFT may provide a powerful and practical 
framework for treating the low-energy NN potential in a manner
consistent with the underlying QCD. 
The basic idea of the EFT approach is to shrink
the full-space theory  to a small-space one which contains only
the low-momentum modes. This is accomplished by integrating out the high
momentum modes, thus generating effective couplings which implicitly
contain the effects of the high-momentum modes. The EFT  approach was 
originally developed by Wilson and Kadanoff \cite{wilson74}
for condensed matter systems.
  Very similar ideas have been employed in nuclear effective 
interaction theory, and in fact many such shrinking methods 
\cite{klr71,ko90,krmku74,suzuki80,suzuki83,kuo93,suzuki94,kuo95,andre96} 
have been developed.  We feel that some of these
methods may be useful for EFT problems. 
To explore this possibility , a first
step is perhaps to apply them to a simple nuclear system, namely
the two-nucleon problem. Epelbaoum et al.
\cite{epel98} have studied a low-momentum effective theory for the 
two-nucleon system based on the unitary transformation method 
of Okubo and Fukuda.
In the present work we would like to study a low-momentum reduction
of realistic two-nucleon potentials, such as the Bonn \cite{bonn} and Paris
\cite{paris} potential, using the Kuo-Lee-Ratcliff (KLR) folded-diagram
approach \cite{klr71,ko90} in conjunction with the Andreozzi-Lee-Suzuki
 iteration method \cite{andre96,suzuki80}. The model-space effective
Hamiltonian given by the
ALS method reproduces the lowest $d$ eigenvalues of the full-space
Hamiltonian, d being the dimension of the model space.
 As to be discussed later, the deuteron binding energy
as well as the model-space half-on-shell T-matrix given by 
our reduced potential is the same as those given by the original potential.
Let us first briefly review the KLR folded-diagram approach method. One starts
from the full-space nuclear Schroedinger equation
\begin{equation}
H \Psi _n = E_n \Psi _n;~ H=H_0 + V_{NN},
\end{equation}
where $H_0$ is the unperturbed Hamiltonian and $V_{NN}$ 
a realistic NN potential.
For simplicity, we shall from now on abbreviate $V_{NN}$ as $V$.
 The above equation can be reduced
using the KLR  folded diagram method \cite{klr71,ko90}
 to a model-space one of the form
\begin{equation}
PH_{eff}P \chi _m = E_m \chi _m;~ H=H_0 + PV_{eff}P,
\end{equation}
where P denotes the projection operator for the model space, which
can be written as 
$P=\sum _{k\leq d}\mid \phi _k  \rangle \langle \phi _k \mid$.
Here $\phi_k $ is an eigenstate of $H_0$, namely $H_0 \phi _k=\varepsilon _k
\phi _k$, and $d$ denotes the dimension (or size) of the model space.
In the present work we shall apply the above approach
to the two-nucleon system, and
for this case $k$ denotes the relative momentum. We shall use
a model space specified by $k \leq k_{cut}$ where $k_{cut}$ is the cut-off
momentum, a typical value of it being about 2 $fm^{-1}$. 
We shall from now on abbreviate $PV_{eff}P$ as $V_{eff}$ and similarly for
$PH_{eff}P$.
Certain properties of the original Schroedinger equation defined by 
$H$ for the full space are preserved by the model-space one defined
by $H_{eff}$, namely
$H_{eff}$ reproduces a subset of the eigenvalues
and the P-space projections of the corresponding 
 eigenfunctions of the full $H$.  Note that $\chi _m=P\chi _m$.
In the above approach 
 the effective interaction $V_{eff}$ is given as
\begin{equation}
V_{eff} = \hat{Q} - \hat{Q'} \int \hat{Q} + \hat{Q'} \int \hat{Q} \int 
\hat{Q} - \hat{Q'} \int \hat{Q} \int \hat{Q} \int \hat{Q} + ~...~~,
\end{equation}
where $\hat{Q}$ is an irreducible vertex function often referred to as 
the $\hat{Q}$-box, and the integral sign represents a 
generalized folding operation \cite{klr71,ko90}. 
$\hat{Q'}$ is obtained from $\hat{Q}$ by removing terms of first order in the 
interaction $V$.
The above effective interaction has been extensively applied to 
nuclear bound state problems \cite{jensen,covello} and has led to 
rather encouraging results.  As to be presented in more detail
later, we have here applied it to the deuteron problem.
We have found that the deuteron binding energy given 
by $V$ is exactly reproduced by $V_{eff}$ for various various
choices for $k_{cut}$ such as $2 fm^{-1}$. 
Can the above folded-diagram effective potential also 
reproduce the phase shifts given by the original potential $V$? 
 This is an important question, as empirical NN potentials are all constructed
such that they reproduce not only the deuteron
binding energy but also  the phase shifts deduced from two-nucleon scattering
experiments. It would be desirable if our $V_{eff}$ can also reproduce the
phase shifts. 
To study this question, we
 start from the half-on-shell T-matrix
\begin{equation}
 \langle p' \mid T(\omega) \mid p \rangle 
 =\langle p' \mid V \mid p \rangle 
 +\sum _{p",all} \langle p' \mid V \mid p" \rangle 
\frac{1}{\omega - \varepsilon _p"}
 \langle p" \mid T(\omega) \mid p \rangle 
 ;~\omega=\varepsilon _p. 
\end{equation}
This T-matrix is just
 $\langle p' \mid VU(0,-\infty ) \mid p \rangle$ 
where $U$ is the time evolution operator. Writing the T-matrix in this way
is convenient for making a 
diagramatic analysis, as we shall do in Fig. 1,  of the T-matrix.
For simplicity, we have not written out the 
boundary conditions (for outgoing, incoming or standing waves)
in the above equation.
A general term of the above T-matrix can be written as 
 $\langle p' \mid (V +V\frac{1}{e(p)}V +V\frac{1}{e(p)}V\frac{1}{e(p)}V+
\cdots) \mid p \rangle $ where $e(p)\equiv (\varepsilon _p- H_o)$.
 Note that the intermediate states (represented by 1 in the numerator)
cover the entire space, and 1=P+Q where P denotes the model space and Q its
complement. Writing it out in terms of P and Q, a typical term of T 
is  $V\frac{Q}{e} V\frac{Q}{e}V\frac{P}{e}V\frac{Q}{e}V
\frac{P}{e}V$. Note it has three segments partitioned by two $\frac{P}{e}$
propagators.
Let us define a $\hat Q$-box as $\hat Q=V+V\frac{Q}{e}V+V\frac{Q}{e}
V\frac{Q}{e}V+ \cdot \cdot \cdot$, where all intermediate states belong
to Q. One readily sees that the previous term 
is just a part of the three-$\hat Q$-box term,
and  in general we have $T=\hat Q+\hat Q \frac{P}{e} \hat Q
+ \hat Q \frac{P}{e} \hat Q  \frac{P}{e} \hat Q +\cdot \cdot \cdot$. 
This regrouping is depicted
in Fig. 1, where each $\hat Q$-box is denoted by a circle and the solid
line represents the propagator $\frac{P}{e}$. The diagrams
A, B and C are respectively the one- and two- and three-$\hat Q$-box
terms of T, and clearly T=A+B+C+$\cdots$.
 Note the  dashed vertical line is not a propagator; 
it is just a "ghost" line
to indicate the external indices.
 We now perform a folded-diagram factorization for the T matrix, 
following  closely the folded-diagram method of Kuo-Lee-Ratcliff
\cite{klr71,ko90}. Diagram B of Fig. 1 is factorized into the product
of two parts (see B1) where the time integrations of the two parts
are independent from each other, each integrating from $-\infty$ to
0. In this way we have introduced a time-incorrect contribution
which must be corrected. In other words B is not equal to B1, rather
it is equal to B1 plus the folded-diagram correction B2.
 Recall that the integral
sign reperesents a generalized folding \cite {klr71,ko90}. 
Let us give one example to illustrate the above folding factorization.
Consider a third-order term of diagram B:
 $\langle p' \mid V\frac{Q}{e(p)}V\frac{P}{e(p)}V \mid p \rangle $.
The factorization indicated by diagram B1 is to factorize this term as
 $\sum _{p"} \langle p' \mid V\frac{Q}{e(p")}V \mid p" \rangle
 \langle p" \mid \frac{P}{e(p)}V) \mid p \rangle $. Note the difference
in energy denominators for the two diagrams. This clearly illutrates that
 B1 is not equal to B.
Similarly we factorize the 
three-$\hat Q$-box term C as shown in the third line of Fig. 1.
Higher-order $\hat Q$-box terms are also factorized following the same
folded-diagram procedure.
 Let us now collecting terms in the figure in a "slanted" way. The sum
of terms A1, B2, C3... is just the effective interaction of Eq. (3).
(Note that the leading $\hat Q$-box of any folded term must be at least
second order in V, and hence it is the $\hat Q'$-box mentioned earlier.)
The sum B1, C2, D3.... is $V_{eff}\frac{P}{e}\hat Q$. 
Similarly the sum C1+D2+E3+$\cdots$
is just $V_{eff}\frac{P}{e}\hat Q\frac{P}{e}\hat Q$. (Note diagrams
D1, D2, $\cdots$, E1, E2, $\cdots$ are not shown in the figure.)
Continuing 
this way, we have from
Fig. 1 that for p and p' belonging to the P space
\begin{equation}
 \langle p' \mid T(\omega=\varepsilon _p) \mid p \rangle 
 =\langle p' \mid T_{eff}(\omega=\varepsilon _p) \mid p \rangle, 
\end{equation}
with
\begin{equation}
 \langle p' \mid T_{eff}(\omega) \mid p \rangle 
 =\langle p' \mid V_{eff} \mid p \rangle 
 +\sum _{p"\leq k_{cut}} \langle p' \mid V_{eff} \mid p" \rangle 
\frac{1}{\omega - \varepsilon _p"}
 \langle p" \mid T_{eff}(\omega) \mid p \rangle 
 ;~\omega=\varepsilon _p. 
\end{equation}
Since phase shifts are given by the fully on-shell (p'=p and 
$\omega = \varepsilon _p$) T-matrix, clearly the phase shifts given
by $V$ and $V_{eff}$ are the same up to energy 
$E_{lab}=2k_{cut}^2\hbar^2/M^2$, M being the nucleon mass. 
It has long been known that the effective interaction $V_{eff}$ of Eq.(3)
preserves certain discrete (bound-state) eigenvalues  given by $V$.
We have just shown in the above that this $V_{eff}$ also preserves
the low-momentum half-on-shell T-matrix and consequently the corresponding
low-energy phase shifts
given by $V$. 
 The above preservation of the half-on-shell T-matrix by $V_{eff}$
is a new (to the best of our knowledge) and interesting result, and  it 
is necessary to numerically
check this preservation. To do this we need to calculate the effective
interaction as given by the folded-diagram expansion of Eq.(3). 
A number of methods for this calculation have been developed;
the Krenciglowa-Kuo \cite{krmku74}  
and the Lee-Suzuki iteration methods
\cite{suzuki80} are two well-known examples.
These methods were formulated
for the case that $PH_oP$ is degenerate. (By way of an energy shift,
they could also treat nodegenerate $PH_oP$ cases, but were inefficient.)
For our present two-nucleon problem,
$PH_oP$ (the kinetic energy)is obviously non-degenerate. Non-degenerate Lee-Suzuki
\cite{suzuki94} and Krenciglowa-Kuo \cite{kuo95} methods were
subsequently developed. Both methods can be employed for our 
present calculation, but they are still rather involved.
Recently Andreozzi \cite{andre96} has proposed  new iteration
methods for the derivation of the model-space effective interaction,
and among them is a much improved Lee-Suzuki
method which we refer to as the Andreozzi-Lee-Suzuki (ALS) method.
This method is particularly suitable for  non-degenerate
$PH_oP$.  In the present work, we have  
found this method to be very efficient for our calculation. 
Here the effective interaction
is calculated in terms of the wave operator $\Omega$.
We divide the entire momentum space into two parts, P and
Q with P+Q=1. The full Hamiltonian is split into four parts: PHP, PHQ,
QHP and QHQ. The wave operator is obtained from the decoupling
relation $0=Q(H-\Omega PH +HQ\Omega +\Omega PHQ\Omega)P$, using
the ALS method. Then the effective interaction is given as
$PV_{eff}P=PVP+PHQ\Omega$. Note that the above decoupling equation is
non-linear, and different methods for its solution can give different
answers. The ALS method   
reproduces the lowest d eigenvalues of the full Hamiltonian, d being
the dimension of the model space \cite{andre96}.
We use a set of Gauss mesh points to discretize the momentum space, and in
this way $V(k,k')$, $T(k,k')$ and so forth all become finite-dimensional
matrices. The deuteron binding energy $BE_{d}$ is obtained from a 
diagonalization of the $H(k,k')$ matrix, $H=K+V$ where $K$ is  the kinetic
energy. Phase shifts are calculated from the half-on-shell
principal-value T-matrix given by  
\begin{equation}
 \langle p' \mid T(\omega) \mid p \rangle 
=\langle p' \mid V \mid p \rangle 
 +\it P^r \int _0 ^{\infty} k^2 dk \langle p' \mid V \mid k \rangle
 \frac{1}{\omega -H_0(k)}\langle k \mid T(\omega) \mid p \rangle
;~\omega=\varepsilon _p,
\end{equation}
where $\it P^r$ denotes the principal-value integration. The phase shifts
so-calculated from the full-space potential are plotted
as the solid lines in Fig. 2. Note the $^1S_0$ phase shifts change
sign at  $E_{lab}\approx~250~MeV$.
The deuteron binding energy
$BE_d^{eff}$ is obtained by diagonalizing $H_{eff}=K+V_{eff}$ within the
model space defined by $k_{cut}$.  We have  employed various values for
$k_{cut}$, ranging from 1 to 3 $fm^{-1}$. For all cases, $BE_d^{eff}$
agrees to high precision (3 places after the decimal point) with $BE_d$. 
Using $V_{eff}$ we then calculate the half-on-shell T-matrix
\begin{equation}
 \langle p' \mid T_{eff} (\omega) \mid p \rangle 
=\langle p' \mid V_{eff} \mid p \rangle 
 +\it P^r \int _0 ^{k_{cut}} k^2 dk \langle p' \mid V_{eff} \mid k \rangle
 \frac{1}{\omega -H_0(k)}\langle k \mid T_{eff}(\omega) \mid p \rangle
;~\omega=\varepsilon _p,
\end{equation}
within the model space ($p',k,p \le k_{kcut}$). Note the upper integration 
limit here is $k_{cut}$, while in Eq.(7) it is $\infty$.
From the above $T_{eff}$-matrix, we obtain the phase shifts, 
which are also shown in Fig. 2. They agree very well with those calculated 
from $V$. 
Note that $V_{eff}$ can only give low-energy phase shifts up to $E_{lab}=
2k_{cut}^2 \hbar ^2 / M$, M being the nucleon mass. 
For checking the preservation of the model-space half-on-shell
T-matrix, we have compared the principal-value T-matrices 
 $\langle p' \mid T_{eff} (\omega=\varepsilon _p) \mid p \rangle$ 
calculated from $V_{eff}$ with
  $\langle p' \mid T (\omega=\varepsilon _p) \mid p \rangle$
calculated from $V$.
They agree quite well, as illustrated in Fig. 3.
 In Fig. 4 we compare $V_{eff}$ with $V$. As seen, they are quite different.
The k-space matrix elements of $V$ are generally repulsive (particularly
for the Paris potential), a clear reflection of the strong repulsive core
contained in $V$. In contrast, those of $V_{eff}$ are largely attractive
(for $^1S_0$ and $^3S_1$ channels) and they diminish gradually as k increases.
It appears that the low-momentum $V_{eff}$ is a smooth potential, and it may
be possible to use it directly in nuclear calculations, without
first calculating the Brueckner G-matrix. An important point is that  $V_{eff}$
is energy independent, while the Brueckner G-matrix is energy
dependent and this energy dependence  has introduced a lot of difficulties
in nuclear calculations. We have performed an exploratory
 shell model calculation for 
$^{18}O$ following the same procedure as in Ref.\cite{jiang92}
except that the G-matrix elements used there are replaced by our present 
$V_{eff}$ calculated with the Paris potential and a momentum cut-off
of $k_{cut}=2.0fm^{-1}$, which is a reasonable choice \cite{kaplan98}.
Our calculated results for the lowest $(0^+,2^+,4^+,0^+,2^+,3^+)$
states are respectively  (-4.00,-1.93,-0.47,-0.06,0.07,0.96) MeV, in
fairly good agreement with the corresponding experimental results of 
(-3.90,-1.92,-0.35,-0.27,0.02,1.47) MeV. These results are promising,
although we must emphasize that further work in this direction remains
to be done, to check out the various aspects of using the low-momentum
$V_{eff}$ directly in nuclear many-body calculations.
It may be pointed out that our present $V_{eff}$ is not Hermitian, although
it could be made Hermitian by a further transformation. In terms of the
wave operator $\Omega$, our present $V_{eff}$ is given by  $(1-\Omega)
H(1+\Omega)-H_0$. Suzuki et al. \cite{suzuki83,kuo93} have discussed how to
obtain a Hermitian effective interaction $V_{eff}^h$, namely
$V_{eff}^h=UHU^{\dag}-H_0$ with $U=(1+\Omega ^{\dag}\Omega 
+\Omega \Omega^ {\dag})^{-1/2}(1+\Omega ^{\dag}-\Omega)$, U being unitary.
 In so doing, we are however paying a price,
namely this $V_{eff}^h$ no  longer preserves
the half-on-shell T-matrix. Our proof of the equivalence between the
half-on-shell model-space $T_{eff}$ given by $V_{eff}$ 
and $T$ given by $V$ is based 
on  folded-diagram factorization illutrated in Fig. 1. Clearly we can no longer 
obtain $V^h_{eff}$ from this factorization, and $T^h_{eff}$ generated by
$V^h_{eff}$ is in fact related to $T$ in a complicated way. 
Epelbaum et al. \cite{epel98} have used the above unitary operator approach
 to calculate $V^h _{eff}$ for the $^1S_0$ channel, using  
 the Mafleit-Tjon potential.  They did not employ the
Lee-Suzuki method to obtain the wave operator $\Omega$, as we did. 
Instead they calculated $\Omega$ by way of a perturbation expansion.
They have found that $V^h_{eff}$ preserves the S-matrix, and consequently
the low-energy phase shifts. In contrast, we have obtained 
$\Omega$ using the non-perturbative ALS iteration method, which converges to 
the lowest d eigenstates, d being the dimension of the model space.  
For all partial wave
channels (up to J=5)  our $V_{eff}$ preserves not only the
low-energy phase shifts but also the model-space half-on-shell
T-matrix. 
Following the method 
described in Ref.\cite{kuo93}, we have
calculated $V_{eff}^h$ starting from our $V_{eff}$ obtained with the ALS method.
For both Bonn-A and Paris potentials, $V_{eff}^h$ and $V_{eff}$ 
are found to be numerically quite close to each other.
There are important advantages in preserving the half-on-shell T-matrices.
For example the scattering wave function obtained with $V$ is
\begin{equation}
 \Psi _p ^{\pm}  = \phi _p + \frac{1}{\varepsilon _p -H_0 \pm i0^+}
T( \varepsilon _p) \phi _p,
\end{equation}
with the T-matrix given by Eq.(5).  The scattering wave function
given by $V_{eff}$ is 
\begin{equation}
 \chi _p ^{\pm}  = \phi _p + \frac{1}{\varepsilon _p -H_0 \pm i0^+}
T_{eff}( \varepsilon _p) \phi _p,
\end{equation}
with $T_{eff}$ given by Eq.(7). Then from Eq.(6) we have 
$\chi _p ^{\pm} =P\psi _p ^{\pm}$ for $\phi _p=P\phi _p$,
 namely $V_{eff}$ preserves the P-space
projection of the scattering wave function given by $V$. 
  Let us go back to Fig. 4. As shown the $V$ matrix elements
of the Paris and Bonn-A potentials are very different, the former being
much more repulsive indicating a stronger repulsive core. However,
the low-momentum $V_{eff}$'s given by them are almost identical! 
This suggests the
following scenerio: The ingredients of realistic NN potentials may be
roughly classified into two parts, one responsible for the low-energy 
and the other for high energy physics. It seems that 
our low-momentum reduction has 
extracted out the physically relevant low-energy part of the NN potentials. It is encouraging
that this low-energy part of the two NN potentials appears to be nearly the 
same, indicating that one may have a common low-momentum NN potential,
which may be obtained from an underlying EFT.
To summarize, using the
KLR folded diagram approach in conjunction with the ALS iteration method
we have reduced realistic NN potentials $V$ (Bonn-A
and Paris) to corresponding effective low-momentum potentials 
$V_{eff}$, whose k-space matrix elements are non-vanishing only
within a cutoff momentum $k_{cut}$. The deuteron binding energy given by
$V$ is exactly reproduced by $V_{eff}$. By way of a folded-diagram
factorization of the half-on-shell T-matrix, we have shown that
our $V_{eff}$ preserves the model-space    half-on-shell
T-matrix given by $V$ and consequently the low-energy phase shifts. 
Numerical calculations using the Bonn-A and Paris potentials (all
partial waves up to J=5) have verified this preservation.
For a cutoff momentum $k_{cut}\approx 2 fm^{-1}$, the reduced potential
is generally smooth and may be used directly in nuclear many-body
calculations, as indicated by our  preliminary shell-model calculation.
Although the k-space matrix elements of the Bonn-A and Paris potentials
are quite different, the low-momentum effective potentials derived
from them are remarkably similar to each other.
{\bf Acknowledgement}: We thank Prof. G.E. Brown for encouragement and
many stimulating discussions.
 This work was supported in part by the U.S. DOE Grant No. DE-FG02-88ER40388, 
and the Italian Ministero dell'Universit\`ae della Ricerca Scientifica e 
Tecnologica (murst).
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/VM? {vmstatus exch sub exch pop gt { [
(This job requires more memory than is available in this printer.) 100 500
(Try one or more of the following, and then print again:) 100 485
(In the PostScript dialog box, click Optimize For Portability.) 115 470
(In the Device Options dialog box, make sure the Available Printer Memory is accurate.) 115 455
(Reduce the number of fonts in the document.) 115 440
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12 /Times-Roman showpage
(%%[ PrinterError: Low Printer VM ]%%) =
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/|/def load def/,/load load |/~/exch , |/?/ifelse , |/!/pop , |/`/begin , |/^
/index , |/@/dup , |/+/translate , |/$/roll , |/U/userdict , |/M/moveto , |/-
/rlineto , |/&/currentdict , |/:/gsave , |/;/grestore , |/F/false , |/T/true ,
|/N/newpath , |/E/end , |/Ac/arc , |/An/arcn , |/A/ashow , |/D/awidthshow , |
/C/closepath , |/V/div , |/O/eofill , |/L/fill , |/I/lineto , |/-C/rcurveto ,
|/-M/rmoveto , |/+S/scale , |/Ji/setfont , |/Lc/setlinecap , |/Lj/setlinejoin
, |/Lw/setlinewidth , |/S/show , |/LH/showpage , |/K/stroke , |/W/widthshow ,
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/lc/Lc ld/lj/Lj ld/sg/setgray ld/L2? F/languagelevel where{! languagelevel 2
ge{! T}if}if |/g{@ not{U/DefIf_save save put}if U/DefIf_bool 2 ^ put}b
/DefIf_El{if U/DefIf_bool get not @{U/DefIf_save get restore}if}b/e{DefIf_El !
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self eq}{F}?{cvx exec}{!}?}repeat cleartomark}b/initialize{`{/ADO_mxRot ~ |
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/ADO_mxRot ~ |/TextInitialised? F | reinitialize}?}b/terminate{!{& self eq{
exit}{E}?}loop E}b/suspend/terminate , |/resume{` Pscript_Win_Data `}b/snap{
transform 0.25 sub round 0.25 add ~ 0.25 sub round 0.25 add ~ itransform}b
/dsnap{dtransform round ~ round ~ idtransform}b<04>cvn{}|/setjn{{statusdict
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0 setdash}b/setdsh{0 setdash}b/colspRefresh{}b/rp{4 2 $ M 1 ^ 0 - 0 ~ - neg 0
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L2? g{/colspA/DeviceGray |/colspABC/DeviceRGB |/setAorABC{{colspA}{colspABC}?
setcolorspace}b/rf/rectfill , |/fx{1 1 dtransform @ 0 ge{1 sub 0.5}{1 add -0.5
}? 3 -1 $ @ 0 ge{1 sub 0.5}{1 add -0.5}? 3 1 $ 4 1 $ idtransform 4 -2 $
idtransform}b/BZ{4 -2 $ snap + +S fx rf}b/rs/rectstroke , |/rc/rectclip , |/sg
{@ @ setcolor}b/sco{setcolor}b/colspRefresh{colspABC setcolorspace}b/sgco{{sg
}{sco}?}b/UtilsInit{F setglobal}b/definecolorrendering{/ColorRendering
defineresource !}b/findcolorrendering{@/ColorRendering resourcestatus{! !
/ColorRendering findresource T}{! F}?}b/selectcolorrendering{@/ColorRendering
resourcestatus{! !/ColorRendering}{!/DefaultColorRendering/ColorRendering}?
findresource setcolorrendering}b}e
%%EndResource
end
%%EndProlog

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[{0 
/languagelevel where{pop languagelevel}{1}ifelse 2 ge{1 dict dup/JobTimeout  4 -1 roll put setuserparams}{statusdict/setjobtimeout get exec}ifelse 
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[{240 
/languagelevel where{pop languagelevel}{1}ifelse 2 ge{1 dict dup/WaitTimeout 4 -1 roll put setuserparams}{statusdict/waittimeout 3 -1 roll put}ifelse 
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[{
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[{

             <</ManualFeed false>> setpagedevice
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[{
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[{
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<< /PostRenderingEnhance true
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1 setlinecap 1 setlinejoin 
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[{
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[{
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Pscript_Win_Driver_Incr begin
%%BeginResource: file Pscript_Win_Encoding 2.0 0
/bullets{{/bullet}repeat}b/ANSIEncoding[/grave/acute/circumflex/tilde/macron
/breve/dotaccent/dieresis/ring/cedilla/hungarumlaut/ogonek/caron/dotlessi 18
bullets StandardEncoding 32 95 getinterval aload ! 3 bullets/quotesinglbase
/florin/quotedblbase/ellipsis/dagger/daggerdbl/circumflex/perthousand/Scaron
/guilsinglleft/OE 4 bullets/quoteleft/quoteright/quotedblleft/quotedblright
/bullet/endash/emdash/tilde/trademark/scaron/guilsinglright/oe 2 bullets
/Ydieresis/space/exclamdown/cent/sterling/currency/yen/brokenbar/section
/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered
/macron/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph
/periodcentered/cedilla/onesuperior/ordmasculine/guillemotright/onequarter
/onehalf/threequarters/questiondown/Agrave/Aacute/Acircumflex/Atilde/Adieresis
/Aring/AE/Ccedilla/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute
/Icircumflex/Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis
/multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls
/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla/egrave/eacute
/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis/eth/ntilde/ograve
/oacute/ocircumflex/otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex
/udieresis/yacute/thorn/ydieresis]| ANSIEncoding @ 39/quotesingle put 96/grave
put/ANSIEncodingOld ANSIEncoding 256 array copy | ANSIEncodingOld @[138 153
154 169 172 174 177 178 179 181 185 188 189 190 208 215 221 222 240 247 253
254]{/bullet put @}forall 166/bar put 176/ring put
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%%BeginResource: file Pscript_Win_Text 2.0 0
/TextInit{TextInitialised? not{/Pscript_Windows_Font & |/TextInitialised? T |
/fM[1 0 0 -1 0 0]|/mFM matrix |/iMat[1 0 0.212557 neg 1 0 0]|}if}b/xUP null |
/yUP null |/uW null |/xSP null |/ySP null |/sW null |/copyfont{1 ^ length add
dict `{1 ^/FID ne{|}{! !}?}forall & E}b/rF{3 copyfont @ `/Encoding
ANSIEncoding &/CharStrings known{CharStrings/Eth known not{! ANSIEncodingOld}
if}if | E}b/mF{findfont ~{@/Encoding get @ StandardEncoding eq{! T}{{
ISOLatin1Encoding}stopped{! F}{eq}?{T}{@ ` T 32 1 127{Encoding 1 ^ get
StandardEncoding 3 -1 $ get eq and}for E}?}?}{F}?{rF}{3 copyfont}? `
/OrigFontType ~ |/OrigFontName ~ | & E 2 ^ ~ definefont fM 5 4 -1 $ put fM 4 0
put fM makefont Pscript_Windows_Font 3 1 $ put}b/xF{scalefont
Pscript_Windows_Font 3 1 $ put}b/xMF{mFM astore makefont Pscript_Windows_Font
3 1 $ put}b/xF2/scalefont , |/xMF2{mFM astore makefont}b/sLT{: Lw -M
currentpoint snap M 0 - 0 Lc K ;}b/sSU{N/uW ~ |/yUP ~ |/xUP ~ |}b/sU{xUP yUP
uW sLT}b/sST{N/sW ~ |/ySP ~ |/xSP ~ |}b/sT{xSP ySP sW sLT}b/sR{: + R 0 0 M}b
/sRxy{: matrix astore concat 0 0 M}b/eR/; , |
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end reinitialize
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(F0) cvn
0.892 
 (Times-Roman) cvn /Type1 
T 
(Times-Roman) cvn
mF 
(F0_83) cvn
F0
83 
xF 
F0_83
Ji 
492 969 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 ( )W 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
951 969 M 
1.176 0 32 -0.926 0 (A           )D 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.926 0 (A)A 
0.500 0 (1)A 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
492 2378 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
930 2378 M 
-0.389 0 32 0.639 0 (B  )D 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.639 0 (B)A 
-1.250 0 32 0.500 0 (1 )D 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (   )W 
-0.750 0 32 ( )W 
0.639 0 (B)A 
0.500 0 (2)A 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
972 3623 M 
0.250 0 32 (   )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
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0.250 0 32 (     )W 
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0.250 0 32 (     )W 
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0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (  )W 
16.250 0 32 ( )W 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
492 3815 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
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0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (   )W 
-0.750 0 32 ( )W 
0.250 0 32 (   )W 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
2889 3815 M 
-0.361 0 (C)A 
-0.250 0 32 0.500 0 (2   )D 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
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0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (  )W 
-0.361 0 (C)A 
0.500 0 (3)A 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
492 4294 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.361 0 (C)A 
0.500 0 (1)A 
; : 0 0 6392 4883 rc 0 0 0 sco F0_83
Ji 
492 4390 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (      )W 
-0.361 0 (C)A 
; : 0 0 6392 4883 rc pagesave restore
Pscript_Win_Driver_Incr begin
%%BeginResource: file Pscript_Win_Graphics 2.0 0
/SavedCTM null |/CTMsave{/SavedCTM SavedCTM currentmatrix |}b/CTMrestore{
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/GDIHPatternDict 22 dict | GDIHPatternDict `/PatternType 1 |/PaintType 2 |
/Reps L2?{1}{5}? |/XStep 8 Reps mul |/YStep XStep |/BBox[0 0 XStep YStep]|
/TilingType 1 |/PaintProc{` 1 Lw[]0 setdash PaintData , exec E}b/FGnd null |
/BGnd null |/HS_Horizontal{horiz}b/HS_Vertical{vert}b/HS_FDiagonal{fdiag}b
/HS_BDiagonal{biag}b/HS_Cross{horiz vert}b/HS_DiagCross{fdiag biag}b/MaxXYStep
XStep YStep gt{XStep}{YStep}? |/horiz{Reps{0 4 M XStep 0 - 0 8 +}repeat 0 -8
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{Reps{0 0 M MaxXYStep @ - 0 YStep neg M MaxXYStep @ - 0 8 +}repeat 0 -8 Reps
mul + 0 YStep M 8 8 - K}b/fdiag{Reps{0 0 M MaxXYStep @ neg - 0 YStep M
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/makehatch{GDIHPatternDict/PaintData 3 -1 $ put CTMsave GDIHMatrix setmatrix
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makehatch}b/h1{/h1/HS_Vertical makehatch}b/h2{/h2/HS_FDiagonal makehatch}b/h3{
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makehatch}b/GDIBWPatternDict 17 dict @ `/PatternType 1 |/PaintType L2?{1}{2}?
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/XStep 1 |/YStep 1 |/Height 8 RepsV mul |/Width 8 |/mx[Width 0 0 Height neg 0
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}{{}}? b/PaintProc{` SetBGndFGnd RepsH{Width Height F mx PaintData imagemask
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/PatternOfTheDay 4 1 $ GDIBWPatternDict `/PaintData ~ |/BGnd ~ |/FGnd ~ | E
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/hrf null |/prf{pfprep ~ 6 1 $ 5 hrf restore}b/GraphInit{GDIHMatrix null eq{
/SavedCTM matrix | : ADO_mxRot concat 0 0 snap + : 0.48 @ GDIHPatternDict `
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~ |/x2 ~ |/y1 ~ |/x1 ~ | x2 x1 add 2 V y1 M x1 y1 x1 y2 rad arct x1 y2 x2 y2
rad arct x2 y2 x2 y1 rad arct x2 y1 x1 y1 rad arct C}b/RRp{CTMsave + +S/dyS ~
|/dxS ~ | dxS 2 V 0 M 0 0 0 dyS 0.5 arct 0 dyS dxS dyS 0.5 arct dxS dyS dxS 0
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%%BeginResource: file Pscript_Win_Graphics_L1 2.0 0
L2? not g{/arct{arcto ! ! ! !}b/GDIpattfill{@ ` BGnd null ne PaintType 2 eq
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-1 $}if E @ patterncalc : 4 ^/PaintType get 2 eq{fGray{6 -1 $ sg}{8 -3 $ sco}?
}if fEOFill{eoclip}{clip}? N patternfill ; N}b/hrf{/fGray 1 ^ 6 eq | -4 $ N rp
C/fEOFill F | GDIpattfill}b/hfMain{/fEOFill ~ |/fGray ~ | GDIpattfill}b/hf{T
hfMain}b/hfW{F hfMain}b/hs{currentpoint strokepath M hfW}b/pfMain{/fEOFill ~ |
pfprep GDIpattfill restore N}b/pf{T pfMain}b/pfW{F pfMain}b/ps{currentpoint
strokepath M pfW}b/mpstr 1 string |/mp{~ @ length 12 add dict copy `
/PatternCTM matrix currentmatrix |/PatternMatrix ~ |/PatWidth XStep mpstr
length mul |/PatHeight YStep |/FontType 3 |/Encoding 256 array | 3 string 0 1
255{Encoding ~ @ 3 ^ cvs cvn put}for !/FontMatrix matrix |/FontBBox BBox |
/BuildChar{! @ ` XStep 0 FontBBox aload ! setcachedevice/PaintProc , E : exec
;}b & E ~ @ 3 -1 $ definefont}b/patterncalc{` : PatternCTM setmatrix
PatternMatrix concat BBox aload ! ! ! + pathbbox ; PatHeight V ceiling 4 1 $
PatWidth V ceiling 4 1 $ PatHeight V floor 4 1 $ PatWidth V floor 4 1 $ 2 ^
sub cvi abs ~ 3 ^ sub cvi abs ~ 4 2 $ PatHeight mul ~ PatWidth mul ~ E}b
/patternfill{5 -1 $ @ ` Ji PatternCTM setmatrix PatternMatrix concat 0 2 ^ 2 ^
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%%BeginResource: file Pscript_Win_Graphics_L2 2.0 0
L2? g{/mp/makepattern , |/hrf{6 eq setAorABC setpattern rectfill}b/hf{
setAorABC setpattern O}b/hfW{setAorABC setpattern L}b/hs{setAorABC setpattern
K}b/pf{pfprep setpattern O restore N}b/pfW{pfprep setpattern L restore N}b/ps{
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%%EndResource

end reinitialize
userdict begin /pagesave save def end mysetup concat colspRefresh 
: 0 0 0 sco 1 Lc 1 Lj solid 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 717 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc 1 Lj solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
788 597 N M 8 -32 rr : 0 0 0 sco O ; 
788 541 N M 8 -32 rr : 0 0 0 sco O ; 
788 485 N M 8 -32 rr : 0 0 0 sco O ; 
788 429 N M 8 -32 rr : 0 0 0 sco O ; 
788 373 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
788 1083 N M 8 -8 rr : 0 0 0 sco O ; 
788 1067 N M 8 -8 rr : 0 0 0 sco O ; 
788 1051 N M 8 -8 rr : 0 0 0 sco O ; 
788 1035 N M 8 -8 rr : 0 0 0 sco O ; 
788 1019 N M 8 -8 rr : 0 0 0 sco O ; 
788 1003 N M 8 -8 rr : 0 0 0 sco O ; 
788 987 N M 8 -8 rr : 0 0 0 sco O ; 
788 971 N M 8 -8 rr : 0 0 0 sco O ; 
788 955 N M 8 -8 rr : 0 0 0 sco O ; 
788 939 N M 8 -8 rr : 0 0 0 sco O ; 
788 923 N M 8 -8 rr : 0 0 0 sco O ; 
788 907 N M 8 -8 rr : 0 0 0 sco O ; 
788 891 N M 8 -8 rr : 0 0 0 sco O ; 
788 875 N M 8 -8 rr : 0 0 0 sco O ; 
788 859 N M 8 -8 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
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(F0) cvn
1.008 
 (Symbol) cvn /Type1 
F 
(Symbol) cvn
mF 
(F0_126) cvn
F0
126 
xF 
F0_126
Ji 
1047 544 M 
-0.174 0 (=)A 
; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 1392 717 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1388 597 N M 8 -32 rr : 0 0 0 sco O ; 
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-108 -108 N M 1 1 rr ; 
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1388 859 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 1623 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
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788 1447 N M 8 -32 rr : 0 0 0 sco O ; 
788 1391 N M 8 -32 rr : 0 0 0 sco O ; 
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792 1983 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 2103 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
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788 2239 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 1032 1872 83 84 rc 0 0 0 sco F0_126
Ji 
1047 1819 M 
-0.174 0 (=)A 
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N 1392 1932 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 0 0 0 sco K : 0 0 6392 4883 rc 
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1392 2292 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 1572 1872 75 105 rc 0 0 0 sco F0_126
Ji 
1581 1829 M 
-0.174 0 (\264)A 
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-108 -108 N M 1 1 rr ; 
N 1872 1932 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1872 1812 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1868 2292 N M 8 -32 rr : 0 0 0 sco O ; 
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-108 -108 N M 1 1 rr ; : 2172 1872 83 91 rc 0 0 0 sco (F0_182) cvn
F0
182 
xF 
F0_182
Ji 
2187 1780 M 
0.082 0 (-)A 
; : 0 0 6392 4883 rc 
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-108 -108 N M 1 1 rr ; 
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-108 -108 N M 1 1 rr ; 
2622 2295 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 2862 1759 133 331 rc 0 0 0 sco (F0_274) cvn
F0
274 
xF 
F0_274
Ji 
2879 1759 M 
-0.076 0 (\362)A 
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N 3162 1936 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
3162 1815 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
3158 2295 N M 8 -32 rr : 0 0 0 sco O ; 
3158 2239 N M 8 -32 rr : 0 0 0 sco O ; 
3158 2183 N M 8 -32 rr : 0 0 0 sco O ; 
3158 2127 N M 8 -32 rr : 0 0 0 sco O ; 
3158 2071 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 2997 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
788 2877 N M 8 -32 rr : 0 0 0 sco O ; 
788 2821 N M 8 -32 rr : 0 0 0 sco O ; 
788 2765 N M 8 -32 rr : 0 0 0 sco O ; 
788 2709 N M 8 -32 rr : 0 0 0 sco O ; 
788 2653 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
792 3357 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 3972 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
792 3357 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 792 3477 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
788 4332 N M 8 -32 rr : 0 0 0 sco O ; 
788 4276 N M 8 -32 rr : 0 0 0 sco O ; 
788 4220 N M 8 -32 rr : 0 0 0 sco O ; 
788 4164 N M 8 -32 rr : 0 0 0 sco O ; 
788 4108 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 1032 3312 83 84 rc 0 0 0 sco F0_126
Ji 
1047 3259 M 
-0.174 0 (=)A 
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-108 -108 N M 1 1 rr ; 
N 1392 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1388 3192 N M 8 -32 rr : 0 0 0 sco O ; 
1388 3136 N M 8 -32 rr : 0 0 0 sco O ; 
1388 3080 N M 8 -32 rr : 0 0 0 sco O ; 
1388 3024 N M 8 -32 rr : 0 0 0 sco O ; 
1388 2968 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1392 3672 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 1572 3252 75 105 rc 0 0 0 sco F0_126
Ji 
1581 3209 M 
-0.174 0 (\264)A 
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-108 -108 N M 1 1 rr ; 
N 1872 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1872 3192 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
1872 3672 N M 0 -240 - 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 1872 3807 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
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-108 -108 N M 1 1 rr ; : 2172 3252 83 91 rc 0 0 0 sco F0_182
Ji 
2187 3160 M 
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-108 -108 N M 1 1 rr ; 
N 2652 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
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2648 3136 N M 8 -32 rr : 0 0 0 sco O ; 
2648 3080 N M 8 -32 rr : 0 0 0 sco O ; 
2648 3024 N M 8 -32 rr : 0 0 0 sco O ; 
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-108 -108 N M 1 1 rr ; 
2652 3672 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 2832 3196 133 331 rc 0 0 0 sco F0_274
Ji 
2849 3196 M 
-0.076 0 (\362)A 
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-108 -108 N M 1 1 rr ; 
N 3072 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
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3488 3616 N M 8 -32 rr : 0 0 0 sco O ; 
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3488 3448 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
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-108 -108 N M 1 1 rr ; 
3072 3672 N M 0 -240 - 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 3252 3252 75 105 rc 0 0 0 sco F0_126
Ji 
3261 3209 M 
-0.174 0 (\264)A 
; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 3492 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
3492 3192 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
792 3837 N M 0 -240 - 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 3792 3252 83 91 rc 0 0 0 sco F0_182
Ji 
3807 3160 M 
0.082 0 (-)A 
; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 4152 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
4148 3192 N M 8 -32 rr : 0 0 0 sco O ; 
4148 3136 N M 8 -32 rr : 0 0 0 sco O ; 
4148 3080 N M 8 -32 rr : 0 0 0 sco O ; 
4148 3024 N M 8 -32 rr : 0 0 0 sco O ; 
4148 2968 N M 8 -16 rr : 0 0 0 sco O ; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
4152 3672 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 4632 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 4332 3196 133 331 rc 0 0 0 sco F0_274
Ji 
4349 3196 M 
-0.076 0 (\362)A 
; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; : 4812 3196 133 331 rc 0 0 0 sco F0_274
Ji 
4829 3196 M 
-0.076 0 (\362)A 
; : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
4632 3192 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
4632 3672 N M 0 -240 - 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
N 5052 3312 120.000 360 0 An C : 1.000 1.000 1.000 sco O ; 6 Lw 1 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
5052 3192 N M 0 -240 - 8 Lw 0 Lc solid 0 0 0 sco K : 0 0 6392 4883 rc 
-108 -108 N M 1 1 rr ; 
5048 3672 N M 8 -32 rr : 0 0 0 sco O ; 
5048 3616 N M 8 -32 rr : 0 0 0 sco O ; 
5048 3560 N M 8 -32 rr : 0 0 0 sco O ; 
5048 3504 N M 8 -32 rr : 0 0 0 sco O ; 
5048 3448 N M 8 -16 rr : 0 0 0 sco O ; 
LH 
pagesave restore



} stopped cleartomark
[{
%%BeginFeature: *Duplex 

%%EndFeature
} stopped cleartomark
userdict begin /pagesave save def end mysetup concat colspRefresh : 1.000 1.000 1.000 sco 0 0 6600 5100 rf ; 
%%EndPageSetup

: 0 0 6392 4883 rc 0 0 0 sco %%IncludeFont: Times-Roman
(F0) cvn
0.892 
 (Times-Roman) cvn /Type1 
T 
(Times-Roman) cvn
mF 
(F0_83) cvn
F0
83 
xF 
F0_83
Ji 
492 747 M 
0.250 0 32 (        )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (     )W 
-0.750 0 32 ( )W 
0.250 0 32 (  )W 
-0.361 0 (C)A 
; 
LH 
pagesave restore


%%DocumentNeededFonts: Times-Roman
%%+ Symbol
%%DocumentSuppliedFonts: 
Pscript_Win_Driver_Incr dup /terminate get exec
savelevel0 restore 
%%Pages: 2
(%%[ LastPage ]%%)  =
%%EOF
%-12345X@PJL EOJ
%-12345X
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%%BoundingBox: 40 43 520 461
%%Title: bonn.eps
%%Creator: visitor1@tibidabo.ecm.ub.es with xmgr v4.00 (Beta)
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80 dict begin
/languagelevel where
{pop /gs_languagelevel languagelevel def}
{/gs_languagelevel 1 def} ifelse
gs_languagelevel 1 gt {
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<0000000000000000000000000000000000000000000000000000000000000000>} image end }
			>> 	matrix makepattern /Pat0 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<1111000044440000111100004444000011110000444400001111000044440000>} image end }
			>> 	matrix makepattern /Pat1 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<1111444411114444111144441111444411114444111144441111444411114444>} image end }
			>> 	matrix makepattern /Pat2 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<aaaa5555aaaa5555aaaa5555aaaa5555aaaa5555aaaa5555aaaa5555aaaa5555>} image end }
			>> 	matrix makepattern /Pat3 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<eeeebbbbeeeebbbbeeeebbbbeeeebbbbeeeebbbbeeeebbbbeeeebbbbeeeebbbb>} image end }
			>> 	matrix makepattern /Pat4 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<eeeeffffbbbbffffeeeeffffbbbbffffeeeeffffbbbbffffeeeeffffbbbbffff>} image end }
			>> 	matrix makepattern /Pat5 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<f77dffffbeeffffff77dffffbeeffffff77dffffbeeffffff77dffffbeefffff>} image end }
			>> 	matrix makepattern /Pat6 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff>} image end }
			>> 	matrix makepattern /Pat7 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<e1e1f0f078783c3c1e1e0f0f8787c3c3e1e1f0f078783c3c1e1e0f0f8787c3c3>} image end }
			>> 	matrix makepattern /Pat8 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<87870f0f1e1e3c3c7878f0f0e1e1c3c387870f0f1e1e3c3c7878f0f0e1e1c3c3>} image end }
			>> 	matrix makepattern /Pat9 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc>} image end }
			>> 	matrix makepattern /Pat10 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<00000000ffffffff00000000ffffffff00000000ffffffff00000000ffffffff>} image end }
			>> 	matrix makepattern /Pat11 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<7e7ebdbddbdbe7e7e7e7dbdbbdbd7e7e7e7ebdbddbdbe7e7e7e7dbdbbdbd7e7e>} image end }
			>> 	matrix makepattern /Pat12 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<fd7e7ebfbfdfdfefeff7f7fbfbfdfd7e7ebfbfdfdfefeff7f7fbfbfdfd7e7ebf>} image end }
			>> 	matrix makepattern /Pat13 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<7ebffd7efbfdf7fbeff7dfefbfdf7ebffd7efbfdf7fbeff7dfefbfdf7ebffd7e>} image end }
			>> 	matrix makepattern /Pat14 exch def 
<</PaintType 1 /PatternType 1 /TilingType 1
			/BBox [0 0 12 12] /XStep 12 /YStep 12 /PaintProc{ begin 
			12 12 scale 16 16 1 [16 0 0 16 0 0] 
{<fffffbdff3dfebdfdbdfbbdf7bdffbdefbddfbdbfbd7fbcffbdfffffffffffff>} image end }
			>> 	matrix makepattern /Pat15 exch def 
}{
/Pat0 { 0.000000 setgray } def
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/Pat2 { 0.125000 setgray } def
/Pat3 { 0.187500 setgray } def
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0.24 0.24 scale
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nucl-th0009077.extracted_bib
\bibitem{Wein} S. Weinberg, Phys. Lett. {\bf B251}, 288 (1990).

\bibitem{ksw} D.B. Kaplan, M.J. Savage and M.B. Wise,
Phys. Lett. {\bf B424} 390 (1998).

\bibitem{orv} C. Ordonez, L. Ray, U. van Kolck, Phys. Rev. Lett. {\bf
72}, 1982 (1995).
%
\bibitem{lynn} B. Lynn, Nucl. Phys.  {\bf B402}, 281 (1993).

\bibitem{fsteele} J.V. Steele and R.J. Furnstahl, Nucl. Phys. {\bf A663},
999c (2000).

\bibitem{polch} J. Polchinski, "Effective Field Theory and the Fermi Surface," Proceedings of 1992 Theoretical Advanced Studies Institute in Elementary Particle Physics, edited by J. Harvey and J. Polchinski (World Scientific, Singapore, 1993).

\bibitem{lepage} G.P. Lepage, "How to Renormalize the Schr$\ddot o$dinger Equation", [nucl-th/9706029].

\bibitem{epel98} E. Epelbaum, W. Gl$\ddot o$ckle, and Ulf-G Meiszner,
Phys. Lett. {\bf B439} 1 (1998); Nucl. Phys. {\bf A637} 107 (1998).

\bibitem{Wilson} K. Wilson and J. Kogut, Phys. Rep. {\bf 12}, 75
(1974).

\bibitem{Birse} M. Birse, J. McGovern, and K. Richardson, Phys. Lett.
{\bf B464} 169 (1999).

\bibitem{Bogner} S. Bogner and T.T.S. Kuo, nucl-th/9912056.

\bibitem{ko90} T.T.S. Kuo and E. Osnes, Springer Lecture Notes of Physics,
{\bf Vol. 364}, p.1 (1990).

\bibitem{krmku74} E.M. Krenciglowa and T.T.S. Kuo, Nucl. Phys. {\bf A235} 171
(1974).

\bibitem{suzuki80} K. Suzuki and S. Y. Lee, Prog. Theor. Phys. {\bf 64}, 2091 
(1980).

\bibitem{suzuki94}K. Suzuki, R. Okamoto, P.J. Ellis and T.T.S. Kuo,
Nucl. Phys. {\bf A567} 576 (1994).

\bibitem{andre96} F. Andreozzi, Phys. Rev. {\bf C54}, 684 (1996).

\bibitem{jensen}M. Jorth-Jensen, T.T.S. Kuo and E. Osnes, Physis Report
{\bf 261} 126 (1995).

\bibitem{fields} T.J. Fields, J.P. Vary, and K.S. Gupta, Mod. Phys. Lett. 
{\bf A11}, 2233 (1996). 

\bibitem{haxton} W. Haxton and C.-L. Song, Phys. Rev. Lett. {\bf 84}, 5484 (2000
).  
