Final-state interactions in the response of nuclear matter
M. Petraki$^1$
E. Mavrommatis$^1$
O. Benhar$^2$
J.W. Clark$^3$
%#A. Fabrocini$^4$
S. Fantoni$^5$
Final-state interactions in the response of a many-body system to an %#external probe delivering large momentum are generally described %#using the eikonal aproximation, for the trajectory of the struck particle, external probe delivering large momentum are normally described using the eikonal approximation, for the trajectory of the struck particle, %# and the frozen approximation, for the positions of the spectators. We propose a generalization of this scheme, in which the initial momentum of %#the struck particle is explicitly taken into account. The results %#of numerical the struck particle is explicitly taken into account. Numerical %# calculations of the nuclear matter response at 1 $< |{q}| <$ 2 GeV/c show that the inclusion of this momentum dependence leads to a sizable effect in the low energy tail. Possible implications for the analysis of existing electron-nucleus scattering data are discussed.
benhar2.tex

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\def\ci{\cite}
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\def\lab{\label}
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\def\PkE{P(|{\bf k}|,E)}
\def\magk{|{\bf k}|}
\def\magq{|{\bf q}|}
\def\vecq{{\bf q}}
\def\primo{^\prime}
\def\shat{{\widehat S}({\bf q},t)}  
\def\bm #1{ \bbox{#1} }    
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\begin{document}   

\draft

%#Hyphen everywhere in "final-state" when used as an adjective.
\title{Final-state interactions in the response of nuclear matter}

\author
%#{M. Petraki$^1$,  E. Mavrommatis$^1$, O. Benhar$^2$, J.W. Clark$^3$, 
%#A. Fabrocini$^4$,  S. Fantoni$^5$}
{M. Petraki,$^1$  E. Mavrommatis,$^1$ O. Benhar,$^2$ J.~W. Clark,$^3$ 
A. Fabrocini,$^4$ and S. Fantoni$^5$}
%#

\address
%#{ $^1$ Physics Department, Division of Nuclear and Particle Physics\\
%#University of Athens. GR-15771 Athens, Greece\\
%#$^2$ INFN, Sezione Roma 1. I-00185 Roma, Italy\\
%#$^3$ Department of Physics, Washington University. St. Louis MO 633130, USA \\
%#$^4$ Department of Physics {\em E. Fermi}, University of Pisa, and INFN. I-56100 Pisa, Italy \\
%#$^5$ International School for Advanced Studies (SISSA). I-30014 Trieste, 
%#Italy\\ }
{$^1$Physics Department, Division of Nuclear and Particle Physics\\
University of Athens, GR-15771 Athens, Greece\\
$^2$INFN, Sezione di Roma 1, I-00185 Roma, Italy\\
$^3$Physics Department, Washington University, St. Louis MO 633130, USA \\
$^4$Department of Physics "E. Fermi", University of Pisa, 
 and INFN, Sezione di Pisa, I-56100 Pisa, Italy \\
$^5$International School for Advanced Studies (SISSA), I-30014 Trieste, 
Italy\\ }
%#

\date{\today}

\maketitle


\begin{abstract}

Final-state interactions in the response of a many-body system to an 
%#external probe delivering large momentum are generally described 
%#using the eikonal aproximation, for the trajectory of the struck particle,  
external probe delivering large momentum are normally described 
using the eikonal approximation, for the trajectory of the struck particle,  
%#
and the frozen approximation, for the positions of the spectators. We 
propose a generalization of this scheme, in which the initial momentum of 
%#the struck particle is explicitly taken into account. The results 
%#of numerical
the struck particle is explicitly taken into account. Numerical
%#
calculations of the nuclear matter response at 1 $< |{\bf q}| <$ 2 GeV/c
show that the inclusion of this momentum dependence leads to a sizable effect
in the low energy tail. Possible implications for the analysis of 
existing electron-nucleus scattering data are discussed. 

\end{abstract}

\pacs{PACS numbers: 13.6.Le, 25.30Fj, 25.30Rw}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}   

Final state interactions (FSI) of fast nucleons produced in electron-nucleus 
%#scattering at large momentum transfer have long been known to sizably affect 
%#the coincidence $(e,e\primo p)$ cross section, as well as provide most of the 
%#strength observed in the low energy loss tail of the inclusive 
scattering at large momentum transfer have long been known to 
exert a significant effect on the coincidence $(e,e\primo p)$ cross section; 
moreover, they provide most of the strength observed in the low-energy loss 
%#
%#tail of the inclusive $(e,e\primo)$ cross section (see, e.g., ref.\ci{book}).
tail of the inclusive $(e,e\primo)$ cross section (see, e.g., Ref.\ci{book}).
%#
%#Join these two paragraphs.  PR doesn't like very short paragraphs. - JWC
%#
%#The main effect of FSI is the damping of the motion of the struck particle,
The main effect of FSI is a damping of the motion of the struck particle,
%#
%#that can be qualitatively described in terms of the imaginary part
which can be qualitatively described in terms of the imaginary part
%#
%#of the nuclear optical potential. However, as nucleons in nuclei 
%#are strongly correlated, in order to develop a fully quantitative treatment 
%#of FSI one has to improve upon the optical potential approach, 
%#that is based on a simple mean field description of nuclear dynamics. 
%#It is very important to realize that nucleon-nucleon (NN) correlations, 
of the nuclear optical potential.  However, since nucleons in nuclei 
are strongly correlated, one must improve upon the optical potential 
approach rooted in a simple mean-field description of nuclear dynamics,
if one hopes to develop a fully quantitative treatment of FSI. 
It is very important to realize that nucleon-nucleon ($NN$) correlations, 
%#
leading to large density fluctuations and to the appearance of 
high momentum components in the nuclear wave function, 
%#heavily affect both the initial and the final state in electron nucleus
strongly affect both the initial and the final state in electron-nucleus
%#
scattering and must be consistently taken into account.

A theoretical description of $(e,e\primo)$ processes including 
correlation effects has been developed in
%#ref.\cite{Benhar91} (hereafter referred to as I), and succesfully 
Ref.\cite{Benhar91} (hereafter referred to as I) and successfully 
%#
%#applied to describe $(e,e\primo)$ data with momentum transfer in 
employed to analyze $(e,e\primo)$ data for momentum transfer in 
%#
the range 1 GeV/c $< \magq <$ 2 GeV/c \cite{Benhar91,Benhar93,Benhar94}.
%#A similar approach, based on an extension of the theoretical 
%#treatment of FSI in neutron scattering off quantum liquids, has been 
%#proposed in ref.\ci{silcla}.
A similar approach, formulated in analogy with the theoretical
treatment of FSI in neutron scattering from quantum liquids, has been 
proposed in Ref.\ci{silcla}.
%#

The treatment of FSI discussed in I, commonly referred to as Correlated
%#Glauber Approximation (CGA), is based on the assumptions that 
%#i) the struck nucleon moves along a straight line with constant velocity 
%#(eikonal approximation), and ii) the spectator nucleons are seen by the 
Glauber Approximation (CGA), rests on the assumptions that 
(i) the struck nucleon moves along a straight line with constant velocity 
(eikonal approximation), and (ii) the spectator nucleons are seen by the 
%#
fast struck particle as a collection of fixed scattering centers 
(frozen approximation). 
The resulting inclusive cross section can be written as
a convolution integral, involving the cross section evaluated within the 
%#Plane Wave Impulse Approximation (PWIA), i.e. in absence of FSI, and a 
Plane-Wave Impulse Approximation (PWIA), i.e., evaluated in the absence 
of FSI, and a 
%#
%#folding function embodying FSI effects.  CGA implies 
%#the same set of approximations underlying Glauber theory 
%#of high energy proton scattering off nuclei \ci{Glauber}, which has been 
%#successfully applied for over fourty years. 
folding function embodying FSI effects.  The CGA entails
the same set of approximations as the Glauber theory 
of high-energy proton scattering off nuclei \ci{Glauber}, which has been 
successfully applied for over forty years. 
%#

The results of I show that FSI produce a huge enhancement of the
inclusive cross section in the region of 
$\omega << \omega_{QE}$, where $\omega_{QE}$ is the energy transfer 
corresponding to elastic scattering off an isolated stationary nucleon. 
While this enhancement brings theory and experiment into agreement over a 
%#broad $\omega$ range, in the extreme low $\omega$ tail (roughly corresponding 
%#to values of the Bjorken scaling variable $x=(Q^2/2m\omega)>$ 2, 
%#where $Q^2=\magq^2-\omega^2$ and $m$ denotes the nucleon mass) the 
%#calculated cross section sizably overestimates the data. 
broad range in $\omega$, the calculated cross section substantially 
overestimates the data in the extreme low-$\omega$ tail (roughly corresponding 
to values of the Bjorken scaling variable $x=(Q^2/2m\omega)>$ 2, 
where $Q^2=\magq^2-\omega^2$ and $m$ denotes the nucleon mass). 
%#
%#In order to reproduce the tail of the measured cross sections the imaginary 
%#part of the free space NN scatering amplitude, determining the shape of 
%#the CGA folding function, has to be modified in such a way as to reduce
In order to reproduce the tail of the measured cross sections, the imaginary 
part of the free-space $NN$ scattering amplitude, which determines the shape of 
the CGA folding function, must be modified in such a way as to reduce
%#
the effect of FSI. 

%#As pointed out in I, NN scattering in the nuclear medium may in principle be
%#substantially different from scattering in free space. For example, Pauli 
As pointed out in I, $NN$ scattering in the nuclear medium may in principle 
differ markedly from scattering in free space.  For example, Pauli 
%#
blocking and dispersive corrections are known to be important at moderate 
energies \ci{papi}. However, their effects on the calculated cross sections 
have been 
%#found to be small in the kinematical region covered by the data analyzed in 
%#refs.\ci{Benhar91,Benhar93,Benhar94}. Corrections to the NN
found to be small in the kinematical region spanned by the data analyzed in
Refs.\ci{Benhar91,Benhar93,Benhar94}.  Corrections to the $NN$
%#
amplitude associated with the extrapolation to off-shell energies are also
expected to be small\ci{bl}.

%#A different type of modification of the NN cross section, originating from 
A different type of modification of the $NN$ cross section, originating from 
%#
the internal structure of the nucleon, may play a more significant role. 
It has been suggested \ci{CT1,CT2} that
elastic scattering on a nucleon at high momentum transfer can only 
occur if the nucleon is found in the Fock state having the lowest number of 
constituents, so that the momentum transfer can be most effectively shared 
among them.
%#Within this picture a nucleon, after absorbing a large momentum ${\bf q}$, 
%#is in a very compact configuration, and travels through nuclear matter 
%#experiencing very little FSI, until its standard size is recovered
Within this picture, a nucleon is in a very compact configuration
after absorbing a large momentum ${\bf q}$.  It then travels through 
nuclear matter
%#
experiencing very little FSI, until its standard size is recovered
%#with a characteristic timescale that increases with $|{\bf q}|$. 
%#In the limit of infinite momentum transfer 
on a characteristic time scale that increases with $|{\bf q}|$. 
In the limit of infinite momentum transfer, 
%#
FSI are totally suppressed, and the nuclear medium is said to exhibit 
%#{\it color transparency} (CT).
{\it color transparency}.
%#

%#The results of the calculations of refs.\ci{Benhar91,Benhar93,Benhar94} 
%#show that inclusion of the effects of CT according to the model of
%#ref.\ci{CT}, {\it involving no adjustable parameters}, greatly improves the 
The results of the calculations of Refs.\ci{Benhar91,Benhar93,Benhar94}
show that inclusion of the effects of color transparency according to 
the model of 
Ref.\ci{CT}, {\it with no adjustable parameters}, greatly improves the
%#
%#agreement between theory and data, resulting in a satisfactory description of 
%#the low energy loss tail of the nuclear inclusive cross sections 
agreement between theory and data, yielding a satisfactory description of 
the low-energy loss tail of the nuclear inclusive cross sections 
%#
for $Q^2>1.5$ (GeV/c)$^2$.     

%#In order to firmly establish the occurrence of CT in $(e,e\primo)$ processes,
%#the accuracy of the approximations underlying CGA must be carefully 
%#investigated.
To firmly establish the occurrence of color transparency in $(e,e\primo)$ 
processes, the accuracy of the approximations underlying the CGA must be 
carefully investigated and either validated or transcended.
%#
%#In this paper we discuss a treatmet of FSI which improves upon CGA, in that 
%#it allows one to take into account the initial momentum of 
%#the struck nucleon.  Within this approach the response
%#can no longer be written as a convolution integral, but can still 
In this paper, we introduce a treatment of FSI which improves upon the CGA,  
in that it allows one to take account of the initial momentum of 
the struck nucleon.  Within this approach, the response can no longer 
be written as a simple convolution integral.  However, it can still 
%#
be expressed in terms of the spectral function and a generalized folding 
%#function, exhibiting an explicit dependence on both the initial and final 
%#and final momentum of the struck particle.
function, in a form displaying explicit dependence on both the initial 
and final momenta of the struck particle.
%#

%#The theoretical description of nuclear matter response is discussed
The theoretical description of nuclear-matter response is discussed
%#
in Section II, where we outline the development of a systematic scheme 
%#to improve upon PWIA and include FSI effects. The details of the many-body 
%#of the generalized folding function within the Fermi-HyperNetted Chain
%(FHNC) approach are discussed in Section III, while Section IV is 
%devoted to the presentation of numerical results, including the 
that improves upon the PWIA and includes FSI effects.  The details
of the many-body calculation of the generalized folding function within 
the Fermi-HyperNetted Chain (FHNC) approach are traced in Section III.
The ensuing numerical results are presented and analyzed in Section IV,
with particular attention to the 
%#
generalized folding functions and the nuclear matter $y$-scaling functions
%#at different values of $|{\bf q}|$.  Finally, a summary and the 
%#conclusions are contained in Section V.
at different values of $|{\bf q}|$.  The final section summarizes 
our findings and states our conclusions.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%#\section{Nuclear matter response}
\section{Nuclear-matter response}
%#

%#\subsection{Plane Wave Impulse Approximation}        
\subsection{Plane-Wave Impulse Approximation}        
%#

%#The analysis carried out in refs.\ci{Benhar91,Benhar93,Benhar94} required 
%#the full calculation of the nuclear cross section, including the 
%#electromagnetic
%#vertex, and the use of spectral functions adequate to describe finite targets,
%#such as those obtained within the local density approximation \ci{Benhar94}.
%#In addition, the typical momentum transfer being in the 1-2 GeV/c range, 
%#relativistic kinematics had to be consistently used. 
The analysis carried out in Refs.\ci{Benhar91,Benhar93,Benhar94} required 
a full calculation of the nuclear cross section, including the electromagnetic 
vertex, as well as the use of spectral functions adapted to finite targets 
(such as those obtained within a local-density approximation \ci{Benhar94}).
In addition, since the typical momentum transfers lie in the 1--2 GeV/c
range, consistent use of relativistic kinematics was essential.
%#

%#In this paper we will avoid these complications, and focus on the 
In this paper, we will avoid these complications and focus on the 
nonrelativistic response of infinite nuclear matter to a scalar 
%#probe, defined as
probe, defined by
%#
\be
\nonumber
%#{\rm H} replaced by H, for consistency; {\rm E} replaced by e. 
%#{\rm e} replaced by e (standard).
%#S({\bf q},\omega) & = & \frac{1}{A}\ \int\ \frac{dt}{2\pi}\  {\rm e}^{i\omega t} \ 
%#\langle 0 | \rho^\dagger_{\vecq}(t)\rho_{\vecq}(0) | 0 \rangle \\
%#& = & \frac{1}{A}\ \int\ \frac{dt}{2\pi}\ {\rm e}^{i(\omega+{\rm E}_0)t} \ 
%#\langle 0 | \rho^\dagger_{\vecq} {\rm e}^{-i{\rm H}t} \rho_{\vecq} 
%#| 0 \rangle \ .
S({\bf q},\omega) & = & \frac{1}{A} \int \frac{dt}{2\pi}  e^{i\omega t} 
\langle 0 | \rho^\dagger_{\vecq}(t)\rho_{\vecq}(0) | 0 \rangle \\
& = & \frac{1}{A} \int \frac{dt}{2\pi}\ e^{i(\omega+E_0)t} 
\langle 0 | \rho^\dagger_{\vecq} e^{-iHt} \rho_{\vecq} 
| 0 \rangle \ .
%#
\label{resp:def}
\ee
%#In the above equation H and $|0\rangle$ denote the nuclear Hamiltonian and 
%#ground state,
%#satisfying the Schr\"odinger equation H$|0\rangle$ = E$_0 |0\rangle$, while
%#the time-dependent density fluctuation operator reads
Here $H$ and $|0\rangle$ denote the nuclear Hamiltonian and the
corresponding ground state satisfying the Schr\"odinger equation 
$H|0\rangle = E_0 |0\rangle$.  The time-dependent density fluctuation 
operator $\rho_{\vecq}(t)$ is constructed as
%#
\beq
%#{\rm e} replaced by e (standard).
%#\rho_{\vecq}(t) = {\rm e}^{iHt}\ \rho_{\vecq}\ {\rm e}^{-iHt}
%# = {\rm e}^{iHt}\ \sum_{{\bf k}}\ a^\dagger_{{\bf k}+\vecq}a_{{\bf k}}\ \ 
%#{\rm e}^{-iHt}\ ,
\rho_{\vecq}(t) = e^{iHt} \rho_{\vecq} e^{-iHt}
 = e^{iHt} \sum_{{\bf k}} a^\dagger_{{\bf k}+\vecq}a_{{\bf k}}\, e^{-iHt}\, ,
%#
\eeq
%#$a^\dagger_{{\bf k}}$ and $a_{{\bf k}}$ being nucleon creation and 
where $a^\dagger_{{\bf k}}$ and $a_{{\bf k}}$ are nucleon creation and 
%#
annihilation 
%#operators, respectively. Note that the definition of Eq.(\ref{resp:def}) can 
operators, respectively.  Note that the definition given 
in Eq.~(\ref{resp:def}) can 
%#
be readily generalized to describe the electromagnetic response
%#replacing $\rho_{\vecq}$ with the appropriate current operator.
by replacing $\rho_{\vecq}$ with the appropriate current operator.
%#


%#Keeping only the incoherent contribution, which is known to be dominant
%#at large $\magq$, Eq.(\ref{resp:def}) can be rewritten in the form
Retaining only the incoherent contribution to the response, which is known 
to be dominant at large $\magq$, Eq.~(\ref{resp:def}) can be rewritten 
in the form
\beq
%#S(\vecq,\omega) = \int \frac{dt}{2\pi}\ {\rm e}^{i(\omega+{\rm E}_0)t} \ 
S(\vecq,\omega) = \int \frac{dt}{2\pi} e^{i(\omega+E_0)t} 
%#
\shat \, ,
\label{sqw:ft}
\eeq
with
\beq
%#\shat = \int dR dR^\prime\  \Psi_{0}^{*}(R\primo)\  
%#{\rm e}^{-i{\vecq}\cdot{\bf r}\primo_1}\ 
%#\langle R\primo | {\rm e}^{-iHt} | R \rangle \ 
%#e^{i{\bf q}\cdot{\bf r}_1}\  \Psi_{0}(R)\ ,
\shat = \int dR \, dR^\prime\, \Psi_{0}^{*}(R\primo) 
e^{-i{\vecq}\cdot{\bf r}\primo_1} 
\langle R\primo | e^{-iHt} | R \rangle  
e^{i{\bf q}\cdot{\bf r}_1}\Psi_{0}(R)\ ,
\label{incoh:resp}
\eeq
where $\{R\}=\{ {\bf r}_1,{\bf r}_2,...,{\bf r}_A \}$ specifies
%#the configuration of the A-nucleon system and
%#$\Psi_{0}(R)=\langle R | 0 \rangle$ is its ground state wave function, that
%#can in principle be evaluated within nonrelativistic nuclear many-body theory.
%#On the other hand, the nonrelativistic approach cannot be used to
%#obtain the A-particle propagator 
%#$U_A(R,R^\prime;t)=\langle R\primo | {\rm e}^{-iHt} | R \rangle$, 
%#yielding the amplitude for the system to evolve from the configuration ${R}$ 
%#to the configuration ${R\primo}$ during the time $t$, as in 
%#the kinematical regime under consideration the struck nucleon
%#typically carries a momentum larger than the nucleon mass.
%#A fully realistic and consistent calculation of $U_A(R,R^\prime;t)$ being 
%#still out of reach of the existing relativistic approaches to the nuclear 
%#many-body problem, one has therefore to resort to simplifiyng assumptions.
the spatial configuration of the $A$-nucleon system,   
$\Psi_{0}(R)=\langle R | 0 \rangle$ is its ground-state wave function, 
and the propagator 
$U_A(R,R^\prime;t)=\langle R\primo |  e^{-iHt} | R \rangle$
represents the amplitude for the system to evolve from configuration ${R}$ 
to configuration ${R\primo}$ during the time $t$.  The wave function 
$\Psi_0(R)$ can in principle be evaluated within nonrelativistic
nuclear many-body theory.  On the other hand, the nonrelativistic approach 
cannot be used to obtain the $A$-particle propagator $U_A(R,R^\prime;t)$, 
since -- in the kinematical regime under consideration -- the struck 
nucleon typically carries a momentum larger than the nucleon mass.  
In view of the fact that a fully realistic and consistent calculation 
of $U_A(R,R^\prime;t)$ remains intractable, one must resort to
simplifying assumptions.
%#

%#A systematic approximation scheme can be developed rewriting the nuclear
%#hamiltonian in the form
A systematic approximation scheme can be developed by first decomposing
the Hamiltonian according to 
%#
\beq
%#H = H_{{\rm A}-1} + T_{1} + H_{I}\ ,
H = H_{A-1} + T_{1} + H_{I}\, ,
%#
\label{split:H}
\eeq
%#where $H_{{\rm A}-1}$ is the nonrelativistic hamiltonian of the
%#{\it fully interacting} $A-1$-particle spectator system, $T_1$
%#denotes the hamiltonian describing a {\it free }
%#nucleon and
where $H_{{\rm A}-1}$ is the nonrelativistic Hamiltonian of the
{\it fully interacting} $(A-1)$-particle spectator system and $T_1$
denotes the Hamiltonian describing a {\it free} nucleon. The term 
%#
\beq
%#H_{I} = \sum_{j=2}^{{\rm A}} v_{1j}\ ,
H_{I} = \sum_{j=2}^A v_{1j}\, ,
%#
\eeq
%#$v_{ij}$ being the NN potential, accounts for the interactions between the
where $v_{ij}$ is the $NN$ potential, accounts for the interactions between the
%#
struck particle and the spectators.       

%#PWIA amounts to setting $H_{I}=0$ in Eq.(\ref{split:H}), thus disregarding FSI
%#altogether. The resulting A-particle propagator factorizes into the product
%#of the interacting (A-1)-particle propagator and the free space one-body 
The
PWIA amounts to setting $H_{I}=0$ in Eq.~(\ref{split:H}), thus disregarding FSI
altogether. The resulting $A$-particle propagator factorizes into the product
of the interacting $(A-1)$-particle propagator and the free-space one-body 
%#
propagator describing the struck nucleon:
\beq
U_{\rm PWIA}(R,R^\prime;t) = U_{A-1}(\widetilde{R},\widetilde{R}^\prime;t) 
U_0({\bf r}_1,{\bf r}^\prime_1;t)\, ,
\label{PWIA:prop}
\eeq
where $\{\widetilde{R}\}=\{{\bf r}_2,...,{\bf r}_A\}$ specifies the
%#configuration of the spectator system. Eq.(\ref{PWIA:prop}) clearly shows
%#that within PWIA nuclear dynamics only appears through $U_{A-1}$, while the 
configuration of the spectator system. Eq~.(\ref{PWIA:prop}) clearly shows
that within the PWIA, nuclear dynamics only appears through $U_{A-1}$, 
while the 
%#
treatment of the relativistic motion of the struck nucleon reduces to a 
%#trivial kinematical problem.      
trivial kinematic problem.      
%#
                                                   
%#Using spectral representations for both $U_{A-1}$ and $U_0$ 
%#(see, e.g., ref.\ci{PRL2001})
%#and introducing the nucleon spectral function $P({\bf k},E)$, 
%#yielding the probability of removing a nucleon with momentum ${\bf k}$
%#from the nuclear ground state leaving the residual system
%#with excitation energy $E$, one obtains 
We next express the PWIA to $S({\bf q},t)$ (and thereby the response)
in terms of the nucleon spectral function $P({\bf k},E)$, which by definition
gives the probability of removing a nucleon with momentum ${\bf k}$
from the nuclear ground state, leaving the residual system
with excitation energy $E$.  Introducing spectral representations for 
both $U_{A-1}$ and $U_0$ (see, e.g., Ref.\ci{PRL2001}), we obtain
%#
\beq
\widehat{S}_{\rm PWIA}({\bf q},t) = \int \frac{d^3p}{(2\pi)^3} \int dE \,
P({\bf p}-{\bf q},E) {\rm e}^{-i(E-E_0+E_p)t}\, ,
\label{PWIA:shat}
\eeq
%#leading to the familiar result \ci{PRL2001}
which leads to the familiar result \ci{PRL2001}
%#
\beq
S_{\rm PWIA}({\bf q},\omega) =
\int \frac{d^3p}{(2\pi)^3}  \int dE\,
 P({\bf p}-{\bf q},E) \delta(\omega - E - E_p)\, ,
\eeq
where $E_p=|{\bf p}|^2/2m$ denotes the kinetic energy of a nucleon carrying 
momentum ${\bf p}$.

%#\subsection{Inclusion of FSI}
\subsection{Inclusion of Final-State Effects}
%#

%#In order to improve upon PWIA, one needs to devise
In order to improve upon the PWIA, one needs to devise
%#
a set of sensible approximations to treat the contributions to the
%#A-particle propagator coming from the FSI hamiltonian $H_I$. As a 
%#first step, we replace
$A$-particle propagator coming from the FSI Hamiltonian $H_I$. As a 
first step, we make the replacement
%#
\beq
%#{\rm e}^{-i(H_{A-1} + T_1 + H_I)t} \rightarrow 
%#{\rm e}^{-iH_{A-1}t}\ {\rm e}^{-i(T_1 + H_I)t}\ ,
e^{-i(H_{A-1} + T_1 + H_I)t} \rightarrow 
e^{-iH_{A-1}t} e^{-i(T_1 + H_I)t}\, ,
%#
\eeq
which essentially amounts to assuming that the internal dynamics of
the spectator system and its FSI with the struck particle do not
affect one another, and can therefore be completely decoupled.
%#Within this picture, during the
%#time $t$ the spectator system evolves as if there was
Within this picture, the spectator system evolves during the
time $t$ as if there were
%#
no struck particle moving around, while the fast struck particle
sees the spectator system as if it were {\it frozen} at time $t=0$. 

%#Under this assumption, implying that the
%#configuration of the spectator system does not change on account of
Under this assumption, which implies that the
configuration of the spectator system does not change due to
%#
interactions with the fast struck nucleon, we can use
%#completeness of the (A-1)-particle position 
completeness of the $(A-1)$-particle position 
%#
eigenstates to rewrite the propagator in the simple factorized form
\begin{eqnarray}
\nonumber
U_A(R,R^\prime;t) & = & \int d\widetilde{R}^{\prime\prime}
\langle \widetilde{R}^\prime | {\rm e}^{-iH_{A-1}t} | 
\widetilde{R}^{\prime\prime} \rangle
\langle {\bf r}\primo_1, \widetilde{R}^{\prime\prime} | 
%#{\rm e}^{-i(T_1 + H_I)t}| {\bf r}_1,\widetilde{R} \rangle\ 
e^{-i(T_1 + H_I)t}| {\bf r}_1,\widetilde{R} \rangle 
%#
\delta^{3(A-1)}(\widetilde{R}-\widetilde{R}^{\prime\prime}) \\
\nonumber
& = & U_{A-1}(\widetilde{R},\widetilde{R}\primo;t)
\langle {\bf r}\primo_1, \widetilde{R} | 
%#{\rm e}^{-i(T_1 + H_I)t}| {\bf r}_1, \widetilde{R} \rangle \\
e^{-i(T_1 + H_I)t}| {\bf r}_1, \widetilde{R} \rangle \\
%#
& = & U_{A-1}(\widetilde{R},\widetilde{R}\primo;t)\, 
U_1( {\bf r}_1 \widetilde{R}, {\bf r}\primo_1 \widetilde{R}; t)\, .
\label{FSI:prop}
\end{eqnarray}
%# Join these two paragraphs. PR does not like single-sentence paragraphs.
%#
%#The calculation of $U_{1}({\bf r}_1\ \widetilde{R},
%#{\bf r}\primo_1\ \widetilde{R};t)$ in general requires a 
%#functional integration, extended to the space of the 
Evaluation of $U_{1}({\bf r}_1 \widetilde{R},
{\bf r}\primo_1 \widetilde{R};t)$ in general requires a 
functional integration over the set of 
%#
trajectories ${\bf r}_1(\tau)$ satisfying the boundary
conditions ${\bf r}_1(0)={\bf r}_1$ and ${\bf r}_1(t)={\bf r}^{\prime}_1$
%#(see, e.g., ref.\ci{Negele}). 
%#For large momenta of the struck nucleon, however, it can be
%#drastically simplified making use of the {\it eikonal approximation}, i.e. 
(see, e.g., Ref.\ci{Negele}). 
However, for large momenta of the struck nucleon, the evaluation can be
drastically simplified by invoking the {\it eikonal approximation}, i.e. 
by
%#
assuming that 
the particle moves along a straight trajectory with constant velocity
${\bf v}=({\bf r}\primo_1 - {\bf r}_1)/t$, so that 
${\bf r}_1(\tau) = {\bf r}_1 + {\bf v}\tau\ $.
%#Under this assumption, $U_{1}({\bf r}_1\ \widetilde{R},
%#{\bf r}\primo_1\ \widetilde{R};t)$ can be shown to take the factorized form 
%#\cite{Carraro91}:
Within this approximation, the propagator $U_{1}({\bf r}_1 \widetilde{R},
{\bf r}\primo_1 \widetilde{R};t)$ takes the factorized form 
\cite{Carraro91}
%#
\beq
U_{1}({\bf r}_1 \widetilde{R}, {\bf r}\primo_1 \widetilde{R};t) =
U_0({\bf r}_1,{\bf r}^\prime_1;t) U_{\bf p}({\bf r}_1,\widetilde{R};t) \, ,
\eeq  
where the eikonal propagator $U_{\bf p}({\bf r}_1,\widetilde{R};t)$ is 
given by
\beq
U_{\bf p}({\bf r}_1,\widetilde{R};t) = \exp { \left[ {- i \int_0^t d\tau
\sum_{j=2}^A  v({\bf r}_1 + {\bf v}\tau - {\bf r}_j)}  \right] }\, ,
\label{eikonal:prop}
\eeq
%#$v$ being the NN potential and $U_0({\bf r}_1,{\bf r}^\prime_1;t)$ the 
%#free space nucleon propagator. 
$v$ being the $NN$ potential and $U_0({\bf r}_1,{\bf r}^\prime_1;t)$ the 
free-space nucleon propagator. 
%#

%#Expanding the exponential appearing in the right hand side of 
%#Eq.({\ref{eikonal:prop}) one obtains a series, whose terms are associated with
Expanding the exponential appearing in the right-hand side of 
Eq.~({\ref{eikonal:prop}), one obtains a series whose terms are associated with
%#
processes involving an increasing number of interactions between the struck 
nucleon and the spectators. The terms corresponding to repeated
interactions with the same spectator can be summed up to all orders 
%#replacing the bare NN interaction
by replacing the bare $NN$ interaction
%#$v$ with the coordinate space $t$-matrix, $\Gamma_{\bf q}$, related to the 
%#NN scattering amplitude at incident momentum ${\bf q}$ 
%#and momentum transfer ${\bf k}$, $f_{\bf q}({\bf k})$, through
$v$ with the coordinate space $t$-matrix $\Gamma_{\bf q}({\bf r})$, 
which is related to the $NN$ scattering amplitude $f_{\bf q}({\bf k})$ 
at incident momentum ${\bf q}$ and momentum transfer 
${\bf k}$ through
%#
\beq
%#\Gamma_{\bf q}({\bf r}) = -\ \frac{2 \pi}{m}\ \int\ \frac{d^3k}{(2 \pi)^3}\, 
%#{\rm e}^{i{\bf k}\cdot{\bf r}}\ f_{\bf q}({\bf k})\ .
\Gamma_{\bf q}({\bf r}) = - \frac{2 \pi}{m} \int \frac{d^3k}{(2 \pi)^3}
e^{i{\bf k}\cdot{\bf r}} f_{\bf q}({\bf k})\, .
%#
\label{def:Gamma}
\eeq

%#Using the above results and spectral representation for both 
Using the above results together with spectral representations of both 
%#
$U_0({\bf r}_1,{\bf r}^\prime_1;t)$ and 
%#$U_{A-1}(\widetilde{R},\widetilde{R}^{\prime};t)$ 
%#the response can be finally cast in the form
$U_{A-1}(\widetilde{R},\widetilde{R}^{\prime};t)$, 
the response can finally be expressed as
%#
\beq
%#S({\bf q},\omega)  = \int \ \frac{dt}{2\pi}\ 
%#{\rm e}^{i(\omega + E_0) t} \int \ \frac{d^3p}{(2\pi)^3}\ {\rm e}^{-iE_pt} 
%# \sum_n {\rm e}^{-iE_nt}\ M_{0n}^*({\bf p} - {\bf q})
S({\bf q},\omega)  = \int  \frac{dt}{2\pi} 
e^{i(\omega + E_0) t} \int  \frac{d^3p}{(2\pi)^3} e^{-iE_pt} 
 \sum_n e^{-iE_nt} M_{0n}^*({\bf p} - {\bf q})
%#
\widetilde{M}_{0n}({\bf p},{\bf q};t) \, ,
\label{resp:over}
\eeq
%#where the sum is extended over the (A-1)-particle states satisfying the
%#Schr\"odinger equations $H_{A-1}|n\rangle=E_n|n\rangle$, 
where the sum extends over the $(A-1)$-particle states satisfying the
Schr\"odinger equations $H_{A-1}|n\rangle=E_n|n\rangle$.  We have
introduced the definitions 
%#
\beq
%#M_{0n}({\bf k}) = \int \ dR \ {\rm e}^{i{\bf k}\cdot{\bf r}_1}
M_{0n}({\bf k}) = \int dR\, e^{i{\bf k}\cdot{\bf r}_1}
%#
\Psi_{0}^{*}(R) \Phi_{n}(\widetilde{R})\, ,
\label{mon:def}   
\eeq
with $\Phi_{n}(\widetilde{R}) = \langle \widetilde{R} | n \rangle$, and
\beq
%#\widetilde{M}_{0n}({\bf p},{\bf q};t) =  \int \ dR\ 
%#{\rm e}^{-i({\bf p} - {\bf q})\cdot{\bf r}_1}
%#\Phi_{n}^{*}(\widetilde{R})\Psi_{0}(R)\ 
\widetilde{M}_{0n}({\bf p},{\bf q};t) =  \int  dR\, 
e^{-i({\bf p} - {\bf q})\cdot{\bf r}_1}
\Phi_{n}^{*}(\widetilde{R})\Psi_{0}(R)
%#
U_{\bf p}({\bf r}_1,\widetilde{R};t)\, .
\label{mtilde:def}
\eeq
%#In the limit $U_{\bf p}({\bf r}_1,\widetilde{R};t) \rightarrow 1$, 
%#$\widetilde{M}_{0n}({\bf p},{\bf q};t) \rightarrow M_{0n}({\bf p}-{\bf q})$
%#and the response $S({\bf q},\omega)$ given by Eq.(\ref{resp:over}) 
In the limit $U_{\bf p}({\bf r}_1,\widetilde{R};t) \rightarrow 1$, we have
$\widetilde{M}_{0n}({\bf p},{\bf q};t) \rightarrow M_{0n}({\bf p}-{\bf q})$,
and the response $S({\bf q},\omega)$ given by Eq.~(\ref{resp:over}) 
%#
reduces to the PWIA result.
           
%#It has to be emphasized that the calculation of the response according 
%#to Eq.(\ref{resp:over}) only implies two approximations: (i) the frozen 
It is to be emphasized that the calculation of the response according 
to Eq.~(\ref{resp:over}) involves only two approximations: (i) the frozen 
%#
approximation for the configuration of the spectator 
%#system, and (ii) the eikonal approximation for the trajectory of the struck 
system and (ii) the eikonal approximation for the trajectory of the struck 
%#
particle. 
%#Join these two paragraphs.  PR does not like single-sentence paragraphs.
%#
%#Although the explicit calculations of the relevant 
%#$\widetilde{M}_{0n}({\bf p},{\bf q};t)$'s
%#using nonrelativistic many-body theory appears to be feasible, at least for
%#the few-nucleon systems and infinite nuclear matter, to make a clear 
%#connection with 
%#the PWIA picture it is useful to devise approximations that allow to express 
%#$S({\bf q},\omega)$ of Eq.~(\ref{resp:over}) in terms 
%#of either the spectral function  $P({\bf k},E)$ or 
Explicit calculation of the relevant 
$\widetilde{M}_{0n}({\bf p},{\bf q};t)$ integrals
within nonrelativistic many-body theory appears to be feasible, at least for
few-nucleon systems and infinite nuclear matter. 
However, to establish a clear connection with 
the PWIA picture, it is useful to devise approximations that permit
$S({\bf q},\omega)$ of Eq.~(\ref{resp:over}) to be expressed in terms 
of either the spectral function  $P({\bf k},E)$ or 
%#
the PWIA response $S_{\rm PWIA}({\bf q},\omega)$.           

%#To recover the spectral function, whose definition can be written
%#\beq
%#P({\bf k},E) = \sum_n | {M}_{0n}({\bf k}) |^2 \delta(E+E_0-E_n)\ ,
%#\eeq
%#we have to require that $\widetilde{M}_{0n}$ of eq.(\ref{mtilde:def}) 
%#can be cast in the form
%#\beq
%#widetilde{M}_{0n}({\bf p},{\bf q};t) = {M}_{0n}({\bf p}-{\bf q})
%#{\cal U}({\bf p},{\bf q};t)\ ,
%#\label{mtilup:def}
%#\eeq
%#where the function ${\cal U}$ is independent of the state of the
%#spectator system, labelled by the index $n$.
The spectral function is defined as
\beq
P({\bf k},E) = \sum_n | {M}_{0n}({\bf k}) |^2 \delta(E+E_0-E_n)\, .
\eeq
To recover this quantity within the formula (15), 
the integrals $\widetilde{M}_{0n}$ of Eq.~(\ref{mtilde:def}) 
are required to take the form
\beq
\widetilde{M}_{0n}({\bf p},{\bf q};t) = {M}_{0n}({\bf p}-{\bf q})\,
 {\cal U}({\bf p},{\bf q};t)\, ,
\label{mtilup:def}
\eeq
where the function ${\cal U}$ is to be independent of the state of the
spectator system, labeled by the index $n$.
%#
%#An even more drastic simplification can be achieved requiring that the time 
An even more drastic simplification is achieved upon requiring that the time 
%#
dependence of $\widetilde{M}_{0n}({\bf p},{\bf q};t)$ be factorizable 
according to
\beq
\widetilde{M}_{0n}({\bf p},{\bf q};t) = 
%#M_{0n}({\bf p}-{\bf q})\ {\overline U}_{\bf q}(t)\ ,
M_{0n}({\bf p}-{\bf q}) {\overline U}_{\bf q}(t)\, ,
%#
\label{fact:t}
\eeq
%#i.e. assuming that the function ${\cal U}$ defined by
i.e.\ upon assuming that the function ${\cal U}$ defined by
%#
%#Eq.(\ref{mtilup:def}) does not depend upon ${\bf p}$, which in turn amounts 
%#to using the approximation ${\bf p} \sim {\bf q}$ in ${\cal U}$.    
Eq.~(\ref{mtilup:def}) does not depend upon ${\bf p}$, which in turn 
corresponds to making the approximation ${\bf p} \simeq {\bf q}$ in 
${\cal U}$.    
%#

%#Substitution of Eq.(\ref{fact:t}) into Eq.(\ref{resp:over}) allows one to 
%# rewrite the response as a convolution integral:
%#\beq
%#S({\bf q},\omega) = \int\ d\omega^\prime\ S_{\rm PWIA}({\bf q},\omega^\prime)\
%#F_{\bf q}(\omega-\omega^\prime)\ ,
%#\label{convolution}
%#\eeq
%#the folding function $F_{\bf q}(\omega)$ being given by
%#the folding function $F_{\bf q}(\omega)$ being given by
%#\beq
%#F_{\bf q}(\omega) = \int\  \frac{dt}{2\pi}\  {\rm e}^{i \omega t}\ 
%#{\overline U}_{\bf q}(t)\ .
%#\eeq
Substitution of Eq.~(\ref{fact:t}) into Eq.~(\ref{resp:over}) allows one to 
rewrite the response as a convolution integral,
\beq
S({\bf q},\omega) = \int  d\omega^\prime S_{\rm PWIA}({\bf q},\omega^\prime) 
F_{\bf q}(\omega-\omega^\prime)\, ,
\label{convolution}
\eeq
the folding function $F_{\bf q}(\omega)$ being given by
\beq
F_{\bf q}(\omega) = \int  \frac{dt}{2\pi}  e^{i \omega t}
{\overline U}_{\bf q}(t)\, .
\eeq
%#
%#To obtain the function ${\overline U}_{\bf q}(t)$, embodying all 
%#FSI effects 
%#(note that PWIA can be recovered setting ${\overline U}_{\bf q}(t) \equiv 1$, 
%#i.e. $F(\omega) = \delta(\omega)$), one has to replace
To obtain the function ${\overline U}_{\bf q}(t)$ embodying all FSI 
effects ({\it N.B.} the PWIA can be regained by setting
${\overline U}_{\bf q}(t) \equiv 1$, i.e., $F(\omega) = \delta(\omega)$), 
one makes the replacement 
%#
\beq
\sum_{j=2}^A \Gamma_{\bf q}({\bf r}_1 + {\bf v}\tau - {\bf r}_j) 
%#\rightarrow  \frac{ \int \ dR\ |\Psi_0(R)|^2 \sum_{j=2}^A \
%#\Gamma_{\bf q}({\bf r}_1 + {\bf v}\tau - {\bf r}_j) }
%#{ \int \ dR\ |\Psi_0(R)|^2 } = {\overline V}_{\bf q}(\tau)\ , 
\rightarrow  \frac{ \int   dR\, |\Psi_0(R)|^2 \sum_{j=2}^A
\Gamma_{\bf q}({\bf r}_1 + {\bf v}\tau - {\bf r}_j) }
{ \int dR\, |\Psi_0(R)|^2 } = {\overline V}_{\bf q}(\tau)\, , 
%#
\label{average}
\eeq
%#which amounts to averaging FSI over the ground state configurations, 
%#weighted with the squared ground state wave function. 
which amounts to averaging FSI with the ground-state configuration
weights.
%#
%#In infinite nuclear matter at uniform density $\rho$ the average involved
%#in Eq.(\ref{average}) takes the simple form
In infinite nuclear matter at uniform density $\rho$, the average involved
in Eq.~(\ref{average}) takes the simple form
%#
\beq
%#{\overline V}_{\bf q}(\tau) = \rho \int \ d^3r\ g(r)\,
{\overline V}_{\bf q}(\tau) = \rho \int d^3r\, g(r)
%#
\Gamma_{\bf q}({\bf r} + {\bf v}\tau)\, ,
\label{Vbar:nm}
\eeq
%#where the radial distribution function $g(r)$ yields the probability of
where the radial distribution function $g(r)$ measures the probability of
%#
finding two nucleons separated by a distance $r=|{\bf r}|$.
%#Join these 2 paragraphs - PRL doesn't like 1-sentence paragraphs.
%#
Keeping only the contributions associated with the imaginary part of the 
%#NN amplitude, that is known to be dominant 
$NN$ amplitude, which is known to be dominant 
%#
at large incident momentum, we can finally write the eikonal propagator
%#in the form
as
%#
\beq
%#{\overline U}_{\bf q}(t) = \exp { \int d\tau\ 
%#Im}\  {\overline V}_{\bf q}(\tau) }\ .
{\overline U}_{\bf q}(t) = \exp { \int d\tau \,
{\rm Im}  {\overline V}_{\bf q}(\tau) }\, .
%#
\label{averaged:prop}
\eeq

%#The approach developed in I and employed in 
%#refs.\cite{Benhar91,Benhar93,Benhar94} is based on the assumptions implied 
%#by Eqs.(\ref{fact:t})-(\ref{averaged:prop}). A different approximation scheme 
The approach developed in I and employed in 
Refs.\cite{Benhar91,Benhar93,Benhar94} is based on the assumptions underlying
Eqs.~(\ref{fact:t})--(\ref{averaged:prop}).  A different approximation scheme 
%#
%#leading to the factorization of $M_{0n}$ can be obtained inserting 
%#into Eq.(\ref{mtilde:def}) the relation
leading to the factorization of $\widetilde{M}_{0n}$
can be obtained by inserting into Eq.~(\ref{mtilde:def}) the identity
%# - presumably the tilde-M is meant here - JWC
%#
\beq
\int d\tilde{R}\primo d^3r\primo_1
\delta({\bf r}_1-{\bf r}\primo_1)\, \delta^{3(A-1)} (R - R\primo) =
\sum_N \int dR\primo\, \Psi^{*}_N(R\primo) \Psi_N (R)\, ,
\label{deltaR}
\eeq 
%#where the sum includes the complete set of eigenstates of the
%#$A$-particle hamiltonian $H$. This procedure is not univoque, 
%#as the $\delta$-function insertion allows for different
%#assignements of the arguments of the functions entering the right hand side of
%#eq.(\ref{mtilde:def}), leading to different representations which are, of
%#course, all equivalent. By equating two such representations, we obtain 
where the sum includes a complete set of eigenstates of the
$A$-particle Hamiltonian $H$.  This procedure is not unique,
because the $\delta$-function insertion allows for different
assignments of the arguments of the functions entering the right-hand side of
Eq.~(\ref{mtilde:def}), leading to different but ultimately equivalent
representations.  Equating two such representations, we have
%#
%\begin{eqnarray}
%\nonumber
%# & & \sum_{N} \int d\tilde{R}\  d^3r_1\ {\rm e}^{-i{\bf k}\cdot{\bf r}_1}
%# \ \Phi^*_n(\widetilde{R})\Psi_N({\bf r}_1,\widetilde{R})\\
%# \nonumber
%# & \times & \int d\widetilde{R}\primo\  d^3r\primo_1\ 
%# {\rm e}^{i{\bf k}\cdot({\bf r}_1 - {\bf r}\primo_1)}\ 
%#\Psi_0({\bf r}\primo_1,\widetilde{R}\primo)
%#\Psi_N^*({\bf r}_1,\widetilde{R}\primo)\ 
%#U_{\bf p}({\bf r}\primo_1,\widetilde{R}\primo;t)\\
%#\nonumber
%# & = & \sum_{N} \int d\widetilde{R}\  d^3r_1\ 
%# {\rm e}^{-i{\bf k}\cdot{\bf r}_1}
%# \ \Phi^*_n(\widetilde{R})\Psi_N(R)\ U_{\bf p}({\bf r}_1, \widetilde{R};t)\\
%# & \times & \int d\widetilde{R}\primo\  d^3r\primo_1\
%# {\rm e}^{i{\bf k}\cdot({\bf r}_1 - {\bf r}\primo_1)}\ 
%#\Psi_0({\bf r}\primo_1,\widetilde{R}\primo) 
%#\Psi_N^*({\bf r}_1,\widetilde{R}\primo)\ 
$$
  \sum_{N} \int d\tilde{R}\, d^3r_1\, e^{-i{\bf k}\cdot{\bf r}_1}
  \Phi^*_n(\widetilde{R})\Psi_N({\bf r}_1,\widetilde{R})
 \int d\widetilde{R}\primo  d^3r\primo_1 \,
 e^{i{\bf k}\cdot({\bf r}_1 - {\bf r}\primo_1)}
\Psi_0({\bf r}\primo_1,\widetilde{R}\primo)
\Psi_N^*({\bf r}_1,\widetilde{R}\primo) 
U_{\bf p}({\bf r}\primo_1,\widetilde{R}\primo;t)
$$
\beq
  =  \sum_{N} \int d\widetilde{R}\,  d^3r_1 \,
 e^{-i{\bf k}\cdot{\bf r}_1}
  \Phi^*_n(\widetilde{R})\Psi_N(R) U_{\bf p}({\bf r}_1, \widetilde{R};t)
   \int d\widetilde{R}\primo   d^3r\primo_1 \,
 e^{i{\bf k}\cdot({\bf r}_1 - {\bf r}\primo_1)}
\Psi_0({\bf r}\primo_1,\widetilde{R}\primo) 
\Psi_N^*({\bf r}_1,\widetilde{R}\primo)\, ,
\label{eq:5}
\eeq
%\end{eqnarray}  
%#with ${\bf k}={\bf p}-{\bf q}$. Assuming now translation invariance, 
with ${\bf k}={\bf p}-{\bf q}$.  Now imposing translation invariance, 
%#
as appropriate for the case of infinite nuclear matter, and
retaining only the term corresponding to $|N\rangle = |0\rangle$ in
%#the above sums, Eq.(\ref{eq:5}) becomes   
the above sums, Eq.~(\ref{eq:5}) becomes   
%#
\beq
%#\widetilde{M}_{0n}({\bf p},{\bf q};t) \sim M_{0n}({\bf k})\ \frac
%#{{\cal U}_0({\bf k},{\bf p};t)}{n({\bf k})}\ ,
\widetilde{M}_{0n}({\bf p},{\bf q};t) = M_{0n}({\bf k})\ \frac 
{{\cal U}_0({\bf k},{\bf p};t)}{n({\bf k})}\, .
%# Within the stated approximation, = rather than \sim is appropriate.
%#
\label{mtilapp}
\eeq
%#where $n({\bf k})$ denotes the nucleon momentum distribution,       
Here $n({\bf k})$ is the nucleon momentum distribution,       
%#
defined in terms of the spectral function as 
\beq 
%#n({\bf k}) = \int\ dE\ P({\bf k},E)\ ,
n({\bf k}) = \int dE\, P({\bf k},E)\, ,
%#
\eeq
while 
\beq
{\cal U}_0({\bf k},{\bf p};t) =  \int d^3 r_{11\primo}\, 
{\rm e}^{i{\bf k}\cdot{\bf r}_{11\primo}}
%#{\cal P}({\bf p},{\bf r}_{11\primo};t),
{\cal P}({\bf p},{\bf r}_{11\primo};t) \, ,
%#
\label{def:u0}
\eeq
with ${\bf r}_{11\primo}={\bf r}_1-{\bf r}_{1\primo}$ and
\beq
{\cal P}({\bf p},{\bf r}_{11\primo};t) = \int d\widetilde{R}\,
 \Psi_{0}^{*}({\bf r}\primo_1,\widetilde{R})\Psi_{0}({\bf r}_1,
%#\widetilde{R})\, U_{\bf p}({\bf r}_1,\widetilde{R};t).
\widetilde{R}) U_{\bf p}({\bf r}_1,\widetilde{R};t)\, .
%#
\label{def:p}
\eeq               
It can be readily seen that if $U_{\bf p}(\tilde{R},{\bf r}_1;t)\equiv 1$, 
%#i.e. in absence of FSI, the function ${\cal P}({\bf p},{\bf r}_{11\primo};t)$ 
i.e.\ if FSI are absent, the function ${\cal P}({\bf p},{\bf r}_{11\primo};t)$ 
%#
reduces to $\rho({\bf r}_1;{\bf r}_{1\primo})/A$, where 
\beq
%#\rho_{1}({\bf r}_{1};{\bf r}_{1}^{\prime}) = A \int d\widetilde{R}\
%#\Psi^*_0({\bf r}\primo_1,\widetilde{R})
%#\Psi_0({\bf r}_1,\widetilde{R})\ ,
\rho_{1}({\bf r}_{1};{\bf r}_{1}^{\prime}) = A \int d\widetilde{R}\,
\Psi^*_0({\bf r}\primo_1,\widetilde{R})
\Psi_0({\bf r}_1,\widetilde{R})
%#
\label{obdm}
\eeq
is the one-body density matrix, whose Fourier transform is $n({\bf k})$. 
%#As a consequence, 
%#$\widetilde{M}_{0n}({\bf p},{\bf q};t) = M_{0n}({\bf k})$ and PWIA is
As a consequence, we have
$\widetilde{M}_{0n}({\bf p},{\bf q};t) = M_{0n}({\bf k})$ and the PWIA is
%#
recovered. In addition, the relation (\ref{mtilapp}) can be shown to 
hold as an equality at the lowest order of the cluster expansion.           

%#Substitution of Eq.(\ref{mtilapp}) into the definition of the response 
%#leads to
%#\beq
%#S({\bf q},\omega) = \int \ d\omega\primo \int \
%#\frac {d^3k}{(2 \pi)^3}\ {F}_{{\bf k}{\bf q}}(\omega\primo - \omega)
%#\int dE\ P({\bf k},E)\ \delta(\omega\primo-E-E_{|{\bf k}+{\bf q}|})\ ,
Substitution of Eq.~(\ref{mtilapp}) into the definition of the response 
leads to
\beq
S({\bf q},\omega) = \int  d\omega\primo \int 
\frac {d^3k}{(2 \pi)^3} {F}_{{\bf k},{\bf q}}(\omega\primo - \omega)
 \int  dE\, P({\bf k},E) \delta(\omega\primo-E-E_{|{\bf k}+{\bf q}|})\, ,
%#
\eeq
where the generalized folding function ${F}_{{\bf k},{\bf q}}(\omega)$
is defined as
\beq
{F}_{{\bf k},{\bf q}}(\omega) =
\frac{1}{n({\bf k})}\int  \frac{dt}{2\pi} {\rm e}^{i\omega t} \,
{\cal U}_0({\bf k},{\bf k}+{\bf q};t)\, .
\label{def:genf}
\eeq                                         
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Many-Body Calculation of the Generalized Folding Function}

%#Eqs.(\ref{def:u0}) and (\ref{def:p}) show that the calculation of the 
%#generalized folding function of Eq.(\ref{def:genf}) requires 
%#the knowledge of the partially diagonal $n$-body density matrices of 
%#the target nucleus, defined as
%#\beq
%#\rho_{n}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n};
%#{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n})
%#= \frac{A!}{(A-n)!}\ 
%#\int\ d^3r_{n+1}\ldots d^3r_A \Psi^*_0({\bf r}\primo_1,\widetilde{R})
%#\Psi_0({\bf r}_1,\widetilde{R})
%#\label{def:densmat}
%#\eeq
Eqs.~(\ref{def:u0}) and (\ref{def:p}) show that calculation of the 
generalized folding function of Eq.~(\ref{def:genf}) requires 
a knowledge of the partially diagonal $n$-body density matrices 
\beq
\rho_{n}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n};
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n})
= \frac{A!}{(A-n)!}\, 
\int d^3r_{n+1}\ldots d^3r_A \Psi^*_0({\bf r}\primo_1,\widetilde{R})
\Psi_0({\bf r}_1,\widetilde{R})
\label{def:densmat}
\eeq
of the target nucleus, for all $n\leq$A. 
%#

The numerical calculation of $\rho_{n}$ within 
an {\it ab initio} microscopic approach involves 
prohibitive difficulties, even for the 
%#case of infinite nuclear matter considered in this work. 
%#In view of this problem, $\rho_{n}$ has to be modeled in terms 
%#of quantities which consistently incorporate the relevant physics
case of infinite nuclear matter considered here.
In view of this problem, we need to model $\rho_{n}$ in terms 
of quantities that consistently incorporate the relevant physics
%#
and can still be reliably calculated. The results presented in this 
paper have been obtained using an approximation scheme 
%#(hereafter referred to as {\it hole approximation} (HA)), 
(hereafter referred to as the {\it hole approximation}), 
%#
%#Explain meaning of name "hole approximation?? 
%#
in which $\rho_{n}$ is written in terms of the one-body density matrix
%#of Eq.(\ref{obdm}) and the half-diagonal two-body density matrix
of Eq.~(\ref{obdm}) and the half-diagonal two-body density matrix
%#
\beq
\rho_{2}({\bf r}_{1},{\bf r}_{2};{\bf r}_{1}^{\prime},{\bf r}_{2}) 
= A(A-1) \int d^3r_3 \ldots d^3r_A
\Psi^*_0({\bf r}\primo_1,\widetilde{R})
\Psi_0({\bf r}_1,\widetilde{R})\, .
\label{2hdm}
\eeq
%#The definition of the resulting $n$-body density matrix, 
The explicit formula for the resulting $n$-body density matrix, 
%#
\begin{equation}
\rho_{n}^{\rm HA}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n};
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n})
%#\  = \ 
 =  
%#
\frac{1}{[\rho_{1}({\bf r}_{1};{\bf r}_{1}^{\prime})]^{n-2}}
\prod_{i=2}^{n}\rho_{2}({\bf r}_{1},{\bf r}_{i};
{\bf r}_{1}^{\prime},{\bf r}_{i}) \, ,
\label{eq:HAI}
\end{equation}
%#shows that HA basically amounts to representing 
shows that the hole approximation represents
%#
the multinucleon spatial correlations involved in the definition of 
$\rho_{n}$ as a superposition of two-nucleon correlations. 

%#Eq.(\ref{eq:HAI})
%#was chosen among other expressions, which can be constructed 
%#using the same building blocks, mainly because it fulfills some basic 
Among other expressions that can be constructed from the same
building blocks, Eq.~(\ref{eq:HAI}) was chosen primarily because
it fulfills some basic 
%#
properties of the exact density matrices. 
In particular, $\rho_{n}^{\rm HA}$, which is obviously real, satisfies exactly 
the asymptotic factorization requirement
\beq
\lim_{|{\bf r}_n| \rightarrow \infty}
\rho_{n}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n};
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n})
= \rho \rho_{n-1}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n-1};
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n-1})\, ,
\eeq
while violating, although not severely, the sequential relation 
\begin{eqnarray}
\nonumber
%# & & \int d^3r_n\ \rho_{n}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n};
 & & \int d^3r_n\, \rho_{n}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n};
%#
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n}) \\
 & & \ \ \ \ \ \ \ \ \ \ \ \ \ \  = 
\left[A-(n-1)\right] 
\rho_{n-1}({\bf r}_{1},{\bf r}_{2},\ldots ,{\bf r}_{n-1};
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n-1})\, .
\label{seq}
\end{eqnarray}
%#Within HA Eq.(\ref{seq}) translates into
Within the hole approximation, Eq.~(\ref{seq}) translates into
%#
\begin{eqnarray}
\nonumber
%#& & \int d^3r_n\ \rho_{n}^{HA}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n};
& & \int d^3r_n\, \rho_{n}^{\rm HA}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n};
%#
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n}) \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 
%#(A-1)\ \rho_{n-1}^{HA}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n-1};
(A-1) \rho_{n-1}^{\rm HA}({\bf r}_{1},{\bf r}_{2},\ldots,{\bf r}_{n-1};
%#
{\bf r}_{1}^{\prime},{\bf r}_{2},\ldots,{\bf r}_{n-1}) \\ 
& & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 
%#[A-(n-1)]\ \rho_{n-1}^{HA} + 0(n-2)\ .
[A-(n-1)]\rho_{n-1}^{\rm HA} + O(n-2)\, .
%#
\end{eqnarray}

%#Besides $\rho_{2}$, the calculation of the 
%#HA generalized folding function requires the knowledge of the
In addition to $\rho_2$, calculation of the generalized folding function  
in the hole approximation calls for a knowledge of the 
%#
imaginary part of the quantity $\Gamma_{\bf q}$ of 
%#Eq.(\ref{def:Gamma}), i.e. of the imaginary part 
%#of the NN scattering amplitude $f_{\bf q}$. In this work, 
Eq.~(\ref{def:Gamma}), i.e.\ of the imaginary part 
of the $NN$ scattering amplitude $f_{\bf q}$. In this work, 
%#
we have employed the simple parametrization 
%#originally proposed in ref.\cite{Bassel-Wilkin67}
originally proposed in Ref.\cite{Bassel-Wilkin67}, namely
%#
\beq
%#Im\ f_{\bf q}({\bf k}) = \frac{|{\bf q}|}{4 \pi}\ \sigma_{NN}\ 
%#\exp{ \left[ -\beta^2 |{\bf k}|^2 \right] }\ .
{\rm Im} f_{\bf q}({\bf k}) = \frac{|{\bf q}|}{4 \pi}\, \sigma_{NN}
\exp{ \left( -\beta^2 |{\bf k}|^2 \right) }\, .
%#
\eeq
%#Numerical values of the total NN cross section $\sigma_{NN}$ and the
%#slope parameter $\beta$, resulting from fits to NN scattering data,  
%#are given in refs.\cite{Dobrovolsky-et-al83} and \cite{Silverman-et-al89}.
Numerical values of the total $NN$ cross section $\sigma_{NN}$ and the
slope parameter $\beta$ resulting from fits to $NN$ scattering data
are given in Refs.\cite{Dobrovolsky-et-al83} and \cite{Silverman-et-al89}.
%#

%#Using HA and the above equation to obtain $Im\ \Gamma_{\bf q}$ we can 
%#finally write the expression of the generalized folding function
Using the hole approximation (\ref{eq:HAI}) and together with the 
parametrization (42) of ${\rm Im} \Gamma_{\bf q}$, we can finally 
assemble the working expression
%#
\beq
%#{F}_{{\bf k},{\bf q}}(\omega) = \frac{1}{n({\bf k})}\ 2 Re\ 
%# \frac{1}{v}\ \int_{0}^{\infty} \frac{dz}{2\pi}\  
%#\exp{ \left[ i\omega \frac{z}{v} \right] }\  \int d{\bf r}_{11^\prime}\ 
%#{\rm e}^{i{\bf k}\cdot{\bf r}_{11^\prime}}\ E_{\bf q}(z,{\bf r}_{11^\prime})
{F}_{{\bf k},{\bf q}}(\omega) = \frac{1}{n({\bf k})} 2 {\rm Re}\, 
 \frac{1}{v} \int_{0}^{\infty}  \frac{dz}{2\pi}  
\exp{ \left( i\omega \frac{z}{v} \right) }  \int d{\bf r}_{11^\prime}\, 
e^{i{\bf k}\cdot{\bf r}_{11^\prime}} E_{\bf q}(z,{\bf r}_{11^\prime})
%#
\label{eq:fol5}
\eeq
%#with 
for the generalized folding function, with
%#
\beq
E_{\bf q}(z,{\bf r}_{11^\prime}) = 
%#\exp{ \left\{ -\frac{\sigma_{NN}}{16\pi \beta^{2}}\ 
%#J_{\bf q}(z,{\bf r}_{11^\prime}) \right\} }
\exp{ \left[ -\frac{\sigma_{NN}}{16\pi \beta^{2}}
J_{\bf q}(z,{\bf r}_{11^\prime}) \right] }
%#
\label{eq:fol5b}
\eeq
and 
\begin{eqnarray}
\nonumber
J_{\bf q}(z,{\bf r}_{11^\prime}) & = & \frac{1}{\rho_{1}(r_{11^\prime})}
\int_{0}^{\infty} r^2 dr \int_{-1}^{1}
d\cos \theta\ \rho_{2}(r_{11^\prime},r,r^\prime) \\
%#& \times & \exp{ \left[ -\frac{r^{2}\sin^{2}\theta}{4\beta^{2}} \right] }
& \qquad & \times
\exp{ \left[ -\frac{r^{2}\sin^{2}\theta}{4\beta^{2}} \right] }
%#
\left[ {\rm erf} \left( \frac{z + r\cos\theta}{2 \beta} \right) 
     - {\rm erf} \left( \frac{r\cos\theta}{2 \beta} \right)   \right] \, .
\label{eq:fol5c}
\end{eqnarray}
In the above equations, $\rho_{1}(r_{11^\prime})$ and 
$\rho_{2}(r_{11^\prime},r,r^\prime)$, with 
$r = |{\bf r}| = |{\bf r}_1 - {\bf r}_2|$ and 
$r^\prime = |{\bf r}^\prime| = |{\bf r}_{1}^{\prime} -
%#{\bf r}_2|$, are the nuclear matter 
{\bf r}_2|$, are the nuclear-matter 
%#
one-body and half-diagonal two-body density matrices, respectively, 
$v$ is the speed of the struck particle, the $z$ axis is 
%#chosen along the direction of ${\bf q}$ and $\theta$ is the angle between
%#${\bf r}$ and ${\bf q}$. It can be shown that the 
%#${\bf k}$-independent CGA folding function of ref.\cite{Benhar91} 
%#can be recovered from the above expression by substituting  
chosen along the direction of ${\bf q}$, and $\theta$ is the angle between
${\bf r}$ and ${\bf q}$.  It can be shown that the 
${\bf k}$-independent CGA folding function of Ref.\cite{Benhar91} 
can be recaptured from Eqs.~(43)--(45) by replacing
%#
$\rho_{2}({\bf r}_{1},{\bf r}_{2};{\bf r}_{1}^{\prime},{\bf r}_{2})$ 
with its fully diagonal part 
$\rho_{2}({\bf r}_{1},{\bf r}_{2};{\bf r}_{1},{\bf r}_{2})$.  
The numerical calculation of ${F}_{{\bf k},{\bf q}}(\omega)$ has
%#been carried out following the scheme proposed in ref.\cite{Flynn-et-al84} 
%#to obtain $n(k)$ and $\rho_{1}(r_{11^{'}})$, while the formalism 
%#employed to evaluate the nuclear matter half-diagonal two-body density matrix
%#has been developed in ref.\cite{Petraki-et-al00}. Both approaches use 
been performed by applying (i) the scheme proposed in Ref.\cite{Flynn-et-al84} 
to obtain $n(k)$ and $\rho_{1}(r_{11^{'}})$ and (ii) the formalism 
developed in Ref.\cite{Petraki-et-al00} to evaluate the nuclear-matter 
half-diagonal two-body density matrix.  Both approaches use 
%#
correlated many-body wave functions and FHNC integral equations to sum up 
selected cluster contributions to all orders in the relevant expansions.

In particular, $\rho_{2}({\bf r}_{1},{\bf r}_{2};
{\bf r}_{1}^{\prime},{\bf r}_{2})$ has been approximated by its 
leading $dd$ part in the FHNC formalism according to 
%#\ci{Petraki-et-al00}
\ci{Petraki-et-al00}; thus we set
%#
\begin{equation}
\rho_{2}({\bf r}_{1},{\bf r}_{2};{\bf r}_{1}^{\prime},{\bf r}_{2})
%#\  \simeq \ 
=
%#
\rho_{1}(r_{11^\prime})
g_{Qdd}(r) g_{Qdd}(r^\prime)\, ,
\end{equation}
where $g_{Qdd}(r)$ consists of the  $dd$ nodal and non-nodal
components of the FHNC expression for 
%#$\rho_{1}({\bf r}_{1};{\bf r}_{1}^{\prime})$ (the notation $dd$ 
$\rho_{1}({\bf r}_{1};{\bf r}_{1}^{\prime})$.  (The notation $dd$ 
%#
refers to the topological classification of the diagrams associated 
%#to the corresponding terms in the cluster expansion. 
%#See ref.\cite{Petraki-et-al00} for details).
with the corresponding terms in the cluster expansion. 
See Ref.\cite{Petraki-et-al00} for details.)
%#

Due to its weak dependence upon $\cos\phi = 
{\widehat {\bf r}}\cdot{\widehat {\bf r}}_{11^\prime}$, 
%#the quantity $g_{Qdd}(r^\prime)$ has been replaced with its angular
%#average according to 
the quantity $g_{Qdd}(r^\prime)$ is approximated by its angular
average, according to 
%#
\begin{equation}
g_{Qdd}(r_1^\prime) \rightarrow 
%#\overline{g}_{Qdd}(r^\prime) \
\overline{g}_{Qdd}(r^\prime) 
%#
\equiv \frac{1}{2}
%#\int_{-1}^{1} d\cos\phi \ g_{Qdd}\left[ (r^{2}+r_{11^{'}}^{2}
%#-2rr_{11^{'}}\cos\phi)^{1/2} \right]\ , 
\int_{-1}^{1} d(\cos\phi) \, g_{Qdd}\left[ (r^{2}+r_{11^{'}}^{2}
-2rr_{11^{'}}\cos\phi)^{1/2} \right]\, . 
%#
\label{eq:approxg}
\end{equation}
%#the resulting expression for the generalized folding function being
The resulting expression for the generalized folding function is
%#
\beq
%#{F}_{{\bf k}{\bf q}}(\omega) = \frac{4}{n({\bf k})}\ 
%#\int_{0}^{\infty} \frac{dz}{v} \cos \left( \omega \frac{z}{v} \right)\ 
%#\int_{0}^{\infty}\ r^2_{11^\prime} dr_{11^\prime}
%#\ j_0(kr_{11^\prime})\ \rho_{1}(r_{11^\prime})\ 
%#{\bar E}_{\bf q}(z,r_{11^\prime}) 
{F}_{{\bf k},{\bf q}}(\omega) = \frac{4}{n({\bf k})}
\int_{0}^{\infty} \frac{dz}{v} \cos \left( \omega \frac{z}{v} \right)
\int_{0}^{\infty}\ r^2_{11^\prime} dr_{11^\prime} \,
j_0(kr_{11^\prime}) \rho_{1}(r_{11^\prime}) 
{\bar E}_{\bf q}(z,r_{11^\prime}) \, ,
%#
\label{eq:fol8}
\eeq
%#where $j_0(x)$ is the spherical Bessel function and  
where $j_0(x)$ is the zeroth-order spherical Bessel function and  
\beq
{\bar E}_{\bf q}(z,{\bf r}_{11^\prime})  =  
%#\exp{ \left\{ -\frac{\sigma_{NN}}{16\pi \beta^{2}} 
%#{\bar J}_{\bf q}(z,{\bf r}_{11^\prime}) \right\} }
\exp{ \left[ -\frac{\sigma_{NN}}{16\pi \beta^{2}} 
{\bar J}_{\bf q}(z,{\bf r}_{11^\prime}) \right] }
%#
\label{eq:fol5bn}
\eeq
with 
\begin{eqnarray}
\nonumber
%#{\bar J}_{\bf q}(z,{\bf r}_{11^\prime}) & = & 
%#\ 2 \pi \rho \int_{0}^{\infty} r^2\ dr\ \int_{-1}^{1}
%#d\cos \theta\ g_{Qdd}(r) {\overline g}_{Qdd}(r^\prime) \\
%#& \times & \exp{ \left[ -\frac{r^{2}\sin^{2}\theta}{4\beta^{2}} \right] }
%#\left[ {\rm erf} \left( \frac{z + r\cos\theta}{2 \beta} \right)
%#     - {\rm erf} \left( \frac{r\cos\theta}{2 \beta} \right)   \right] \ .
{\bar J}_{\bf q}(z,{\bf r}_{11^\prime}) & = & 
\ 2 \pi \rho \int_{0}^{\infty} r^2 dr \int_{-1}^{1}
d\cos \theta\, g_{Qdd}(r) {\overline g}_{Qdd}(r^\prime) \\
& \qquad & \times \exp{ \left[ -\frac{r^{2}\sin^{2}\theta}{4\beta^{2}} \right] }
\left[ {\rm erf} \left( \frac{z + r\cos\theta}{2 \beta} \right)
     - {\rm erf} \left( \frac{r\cos\theta}{2 \beta} \right)   \right] \, .
%#
\label{eq:calE-last}
\end{eqnarray}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%#We have carried out the calculation of the norelativistic response of 
%#a realistic model of nuclear matter, based on the hamiltonian
We have carried out the calculation of the nonrelativistic response of 
a realistic model of nuclear matter, based on the Hamiltonian
%#
\beq
H = \sum_{i=1}^A \frac{{\bf p}^2}{2m} + \sum_{j>i=1}^A v_{ij} 
 + \sum_{k>j>i=1}^A\ V_{ijk}\, ,
\label{nucl:ham}
\eeq
where $v_{ij}$ and $V_{ijk}$ are potentials describing two- and 
%#three-nucleon interactions, and on the ground state wave function
%#written of the form
three-nucleon interactions, and on a variational ground-state wave function
%#
\beq
\Psi_{0}(R) = {\cal S} \bigl[   \prod_{i<j} F_{ij}  \bigr]  \chi_0(R)\, .
\label{def:NMWF}
\eeq 
%#In the above equation, ${\cal S}$ is a symmetrization operator, acting on 
In this trial form, ${\cal S}$ is a symmetrization operator acting on 
%#
the product of two-nucleon correlation operators, $F_{ij}$, and 
$\chi_0$ is the Slater determinant describing a noninteracting Fermi gas 
%#of nucleons carrying momenta ${\bf k}$ with 
%#$|{\bf k}|\le k_F = (3 \pi^2 \rho/2)^{1/3}$. The operator $F_{ij}$,
%#accounting for the correlation structure induced by
%#the nuclear hamiltonian, has been chosen in the form \cite{Pan79}
of nucleons with momenta ${\bf k}$ filling the Fermi sea, i.e., with
$|{\bf k}|\le k_F = (3 \pi^2 \rho/2)^{1/3}$. The operator $F_{ij}$,
which should reflect the correlation structure induced by
the nuclear Hamiltonian, has been chosen as \cite{Pan79}
%#
\begin{eqnarray}
F_{ij} & = & f_c(r_{ij}) + f_\sigma(r_{ij}) (\bm\sigma_i\cdot\bm\sigma_j)
 + f_\tau(r_{ij}) (\bm\tau_i\cdot\bm\tau_j)
 + f_{\sigma\tau}(r_{ij}) (\bm\sigma_i\cdot\bm\sigma_j)
( \bm\tau_i\cdot\bm\tau_j)
 \nonumber \\
 &   & \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \ f_t(r_{ij})\ S_{ij}
+ f_{t\tau}(r_{ij})\  S_{ij}\ (\bm\tau_i\cdot\bm\tau_j)\, .
\label{corr.op.-coord.}
\end{eqnarray}
%#where $S{ij} = 3(\bm\sigma_i \cdot {\bf r}_{ij})
%#(\bm\sigma_j \cdot {\bf r}_{ij})/|{\bf r}_{ij}|^2 - 
%#(\bm\sigma_i\cdot\bm\sigma_j)$, while 
%#$f_c(r), f_\sigma(r), f_\tau(r), f_{\sigma\tau}(r), f_t(r)$ and
Here $S_{ij} = 3(\bm\sigma_i \cdot {\bf r}_{ij})
(\bm\sigma_j \cdot {\bf r}_{ij})/|{\bf r}_{ij}|^2 - 
(\bm\sigma_i\cdot\bm\sigma_j)$ is the usual tensor operator, while
$f_c(r), f_\sigma(r), f_\tau(r), f_{\sigma\tau}(r), f_t(r)$, and
%# 
$f_{t\tau}(r)$ are correlation functions whose radial shapes
%#are determined minimizing the expectation value of the hamiltonian
%#(\ref{nucl:ham}) in the ground state described by Eq.(\ref{def:NMWF}) 
are determined by minimizing the expectation value of the Hamiltonian
(\ref{nucl:ham}) in the ground state described by Eq.~(\ref{def:NMWF}) 
%#
\cite{Pan79}. 

The PWIA response has been calculated using the nucleon spectral function
%#of ref.\ci{BFF89}, obtained from a nuclear hamiltonian including
%#the Urbana $v_{14}$ NN potential supplemented by the TNI model of
%#many-body interaction \cite{Lag81}. The same ingredients entering the
of Ref.\ci{BFF89}, obtained from a nuclear Hamiltonian including
the Urbana $v_{14}$ $NN$ potential supplemented by the TNI model of the
three-body interaction \cite{Lag81}. The same ingredients entering the
%# 
calculation of the spectral function have been employed in the calculation
of the density matrices needed to obtain the generalized folding function
%#defined by Eqs.(\ref{eq:fol8})-(\ref{eq:calE-last}).
defined by Eqs.~(\ref{eq:fol8})--(\ref{eq:calE-last}).
%#

%#The $|{\bf k}|$- and $\omega$-dependence of $F_{{\bf k}{\bf q}}(\omega)$ 
%#at $|{\bf q}| =$ 1.31 and 1.94 GeV/c, two values representative of the range 
%#covered by the data analyzed in ref.\ci{Benhar91,Benhar93,Benhar94}, are 
%#displayed in 
%#figs.\ref{GFF1} and \ref{GFF2}. For comparison, we also show the 
%#$|{\bf k}|$-independent folding functions obtained using the approach 
%#developed in I.
The dependence of the folding function $F_{{\bf k},{\bf q}}(\omega)$
on $k \equiv |{\bf k}|$ and $\omega$ is illustrated in Figs.~1 and 2 at  
$|{\bf q}| =$ 1.31 and 1.94 GeV/c, two values representative of the range 
covered by the data analyzed in Refs.\ci{Benhar91,Benhar93,Benhar94}.
%#
For comparison, we also show the 
$k$-independent folding functions obtained using the approach 
developed in I.
%#It clearly appears that in the region $k<k_{F}$ (in symmetric nuclear matter 
%#at equilibrium density $\rho=\rho_0=.16$ fm$^{-3}$ the Fermi momentum is 
%#$k_F$ = 1.33 fm$^{-1}$) $F_{{\bf k}{\bf q}}(\omega)$ is very close to the 
%#CGA folding function, whereas a strong $k$-dependence is observed at 
%#$k>k_{F}$.
We note that in symmetric nuclear matter at its equilibrium density
$\rho=\rho_0= 0.16$ fm$^{-3}$, the Fermi momentum is $k_F=1.33$ fm$^{-1}$.
It is apparent that in the region $k<k_{F}$, the folding function
$F_{{\bf k},{\bf q}}(\omega)$ is very close to its CGA counterpart,
whereas a strong $k$-dependence is observed at $k>k_{F}$.
%#
%#As $|{\bf k}|$ increases, the generalized folding function shrinks, and its
%#tail begins to oscillate, implying that larger $|{\bf k}|$ correspond to
%#weaker FSI. The results of figs.\ref{GFF1} and \ref{GFF2} confirm the naive 
%#expectation that, while the averaging procedure involved in the approach 
%#of ref.\ci{Benhar91} is quite resasonable for slow nucleons, 
%# the momentum dependence of the folding function associated with a fast 
%#nucleon, carrying momentum larger than $k_F$, must be explicitely 
%#taken into account.
%As $k$ increases, the generalized folding function shrinks, and its
%tail begins to oscillate, implying that larger $k$ values correspond
%to weaker FSI. The results of Figs.~\ref{GFF1} and \ref{GFF2} confirm the naive 
%expectation that while the averaging procedure involved in the approach 
%of Ref.\ci{Benhar91} is quite reasonable for slow nucleons, 
%the momentum dependence of the folding function associated with a fast 
%nucleon, carrying momentum larger than $k_F$, must be treated
%explicitly.
%#
At $|{\bf k}| \gtrsim k_F$ the generalized folding function shrinks, and its
tail begins to oscillate, implying that these $k$ values correspond
to weaker FSI.  On the other hand, for larger momenta, well
above the Fermi level, ${F}_{{\bf k},{\bf q}}(\omega)$ gets broader again.    
The results of Figs.~\ref{GFF1} and \ref{GFF2} confirm the naive
expectation that while the averaging procedure involved in the approach
of Ref.\ci{Benhar91} is quite reasonable in the region of
$|{\bf k}| < k_F$, where the nucleon momentum distribution is nearly
constant, the momentum dependence of the folding function associated with
fast nucleons, carrying momenta larger than $k_F$, must be treated 
explicitly.                                

%#Figs.\ref{FY1} and \ref{FY2} show the $y$-dependence of the nuclear 
%#matter $y$-scaling function, 
%#defined as
Figs.~\ref{FY1} and \ref{FY2} show the dependence of the nuclear 
matter $y$-scaling function
%# 
\beq
%#F({\bf q},y) = \frac{|{\bf q}|}{m}\ S({\bf q},y)
F({\bf q},y) = \frac{|{\bf q}|}{m} S({\bf q},y)
%# 
\label{def:Fy}
\eeq
%#with
on the scaling variable
%#
\beq
y = \frac{m}{|{\bf q}|} \left( \omega - \frac{|{\bf q}|^2}{2m} \right)\, , 
\label{def:y}
\eeq
%#evaluated at $|{\bf q}| =$ 1.94 and 1.31 GeV/c, respectively.
evaluated for the respective choices $|{\bf q}| =$ 1.94 and 1.31 GeV/c 
of the momentum transfer.
%#
Using the $y$-scaling function rather than the response makes it 
easier to directly compare FSI effects at different values of $|{\bf q}|$, 
as measured by the deviation from the PWIA results at fixed $y$.

%#Comparison of the PWIA scaling functions (dash-dot lines), 
%#to the results obtained from CGA (dashed lines) and the approach 
%#described in the previous Sections (solid lines) clearly shows that 
%#using the ${\bf k}$-dependent generalized folding
Comparison of the PWIA scaling functions (dash-dot lines) 
to the results obtained from the CGA (dashed lines) and the approach 
described in the previous sections (solid lines) clearly demonstrates that 
use of the ${\bf k}$-dependent generalized folding 
%#
function leads to a suppression of FSI effects in the region of large negative 
%#$y$, corresponding to the low energy tail of the response 
%#(see Eq.(\ref{def:y})). For example, at $y$ = 600 MeV/c the
%#differences between the dashed and solid lines are $\sim$ 70 \%
%#and $\sim$ 20 \% at $|{\bf q}|$ = 1.31 and 1.94 GeV/c, respectively.
$y$, corresponding to the low-energy tail of the response 
[see Eq.~(\ref{def:y})].  For example, at $y$ = 600 MeV/c the
differences between the dashed and solid lines are $\simeq$ 70 \%
and $\simeq$ 20 \% at $|{\bf q}|$ = 1.31 and 1.94 GeV/c, respectively.
%# 

The fact that the suppression of FSI appears to be more 
%#pronounced at the lower value of $|{\bf q}|$ indicates that a broader
%#
pronounced at the lower values of $|{\bf q}|$ indicates that a broader
CGA folding function is associated with a smaller effect of 
%#${\bf k}$-dependence. In fact, the larger value of the NN cross section at
%#$|{\bf q}|$ = 1.94 GeV/c, $\sigma_{NN} \sim $ 43 mb, compared to 
%#$\sigma_{NN} \sim $ 35 mb at $|{\bf q}|$ = 1.31 GeV/c, makes the 
${\bf k}$-dependence. In fact, the larger value of the $NN$ cross section at
$|{\bf q}|$ = 1.94 GeV/c, namely $\sigma_{NN} \simeq $ 43 mb compared to
$\sigma_{NN} \simeq $ 35 mb at $|{\bf q}|$ = 1.31 GeV/c, makes the 
%# 
CGA folding function broader at the higher momentum transfer (see
%#figs.\ref{GFF1} and \ref{GFF2}). 
Figs.\ref{GFF1} and \ref{GFF2}).
%# 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have carried out a calculation of the nuclear matter response in which 
%#NN correlations, that are known to play an important role
%#in the low energy tail of the reponse, have been taken into account both 
$NN$ correlations, which are known to play an important role
in the low-energy tail of the response, have been taken into account both 
%#
in the initial and final states.
The effects of dynamical correlations in the initial state have been 
consistently incorporated into the PWIA calculation, based on a realistic 
%#spectral function obteined from an {\it ab initio} microscopic many-body
spectral function obtained from an {\it ab initio} microscopic many-body 
%#
approach \ci{BFF89}. 

The same ingredients entering the calculations of the spectral function 
%#of ref.\ci{BFF89} have been employed in the calculation of corrections
%#to PWIA arising from FSI between the struck nucleon and the spectator system.
of Ref.\ci{BFF89} have been employed in the calculation of corrections
to the PWIA arising from FSI between the struck nucleon and the spectator
system.
%#
%#Our many-body treatment of FSI, while being based on the same 
%#scheme employed in I, involving the eikonal and frozen approximations,
%#allows one to to go beyond the convolution expression of the response and
Our many-body treatment of FSI is based on much the same 
scheme as applied in I and is therefore predicated on the eikonal and 
frozen approximations.  However, the new treatment allows us
to go beyond the simple convolution expression for the response and
%# 
explicitly take into account the dependence on the initial momentum of 
%#the struck nucleon, which is averaged over in the CGA.  
the struck nucleon, which is averaged over in the CGA.
%#  

Numerical results show that the momentum dependence of the generalized 
%#folding function produces a sizable effect in the low energy tail
folding function produces a sizable effect in the low-energy tail
%#  
of the $y$-scaling function in the range of momentum transfer 
%#1 $< |{\bf q}| <$ 2 GeV/c, covered by the inclusive electron-nucleus
%#scattering data analyzed in refs.\ci{Benhar91,Benhar93,Benhar94}.
%#While repeating the analysis of refs.\ci{Benhar91,Benhar93,Benhar94}
%#within the approach proposed in 
%#this paper would certainly be of great interest, it must be pointed out 
1 $< |{\bf q}| <$ 2 GeV/c covered by the inclusive electron-nucleus 
scattering data analyzed in Refs.\ci{Benhar91,Benhar93,Benhar94}. 
While repeating the analysis of Refs.\ci{Benhar91,Benhar93,Benhar94}
within the approach proposed here
%#
would certainly be of great interest, it must be pointed out 
%#that using a momentum dependent generalized folding function 
%#would in no way help to improve the agreement between CGA and data. 
that using a momentum-dependent generalized folding function
would in no way help to improve the agreement between the CGA and 
the data. 
%#
%#Best to join the 2 paragraphs, not to lose the connections.
%#
As shown in I, the discrepancy between the CGA results and the measured 
cross sections
increases as $|{\bf q}|$ increases, while the suppression of FSI 
%#due to the momentum dependence of the folding function appers to be
due to the momentum dependence of the folding function appears to be
%# 
larger at lower momentum transfer. A different mechanism leading 
%#to a quenching of FSI and exhibiting the opposite momentum transfer 
%#dependence, as the one associated with the CT model employed in 
%#refs.\ci{Benhar91,Benhar93,Benhar94},
to a quenching of FSI and exhibiting the {\it opposite} momentum-transfer 
dependence, such as the one associated with the color-transparency model 
employed in Refs.\ci{Benhar91,Benhar93,Benhar94},  
%#
still seems to be needed to reconcile theory and data.

\section{acknowledgments}
This research was supported in part by the U. S. National Science
Foundation under Grant No.~PHY-9900713 (JWC) and by the Italian MIUR 
through the {\em Progetto di Ricerca di Interesse Nazionale: Fisica 
Teorica del Nucleo Atomico e dei Sistemi a Molti Corpi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{references}
%#Format adjusted to PRC - jwc.
%
\bibitem{book}
{\it Modern Topics in Electron Scattering}, edited by B. Frois and
I. Sick (World Scientific, Singapore, 1991).              

\bibitem{Benhar91} O. Benhar, A. Fabrocini, S. Fantoni, G.~A. Miller, V.~R.
Pandharipande, and I. Sick, Phys. Rev. C {\bf 44}(1991)2328.

\bibitem{Benhar93} O. Benhar and V.~R. Pandharipande, Phys. Rev. C {\bf 
47}, 2218 (1993).
 
\bibitem{Benhar94} O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nucl. Phys.
{\bf A579}, 493 (1994).

\bibitem{silcla}
J. W. Clark and R. N. Silver, in {\it Proceedings of the Fifth International
%#Conference on Nuclear Reaction Mechenisms}, edited by E. Gadioli 
Conference on Nuclear Reaction Mechanisms}, edited by E. Gadioli 
%#
(Universit\`a degli Studi di Milano, Milano, 1988), p. 531.

\bibitem{Glauber} R. J. Glauber, in {\it Lectures in Theoretical Physics}, 
edited by W. E. Brittin {\it et al.} (Interscience Publishers, Inc., 
New York, 1959).

\bi{papi}
V. R. Pandharipande and S. C. Pieper, Phys. Rev. C {\bf 45}, 791 (1992).

\bi{bl}
O. Benhar and S. Liuti, Phys. Lett. {\bf B389}, 649 (1996).

\bi{CT1}
S. J. Brodsky, in {\it Proceedings of the Thirteenth International Symposium
on Multiparticle Dynamics}, edited by E. W. Kittel, W. Metzger, and A. Stergion
(World Scientific, Singapore, 1982).

\bi{CT2}
A. Mueller, in {\it Proceedings of the Seventh Rencontre de Moriond},
edited by J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, 1982).    

\bi{CT}
G. R. Farrar, H. Liu, L. L. Frankfurt, and M.I. Strikman, Phys. Rev. Lett.
{\bf 61}, 686 (1988).    

\bi{PRL2001}
O. Benhar, A. Fabrocini, and S. Fantoni, Phys. Rev. Lett. {\bf 87}, 
052501 (2001). 

\bi{Negele} 
J. W. Negele and H. Orland, {\it Quantum Many-Particle Systems}
(Addison-Wesley, New York, 1988).            

\bi{Carraro91} 
C. Carraro and S. E. Koonin, Nucl. Phys. {\bf A524}, 201 (1991).

\bi{Bassel-Wilkin67} 
R. H. Bassel and C. Wilkin, Phys. Rev. Lett. {\bf 18},871 (1967).

\bibitem{Dobrovolsky-et-al83} 
A. V. Dobrovolsky {\it et al.}, Nucl. Phys. {\bf B214}, 1 (1983).

\bi{Silverman-et-al89}
B. H. Silverman {\it et al.}, Nucl. Phys. {\bf A499}, 763 (1989).

\bi{Flynn-et-al84} 
M. F. Flynn, J. W. Clark, R. M. Panoff, O. Bohigas, and S. Stringari, 
Nucl. Phys. {\bf A427}, 253 (1984).

\bi{Petraki-et-al00}
M. Petraki, E. Mavrommatis, and J. W. Clark, Phys. Rev. C 
{\bf 64}, 024301 (2001).

\bi{BFF89}
O. Benhar, A. Fabrocini, and S. Fantoni, Nucl. Phys. 
{\bf A505}, 267 (1989).

\bibitem{Lag81}
I. E. Lagaris and V .R. Pandharipande, Nucl. Phys. {\bf A359}, 331 (1981).     
 
\bibitem{Pan79}
V. R. Pandharipande and R. B. Wiringa, Rev. Mod. Phys. {\bf 51}, 821 (1979).   

\end{references}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace*{1.in}
\centerline
{\epsfig{figure=fig1a.ps}}
\vspace*{.2in}
\caption{
%#$|{\bf k}|$- and $\omega$-dependence of 
%#the generalized folding function, defined as in 
%#Eqs.(\protect\ref{eq:fol8})-(\protect\ref{eq:calE-last}),  
%#at $|{\bf q}| = 1.94$ GeV/c. The diamonds show the results of the approach 
%#of ref.\protect\ci{Benhar91}, in which the dependence of the folding function
Dependence of the generalized folding function $F_{{\bf k},{\bf q}}(\omega)$ 
of Eqs.~(\protect\ref{eq:fol8})--(\protect\ref{eq:calE-last})
on $k=|{\bf k}|$ and $\omega$,
at $q=|{\bf q}| = 1.94$ GeV/c. The diamonds show the results of the approach 
of Ref.\protect\ci{Benhar91}, in which the dependence of the folding function
%# 
upon the initial nucleon momentum ${\bf k}$ is neglected.
}
\label{GFF1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace*{1.in}
\centerline
{\epsfig{figure=fig1b.ps}}
\vspace*{.2in}
\caption{
%#Same as in fig.\protect\ref{GFF1}, but for momentum transfer
Same as in Fig.\protect\ref{GFF1}, but for momentum transfer
%#
$q=|{\bf q}|$ = 1.31 GeV/c.
}
\label{GFF2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace*{1.in}
\centerline
{\epsfig{figure=fig2.ps}}
\vspace*{.2in}
\caption{
%#$y$-dependence of the nuclear matter scaling function, defined by 
%#Eqs.(\protect\ref{def:Fy}) and (\protect\ref{def:y}), at 
%#$|{\bf q}|$ = 1.94 GeV/c. The dash-dot line shows the PWIA result, while
Dependence of the nuclear-matter scaling function $F({\bf q},y)$ defined by 
Eqs.~(\protect\ref{def:Fy})--(\protect\ref{def:y}) on the
scaling variable $y$, at
%# 
$q=|{\bf q}|$ = 1.94 GeV/c. The dash-dot line shows the PWIA result, while 
%#the solid and dashed lines correspond to calculations
the solid and dashed lines correspond respectively to calculations
%#
carried out using the generalized folding function of 
%#Eqs.(\protect\ref{eq:fol8})-(\protect\ref{eq:calE-last}) and the 
%#$|{\bf k}|$-independent CGA folding function of ref.\protect\ci{Benhar91}.
Eqs.~(\protect\ref{eq:fol8})--(\protect\ref{eq:calE-last}) and the 
$|{\bf k}|$-independent CGA folding function of Ref.\protect\ci{Benhar91}.
%#
}
\label{FY1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\vspace*{1.in}
\centerline
{\epsfig{figure=fig3.ps}}
\vspace*{.2in}
\caption{
%#Same as in fig.\protect\ref{FY1}, but for momentum transfer
Same as in Fig.\protect\ref{FY1}, but for momentum transfer
%#
$q=|{\bf q}|$ = 1.31 GeV/c.
}
\label{FY2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}

nucl-th0201019.extracted_bib
\bibitem}
\def\lab{\label}
\def\re{\ref}
\def\PkE{P(|{\bf k}|,E)}
\def\magk{|{\bf k}|}
\def\magq{|{\bf q}|}
\def\vecq{{\bf q}}
\def\primo{^\prime}
\def\shat{{\widehat S}({\bf q},t)}  
\def\bm #1{ \bbox{#1} }    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}   

\draft

%#Hyphen everywhere in "final-state" when used as an adjective.
\title{Final-state interactions in the response of nuclear matter}

\author
%#{M. Petraki$^1$,  E. Mavrommatis$^1$, O. Benhar$^2$, J.W. Clark$^3$, 
%#A. Fabrocini$^4$,  S. Fantoni$^5$}
{M. Petraki,$^1$  E. Mavrommatis,$^1$ O. Benhar,$^2$ J.~W. Clark,$^3$ 
A. Fabrocini,$^4$ and S. Fantoni$^5$}
%#

\address
%#{ $^1$ Physics Department, Division of Nuclear and Particle Physics\\
%#University of Athens. GR-15771 Athens, Greece\\
%#$^2$ INFN, Sezione Roma 1. I-00185 Roma, Italy\\
%#$^3$ Department of Physics, Washington University. St. Louis MO 633130, USA \\
%#$^4$ Department of Physics {\em E. Fermi}, University of Pisa, and INFN. I-56100 Pisa, Italy \\
%#$^5$ International School for Advanced Studies (SISSA). I-30014 Trieste, 
%#Italy\\ }
{$^1$Physics Department, Division of Nuclear and Particle Physics\\
University of Athens, GR-15771 Athens, Greece\\
$^2$INFN, Sezione di Roma 1, I-00185 Roma, Italy\\
$^3$Physics Department, Washington University, St. Louis MO 633130, USA \\
$^4$Department of Physics "E. Fermi", University of Pisa, 
 and INFN, Sezione di Pisa, I-56100 Pisa, Italy \\
$^5$International School for Advanced Studies (SISSA), I-30014 Trieste, 
Italy\\ }
%#

\date{\today}

\maketitle


\begin{abstract}

Final-state interactions in the response of a many-body system to an 
%#external probe delivering large momentum are generally described 
%#using the eikonal aproximation, for the trajectory of the struck particle,  
external probe delivering large momentum are normally described 
using the eikonal approximation, for the trajectory of the struck particle,  
%#
and the frozen approximation, for the positions of the spectators. We 
propose a generalization of this scheme, in which the initial momentum of 
%#the struck particle is explicitly taken into account. The results 
%#of numerical
the struck particle is explicitly taken into account. Numerical
%#
calculations of the nuclear matter response at 1 $< |{\bf q}| <$ 2 GeV/c
show that the inclusion of this momentum dependence leads to a sizable effect
in the low energy tail. Possible implications for the analysis of 
existing electron-nucleus scattering data are discussed. 


\bibitem{book}
{\it Modern Topics in Electron Scattering}, edited by B. Frois and
I. Sick (World Scientific, Singapore, 1991).              


\bibitem{Benhar91} O. Benhar, A. Fabrocini, S. Fantoni, G.~A. Miller, V.~R.
Pandharipande, and I. Sick, Phys. Rev. C {\bf 44}(1991)2328.


\bibitem{Benhar93} O. Benhar and V.~R. Pandharipande, Phys. Rev. C {\bf 
47}, 2218 (1993).
 

\bibitem{Benhar94} O. Benhar, A. Fabrocini, S. Fantoni, and I. Sick, Nucl. Phys.
{\bf A579}, 493 (1994).


\bibitem{silcla}
J. W. Clark and R. N. Silver, in {\it Proceedings of the Fifth International
%#Conference on Nuclear Reaction Mechenisms}, edited by E. Gadioli 
Conference on Nuclear Reaction Mechanisms}, edited by E. Gadioli 
%#
(Universit\`a degli Studi di Milano, Milano, 1988), p. 531.


\bibitem{Glauber} R. J. Glauber, in {\it Lectures in Theoretical Physics}, 
edited by W. E. Brittin {\it et al.} (Interscience Publishers, Inc., 
New York, 1959).

\bi{papi}
V. R. Pandharipande and S. C. Pieper, Phys. Rev. C {\bf 45}, 791 (1992).

\bi{bl}
O. Benhar and S. Liuti, Phys. Lett. {\bf B389}, 649 (1996).

\bi{CT1}
S. J. Brodsky, in {\it Proceedings of the Thirteenth International Symposium
on Multiparticle Dynamics}, edited by E. W. Kittel, W. Metzger, and A. Stergion
(World Scientific, Singapore, 1982).

\bi{CT2}
A. Mueller, in {\it Proceedings of the Seventh Rencontre de Moriond},
edited by J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, 1982).    

\bi{CT}
G. R. Farrar, H. Liu, L. L. Frankfurt, and M.I. Strikman, Phys. Rev. Lett.
{\bf 61}, 686 (1988).    

\bi{PRL2001}
O. Benhar, A. Fabrocini, and S. Fantoni, Phys. Rev. Lett. {\bf 87}, 
052501 (2001). 

\bi{Negele} 
J. W. Negele and H. Orland, {\it Quantum Many-Particle Systems}
(Addison-Wesley, New York, 1988).            

\bi{Carraro91} 
C. Carraro and S. E. Koonin, Nucl. Phys. {\bf A524}, 201 (1991).

\bi{Bassel-Wilkin67} 
R. H. Bassel and C. Wilkin, Phys. Rev. Lett. {\bf 18},871 (1967).


\bibitem{Dobrovolsky-et-al83} 
A. V. Dobrovolsky {\it et al.}, Nucl. Phys. {\bf B214}, 1 (1983).

\bi{Silverman-et-al89}
B. H. Silverman {\it et al.}, Nucl. Phys. {\bf A499}, 763 (1989).

\bi{Flynn-et-al84} 
M. F. Flynn, J. W. Clark, R. M. Panoff, O. Bohigas, and S. Stringari, 
Nucl. Phys. {\bf A427}, 253 (1984).

\bi{Petraki-et-al00}
M. Petraki, E. Mavrommatis, and J. W. Clark, Phys. Rev. C 
{\bf 64}, 024301 (2001).

\bi{BFF89}
O. Benhar, A. Fabrocini, and S. Fantoni, Nucl. Phys. 
{\bf A505}, 267 (1989).


\bibitem{Lag81}
I. E. Lagaris and V .R. Pandharipande, Nucl. Phys. {\bf A359}, 331 (1981).     
 

\bibitem{Pan79}
V. R. Pandharipande and R. B. Wiringa, Rev. Mod. Phys. {\bf 51}, 821 (1979).   

