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\begin{document}
\title{Hadrons in the Nuclear Medium}
\author{M~M~Sargsian$^1$, J~Arrington$^2$, W~Bertozzi$^3$, W~Boeglin$^1$,
C~E~Carlson$^4$, D~B~Day$^5$, L~L~Frankfurt$^{6}$, K~Egiyan$^{7}$,
R~Ent$^{8}$, S~Gilad$^3$, K~Griffioen$^4$, D~W~Higinbotham$^{8}$,
S~Kuhn$^9$, W~Melnitchouk$^8$, G~A~Miller$^{10}$,
E~Piasetzky$^{6}$, S~Stepanyan$^{8,9}$,
M~I~Strikman$^{11}$ and L~B~Weinstein$^{9}$}
\address{$^1$\ Florida International University, University Park, FL, USA}
\address{$^2$\ Argonne National Laboratory, Argonne, IL, USA}
\address{$^3$\ Massachusetts Institute of Technology, Cambridge, MA, USA}
\address{$^4$\ College of William and Mary, Williamsburg, VA, USA}
\address{$^5$\ University of Virginia, Charlottesville, VA, USA}
\address{$^6$\ Tel Aviv University, Tel Aviv, Israel}
\address{$^7$\ Yerevan Physics Institute, Yerevan, Armenia}
\address{$^8$\ Thomas Jefferson National Accelerator Facility,
Newport News, VA, USA}
\address{$^9$\ Old Dominion University, Norfolk, VA, USA}
\address{$^{10}$\ University of Washington, Seattle, WA, USA}
\address{$^{11}$\ Pennsylvania State University, University Park, PA, USA}
%\ead{sargsian@fiu.edu}
\begin{abstract}
Quantum Chromodynamics, the microscopic theory of strong interactions,
has not yet been applied to the calculation of nuclear wave functions.
However, it certainly provokes a number of specific questions
and suggests the existence of novel phenomena in nuclear physics which are
not part of the the traditional framework of the meson-nucleon description of nuclei.
Many of these phenomena are related to high nuclear densities and the role of
color in nucleonic interactions. Quantum fluctuations in the spatial separation between nucleons may lead to
local high density configurations of cold nuclear matter in nuclei, up to
four times larger than typical nuclear densities.
We argue here that experiments utilizing the higher energies available upon completion of the Jefferson Laboratory energy upgrade will be able to probe
the quark-gluon structure of such high density configurations and
therefore elucidate the fundamental nature of nuclear matter.
We review three key experimental programs: quasi-elastic electro-disintegration
of light nuclei, deep inelastic scattering from nuclei at $x>1$, and
the measurement of tagged structure functions. These interrelated programs are all aimed at the exploration
of the quark structure of high density nuclear configurations.
The study of the QCD dynamics of elementary hard processes
is another important research direction and nuclei provide a unique avenue to explore
these dynamics. In particular, we argue that the
use of nuclear targets and large values of momentum transfer
at energies available with the Jefferson Laboratory upgrade
would allow us to determine whether the physics of
the nucleon form factors is dominated by spatially small configurations of three
quarks. Similarly, one could determine if hard two-body processes such as
exclusive vector meson electroproduction are dominated by production of mesons
in small-size $q\bar q $ configurations.
\end{abstract}
\submitto{\JPG}
%\pacs{}
\maketitle
\section{Open questions in our understanding of nuclear structure}
Quantum Chromodynamics (QCD), the only legitimate candidate for a theory of strong
interactions, is a non-Abelian gauge theory with gauge group $SU(3)$ (in color space)
coupled to quarks in the fundamental (triplet) representation. It contains
the remarkable postulate of exact $SU(3)$ color symmetry.
Quarks and gluons carry color and are the fundamental particles that interact
via the color force. An important feature
of this theory is that the $u$ and $d$ quarks, for the purpose of
the strong interaction, are regarded as massless. In this limit,
chiral symmetry is spontaneously broken in the ground state of QCD.
Nuclei should provide an excellent testing ground for this theory because
nuclei are stable systems, made up of quarks and gluons bound together by the
strong force. However, the quarks and gluons are hidden and nuclei seem to
be composed of hadrons bound together by the exchange of evanescent mesons.
The hadrons that are the constituents of nuclei are identified with color
singlet states and have strong interactions very different in nature than that
of gluon exchange by colored quarks and gluons.
There seems at first glance to be a contradiction between the fundamental theory,
QCD, and the very nature of nuclei. This contradiction can be resolved by accounting
for the fact that effective theories of the strong interaction, in which the
degrees of freedom are hadrons, can yield results that are equivalent,
over some range of kinematic resolution, to those of QCD. Indeed, it is widely realized
that the phenomenon of the spontaneously broken chiral symmetry in
QCD is equivalent to the pseudovector pion-nucleon
interaction, which accounts for the long-range, low momentum transfer, aspects of
nuclear physics. The same chiral structure also accounts for the relatively weak
soft pionic fields in nuclei. Furthermore, at low momentum transfer the
relevant degrees of freedom are quasi-particle excitations of the system described
by Landau-Fermi liquid theory.
Thus, while QCD is the fundamental theory of strong interactions, it is not
required to explain the structure of nuclei observed in low-energy processes.
The availability of high energy beams provides the opportunity to
observe features of nuclear structure at small distance scales, which may
reveal the presence of QCD as the ultimate source of the strong interaction.
The central features of QCD, quarks and color, lead to two separate, but related,
avenues of experimental investigation which we examine in this review.
The first avenue is that of high density nuclear matter. What
happens during the brief intervals when two or more nucleons overlap in space?
Can we account for the interactions using meson exchanges, or do we
instead need to consider explicitly quark aspects such as
quark exchanges between nucleons and the kneading of the nucleon's
constituents into six- or nine-quark bags?
At high densities, can we detect the presence of superfast quarks,
i.e. quarks carrying a light cone momentum fraction greater than that
of a nucleon at rest?
The second avenue is concerned with the role of color in high momentum transfer
processes which are exclusive (or sufficiently exclusive) so that the
interference effects of quantum mechanics must be taken into account.
Then, the use of high momentum kinematics offers a remarkable
opportunity to observe the color-singlet fluctuations of a
hadron in configurations of very small spatial extent, {\it viz.},
point-like configurations. If such configurations exist, they do not interact
strongly because the effects of the emitted gluons cancel, just as the
interaction of an electric dipole decreases with size of dipole
due to the cancellation of the electromagnetic interactions.
Such effects are called color-coherent phenomena and include the subject
known as color transparency. Thus, we are concerned with studying the diverse
subjects of high-density cold nuclear matter and color coherence.
While these subjects are connected by their importance in establishing QCD
as the underlying theory of the strong interaction, they may also come
together in the understanding of exotic phenomena in nature.
Studies of the local high-density fluctuations are important to understanding the
equation of state of cold, dense matter which is crucial in understanding the
possibility of the transition of neutron stars to more dense states like
``quark stars'', whose experimental existence was suggested recently~\cite{Drake}.
The phenomenon of color transparency means that for a short period of time
a nucleon can be in a spatially small configuration. In this configuration, the
nucleon can tunnel through the potential barrier given by the repulsive core of the
nucleon-nucleon ($NN$) interaction. This provides a possible mechanism for a phase
transition to a new form of matter for a sufficiently dense nuclear system,
such as in the core of neutron star.
The following is a brief outline of this review. The quark-gluon
physics of high density fluctuations in
nuclei is examined in Sec.~2, which begins with a brief discussion of the
relevant history. The discovery of the nuclear EMC effect almost twenty years ago
brought the subjects of quarks into nuclear physics with great impact. However,
the specific causes of the modifications observed in nuclear structure functions have
not yet been identified with certainty. Thus, the
questions regarding quark dynamics of nuclei
raised by that momentous discovery have not yet been answered. The various
ideas invoked to explain this effect are reviewed. We then argue that an
experimental program focused on discovering
scaling in deep inelastic scattering at high values of
Bjorken-$x$ ($>$1.2) will reveal the nature of high-density fluctuations of cold nuclear
matter. Further deep inelastic scattering
measurements of backward going nucleons in coincidence with the
outgoing electron offer the promise of finally determining the cause of
the EMC effect. Section~3 is concerned with the role of color in nuclear
physics. Electron scattering experiments at high momentum transfer in which one
or two nucleons are knocked out, performed with sufficient precision to verify
that no other particles are produced, could reveal the fundamental nature of color
dynamics: that, for coherent processes, the effects of gluons emitted by a small color
singlet object are canceled. In that case, the dynamical nature of the
nucleon form factor at high momentum transfer will be revealed.
Similarly motivated experiments in which a vector meson is produced in a coherent reaction
with a deuteron target at large momentum transfers are also discussed.
Section~4 presents a summary of the plan proposed to investigate the questions
arising from the study of hadrons in the nuclear medium.
\section{Quark-gluon Properties of Superdense Fluctuations of Nuclear Matter}
\subsection {A Brief History and Short Outlook}
We begin this section with a brief discussion of the history
of experimental lepton-nuclear physics. Prior to the completion of Thomas Jefferson National Accelerator Facility (Jefferson Lab), experimental studies of nuclei using lepton probes
could be discussed in terms of two clearly different classes:
(a) Experiments performed at electron machines with low incident electron energies,
$E_{inc}\leq 1$~GeV, in which the typical energy and momentum transfers, $\nu$ and $\vec q$,
were comparable to the nuclear scale
\begin{equation}
\nu \leq 100~\mbox{MeV}, \vert \vec q \vert \leq 2\;k_F,
\end{equation}
where $k_F\approx 250$~MeV/c is the characteristic Fermi momentum of nuclei.
These reactions were inclusive $(e,e')$ and semi-inclusive $(e,e'N)$ and covered mainly
the quasi-elastic and the low lying resonance regions (the $\Delta$ isobars), corresponding
to relatively large values of Bjorken-$x$ ($x=Q^2/2m_p\nu$, where $Q^2=q^2-\nu^2$).
(b) Deep inelastic scattering (DIS) experiments which probed nuclei at $x<1$ and large $Q^2$
scales, greater than about $4$~GeV$^2$, which resolved the parton constituents of the nucleus.
The first class of experiments are unable to resolve the short range structure of nuclei, and the
second, while having good resolution, typically involved inclusive measurements
which averaged out the fine details and were limited by low luminosities and other factors.
It is interesting to notice that there is a clear gap between
the kinematic regions of these two classes of experiments.
This corresponds exactly to the optimal range for the study of the nucleonic
degrees of freedom in nuclei, $1.5 \leq Q^2 \leq 4$~GeV$^2$, for which short-range
correlations (SRCs) between nucleons can be resolved, and the quark degrees of
freedom are only a small correction.
Work at Jefferson Lab has started to fill this gap in a
series of quasi-elastic $A(e,e'), A(e,e'N)$, and $A(e,e'N_1N_2)$ experiments.
Previously, this range was just touched by inclusive experiments at
SLAC\cite{R1,R2,D1,D2} which also provided the first measurement of $A=2,3,4$ form
factors at large $Q^2$.
A number of these high-energy experiments probe the light-cone
projection of the nuclear wave function and in particular the
light-cone nuclear density matrix, $\rho_A^N(\alpha,p_\perp)$, in the
kinematics where the light-cone momentum fraction $\alpha \geq 1$
($A\geq \alpha \geq 0$) so that short range correlations between nucleons
play an important role.
It is already known that the existence of SRCs gives a natural explanation of
the practically $A$-independent spectrum of the emission of fast backward nucleons and
mesons, as well as the practically $Q^2$- and $x$-independent value of the
ratio of $\sigma_A(x,Q^2)/\sigma_{D(^3He)} (x,Q^2)$ for $Q^2\geq 1$~GeV$^2$ and
$1.5 \leq x
\leq 2$~\cite{fsds93,liuti93,Kim01}. Overall, these and other high-energy data
indicate a significant probability of these correlations ($\sim$25\%
of nucleons in heavy nuclei) which involve momenta larger than the Fermi momentum.
These probabilities are in qualitative agreement with calculations of
nuclear wave functions using realistic $NN$ potentials~\cite{Pa97}.
Most of this probability ($\sim$80\%) is related to two-nucleon SRCs
with the rest involving $N\geq 3 $ correlations (see e.g., Ref.~\cite{FS81}).
The current experiments at Jefferson Lab will allow a detailed study of two-nucleon
SRCs~\cite{Arrin99,e02019}, and take a first look at three-nucleon correlations~\cite{e02019,Larry01}.
We expect that experiments at an electron
energy of up to $11$~GeV~\cite{wp} will allow further explorations of SRCs in the three-nucleon
correlation region, substantially extending the region of initial nucleon momenta
and the recoil nucleus excitation energies that
can be probed in quasi-elastic $A(e,e'), A(e,e'N)$, and $A(e,e'N_1N_2)$ reactions.
The 12~GeV upgrade at Jefferson Lab\footnote{The energy upgrade planned for Jefferson Lab will provide 11~GeV electrons to the experimental Halls A, B, and C, see\cite{wp}.} will also be of a great benefit for studies of deep inelastic
scattering off nuclei (experiments in class (b)). It will enable us to extend the high
$Q^2$ inclusive $A(e,e')$
measurement at $x\geq 1$ to the deep inelastic region where the process is dominated
by the scattering from individual quarks in the nucleus with momenta exceeding the
average momentum of a nucleon in the nucleus (superfast quarks).
Such quarks are likely to originate from configurations
in which two or more nucleons come close together. Thus, these measurements will
complement the studies of SRCs by exploring their sub-nucleonic structure.
The second extension of class (b) reactions concerns the measurement of
nuclear DIS reactions in the $(e,e'N)$ semi-inclusive regime, in which the detection of
the additional nucleon in spectator kinematics will allow us to tag
electron scattering from a bound nucleon.
These studies may provide a number of unexpected results similar to
those in the inclusive studies of parton densities in nuclei, which
yielded the observation of the EMC effect~\cite{EMC1,EMCB1,EMCB2}:
a depletion of the nuclear quark parton density as compared to that
in a free nucleon at $0.4 \leq x \leq 0.8 $, which
demonstrated unambiguously that nuclei cannot be described merely
as a collection of nucleons without any extra constituents.
Among several suggested interpretations of the EMC effect, the idea of mesonic
degrees of freedom (nuclear binding models) was the only one which produced
a natural link to the meson theory of nuclear forces.
This explanation naturally led to a prediction of a large enhancement of the
anti-quark distribution in nuclei~\cite{dyth,dyth1,dyth2}.
A dedicated experiment was performed
at Fermi Lab using the Drell-Yan process to measure the ratio
$R_A^{\bar q}(x,Q^2)\equiv\frac{\bar q_A(x,Q^2)}{\bar A q_N(x,Q^2)}$.
The result of this experiment~\cite{DY} was another major surprise:
instead of a $\sim 10-20$\% enhancement of
$R_A^{\bar q}(x,Q^2)$, a few percent suppression was observed.
Further indications of the unexpected partonic structure of nuclei
come from the studies of the gluon densities in nuclei
using exact QCD sum rules\cite{FS88,FLS} as well as the analysis ~\cite{Pirner} of the
data on the scaling violation of
$R_A^{ q}(x,Q^2)$
indicate
that the gluon densities should be significantly enhanced in nuclei at
$x\sim 0.1$.
However the inclusive nature of these measurements does not provide
insight into the QCD mechanism for the depletion of the DIS structure functions.
The semi-inclusive DIS $(e,e'N)$ reactions with the detection of backward-going nucleons will
test many models of the EMC effect, which was previously impossible due to
the inclusive nature of $(e,e')$ reactions.
Semi-inclusive DIS reactions are ultimately related to the understanding of
the QCD dynamics of multi-nucleon systems at small distances. Thus, it is crucial that
these studies be done in parallel with the above mentioned studies of SRCs.
\subsection{The Big Picture of Small Distance Fluctuations}
The proton electromagnetic radius is $\sim 0.86$ fm, and in the ground state of
infinite nuclear matter the average (center-to-center) distance between
nearby nucleons is $\sim$1.7 fm. Thus, under normal conditions nucleons are
closely packed and nearly overlap. Despite this, quark aspects of
nuclear structure are not evident for most of nuclear physics. A possible
dynamical explanation for this, in terms of the strong-coupling limit of QCD, was
presented in Ref.~\cite{scqcd}. This can also be seen from the fact that
typical nuclear excitations are at much lower energies than
nucleon excitations ($\raisebox{-.4em}{$>$}\atop\sim$~500~MeV). Besides, in low energy
QCD, the pion, being a pseudogoldstone, interacts with the amplitude $\propto k_{\pi}$, leading to
a suppression of the near--threshold pion (multi-pion) production.
However, quantum fluctuations must occur in any quantum system, and well-designed
experiments may expose the physics occurring when nucleons occasionally
encounter each other at smaller than average distances. If such a fluctuation
reduces the center-to-center separation to $1$~fm, then there is a significant
region of overlap. Assuming a uniform nucleon charge distribution, the density
in the region of overlap is twice that of the nucleon, or about four to five
times that of normal nuclear
matter, $\rho_0=0.17$ nucleon/fm$^3$. At these densities, the physics of confinement,
which typifies the strong-coupling limit of QCD, may no longer be applicable and the chiral symmetry
may be (partially) restored, see e.g. \cite{Brown}.
One can also think of these high-density fluctuations as nuclear states with excitation
energies large enough to modify the structure of underlying nucleonic constituents.
Hence, dense nuclear matter
may look very different from a system of closely packed nucleons.
{}From this viewpoint, it is encouraging that experimental data on the
EMC effect indicate that deviations from the expectations of the nucleonic
model of nuclei grow approximately linearly with the nuclear density,
suggesting that the properties of the quark-gluon droplets could indeed
deviate very strongly from those of a collection of nucleons.
It is important to recall that the properties of dense nuclear matter are
closely related to outstanding issues of QCD such as the existence of
chiral symmetry restoration and deconfinement, as well as determining the
nature of the onset of quark-gluon degrees of freedom and the structure
of the phase transition from hadronic to quark-gluon states of matter.
In QCD, transitions to new phases of matter are possible in different regimes
of density and temperature. In particular, it has been suggested~\cite{Wilc}
that nuclear matter could exist in a color superconductivity phase caused by the
condensation of diquarks. Recent estimates suggest that the average nuclear density
could lie in between that of the dilute nucleon phase and the superconducting
phase~\cite{Carter,Rapp}. It is therefore natural to ask whether one can observe precursors
of such a phase transition by studying the quark-gluon properties of superdense
droplets of nuclear matter, i.e., configurations when two or more
nucleons come close together. It should also be remembered that the EMC effect has been interpreted
as a delocalization of quarks in nuclei, which is qualitatively consistent with a
proximity to the phase transition. More recently, measurements of in-medium
proton form factors also hint at such a modification of nucleon structure~\cite{Dieterich}.
So far, the major thrust of studies looking for the phase transitions in hadronic matter
has been focused in the high temperature region (Fig.~\ref{Fig.phase}), which may be realized in
high-energy heavy-ion collisions. The low temperature region of high densities,
critical for building the complete picture of phase transitions
and for determining if the transition from neutron to quark stars is possible,
is practically unexplored.
We wish to argue that this unexplored low temperature region, crucial for the
understanding of the equation of state of neutron stars, is amenable to studies
using high energy lepton probes. Jefferson Lab, upgraded to higher energies~\cite{wp},
would be able to explore this region and provide studies of nuclear fluctuations as dense as
4--5 $\rho_0$.
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.8in,file=phase.eps}
\end{center}
\caption{Phase diagram for baryonic matter.}
\label{Fig.phase}
\end{figure}
\subsection{Outline of the experimental program}
The intellectual problems associated with establishing the existence
of high density fluctuations of nuclear matter are formidable. We cannot
conceive of a single experiment which would be able to answer all of the
interesting questions about the role of non-nucleonic degrees of freedom,
but a series of correlated measurements may succeed.
As mentioned above, we anticipate that studies to be done at Jefferson Lab at
$6$~GeV will provide additional information about nucleonic degrees of
freedom in the $2N$ sector of SRCs. This will certainly help us in refining the
envisioned experimental program.
Our view is that three key experimental programs, for which the use of
$12$~GeV electrons is essential, are needed.
\begin{itemize}
\item Quasi-elastic electro-disintegration of light nuclei at
$1.5\leq Q^2\leq 4$~GeV$^2$, covering a wide range of angles and
missing momenta\footnote{We define the missing momentum as the difference
between momenta of knocked-out nucleon and virtual photon.}
(in excess of $1$~GeV/c), with the goal of
probing the structure of SRCs.
\item Deep inelastic scattering at $x>1$, with the goal of observing
superfast quarks in nuclei.
\item Tagged structure functions (measurement of a nucleon from the target
fragmentation region in coincidence with the outgoing electron) with the goal
of directly observing the presence of non-nucleonic degrees of freedom
in droplets of superdense matter.
\end{itemize}
The first goal of studying SRCs using quasi-elastic electro-disintegration of
light nuclei is based on the capabilities of Jefferson Lab to perform fully
exclusive measurements. The exclusive reactions using a deuteron target in a wide
range
of recoil nucleon momenta and angles will allow us to obtain insight into
many fundamental questions, e.g. the probability of pre-existing non-nucleon
components in the deuteron wave function, the transition of meson currents
to the quark-interchange mechanism in the $NN$ interaction, and the dynamical
structure of the nuclear core.
For heavier ($A=3,4$) nuclei, the capability to measure a recoil
nucleon with light cone momentum fraction $\alpha \geq 2 $ will
provide direct access to the three-nucleon correlation
structure of the nuclear wave function providing the unprecedented opportunity
to study the dynamics of the three-nucleon force.
The second goal of learning about superfast quarks will be accomplished by
studying inclusive electron nucleus scattering in the scaling region at $x\geq
1$, where this process measures, {\it in a model independent way}, the probability
of finding a quark that carries a larger light-cone momentum than
one would expect from a quark in a low-momentum nucleon ($k \leq k_F$).
Obviously, very few such superfast quarks can originate from the effects of the
nuclear mean field, but the uncertainty principle indicates that such quarks
in nuclei could arise from fluctuations of superdense configurations consisting
either of a few overlapping nucleons with large momenta, or of more complicated
multi-quark configurations. The study of the structure of these superdense fluctuations
would be the major goal of deep inelastic scattering measurements at high $x$.
For instance,
high precision measurements of the $A$-dependence of the cross section would permit
the investigation of the density dependence of the superfast quark
probability in the nuclear medium.
The functional dependence of this probability on Bjorken-$x$ would allow
us to disentangle the contributions of two- or multi-nucleon
droplets from more exotic components (e.g., 6-, or 9-quark bags).
The third goal of determining the modification of the quark-gluon
structure of nucleons in a dense nuclear environment could be realized by
measuring tagged structure functions in semi-inclusive DIS reactions.
For example, the observation of a backward-moving
nucleon or nucleons will be used as a tag to select hard scattering
from a short-range correlation of nucleons while the DIS characteristics of the
event will be used to study modifications of the parton density.
These studies primarily require the isolation of a backward-moving nucleon
for which a tagging procedure, rather than reconstruction of a missing nucleon,
will provide the cleanest signature.
Successful completion of these programs will allow significant progress in
understanding such unresolved topics of strong interaction physics as,
{\em interaction of quarks at high densities and low temperatures}, and
{\em the quark-gluon structure of nuclei}.
\subsection{Nucleonic Structure of Short-Range Correlations}
The first step of the program will be to study the structure of SRCs
in terms of nucleonic degrees of freedom, and map out the strength of
two-nucleon and multi-nucleon correlations in nuclei. This can be accomplished
by combining moderate energy ($4-6$~GeV) inclusive scattering experiments from
light and heavy nuclei with moderate and higher energy coincidence reactions on
few-body nuclei, as described in the previous section. Here we present an
experimental overview of these reactions.
\subsubsection{Qualitative Features of High-$Q^2$ Electro-Nuclear Reactions}
The observation of SRCs in nuclei has long been considered one of the most significant aims
of nuclear physics. These correlations, though elusive, are not small:
calculations of nuclear wave functions with realistic $NN$ potentials consistently
indicate that in heavy nuclei about $25\%$ of the nucleons have momenta above the
Fermi surface~\cite{Pa97}. This corresponds to about $50\% $ of the kinetic energy
coming from SRCs. The problem has been a lack of experimental data at high-momentum transfer kinematics
which could decisively discriminate between the effects of the SRCs in the initial state and the
long-range multi-step effects such as final state interactions (FSIs) and meson exchange currents (MECs).
Before we can proceed to study SRCs in nuclei, we must consider these processes. In the following
sections, we will show that some of these effects are suppressed at high $Q^2$, while the remaining effects do not mask the dynamics of the SRCs.
\medskip
\medskip
\noindent {\it Final State Interactions:}\\
Although final state interactions in nucleon knockout reactions do not disappear at
large $Q^2,$ two important simplifications occur which make the
extraction of information about the short-range nuclear structure possible:
\begin{itemize}
\item In high energy kinematics a new (approximate) conservation law exists
- the light-cone momentum fractions of slow nucleons do not change if
they scatter elastically with the ejected nucleon which maintains its
high momentum during the rescattering~\cite{FSS97,SM}.
To demonstrate this feature let us consider the propagation of a fast nucleon with four-momentum
$k_1= (\epsilon_1,0,k_{1z})$ through the nuclear medium. We chose the $z$-axis
in the direction of ${\bf k_1}$ such that
${k_{1-}\over m}\equiv {\epsilon_1-k_{1z}\over m} \approx {m\over 2k_{1z}} \ll 1$.
After this nucleon makes a small angle rescattering in its interaction
with a bound nucleon of four-momentum $p_1=(E_1,p_{1\perp},p_{1z})$, it
maintains its high momentum and leading $z$-direction
with the four-momentum $k_2=(\epsilon_2,k_{2\perp},k_{2z})$, where
${k_{2\perp}^2\over m^2_N}\ll 1$.
The bound nucleon four-momentum becomes $p_2=(E_2,p_{2\perp},p_{2z})$.
The energy momentum conservation for this scattering allows us to write for
the ``$-$'' component ($p_-=E-p_z$):
\begin{equation}
k_{1-} + p_{1-} = k_{2-} + p_{2-}.
\label{claw}
\end{equation}
{}From Eq.(\ref{claw}) for the change of the ``$-$'' component of the bound
nucleon momentum one obtains
\begin{equation}
{\Delta p_{-}\over m} \equiv {p_{2-}-p_{1-}\over m}\equiv \alpha_2- \alpha_1
= {k_{1-}-k_{2-}\over m}\ll 1,
\end{equation}
where we define $\alpha_i = {p_{i-}\over m}$ ($i=1,2$) and use the fact that
${k_{2\perp}^2\over m^2_N},{k_{1\perp}^2\over m^2_N} \ll 1$.
Therefore $\alpha_1 \approx \alpha_2$. The latter indicates that, with an increase of
energy, a new conservation law emerges in which
the light-cone momentum fractions of slow nucleons, $\alpha$,
are conserved. The unique simplification of the high energy rescattering is
that although both the energy and momentum of the bound nucleon are distorted due to
rescattering, the combination $E-p_z$ is not.
\begin{figure}[t]
\vspace{-0.4cm}
\begin{center}
\epsfig{angle=0,width=4.2in,file=alconf.eps}
\end{center}
\caption{The accuracy of the conservation of $\alpha$ as a function of the
propagating nucleon momentum, $k_1$ at different values of average transferred
(during the rescattering) momenta, $$. Note that the eikonal
approximation is theoretically justified for $k_1\geq 2$~GeV/c.}
\label{Fig.alpha_conservation}
\end{figure}
Figure~\ref{Fig.alpha_conservation} demonstrates the accuracy of this conservation law for
a propagating nucleon over a range of four-momenta relevant to our discussion.
It is important to note that the average transferred
momentum in the $NN$ rescattering amplitude for $p_{N}\sim 3-10$~GeV/c
is $\langle k^2_t\rangle \approx 0.25$~(GeV/c)$^2$. Thus, starting from $3$~GeV/c momenta of
the propagating nucleon, the conservation of
$\alpha$ ($\sim {\cal O}(1)$) is accurate to better than $5\%$ and improves
with increasing momentum. Note that the conservation of $\alpha$ to this level is sufficient for studying
SRCs for which the $\alpha$ distribution of the nucleons has a rather
slow, ($\propto \exp (-\lambda \alpha),\lambda \sim 7$) variation.
Indeed, this variation is expected to be much flatter than the
corresponding distribution generated by a mean-field interaction.
\item The small angle rescatterings of high-energy ($2\leq p_N \le 10$~GeV/c) nucleons can be
described by the
generalized eikonal approximation~(GEA) which takes into account the difference
between the space-time picture of the proton-nucleus scattering (a proton
coming from $-\infty$) and $A(e,e'p)$ process (a proton is produced inside the nucleus),
and also accounts for the non-zero Fermi momenta and recoil energy of
rescattered nucleons~\cite{FSS97,SM,FGMSS95}.
{\em Notice that the Glauber theory~\cite{Glauber} can not be applied in this case since it
is derived for the case of the stationary scatterrers
\footnote{In several publications (see e.g.~\cite{Bal,Nal,Bia}) the Glauber approximation has
been applied to $(e,e'N)$ reaction for nonzero values of Fermi momenta and recoil energy,
without taking into account the fact that this approximation is valid only for stationary
scatterrers. In the GEA, the effects of final state interactions are derived using
the eikonal approximation, but without assuming that the scatterrers are stationary.
This causes the presence of the effective longitudinal momentum transfer
(which is proportional to the reaction's recoil energy) in the rescattering amplitude.
Thus this result is analogous to the coherence length effect which was first discussed
for total cross sections in lepton-nucleus scattering, and for diffractive vector meson
production~\cite{Gribov,Yennie}. For the later process this effect was recently observed
by HERMES~\cite{HERMES}. Numerically, the GEA leads to a significant modification of the
cross section as compared to the Glauber approximation. An example is deuteron break-up
at large values of recoil energy~\cite{FSS97}.}.
}
Additionally, the description of small angle rescattering is simplified
due to approximate energy independence of the $pp$ and $pn$ total cross sections in the high $Q^2$
limit (starting at $Q^2\geq 2$~GeV$^2$, which corresponds to $p_N\geq 2$~GeV/c).
\end{itemize}
The above two features of small angle rescattering in the high $Q^2$ domain make it possible
to evaluate FSIs reliably, identifying kinematic requirements which will allow us
to separate SRC effects from long range FSI contributions.
Note that the minimal value of $Q^2$ for which one expects the eikonal approximation to be valid
can be estimated from the application of Glauber theory to $pA$ reactions.
Extensive studies have demonstrated long ago that the
Glauber theory of $pA$ processes can describe the data within a few percent
starting at energies as low as $E^{inc}_p\geq 0.8-1$~GeV, which corresponds to
$Q^2\geq 1.5$~GeV$^2$ in $(e,e'N)$ and ($e,e'N_1N_2$) reactions.
\medskip
\medskip
\noindent{\it Contribution of Meson Exchange Currents:}\\
The major problem we face in the estimation of MECs in $A(e,e'N)$ processes
is that with an increase of energies the virtuality of the exchanged mesons grows
proportional to $Q^2$ ($\gg m_{meson}^2$). Even though the idea of deeply virtual
exchanged mesons is highly complicated or may even be meaningless
(see discussion in Ref.~\cite{Feynman}), one can still estimate its $Q^2$-dependence
as compared to the SRC contribution.
In a kinematic setting typical for studies of SRCs, in which the knocked-out nucleon
carries almost the entire momentum of the virtual photon (while the missing four-momentum
of the recoil system does not change with $Q^2$), the $Q^2$-dependence of the MEC amplitude
can be estimated as follows:
\begin{eqnarray}
A_{MEC}^\mu & \sim & \int d^3p\cdot \Psi(p){J_m^\mu(Q^2)\over
(Q^2+m_{\mathrm{meson}}^2)}\Gamma_{MNN}(Q^2)
\nonumber \\
& \propto & \int d^3p\cdot \Psi(p)\left({1\over (Q^2+m_{\mathrm{meson}}^2)^2
(1+Q^2/\Lambda^2)^2}\right),
\label{MEC}
\end{eqnarray}
where $J_m^\mu(Q^2)$ is the meson electromagnetic current proportional
to the elastic form factor of the meson $\sim {1\over Q^2 +m_{\mathrm{meson}}^2}$, and
$m^2_{\mathrm{meson}}\approx 0.71$~GeV$^2$. For the meson-nucleon vertices,
$\Gamma_{MNN}(Q^2)$, we assume a dipole dependence with $\Lambda\sim 0.8-1$~GeV$^2$.
Since the leading $Q^2$-dependence in the SRC contribution comes from the
nucleon elastic form factors, Eqn.~(\ref{MEC}) will result in
an additional $\sim(1+Q^2/\Lambda^2)^{-2}$ suppression of the MEC amplitude as
compared to the SRC contribution. Note that this gives an upper limit to the
MEC contribution, since for large $Q^2$ the quark counting rule predicts
stronger $Q^2$ suppression for $\Gamma_{MNN}(Q^2)\sim {1\over Q^6}$~\cite{FKAS}.
Thus, one expects that MEC contributions will be strongly
suppressed as soon as $Q^2$ is greater than $m^2_{\mathrm{meson}}$ and $ \Lambda^2$,
both $ \sim 1$~GeV$^2$. This conclusion is relevant only for small angle nucleon knock-out
kinematics (optimal for studies of SRCs). In the large angle kinematics quark-exchange
mechanism may become important similar to the large angle $\gamma d\to pn$ reactions~\cite{gdpn}.
\medskip
\medskip
\noindent
{\it Isobar Current (IC) Contribution:}\\
For the case of IC contributions, the virtual photon produces the
$\Delta$ isobar in the intermediate state which subsequently
rescatters off the spectator nucleon through the $\Delta N \to NN$
channel. There are several factors which contribute to the suppression of
IC contributions at high $Q^2$ as compared to the SRC contributions.
The main factors which should be emphasized are the energy dependence of the
$A_{\Delta N\to NN}$ amplitude and the $Q^2$-dependence of the
electromagnetic $\gamma^*N\Delta$ transition form factors, as compared with the
elastic $NN\to NN$ amplitude and the $\gamma^*N$ form factor respectively.
The $\Delta N\to NN$ amplitude is known to be dominated by the pion Reggeon exchange
with the $\rho$-Reggeon which dominates at very high energies being a small correction
up to the energies $\sqrt{s} \sim 30$~GeV~\cite{ISR}.
Based on the rule that the energy dependence of the Feynman amplitude of the scattering process
is defined by the spin, $J$ of the exchanged particle as: $A\sim s^J$ one observes
that the $\Delta N\to NN$ transition amplitude is suppressed
at least by a factor $\propto 1/Q^2$ (at $Q^2 \geq 2$ GeV$^2$)
as compared to the elastic $NN\to NN$ amplitude leading to a
similar suppression for IC contribution. In addition there is the
experimental indication that the
electromagnetic $\gamma^*N\Delta$ transition is
decreasing faster with $Q^2$ as compared
to the elastic $\gamma^*NN$ transition amplitude~\cite{Stoler}.
\medskip
\medskip
It follows from the above discussions that the smallest value of $Q^2$ required for effective
studies of SRCs in the discussed class of experiments is
$Q^2\geq 1.5$~GeV$^2$. The upper limit for $Q^2$ for studies of SRCs comes
from the onset of color coherence phenomena at $Q^2> 4$~GeV$^2$,
when FSIs will not maintain their energy independent characteristic for small angle
hadronic rescattering (a discussion on color coherence is given in Sec.3).
Hence the optimal range for probing SRCs is
\begin{equation}
1.5 \leq Q^2\leq 4~ \mbox{GeV}^2.
\label{range}
\end{equation}
This range is large enough to check the validity of the $Q^2$-independence of the
extracted parameters of the SRCs. Jefferson Lab experiments at $6$~GeV and very large missing momenta
can reach the lower limit
of this range and they would have only a very limited access to the upper limit of
Eqn.~(\ref{range}). With the upgrade of the beam energy the whole range of Eqn.~(\ref{range}) will
be be easily accessible for Jefferson Lab. This also permits a wider coverage of missing momenta
and excitation energies of the recoil nuclear system.
\subsubsection{Specific Reactions for Studies of Short Range Correlations}
The study of the $(e,e')$, $(e,e'N)$ and $(e,e'NN)$ reactions in the
($1.5 \leq Q^2\leq 4$~GeV$^2$)
range will allow a direct measurements of the nucleon momentum distributions and
spectral functions out to large momenta, between $400$ and $700$~MeV/c.
Here one expects that the nucleon degrees of freedom are dominant and one can explore
short-range correlations.
Although two nucleon correlations are expected to be the dominant
part of the SRCs, triple and higher order SRCs (where more than two nucleons come close together)
are significant as well - they are estimated to constitute
$\sim$20\% of all SRCs~\cite{FS81}.
At initial momenta $> 700$~MeV/c the non-nucleonic degrees of freedom should
play an increasingly dominant role and for the first time one will be able to investigate
them in detail.
The prime reactions for these investigations are:
{\bf $\bullet$} ~~ {\bf Inclusive {\boldmath $A(e,e')X$} reaction at {\boldmath $x>1$}.} In the kinematic
range of Eqn.~(\ref{range}), $A(e,e')X$ reactions at $x>1$ proceed mainly by
quasi-elastic scattering of electrons from bound nucleons. By increasing the values of
$x$ and $Q^2$ for these reactions, one can achieve a better discrimination between
SRCs and long-range multi-step processes. This will allow the measurement of several
average characteristics of SRCs: the probabilities of two-nucleon correlations
in nuclei~\cite{Kim01} and $k_\perp$-averaged longitudinal/light-cone momentum
distributions in SRCs~\cite{John01}. Extending measurements into the $x>2$ region at sufficiently
large values of $Q^2$ will allow us to probe the three-nucleon correlations. The
signature of the dominance of 3N correlations in these reactions will be the
onset of scaling of the cross section ratio of scattering from
nuclei with $A>3$ to that of $A=3$ nucleus in the $x>2$ region. Alternatively,
one can look for multi-nucleon correlations by extending the $Q^2$ range of
measurements for $x<2$, and comparing heavy nuclei to deuterium. QCD evolution
of the structure function leads to a shift of strength from high-$x$
to lower-$x$ values as $Q^2$ increases. If only two-nucleon correlations are
present, then heavy nuclei will have no appreciable strength above $x=2$ and the
evolution should be essentially identical to the evolution of deuterium, and the ratio
will remain constant. If, however, there is significant strength above $x=2$ in
heavy nuclei, coming from multi-nucleon correlations, then this strength will
shift into the $x<2$ region, and the ratio $\sigma_A/\sigma_D$ will increase
with $Q^2$.
{\bf $\bullet$} ~~ {\boldmath $d(e,e'pn)$}. While inclusive reactions have a large kinematic
reach, they cannot probe the details of the structure of the SRCs.
The starting point for these studies is the simplest exclusive
reaction: $e+d \to e+p+n$. It will provide a test of the current understanding of the
dynamics of electro-disintegration processes especially since the wave function of the deuteron
is reasonably well known for a wide range of momenta ($\leq 400$~MeV/c).
Progress in building tensor polarized deuteron targets makes it feasible to study
the polarization degrees of freedom of the disintegration reaction at sufficiently large $Q^2$.
In this case a direct separation of $S$ and $D$ waves is possible. Hence this process will provide
an ultimate test of our understanding of the short-distance $NN$ interactions. In particular,
it will allow us clearly discriminate between
predictions based on approximations to
the Bethe-Salpeter equation and light-cone approaches to the description
of the deuteron as a two-nucleon relativistic system~\cite{FS78,FS83}.
Such studies, using the upgraded energies at Jefferson Lab will be a natural continuation of the
present experimental program of electro-disintegration of the deuteron which is currently
focused on studies in the momentum range of $\leq 400$~MeV/c~\cite{UJ,KG,EGS,WJKUV}.
{\bf $\bullet$} ~~ {\bf {\boldmath $A(e,e'N)$} and {\boldmath $A(e,e'NN)$} with
{\boldmath $A\geq 3$}}.
These reactions, with one nucleon produced along $\vec q$ carrying almost the entire momentum
of the virtual photon, allow measurements of the light-cone density matrix of the nucleus
$\rho_A^N(\alpha,p_\perp)$ for large values of excitation energy, $E_m$, of the residual system.
Within the SRC picture, it is expected that $E_m$ increases with increasing
initial momentum of the ejected nucleon.
In the non-relativistic approximation, the average
excitation energy is $\left\approx p_m^2/2m_N$,
where $\vec p_m$ is the missing momentum of the ejected nucleon.
Measuring the $E_m$--$p_m$ correlation at high $Q^2$ will be one of the
signatures of scattering from SRCs.
Note that polarized $^3$He targets used in $A(e,e'NN)N$ reactions
will play a special role for probing SRCs due to the
relative simplicity of the wave function and the unique
possibility to probe the spin structure of $pp$ and $pn$ correlations.
In particular, there exist kinematic regions where a minimum in
the S-wave $pp$ wave function can be explored, and the P-wave contribution
can be isolated. These measurements will provide stringent tests of the
structure of $A=3$ systems and will test current interpretations
of measurements of the $^3$He form factors at large $Q^2$.
{\bf $\bullet$} ~~{\bf {\boldmath $A(e,e'N_fN_b)$} reactions} with one nucleon ($N_f$)
moving forward and the other ($N_b$) moving backward for nuclei with $A\leq 12$ can be
used to investigate how the excitation energy
is shared between nucleons. It is expected that the dominant contribution will originate from
two-nucleon correlations. In this case $N_b$ should carry most of the excitation energy.
A comparison of the yields of ($pp$), ($pn$) and ($nn$) processes will provide a detailed check
of the mechanisms of the reaction and provide a quantitative comparison between
the wave functions of two nucleon SRCs in the isospin zero and one channels
(the former is expected to dominate by a factor $\geq 4$
for a large range of momenta). In addition, if there is significant strength in
multi-nucleon correlations, it should be manifest in the low excitation tail
of the nuclear spectral function for large momenta of the ejected nucleon.
This is best observed through the $(e,e'NN)$ reaction in which two nucleons
are emitted in the backward direction relative to the virtual photon momentum~\cite{FS88}.
In the high $Q^2$ regime, these reactions will allow us to study the parton
structure of the three nucleon correlations at very high densities.
{\bf $\bullet$} ~~ {\bf Quenching in the {\boldmath $A(e,e'N)$} scattering for {\boldmath $A\geq 10$}.}
The numerous $A(e,e'p)$ experiments at low $Q^2\leq 0.3$~GeV$^2$ have observed
the shell structure of nuclei for the momenta of residual nuclear system $\leq k_F$.
At the same time they observed a significant
($\sim 0.5 $) suppression of the absolute cross sections as compared to
shell model expectations.
Jefferson Lab measurements
on $^{12}$C~\cite{Dutta} and $^{16}$O~\cite{bert} targets at intermediate
$Q^2 \sim 0.6-0.8$~GeV$^2$ suggest a quenching of about a factor of 0.7--0.8.
For large $Q^2$ ($\raisebox{-.4em}{$>$}\atop\sim$~2~GeV$^2$) where virtual photons
resolve individual nucleons the analysis~\cite{Lap,Zhalov} of the current data
including the Jefferson Lab data~\cite{quench}
indicate that this suppression has practically disappeared.
Note however that the current comparison of quenching at different
$Q^2$ should be considered as a semi--quantitative since different models of
nucleon absorptions were used for treating $A(e,e'p)$ reactions at
different $Q^2$.
Hence it is very important to perform similar measurements
including a separation of the different structure functions at
$Q^2\sim 2- 3$~GeV$^2$ to investigate the $Q^2$-dependence of the quenching.
Such studies would be also of importance for the interpretation of the
color transparency searches which will be discussed in Section 3.
Overall, a series of experiments at Jefferson Lab can provide a detailed knowledge of the
nucleon component of the SRCs, mapping out both the strength and structure of two-nucleon
(and multi-nucleon) SRCs for both light and heavy nuclei.
With the proposed upgrade in beam energy, these studies
will probe for the first time the quark substructure of the nucleon configurations
at short space-time separations.
\subsubsection{Experimental Requirements}
Carrying out the scientific program described above involves inclusive $A(e,e')$ and double
coincidence $A(e,e'p)$ measurements as well as triple coincidence measurements of the $(e,e'pN)$
reactions on light nuclei, {\it e.g.} carbon. Inclusive measurements can be performed over a wide
range at $6$~GeV, and extended to the highest $x$-values with $11$~GeV incident electron energies.
The coincidence measurements are to be performed at the highest incident energy ($11$~GeV)
and momentum transfer ($Q^2=4-6$~GeV/c$^2$).
These kinematical conditions are essential for covering the largest possible missing momentum
range (up to $1$~GeV/c). For this we need an electron spectrometer similar to that of the current
high resolution spectrometer (HRS) of Hall A at Jefferson Lab, with extended momentum acceptance
up to about $10$~GeV/c, and a proton spectrometer with momentum acceptance up to about $2.5$~GeV/c.
For the triple coincidence measurements we also need a third large solid angle proton spectrometer
and a neutron array. For protons, the BigBite spectrometer~\cite{BB}, which at its maximum
current can detect particles in the momentum range of $250-900$~MeV/c with moderate momentum
resolution of $\Delta p/p=0.8\%$, can be used. Behind the BigBite spectrometer a neutron counter
array can be installed with a matching solid angle. This is basically the set-up proposed for an
approved $(e,e'pN)$ experiment~\cite{add5:CEBAFpro} with the current Jefferson Lab accelerator
(at $5$~GeV).
The limiting factor for luminosity in these measurements is the singles
rate in the large solid angle detectors used to detect the
recoil particles in coincidence with the knockout proton.
Assuming that for the upgraded energies at $11$~GeV the singles rate
will be similar to that of the current one at $5$~GeV beam, one can use
$100~\mu A$ beam current and $1$~mm carbon target to obtain the nuclear luminosity
${\cal L} = 6\times 10^{36}$ cm$^{-2}$sec$^{-1}$. For the
electro-disintegration of the deuteron we assume a target similar
to the one used in the current Jefferson Lab experiments. The luminosity
in this case (for $100 \mu A$ beam and $15$~cm of $0.16$~gr/cm$^2$ deuteron
target) is: ${\cal L}=3.7\times 10^{37}$cm$^{-2}$sec$^{-1}$.
For the differential cross section for the $d(e,e'p)n$ reaction, estimated in the missing
momentum range of $450-500$~MeV/c in (almost) anti-parallel kinematics and at $Q^2=4$GeV$^2$,
one obtains
\begin{equation}
{{d\sigma \over{dE_e d\Omega_e d\Omega_p}}_d= 1 {pb \over MeV sr^2}}.
\label{m1}
\end{equation}
Assuming $\Delta E_e=50$~MeV and $\Delta \Omega_e=\Delta \Omega_p = 10$~msr,
the counting rate is 650 events/hour.
With a requirement of at least 500~events in 100~hours of beam time one will be able to measure
the differential cross section as low as $0.005{pb\over MeV sr^2}$. Thus
one will be able to measure well beyond $500$ MeV/c region of missing
momentum in the $d(e,e'p)n$ reaction.
To estimate the triple coincidence counting rate for the
$A(e,e'pn)X$ reaction, we assume that events with
high missing momenta ($p_m>500$~MeV/c) originate mainly from
two-nucleon SRCs. Under this condition, the
measured differential cross section, in which the solid angle of the
spectator neutron is integrated around the direction
defined by the deuteron kinematics, can be approximated in the following
way~\cite{add5:CEBAFpro}:
\begin{equation}
{d\sigma^A \over{dE_e d\Omega_{e}\Omega_p}}={ K_0\times a_2\times Z}\times
{d\sigma^d \over{dE_e d\Omega_e \Omega_p}}
\label{eli1}
\end{equation}
where $ {d\sigma^d \over{dE_e d\Omega_e d\Omega_p}}$ is the differential
cross section of the $d(e,e'p)n$ reaction~\cite{FGMSS95}, $a_2$ is
defined through the
ratio of cross sections for inclusive $(e,e')$ scattering from heavy
nuclei to deuterium,
measured at $x>1$ where the scattering from SRCs dominates,
($a_{2}(^{12}C) = 5.0\pm 0.5$~\cite{fsds93}). $K_0$ is a kinematical
factor related to the
center of mass motion of $np$ correlations in the nucleus and defined
by the integration
range of the spectator neutron solid angle.
For $K_0$ we used a conservative estimate of $0.2$ based on the neutron
detector
configuration designed for the experiment of Ref.~\cite{add5:CEBAFpro}.
Assuming additionally that the neutron detection efficiency is $50\%$
one obtains
a triple $(e,e'pn)$ coincidence rate of approximately 250 events/hour.
With a requirement of at least $500$~events in $100$~hours of
beam time a differential cross section as low as
$0.02~{pb\over MeV sr^2}$ can be measured.
The rate for the $A(e,e'pp)X$ reaction is more difficult to estimate since
the $pp$ correlations cannot be approximated by the high momentum part of
the deuteron wave function. The ratio between the $np$ and $pp$ short range
correlation contributions is poorly known and is one of the anticipated outcomes of the
proposed measurements. For estimation purposes we assume that $2 \leq (np)/(pp) \leq 4$. This assumption is
based mainly on counting the isospin degrees of freedom.
\subsection{Quark Structure of Short-Range Correlations -
Study of Superfast Quarks }
The discovery of Bjorken scaling in the late 1960's~\cite{scaling} was one of
the key steps in establishing QCD as the microscopic theory of strong
interactions.
These experiments unambiguously demonstrated that
hadrons contain point-like constituents---quarks
and gluons (see e.g.~\cite{Bjorken,Feynman}).
In the language of quark-partons, the explanation
of the observed approximate scaling was remarkably simple: a
virtual photon knocks out a
point-like quark, and the structure function of the target
nucleon measured experimentally depends on the fraction $x$ of the
nucleon light cone momentum ($\equiv +$) that the quark carries (up to
$\log Q^2$ corrections calculable in pQCD).
For a single free nucleon the
Bjorken variable, $x= {Q^2 / 2 m_N \nu}\leq 1$.
It is crucial that the QCD factorization theorem be valid for this process, in which case the
effects of all initial and final state
interactions are canceled and the deep inelastic
scattering can be described in terms of a light-cone wave function of
the nucleon.
Experimentally such scaling was observed, for a hydrogen
target, for $Q^2 > 4$~GeV$^2$ and $W > 2$~GeV.
Here $W^2 = -Q^2 + 2m_N\nu + m_N^2$ is the invariant mass squared of
the hadronic system produced in the
$\gamma^*$--$N$ interaction.
Since the nucleus is a loosely bound system it is natural to redefine
the Bjorken variable, for a nuclear target,
as $x_A={AQ^2 / 2m_A\nu}$ ($0 \leq x_A \leq A$),
so that for scattering in kinematics allowed for a free
nucleon at rest $x_A \approx x$.
In the case of electron scattering from quarks
in nuclei it is possible to have Bjorken--$x > 1$.
This corresponds to the situation that a knocked-out quark carries a
larger light-cone momentum fraction than a nucleon which is at rest in
the nucleus. Such a situation could occur, for example, if
the quark belongs to a fast nucleon in the nucleus. In the impulse
approximation picture, the DIS structure function of a nucleus, $F_{2A}(x,Q^2)$, which
directly relates to the nuclear quark distribution function, is expressed in terms of
the nucleon structure function and the nuclear light-cone density matrix:
\begin{equation}
F_{2A}(x,Q^2)=\int_x^A
\rho^N_A(\alpha_N,p_\perp)F_{2N}(\frac{x}{\alpha_N},Q^2)
\frac{d\alpha_N d^2p_\perp}{\alpha_N},
\label{conv}
\end{equation}
where $\alpha_N$ is the light-cone momentum fraction of the nucleus carried by
the interacting nucleon.
Choosing $x\geq 1 + k_F/m_N \approx 1.2$ almost completely eliminates the contribution
of scattering by quarks belonging to nucleons with momenta
smaller than the Fermi momentum. Actually, the use of
any realistic nuclear wave function would yield the result that
the contribution of the
component of the wave function with $k\geq k_F$ dominates at large $Q^2$ for values
of $x$ as small as unity. For these values of $x$,
a quark must acquire its momentum from multiple nucleons
with large relative momenta which are significantly closer to each other
than the average inter-nucleon distance~\cite{FS81}. Thus, such
superfast quarks in nuclei could
arise from some kind of superdense configurations consisting either of
a few nearby nucleons with large momenta or of more complicated multi-quark
configurations\footnote{Already at intermediate range of $Q^2$ when A(ee')X cross
section at $x>1$ is dominated by quasielastic scattering, one observes substantial
sensitivity to the short-range behavior of the nuclear wave function
(see e.g.~\cite{fsds93,FS81,CLS,RT}.}
In particular, a comparison with Eqn.~(\ref{conv}), which
builds the nucleus from a distribution of essentially free nucleons, would
provide a quantitative test whether the quarks in a bound nucleon have the
same distribution function as in a free nucleon.
\begin{figure}[htb]
\centerline{
\hbox{\epsfig{figure=pz_x_1.5.eps,height=8cm} } }
\caption[]{\protect\label{fig1}Relation between $p^z_{initial}$ and
$W_N^2$ for $Q^2=10, 15$~(GeV/c)$^2$ and $x = 1, 1.5$}
\label{Fig.kinw}
\end{figure}
The kinematic requirement for detecting the signature of superfast quarks at
$x>1$ is to provide a value of $Q^2$ large enough that the tail of
deep-inelastic scattering would overwhelm the contribution from quasi-elastic
electron scattering from nucleons. One can estimate approximately
the magnitude of the initial momentum of the nucleon, which is relevant
in the DIS channel, expressing its projection in the {\boldmath $q$} direction
through the produced invariant mass, $W_N$ associated with scattering from a
bound nucleon:
\begin{equation}
{p^z_{initial} \over m} = 1-x-x\left[{W_N^2- m^2 \over Q^2}\right].
\end{equation}
For a free nucleon, a given value of $x$ and $Q^2$ automatically fixes the
value of $W_N^2$. In contrast, the internal motion of a bound nucleon
allows a range of values of $W_N^2$ for given values of
$x,Q^2$. When average $W_N\ge 2$~GeV the deep inelastic contribution
becomes dominant and the Bjorken scaling limit is reached.
Figure~\ref{Fig.kinw}
demonstrates that DIS can access extremely large values of initial momenta in nuclei
at large $Q^2$ and $x$.
\begin{figure}[htb]
\begin{center}
\epsfig{width=4.0in,file=inclcrs.eps}
\end{center}
\vspace{-0.4in}
\caption{The differential $^{27}Al(e,e')X$ cross section as
a function of $x$ for fixed beam energy and scattering angle.
The dotted line is the quasi-elastic contribution, the dashed line
is the inelastic contribution, and the solid line is the sum of both contributions.
Values of $Q^2$ are presented for $x=1$. }
\label{Fig.crsal}
\end{figure}
However, in order to probe these values of initial momenta the DIS contributions should
dominate the quasi-elastic contribution.
Figure~\ref{Fig.crsal} shows a calculation of the
$A(e,e')X$ cross section at four different values of $Q^2$. This figure
illustrates that with increasing $Q^2$ the inelastic contribution
remains dominant at increasingly larger values of $x$.
The first signal of the existence of superfast quarks will be the
experimental observation of scaling in the region $x\geq 1$.
Previous experimental attempts to observe such superfast quarks were
inconclusive: the BCDMS collaboration~\cite{BCDMS} has observed a
very small $x\geq 1$ tail ($F_{2A}\propto \exp(-16x)$),
while the CCFR collaboration~\cite{CCFR} observed a tail consistent
with presence of very significant SRCs ($\sim \exp (-8x)$).
A possible explanation for the inconsistencies is that the resolution in
$x$ at $x \geq 1$ of the high energy muon and neutrino experiments
is relatively poor, causing great difficulties in measuring
$F_{2A}$ which is expected to vary rapidly with $x$.
Therefore the energy resolution, intensity and energy of Jefferson Lab
at $11$~GeV may allow it to become the first laboratory to observe the onset of
scaling and thereby confirm the existence of superfast quarks.
To estimate the onset of the Bjorken scaling we should extract the
structure functions $F_{2A}$ and $F_{1A}$ from the cross section of the
inclusive $A(e,e')X$ reaction. The cross section can be represented
as follows:
\begin{equation}
{d\sigma_A\over d\Omega_e dE_e'} =
{\sigma_{\rm Mott}\over \nu} \left[F_{2A}(x,Q^2) +
{2\nu\over m_N} \tan^2(\theta/2)F_{1A}(x,Q^2)\right],
\label{crs2}
\end{equation}
where $F_{2A}(x,Q^2)$ and $F_{1A}(x,Q^2)$ are two invariant
structure functions of nuclei. In the case of scaling, both
structure functions become independent of $Q^2$ (up to $\log Q^2$ terms).
\begin{figure}[ht]
\vspace{-0.4in}
\begin{center}
\epsfig{width=4.0in,file=f2d.eps}
\end{center}
\vspace{-0.6in}
\caption{\protect\label{Fig.f2d}$F_{2d}/A$ as a function of $Q^2$ for $x=1$ and $x=1.5$.
Solid lines correspond to the light-cone calculation of Ref.~\cite{FS88} with
no modification of the nucleon structure function.
Dotted lines account for the binding modification of the nucleon DIS structure
functions within the color screening model discussed in Sec.~\ref{sec:emcmodels}.
No binding modification of elastic structure functions are considered.
The data are from Ref.~\cite{Arrington}.}
\end{figure}
\begin{figure}[ht]
\vspace{-0.6cm}
\begin{center}
\epsfig{width=4.2in,file=fex.eps}
\vspace{-0.8cm}
\end{center}
\caption{\protect\label{Fig.fescale}Prediction of the onset of scaling for $^{56}$Fe$(e,e')X$
scattering. The data are from Ref.~\cite{Arrin99} and the curves are
the mean-field (dotted), two-nucleon SRCs (solid) and multi-nucleon
SRCs (dashed) models, as described in the text.}
\end{figure}
The experimental observable for scaling is the structure function, $F_{2A}$:
\begin{equation}
F_{2A}(x,Q^2) = {{d\sigma_A\over d\Omega_e dE_e'}
\left( {\nu\over \sigma_{\rm Mott}}\right)\left[1 + {1-\epsilon\over \epsilon}
{1\over 1 + R(x,Q^2)}\right]^{-1}},
\label{fnkar}
\end{equation}
where $\epsilon = [1 + 2(1+{\nu^2 / Q^2})\tan^2({\theta / 2})]^{-1}$
and $R \equiv {\sigma_S / \sigma_T}= (F_{2A} / F_{1A})(m_N / \nu)(1+{\nu^2\over Q^2})-1$.
Figures~\ref{Fig.f2d} and ~\ref{Fig.fescale} display calculations~\cite{fsds93,FSS90}
for deuteron and iron targets. Figure~\ref{Fig.f2d} demonstrates that at high $Q^2$
calculations become sensitive to the binding modification of the DIS structure function of
the nucleons. To appreciate the size of the possible modification we used one of the
models (color screening model of Ref.~\cite{FS85}) which describes reasonably well the
nuclear EMC effect at $x<1$ (see Sec.~\ref{sec:emcmodels} for details).
The calculations for $^{56}$Fe show that the onset of scaling
($Q^2$-independence) at $x=1$ is expected at values of $Q^2$
as low as $5-6$~GeV$^2$. At $x=1.5$ the onset of scaling
depends strongly on the underlying model of SRCs and may occur already at
$Q^2\sim 10$~GeV$^2$.
Figure~\ref{Fig.fescale} shows results obtained using three different models
describing the ($A>3$) nuclear state containing the superfast quark.
In the first model, the momentum of the target nucleon in the
nucleus is assumed to be generated by a
mean field nuclear interaction only (dotted line). In the second,
the high momentum component of the
nuclear wave function is calculated using a
two-nucleon short range correlation model
(solid lines). Within this approximation the variations of the structure
functions with $x$ will
be the same for deuteron and $A>2$ targets at large values of $Q^2$.
In the third model, the multi-nucleon correlation model (dashed
lines) of Ref.~\cite{FS80} is used. This model agrees reasonably well with recent
measurements of the nuclear structure functions by the
CCFR collaboration~\cite{CCFR} but yields a significantly larger
quark distribution than the one reported by the BCDMS collaboration~\cite{BCDMS}.
\begin{figure}[ht]
\vspace{-0.6cm}
\begin{center}
\epsfig{width=4.0in,height=3.5in,file=adep.eps}
\end{center}
\vspace{-0.6cm}
\caption{A-dependence of the structure function. The solid curve includes
only two-nucleon SRCs, while the dashed curve includes multi-nucleon SRC contributions.}
\label{Fig.adep}
\end{figure}
Figure~\ref{Fig.adep} represents the $A$-dependence of $F_{2A}$, which
emphasizes that the use of large nuclei and large values of $x$ would
allow the significant study of the models of short-range correlations.
\subsubsection {Experimental Requirements}
The physics program described above involves extending
current inclusive scattering measurements at Jefferson Lab to the
highest possible values of $Q^2$ at $x\geq 1$. The extension
to higher $Q^2$ values requires the detection of high momentum electrons over a
wide angular range $\theta\leq 60^\circ$. To reach the largest possible $x$ values, we need
to detect extremely high energy electrons at angles up to 30$^\circ$.
In both cases the measurements will require high luminosity, excellent pion
rejection and a moderate ($\sim10^{-3}$) momentum resolution.
We use the measurement of the structure function $F_2(x,Q^2)$ for nuclei, as
well as the ratio between Aluminum and Deuterium,
as an example of a possible experiment that can be carried out with the Jefferson Lab
upgrade. The extension to the highest possible $Q^2$ values,
necessary to reach scaling and probe the quark distributions, can be
performed using the equipment proposed for either Halls A or C at Jefferson
Lab. The extension to the highest possible $x$ values, where sensitivity
to multi-nucleon correlations is greatest, requires the detection of
electrons with energies approaching the beam energy, and will require
a very high momentum spectrometer, such as the proposed SHMS in Hall C.
In order to estimate the feasibility of these measurements at large
values of $x$ we have estimated count rates for Deuterium and
Aluminum targets.
The incident beam energy is assumed to be $11$~GeV and a beam
current of $60 \mu A$ is used.
We used a $10$~cm long Deuterium target which corresponds to 1.5\% of a
radiation length. This implies a luminosity of $1.9\cdot
10^{38}$s$^{-1}$cm$^{-2}$. These luminosities are currently used in many
experiments at Jefferson Lab and do not pose any technical problems.
For the Aluminum target we assumed a thickness of $0.5$~cm which
corresponds to a 6\% radiator. This target will also have to be
cooled but this should not pose any special problems. For the
Aluminum target we then obtain a luminosity of
$1.2 \cdot 10^{37}$s$^{-1}$cm$^{-2}$.
We assume a solid angle of 10~msr and an expected momentum resolution of $\approx
10^{-3}$. These properties are certainly satisfied by the spectrometers
proposed as part of the Jefferson Lab energy upgrade.
The pion rates have been estimated using the code EPC~\cite{EPC} and a
parameterization of SLAC experimental data on pion yields. The problem in these
estimates lies in the fact that the kinematics measured at SLAC have
very little overlap with those examined here. Similarly the
parameterizations employed in the code EPC are not optimized for this
kinematical region. We therefore use the yields obtained only as a
rough guide. In a real experiment proposal these models would need
to be refined. Data from the approved $x>1$ measurement at
$6$ GeV~\cite{e02019} will allow us to refine the pion and charge-symmetric
background rate estimates.
The count rates have been evaluated for an bin size of $\Delta x =
\pm 0.1$. The obtained rates for Deuterium are listed in
Table~\ref{tab1} and the ones for Aluminum in Table~\ref{tab2}.
The cross sections come from the calculations shown in Figs.~\ref{Fig.f2d}
and~\ref{Fig.fescale}, (with multi-nucleon correlations included for the Aluminum).
For these estimates, we have not included radiative effects.
\begin{table}[htp]
\begin{center}
\begin{tabular}{cccc}
%
$\theta_{e}$ & $Q^2$(GeV/c)$^2$ & ${d\sigma\over d\Omega d\nu} [{
nb\over sr\cdot GeV/c}]$ & events/hour \\ \hline
%
5 & 1.06 & 1.24E+5 & 4.30E+7 \\
10 & 3.87 & 4.48E+1 & 4.65E+4 \\
20 &11.47 & 3.74E-2 & 6.53E+1 \\
30 &18.01 & 2.07E-3 & 3.35 \\
40 &22.5 & 4.37E-4 & 5.58E-1 \\
%
\hline
\end{tabular}
\end{center}
\caption[]{Cross Sections and Count Rates for Deuterium, $x=1.5$, including correlations.}
\label{tab1}
\end{table}
\begin{table}[htp]
\begin{center}
\begin{tabular}{cccc}
%
$\theta_{e}$ & $Q^2$ (GeV/c)$^2$ & ${d\sigma\over d\Omega d\nu} [{
nb\over sr\cdot GeV/c}]$ & events/hour \\ \hline
%
5 & 1.06 &7.37E+6 & 1.60E+8 \\
10 & 3.87 &4.83E+3 & 3.13E+5 \\
20 &11.47 &2.06E+1 & 2.25E+3 \\
30 &18.01 &2.20 & 2.22E+2 \\
40 &22.5 &6.57E-1 & 5.08E+1 \\
%
\hline
\end{tabular}
\end{center}
\caption[]{Cross Sections and Count Rates for Aluminum, $x=1.5$, including correlations.}
\label{tab2}
\end{table}
The rate estimates indicate that for Deuterium the highest practical $Q^2$
value is about $Q^2 = 18$~(GeV/c)$^2$ while the count rate from Aluminum is still quite large
at $Q^2 = 23$~(GeV/c)$^2$ with the experimental conditions
described above. If we use the cross section with only two-body SRCs included,
the rate will be significantly reduced, but we should still be able to approach
$Q^2 = 23$~(GeV/c)$^2$. In all these cases we will reach $Q^2$ values where we are
clearly dominated by the inelastic processes at $x>1$.
\subsection{Tagged Structure Functions}
Understanding the role of the quark-gluon degrees of freedom in the
hadronic interaction is tied strongly to understanding the
dynamics responsible for modification of the quark-gluon structure
of bound nucleons as compared to
free nucleons. These dynamics at present are far from understood, however.
Almost two decades after the discovery of the nuclear EMC effect~\cite{EMC1}
and increasingly precise measurements~\cite{EMC2,EMC3,EMC3b,EMC4,EMC5}
of the ratios of structure functions of nuclei and the deuteron,
we still know only that this effect requires the presence of {\it some}
non-nucleonic degrees of freedom in nuclei. No consensus has been reached
on the origin of these components.
The $x$-dependence of the effect, while non-trivial, is rather smooth
and has the same basic shape for all nuclei, making it easy to reproduce
using a wide range of models with very different underlying assumptions.
The only additional constraint available to date comes from measurements
of the $A$-dependence of the sea distribution in Drell-Yan reactions\cite{DY},
which poses a problem for several types of models.
The combination of the inclusive DIS and Drell-Yan experiments is still
not sufficient to identify unambiguously the origin of the EMC effect.
Inclusive experiments at Jefferson Lab, after the upgrade, will considerably improve
our knowledge of the EMC effect by (i) measuring the EMC effect for the
lightest nuclei~\cite{e00101}, (ii) studying the isotopic dependence of
the effect, and (iii) separating the different twists in the EMC effect.
Though such experiments will be very important, they are unlikely to lead to
an unambiguous interpretation of the EMC effect.
New experiments involving more kinematical variables accessible to
accurate measurements are necessary to overcome this rather unsatisfactory
situation.
We propose that studying semi-inclusive processes involving a deuteron target,
\begin{equation}
\label{eD}
e + d \to e + N + X,
\end{equation} in which
a nucleon is detected in the target deuteron fragmentation
region, may help to gain insight into the dynamics of the
nuclear structure function modification~\cite{FS85,CL,CLS91,CL95,MSS97}.
Further important
information could be obtained by studying the production
of $\Delta$-isobars and excited baryons in similar kinematics.
Although we focus the discussion here on
the simplest case of a deuteron
target, clearly similar experiments using heavier nuclei ($A=3$, $A=4$) would
provide additional information.
A glimpse of the information that may be obtained from such
experiments has already been provided by analyses of the experimental data on
deep-inelastic neutrino scattering off nuclei~\cite{Ber78,Efr80}.
Even with poor statistics, these experiments have shown that structure functions,
tagged by protons produced in the backward hemisphere,
are different from those determined in inclusive scattering.
The semi-inclusive experiments which we contemplate should
be able to answer the following questions:
What is the smallest inter-nucleon distance (or largest relative momentum)
for which the two nucleons in the deuteron keep their identities?
What new physics occurs when this identity is lost? What is the
signal that an explicit treatment of nuclear quark and gluon degrees
of freedom is necessary? How much do the measured effective bound structure
functions $F_{2}$ differ from those of free nucleons?
Studying this difference will lead us to a better
understanding of the dynamics behind
nuclear binding effects and their relation to QCD.
The tool which allows us to control the relevant distances in the deuteron
is the knowledge of the momentum of the tagged (backward) nucleon.
The dependence of the semi-inclusive cross section on this
variable will test the various assumptions regarding
the lepton interaction with one nucleon while it is in a very
close proximity to another nucleon. These very same assumptions
are the ones that underlie the different models of the EMC effect. The study
of the semi-inclusive reaction using a deuteron target will permit important insights
into the deepest nature of many-baryon physics.
\subsubsection{Theoretical Predictions for the EMC effect}
\label{sec:emcmodels}
The {\bf first group} of models, referred to as binding models, makes
the simplest assumption regarding the lepton-nucleon interaction in a nucleus,
namely, that nucleons maintain their nucleonic character.
In these models, the nuclear EMC effect is caused by the effects of nuclear binding,
and the inclusive structure function data can be understood in terms of
conventional nuclear degrees of freedom---nucleons and pions---responsible for
nuclear binding~\cite{CL,AKV,AKVa,DT,ET,ANL,ANLa,HM,MEC,MECa,BT}.
The nuclear cross section is then expressed as a
convolution of the nuclear spectral function with the structure function
of a {\em free} nucleon (Eqn.~(\ref{conv})).
The off-mass-shell effect (if present at all) in these models constitutes a small
part of the EMC effect, while the pion contribution is simply added to
the contribution given by Eqn.~(\ref{conv}).
The use of the standard, conventional meson-nucleon dynamics of nuclear
physics is not able to explain both the nuclear deep-inelastic and the Drell-Yan
data. One can come to this conclusion using the light-cone sum rules
for the nuclear baryonic charge and the light-cone momentum~\cite{FS81,Jaffe,FS88}.
Another approach~\cite{MillerSmith,MillerSmith2} is to use the Hugenholtz-van Hove
theorem~\cite{HvH} which states that nuclear stability (vanishing of pressure)
causes the energy of the single particle state at the Fermi surface to be
$m_A/A\approx 0.99 m_N$. In light front language, the
vanishing pressure is achieved by setting $P^+=P^-=m_A$.
Since $P^+=\int dk^+f_N(k^+)k^+$ and the Fermi momentum is a relatively small value
the probability $f_N(k^+)$ for a nucleon to have a given value of $k^+$ must be narrowly
peaked about $k^+=0.99 m_N\approx m_N$. Thus, the effects of nuclear binding and Fermi
motion play only a very limited role in the bound nucleon structure function. The
resulting function must be very close to that of a free nucleon unless some quark-gluon
effects are included. In this approach some non-standard explanation involving quark-gluon degrees
of freedom is necessary.
Attempts to go beyond the simplest implementation of pion cloud
effects have been made in Ref.~\cite{KOLTUN}, where recoil effects were taken into
account. As a result it was argued
that the predicted pion excess based on conventional correlated nuclear
theory is not in conflict with data on the Drell Yan process or with the
nuclear longitudinal response function. (Note that the pion enhancement enters in these
two processes in a different way as the
integrands of the integral over the pion light cone fraction, $y_{\pi}$, have different
$y_{\pi}$ dependence.)
What does appear to be ruled out, however, are RPA theories in which there
are strong collective pion modes \cite{KOLTUN}.
At the same time the presence of pions on the level consistent
with the Drell-Yan data does not allow us to reproduce the EMC effect
(in a way consistent with the energy momentum sum rule) without
introducing an ultrarelativistic mesonic component in the nucleus wave function.
Each meson's light-cone fraction in the wave function is
$\leq 0.03$ but, taken together, the mesons carry $\sim 4\%$ of the light-cone fraction
of the nucleus.
The {\bf second group} of models represents
efforts to model the EMC effect in terms of {\em off-mass-shell} structure
functions, which can be defined by taking selective non-relativistic or
on-shell limits of the virtual photon--off-shell nucleon scattering
amplitude~\cite{MST,KPW,GL}. For example, the structure function of a bound nucleon of
four-momentum $p$ can depend on the variable $\gamma\cdot p-m$~\cite{HM}.
Such terms should enter the calculation of the structure function
only in the form of multi-nucleon contact interactions, so these models
represent a parameterization of a variety of dynamical multi-nucleon
effects.
A microscopic, quark-level mechanism which can lead to such a
modification is provided by the quark-meson coupling models~\cite{qmc,qmc1,qmc2},
in which quarks confined in different nucleons interact via the exchange
of scalar and vector mesons.
Here the wave functions of bound nucleons are substantially
different than those of free nucleons, even though the quarks are
localized.
The {\bf third group} of models assume that the main phenomenon
responsible for the nuclear EMC effect is a
modification of the bound state wave function of
individual nucleons. In these models, the presence of
extra constituents in nuclei or clustering of partons from
different nucleons is neglected.
Considerable modification of the bound nucleon structure
functions at different ranges
of $x$ are predicted by these models. Two characteristic approaches are
the color screening model of suppression of point-like configurations in
bound nucleons~\cite{FS88,FS85} and models of quark
delocalization or dynamical rescaling~\cite{CLRR,JRR,CLOSE,NP,GP}.
These models do not include possible effects of partial restoration
of chiral symmetry which would result in modification of the pion
cloud at short internucleon distances.
The {\em color screening} models start from the observation that the
point-like configurations (PLCs) in bound nucleons are suppressed due to
the effects of color transparency~\cite{FS88,FS85,JM96}. Based on the
phenomenological success of the quark counting rules for
$F_{2N}(x,Q^2)$ at large $x$ it is further assumed that three
quark PLCs give the dominant contribution at $x \geq 0.6$ and
$Q^2 \geq 10$~GeV$^2$. The suppression of this component in a bound nucleon
was assumed to be a main source of the EMC effect in inclusive deep-inelastic
scattering in Ref.~\cite{FS85}. The size of the effect was estimated to be
proportional to the internal momentum of the target
$k$ (to the virtuality of the interacting nucleon)~\cite{FS85}:
\begin{equation}
\delta_A({\bf k}^2) = (1+z)^{-2}, \ \ \ \ \ z = ({\bf k}^2/m_N +
2\epsilon_A)/\Delta E_A,
\label{plc}
\end{equation}
where $\Delta E_A = \langle E_i - E_N \rangle \approx m^* - m_N$, is the
energy denominator characteristic of
baryonic excitations. For $m^*=1.5-1.7$~GeV the model reproduces reasonably the
existing data on the EMC effect. Note that the considered
suppression does not lead to a noticeable change in the average
characteristics of nucleons in nuclei~\cite{FS85}: for example, PLC suppression of
Eqn.~(\ref{plc}) will lead to only $2\%$ suppression of the cross section for high $Q^2$
scattering off a bound nucleon in quasi-free kinematics at $x=1$~\cite{FSS90}.
In {\em quark delocalization/rescaling} models it is assumed that
gluon radiation occurs more efficiently in a nucleus than in a free nucleon
(at the same $Q^2$) due to quark delocalization in two
nearby nucleons. It is natural to expect that
such a delocalization will grow with
decreasing inter-nucleon distance. Within this model the
effective structure function of the nucleon can be written as:
\begin{equation}
F_{2N}^{\mathrm{eff}}(x,Q^2) = F_{2N}\left(x,Q^2\xi(Q^2,k)\right),
\label{rsc}
\end{equation}
where $\xi(Q^2)$ is estimated from the observed EMC effect in
inclusive deep inelastic cross sections, and its $Q^2$-dependence is
taken from the generic form of the QCD evolution equations.
The $k$-dependence in $\xi(Q^2,k)$ is modeled based on
the assumption that the quark delocalization grows with increasing
virtuality of a bound nucleon~\cite{MSS97}.
In the above classes of models the cross section for fast backward nucleon
production in the reaction of Eqn.~(\ref{eD}) can be represented as
follows~\cite{MSS97}:
\begin{equation}
{d\sigma^{e D \to e p X} \over
d\phi dx dQ^2 d(\log\alpha) d^2p_\perp }
\approx {2\alpha_{em}^2 \over x Q^4} (1-y) S(\alpha,p_\perp)
F^\mathrm{{eff}}_{2N}\left(\tilde x,\alpha,p_\perp,Q^2\right),
\label{2N}
\end{equation}
where $S(\alpha,p_\perp)$ is the
nucleon spectral function of the
deuteron, and $F^{eff}_{2N}$ is the effective structure
function of the bound nucleon, with $\tilde x \equiv x/(2-\alpha)$.
The variable $\alpha= (E_s - p_{sz}) / m_N$
is the light-cone momentum
fraction of the backward nucleon with $p_{sz}$ negative for backward
nucleons. Both spectral and structure functions can be determined from
the particular models discussed above~\cite{MSS97}.
An alternative scenario to the models discussed above is based on the idea discussed
since the 1970's that two (three) nucleons coming sufficiently close together may form a kneaded
multiquark state
\cite{sixquark1,sixquark2,sixquark3,PV,CH,gm6q,gm6q1,gm6q2,gm6q3,MULDERS1,MULDERS2}.
An example of such a state is a bag of six quarks.
Multiquark cluster models of the EMC effect were developed in a number
of papers~\cite{CH,LS,KS85}).
In {\bf six-quark} ($6q$) models electromagnetic scattering
from a $6q$ configuration is determined from a convolution of
the structure function of the $6q$ system with the
fragmentation functions of the five- (or more, in general) quark
residuum\cite{CLS91}. These types of models cannot be represented through
the convolution of a nucleon structure function and spectral function as in
Eqn.~(\ref{2N}).
Since the quarks in the residuum depend on the flavor of the struck
quark, one finds~\cite{CL95}:
\begin{equation}
{d\sigma^{e D \to e p X} \over
d\phi dx dQ^2 d(\log\alpha) d^2p_\perp }
\approx {2\alpha_{em}^2 \over Q^4} (1-y)
\sum\limits_i x e_i^2 V_i^{(6)}(x)
{\alpha\over 2-x}D_{N/5q}(z,p_\perp)\ ,
\label{6q}
\end{equation}
where the sum is over quark flavors.
Here $V_i^{(6)}$ is the distribution function for a valence quark
in a $6q$ cluster, and $D_{N/5q}(z,p_\perp)$ is the fragmentation
function for the $5q$ residuum, i.e., the probability per unit $z$
and $p_\perp$ for finding a nucleon coming from the $5q$ cluster.
The argument $z$ is the fraction of the residuum's light-cone
longitudinal momentum that goes into the nucleon:
$ z = {\alpha\over 2-x}. $
\subsubsection{Observables}
\begin{figure}[ht]
\vspace{-1cm}
\begin{center}
\epsfig{width=4.4in,file=gratio.eps}
\end{center}
\vspace{-1cm}
\caption{The $\alpha$-dependence of $G(\alpha,p_\perp,x_1,x_2,Q^2)$,
with $x_1=x/(2-\alpha)=0.6$ and $x_2=x/(2-\alpha)=0.2$.
$G^{eff}(\alpha,p_\perp,x_1,x_2,Q^2)$ is normalized to
$G^{eff}(\alpha,p_\perp,x_1,x_2,Q^2)$
calculated with the free nucleon structure function with $p_\perp=0$.
The dashed line is the color screening model~\protect\cite{FS85}, dotted is the
color delocalization model~\protect\cite{CLRR}, and dot-dashed the off-shell
model~\protect\cite{MST}.}
\label{Fig.Gratio}
\end{figure}
Guided by the expectation~\cite{MSS97} that final state interactions
should not strongly depend on $x$, Eqn.(\ref{2N}) suggests it is
advantageous to consider the ratio of cross sections in two different
bins of $\tilde{x}$: one where the EMC effect is small
($\tilde{x}\sim 0.1-0.3$) and one where the EMC effect is large
($\tilde{x}\sim 0.5-0.6$)~\cite{FS88,FS85}.
We suggest therefore measuring the ratio $G$, defined as:
%
\begin{eqnarray}
G(\alpha,p_\perp,x_1,x_2,Q^2)&\equiv&
\left.
{ d\sigma (x_1,\alpha,p_\perp,Q^2) \over
dx dQ^2 d(\log\alpha) d^2p_\perp }
\right/
{ d\sigma (x_2,\alpha,p_\perp,Q^2) \over
dx dQ^2 d(\log\alpha) d^2p_\perp }
\nonumber\\
&=& {F^{eff}_{2N}(x_1/(2-\alpha),\alpha,p_\perp,Q^2) \over
F^{eff}_{2N}(x_2/(2-\alpha),\alpha,p_\perp,Q^2)}\ .
\label{G}
\end{eqnarray}
%
Since the function $G$ is defined by the ratio of cross sections at the
same $\alpha$ and $p_\perp$, any uncertainties in the spectral function
cancel.
This allows one to extend this ratio to larger values of $\alpha$,
thereby increasing the utility of semi-inclusive reactions.
Figure~\ref{Fig.Gratio} shows the $\alpha$-dependence of
$G(\alpha,p_\perp,x_1,x_2,Q^2)$ at $p_\perp=0$ for several different models.
The values of $x_1$ and $x_2$ are selected to fulfill the condition
$x_1/(2-\alpha)=0.6$ (large EMC effect in inclusive measurements)
and $x_2/(2-\alpha)=0.2$ (essentially no EMC effect).
Note that while they yield similar inclusive
DIS cross sections, the models predict significantly different values of
the ratio function $G$. {\em Additionally, the study of the $p_\perp$ dependence of
the ratio function $G$ at fixed $\alpha$ will allow us to study the effects associated
with the final state interaction of the products of DIS scattering
with spectator nucleon (see e.g.\cite{MSS97,CK}}.
\medskip
{}From Eqn.~(\ref{2N}) one further observes that in all models that
do not require the mixing of quarks from different nucleons,
the $x$-dependence of the cross section is confined (up to the kinematic
factor $1/x$) to the argument of the tagged nucleon structure
function, $F_{2N}^{eff}$.
%
On the other hand, for $6q$ models the $x$-dependence reflects the
momentum distribution in the six-quark configuration,
Eqn.~(\ref{6q}).
As a result it is useful to consider the $x$-dependence of the
observable defined as:
%
\begin{equation}
R = \left. {d\sigma\over dxdQ^2d\log(\alpha)d^2p_\perp} \right/
{4\pi\alpha_{em}^2 (1-y)\over x Q^4}\ .
\label{R}
\end{equation}
%
For the convolution-type models discussed above, the ratio $R$ is
just the product of the spectral function, $S(\alpha,p_\perp)$, and the
effective nucleon structure function, $F_{2N}^{eff}(\tilde x,\alpha,p_\perp,Q^2)$.
%
In the on-shell limit the $x$-dependence of $R$ is therefore
identical to that of the free nucleon structure function.
\begin{figure}[ht]
\vspace{-0.4cm}
\begin{center}
\epsfig{
width=4.4in,file=rnorm.eps}
\end{center}
\vspace{-1cm}
\caption{The $x$-dependence of the normalized ratio $R$ defined in
Eqn.~(\ref{R}). The solid, dashed, dotted and dot-dashed curves
represent the predictions of off-shell (covariant spectator)
\protect\cite{MST}, color screening \protect\cite{FS85},
quark delocalization \protect\cite{CLRR} and six-quark
\protect\cite{CL95} models, respectively. The binding models (group one)
predict the ratio of one - thin solid line.}
\label{Fig.Rratio}
\end{figure}
Figure~\ref{Fig.Rratio} presents the $x$
dependence of $R$ normalized by the same ratio calculated using
Eqn.(\ref{2N}) with $F_{2N}^{free}$. The calculations are performed
for different models at $\alpha=1.4$ and $p_\perp=0$.
Within the off-shell/covariant spectator model \cite{MST} $R$ exhibits
a relatively weak dependence on $x$, while $6q$ models predict a
rather distinct $x$-dependence. Note that the models from the first group
(binding models) does not produce any effect for this ratio.
This ratio exposes a large divergence of predictions, all of
which are obtained from models which yield similar EMC effects for
inclusive reactions.
Jefferson Lab at $11$ GeV should therefore have a unique potential
to discriminate between different theoretical approaches to the
EMC effect, and perhaps reveal the possible onset of the $6q$
component of the deuteron wave function.
An important advantage of the reactions considered with
tagged nucleons, in contrast to inclusive reactions, is that any
experimental result will be cross checked by the dependence of the
cross section on $\alpha$ and $p_\perp$.
%
Figure~\ref{Fig.scalingregion} shows the accessibility of the
scaling region as a function of incoming electron energy~\cite{CL95}.
Values of $x$ between the curves labeled by $x_{\rm min}$ and $x_{\rm max}$
can be reached in the scaling region for a given incoming electron energy
$E_e$.
\begin{figure}[ht]
\vspace{-0.4cm}
\begin{center}
\epsfig{width=4.0in,file=kintag.eps}
\end{center}
\vspace{-0.4cm}
\caption{The scaling window for $\alpha=1.4$.
The upper curve is defined by the requirement that
the mass of the produced final hadronic state $W\geq 2$~GeV.}
\label{Fig.scalingregion}
\end{figure}
\subsubsection{Extraction of the ``Free'' Neutron DIS Structure Function}
Aside from providing a qualitatively new insight into the origin of the
nuclear EMC
effect per se, the measurements of the tagged events may also be useful
for extracting the free neutron structure
function from deuteron data (see e.g. ~\cite{FS85,MSS97,Silvano})
By selecting only the slowest recoil protons in the target fragmentation
region, one should be able to isolate the situation whereby the virtual
photon scatters from a nearly on-shell neutron in the deuteron.
In this way one may hope to extract $F_{2n}$ with a minimum of
uncertainties arising from modeling
nuclear effects in the deuteron.
\begin{figure}[ht]
\vspace{-0.4cm}
\begin{center}
\epsfig{width=4.0in,file=virt.eps}
\end{center}
\vspace{-1cm}
\caption{The $E_{kin}$ dependence of the neutron structure function
extracted from $d(e,e'p_{\mathrm{backward}})X$ reactions within PWIA.
The effective structure function is normalized by the on-shell
neutron structure function. Dashed and dotted curves correspond
to the calculation within Color Screening \protect\cite{FS85}
and Color Delocalization \protect\cite{CLRR} models,
respectively.}
\label{Fig.taggedsf}
\end{figure}
One approach to extract the free $F_{2n}$ is to extrapolate the
measured tagged neutron structure function to the region of negative
values of kinetic energy of the spectator proton, where the pole
of the off-shell neutron propagator in the PWIA amplitude is located
($E_{kin}^{pole}=-{|\epsilon_{D}|-(m_{n}-m_{p})\over 2}$).
This method is analogous to the Chew--Low procedure for extraction
of the cross section of scattering off a pion\cite{CL59}.
The advantage of such an approach is that the scattering amplitudes
containing final state interactions do not have singularities
corresponding to on-shell neutron states.
Thus, isolating the singularities through the extrapolation of
effective structure functions into the negative spectator kinetic energy
range will suppress the FSI effects in the extraction of
the free $F_{2n}$\cite{FSSprep}.
Figure \ref{Fig.taggedsf} demonstrates that such an extrapolation can
indeed be done with the introduction of negligible
systematic errors.
\subsubsection{Experimental Objectives}
\begin{figure}[ht]
\center{\epsfig{width=4in,figure=Deeps.eps}}
\caption{
Kinematic coverage in Bjorken--$x$ and proton light-cone fraction
$\alpha_S$ for the high--momentum part of the
proposed experiment. The count rates have
been estimated for a 20 day run with the standard CLAS++ configuration. }
\label{Deeps}
\end{figure}
As described in the previous section, our goal is to measure the
reaction $d(e,e' p_{\mathrm{backward}})X$ over a large range in the electron variables
($x$, $Q^2$) and the backward proton kinematics ($\alpha$, $p_\perp$).
The proposed experiment would use the upgraded ``CLAS$^{++}$'' (CEBAF
Large Acceptance Spectrometer) with an $11$~GeV beam to detect
scattered electrons in coincidence with protons moving backward
relative to the momentum transfer vector, $\bf q$.
A large acceptance spectrometer is
required since the proton is selected in spectator kinematics
(small to moderate momentum up to $700$~MeV/c
anywhere in the backward hemisphere)
and is uncorrelated with the electron direction.
Scattered electrons will be detected by the upgraded forward spectrometer
with two sets of Cerenkov counters, time of flight counters, three tracking
regions and pre-shower and total absorption electromagnetic calorimeters
for highly efficient electron/pion separation. Depending on the momentum
range of interest, two different detector/target arrangements will be used
for the detection of the backward--moving proton.
The first case involves the use of a dedicated integrated target-detector
system with a $5$ atmosphere deuterium gas cell as target ($30$~cm long
and $0.5$~cm diameter) and a multilayer GEM (Gas Electron Multiplier)
detector surrounding the target cell. By minimizing
all materials in the path of large angle tracks, the threshold
for detection of backward--moving protons can be lowered to about $70-80$~MeV/c. One
expects that nucleon structure modifications and off-shell effects will
be small at these momenta, and this method can be used to extract
unambiguously the free neutron structure function $F_{2n}(x)$ up to
very high values of $x$ ($\approx 0.8$).
This measurement is of fundamental importance, since
presently our knowledge of the neutron structure function
at high $x$ is rather poor.
At the same time, it will supply the
``low momentum`` part of the nucleon momentum dependence of the
effective off--shell structure function, $F_{2n}^{eff}$, and thus serve
as a baseline for the non-nucleonic effects
which are expected at higher proton momenta.
This target-detector system is presently under development
and will be used for an exploratory measurement at $6$~GeV beam
energy. Together with CLAS$^{++}$ and an $11$~GeV beam at a luminosity
of $0.5\cdot 10^{34}$ cm$^{-2}$s$^{-1}$, a statistical
precision of better than $\pm 5\%$ on $F_{2n}$
out to the highest values of $x$ would be obtained
with 40 days of data taking.
In the second case, a central detector of CLAS$^{++}$,
with superconducting solenoid, tracking and time-of-flight
detectors would be used to measure backward--going protons
with momenta above $250$~MeV/c.
With these higher momenta, one achieves great sensitivity to
modifications of the neutron structure because of the
proximity of the ``spectator'' proton.
The dependence of the structure function
$F_{2n}^{eff}(x/(2-\alpha), \alpha, p_\perp, Q^2)$ on the proton
momentum from about $250$~MeV/c to over $600$~MeV/c can be extracted
at fixed $x$ and $Q^2$. The experiment will simultaneously cover
a large range in $x$ and $Q^2$, allowing detailed comparisons with
the different models described in the previous section.
Due to the higher momentum threshold, one can use a standard
liquid deuterium target and the full CLAS$^{++}$ luminosity of
10$^{35}$ cm$^{-2}$s$^{-1}$.
Fig.~\ref{Deeps} shows estimates of the
expected number of counts for a $20$ day run
as a function of $x$ for seven bins in the light-cone fraction $\alpha$ of
the backward proton. One can clearly see the kinematic shift
due to the motion of the struck neutron, which we can fully
correct using the proton kinematics.
It is clear that good statistics for a large range in $x$ and in
$\alpha$ (the highest bin corresponds to more than
$600$~ MeV/c momentum opposite to the direction of the $\bf q$ vector)
would be obtained. Together with the low--momentum results, these data
can be used to put the various models described in the previous section
to a stringent test.
\section{Observation of Color Coherent Phenomena at Intermediate Energies }
\subsection{Basic aspects of color coherence and color transparency}
\noindent
QCD displays some of its special characteristics as a theory involving
$SU(3)$-color by its prediction of novel effects in coherent
processes. The basic idea is that the effects of gluons emitted by a
color-singlet which is small-sized (or in a point-like configuration)
are canceled if the process is coherent.
This is because, if the process is coherent, one computes the scattering
amplitude by summing the terms caused by gluon emissions from different quarks.
The implication is that for certain hard exclusive processes, the effects of initial and/or
final state interactions will be absent.
To observe color coherence effects it is necessary to find processes which are
dominated by the scattering of hadrons in a PLC and hence
have amplitudes that can be calculated using pQCD.
A number of such processes have been suggested in the literature,
and the corresponding QCD factorization theorems have been proven for them.
These processes include diffractive pion dissociation into two high transverse momentum
jets~\cite{FMS93} and exclusive production of
vector mesons~\cite{Brod94,CFS97}.
Experiments at HERA which studied exclusive production of vector mesons in deep inelastic
scattering (recently reviewed in~\cite{AC99}), have convincingly confirmed
the basic pQCD predictions -- a fast increase of the cross section
with energy at large $Q^2$, dominance of the longitudinal photon cross section and a weaker
$ t$-dependence of the $\rho$-meson production at large $Q^2$ relative to $J/\psi $ photoproduction.
A distinctive feature of processes such as di--jet and vector meson
production is that, in the case of nuclear targets, the incoming $q\bar q$ pair
does not experience attenuation for the range of $x$ for which gluon shadowing
is small ($x\geq 0.02$). This is the Color Transparency (CT) phenomenon.
As a result, the amplitude of the corresponding nuclear coherent processes
at $t=0$, and the cross section of quasi-elastic processes are each
proportional to the nucleon number $A$, a result which is
vastly different from the results usually obtained
in processes involving soft hadrons.
Color transparency, as predicted by pQCD, was directly observed in
the Fermi Lab experiment E791 which investigated the exclusive
coherent production of two jets in the
process $\pi +A \to 2 {\rm \ jets} + A$ at $E_{\pi}=500$~GeV.
The observed $A$-dependence of the process~\cite{Ashery,Ashery2}
is consistent with the predictions of~\cite{FMS93}, which lead
to a seven times larger platinum/carbon ratio than soft physics would.
The study of this reaction also allowed measuring the pion $q\bar q$
wave function~\cite{Ashery,Ashery2}, which turned out to be close to the
asymptotic one at $k_\perp\geq 1.5$~GeV/c\footnote{Further arguments in favor
of the dominance of the one gluon exchange tail of pion wave function were
given in \cite{FMS2002}. This conclusion was questioned in \cite{Nikolaev,Chernyak,Braun},
while further supported in \cite{FS2002}. It was demonstrated in \cite{FS2002} that in the
simpler case of pion dissociation into two jets off the Coulomb field of target nucleus
the result of \cite{FMS2002} follows from the generalization to hard processes in QCD of the
limiting theorem for photon bremsstrahlung \cite{Gribovbr,Lowbr} which differs from the result
obtained in \cite{IS} using the same approximations as in Ref.~\cite{Braun}.}.
Evidence for color transparency effects was reported also in
incoherent vector meson production in DIS scattering of muons~\cite{E665}.
Hence we conclude that the general concepts of CT in pQCD domain are now
firmly established for high energy processes:
the presence of PLCs in vector mesons and pions and the
form of the small size $q\bar q$ dipole-nucleon
interaction at high energies
are well established experimentally\footnote{At sufficiently small $x (\leq 0.01)$ which
can be achieved at RHIC, HERA, and LHC, the quantum field theory treatment of the CT predicts a
gradual disappearance of the CT and the onset of the color opacity phenomenon.}
It is natural to apply the CT ideas to address the question of interplay of
small and large distances in various high-momentum transfer processes
at intermediate energies. This involves three key elements:
(i) the presence of configurations of small transverse size in hadrons,
(ii) the small size configurations not expanding as they exit the nucleus
(i.e. a large coherence length) at high energies, which leads to the possibility
of considering the small-sized configurations as frozen during the collision, and
(iii) the weakness of the interaction of small color singlet objects at high energies
(for example, for a small color $q\bar q$ dipole of transverse size $r_t$,
the effective cross section is $\sigma_{\rm eff}\sim r_t^2$).
It also is vital that the experiment be performed with sufficient precision to be
certain that no additional pions be created.
This is necessary to maintain the exclusive or nearly exclusive nature of the
experiment, which is required for the necessary quantum interference
effects to dominate the physics.
The considered effect in general is analogous to the reduction of the
electromagnetic interaction strength of the electrically neutral $Q^+Q^-$
dipole at small separations. However the uniqueness of QCD is in the
prediction of the similar phenomena for color-neutral $qqq$ configurations.
Establishing the existence of color coherence effects for three-quark systems
would verify the color SU(3) nature of QCD, and remains an important unmet challenge.
\subsection{Goals For Intermediate Energy Studies}
The major directions for study of CT related phenomena at
Jefferson Lab are determined by the fact that one probes
the transition from soft to hard QCD regime.
These studies include:
\begin{itemize}
\item Determining the interplay between the contribution of large and
small distances for specific processes.
\item
Studying the interaction cross section of small objects in kinematic regions
for which quark exchanges may play the dominant role as compared to
gluon exchanges (at high energies the two-gluon exchange in $t$-channel dominates).
\item Studying the dynamics of wave packet expansion.
\end{itemize}
All of these aspects can be addressed if one considers the fundamental
physics involved with
the nucleon form factor at large $Q^2$ and the scattering amplitude for
large angle hadron-nucleon elastic scattering.
In QCD it is expected that at very large $Q^2$ the form factor and scattering
amplitude are each
dominated by contributions arising from
the minimal Fock space components in the nucleon (hadron) wave function.
Such components, involving the smallest possible number of constituents, are
believed to be of a very small size, or to contain a PLC.
To determine the values of $Q^2$
for which PLCs start to dominate,
Brodsky~\cite{Brodsky82} and Mueller~\cite{Mueller82}
suggested the study of quasi-elastic hard
reactions $l(h) +A \to l(h) + p +(A-1)^*$.
If the energies and momentum transfers are large enough, one expects that
the projectile and ejected nucleon travel through the nucleus as point-like
(small size) configurations, resulting in a cross section proportional to
$A$.
\subsection{Challenges for intermediate energy studies}
To interpret the physics of quasi-elastic reactions one has to address two
questions:
\begin{itemize}
\item Can the PLCs be treated as a frozen during the time that the projectile
is passing through the nucleus?
\item At what momentum transfer do the effects of PLCs dominate in the elementary
amplitude?
\item Do they have small interaction cross sections?
\end{itemize}
These questions must be addressed if color transparency is to be studied at Jefferson Lab.
If the momentum transfer is not large enough for the PLC to
dominate or if the PLC is not frozen and expands,
then there will be strong final state interactions as the PLC
moves through the nucleus.
To appreciate the problems we shall discuss the effects of
expansion in a bit more detail. Current color transparency experiments are
performed in kinematic regions where the expansion of the produced small system
is very important. In other words the length (the coherence length $l_c$)
over which the PLC can move without the effects of time evolution
changing the character of the wave function is too small, and this strongly
suppresses any effects of color transparency~\cite{FLFS88,FSZ90,jm90,jm90a,jm90b,jm90c,boffi}.
The {\em maximal} value of $l_c$ for a given hadron $h$ can be estimated
using the uncertainty principle: $l_c \sim {1\over \Delta M} {p_h\over m_h}$,
in which $\Delta M$ is a characteristic excitation energy
(for a small value of $m_h$ one can write $l_c\approx {2p_h\over \Delta M^2}$
with $\Delta M^2 =(m_{ex}^2-m_h^2)$ where $m_{ex}^2$ is the invariant mass squared
of the closest excited state with the additive quantum numbers of $h$ (cf Eqn.~(\ref{eq2})).
Numerical estimates~\cite{FLFS88,jm90,jm90a,jm90b,jm90c} show that, for the case of a nucleon
ejectile, coherence is completely lost at
distances $l>l_c \sim 0.3-0.5 {\rm\ fm} \times p_h$,
where $p_h$ is measured in GeV/c.
Two complementary languages have been used to describe the effect of the loss of
coherence. Ref.~\cite{FLFS88} used the quark-gluon representation of the
PLC wave function, to argue that the main effect is quantum diffusion of the
wave packet so that
\begin{equation}
\sigma^{PLC,Q^2}(Z,Q^2) =(\sigma_{hard} + {Z\over l_c}[\sigma
-\sigma_{hard}])
\theta(l_c- Z) +\sigma\theta\left(Z-l_c\right),
\label{eq:sigdif}
\label{eq1}
\end{equation}
where $\sigma^{PLC}(Z,Q^{2})$ is the effective total cross section
for the PLC to interact at a distance $Z$ from the hard interaction point.
This equation is justified for the ``hard stage'' of time development
in the leading logarithmic approximation when perturbative QCD can
be applied~\cite{FLFS88,BM88,FS88,DKMT}.
One can expect that Eqn.~(\ref{eq:sigdif}) smoothly interpolates between the hard and
soft regimes, at least for relatively large transverse sizes of the expanding
system ($\geq 1/2$ of the average size) which give the largest contribution to
absorption.
The time development of the PLCs can also be obtained by modeling the
ejectile-nucleus interaction using a baryonic basis for the PLC\cite{GFMS92}:
%
\begin{eqnarray}
\left| \Psi_{PLC}(t)\right> & = & \Sigma_{i=1}^{\infty} a_i \exp(iE_it)\left| \Psi_{i} \right>
\nonumber \\
& = & \exp(iE_1t)\Sigma_{i=1}^{\infty} a_i
\exp\left({i(m_i^2-m_1^2)t\over 2p}\right)\left| \Psi_{i} \right>,
\label{eq2}
\end{eqnarray}
where $\left| \Psi_{i} \right>$ are the eigenstates of the Hamiltonian with
masses $m_i$, and $p$ is the momentum of the
PLC which satisfies $E_i \gg m_i$.
As soon as the relative phases of the different hadronic components
become large (of the order of one) the coherence is likely to be lost.
It is interesting that numerically Eqns.~(\ref{eq1}) and~(\ref{eq2})
lead to similar results if a sufficient number of states is included in
Eqn.~(\ref{eq2})~\cite{GFMS92}. It is worth emphasizing that, though both
approaches model certain aspects of the dynamics of the expansion, a complete
treatment of this phenomenon in QCD is absent.
We next discuss the issue of the necessary momentum transfers required for
PLCs to be prominent. For electromagnetic reactions this question is related to
the dominance of PLCs in the electromagnetic form factors of
interacting hadrons. The later can be studied by considering the applicability
of perturbative QCD in calculating the electromagnetic form factors.
Current analyses indicate that the leading twist approximation for
the pion form factor could become applicable (optimistically) at
$Q^2 \geq 10-20$~GeV$^2$; see e.g. Ref.~\cite{JRS,JRSa}.
For the nucleon case larger values of $Q^2$ may be necessary.
However this does not preclude PLCs from being relevant for smaller values of
$Q^2$. In fact, in a wide range of models of the nucleon, such as constituent
quark models with a singular (gluon exchange type) short-range interaction or
pion cloud models, configurations of sizes substantially smaller
than the average one dominate in the form factor at
$Q^2 \geq 3-4$~GeV$^2$; see Ref.~\cite{fmsplc}. The message
from QCD sum rule model calculations of the nucleon form factor is
more ambiguous.
For hadron-nuclear reactions the question of the dominance of PLCs is related
to theoretical expectations that large angle hadron-hadron scattering is
dominated by the hard scattering of PLCs from each hadron. However this question
is complex because one is concerned with placing larger numbers of quarks
into a small volume. Irregularities in the energy dependence of
$pp$ scattering for $\theta_{c.m.}=90^\circ$, and large spin effects, have led
to suggestions of the presence of two interfering mechanisms in this
process~\cite{rp88,bdet}, corresponding to interactions of the nucleon
in configurations of small and large sizes. See the review~\cite{pire}.
The difficulties involved with using hadron beams in quasi-elastic reactions
can be better appreciated by considering a bit of history.
The very first attempt to observe color transparency effects was
made at the AGS at BNL~\cite{Car88}. The idea was to see if nuclei
become transparent with an increase of momentum transfer in the
$p+A\to p+p+(A-1)$ reaction. As a measure of
transparency, $T$, they measured the ratio:
\begin{equation}
T = {\sigma^{\mathrm{Exp}}\over \sigma^{\mathrm{PWIA}}},
\label{T}
\end{equation}
where $\sigma^{\mathrm{Exp}}$ is the measured cross section and $\sigma^{\mathrm{PWIA}}$ is
the calculated cross section using plane wave impulse approximation
(PWIA) when no final state interaction is taken into account.
Color transparency is indicated if $T$ grows
and approaches unity with increasing energy transfer.
The experiment seems to support an increase of transparency at
incident proton momentum $p_{inc}=6 - 10$~GeV/c as compared to that
at $E_p=1$~GeV; see the discussion in~\cite{FSZ94}.
The magnitude of the effect can be easily described in color transparency
models which include the expansion effect. The surprising result of the
experiment was that with further increase of momentum, ($\geq 11$~GeV/c), $T$ decreases.
The first data from a new $(p,2p)$ experiment, EVA\cite{BNL98}, at $p_{inc}=6 - 7.5$~GeV/c
confirmed the findings of the first experiment~\cite{Car88} and more recently
EVA has reported measurements in a wider momentum range up to $14$~GeV/c.
The data appear to confirm both the increase of
transparency between $6$ and $9$ GeV/c and a drop of transparency at $12$
and $14$~GeV/c~\cite{BNL01}.
The drop in the transparency can be understood as a
peculiarity of the elementary high momentum transfer $pp$ scattering amplitude,
which contains an interplay of contributions of PLCs and large size
configurations as suggested in~\cite{rp88,bdet}. A description of the drop in transparency based on these ideas was presented
in Ref.~\cite{jm94}. However, it is evident that the interpretation of any
experiment would be simplified by using an electron beam.
The most general way to deal with each of the challenges mentioned here is to
perform relevant experiments using electrons at the highest possible values of
energy and momentum transfer.
\subsection{Color Transparency in (e,e'N) and (e,e'NN) Reactions }
The first step is to measure a transparency similar to that of Eqn.~(\ref{T}) using
electroproduction reactions. The first electron $A(e,e'p)$ experiment
looking for color transparency was NE-18 performed at SLAC~\cite{NE18,NE18a}.
The maximum $Q^2$ in this experiment is $\approx 7$~GeV$^2$, which corresponds to
$l_c \leq 2$ fm.
For these kinematics, color transparency models which included expansion effects
predicted a rather small increase of the transparency; see for example~\cite{FS88}.
This prediction is consistent with the NE-18 data. However these data are
not sufficiently accurate either to confirm or to rule out color
transparency on the level predicted by realistic color
transparency models. Recent Jefferson Lab experiments~\cite{quench,garrow} have been
performed up to $Q^2=8$~GeV$^2$, and no effects of color transparency have
been observed (see Fig.~\ref{Fig.transparent}). However, models of color transparency
which predict noticeable effects in the $(p,pp)$ reaction include versions which can also lead
to almost no effects in electron scattering. In those models, the effects of
expansion are strong for the lower energy final state wave functions, but do
allow some color transparency to occur for the initial state wave function.
At $Q^2=8$~GeV$^2$, the momentum of the proton ejected in electron scattering
is about $5$~GeV/c, which is still lower than the lowest momentum, $6$~GeV/c used at BNL.
One needs to achieve a $Q^2$ of about $12$~GeV$^2$ to reach a nucleon momentum for which BNL
experiment observed an increase of the transparency.
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.8in,file=ct2_12gev.eps}
\end{center}
\vspace{-0.9cm}
\caption{\protect The $Q^2$-dependence of $T$ as defined in Eqn.~(\ref{T}).
``Glauber'' - calculation within
Glauber approximation, ``Glauber+CT(I) and Glauber+CT(II)'' - calculations
including CT effects with expansion parameter $\Delta M^2=1.1$~GeV$^2$ and
$\Delta M^2=0.7$~GeV$^2$ respectively~\cite{FMSS}, ``Glauber+CT(III)'' -
CT effects are included according to Ref.~\cite{NSSW}.}
\label{Fig.transparent}
\end{figure}
The recent Jefferson Lab data~\cite{garrow} allow us to put some constraints on the
parameters defining the onset of CT. In Fig.~\ref{Fig.trans12lim} we analyze the lower
limit of $Q^2_0$ at which PLCs are selected in $\gamma^*N$ scattering. Since our interest is
only in the energy dependence of the transparency, we normalized the calculations to the
data at $Q^2=2$~GeV$^2$ to avoid uncertainties related to the $Q^2$ dependence of quenching
\footnote{The transparency is defined in Eqn.(\ref{T}) is inversely proportional to the nuclear
quenching. Hence a decrease of the quenching effect with an increase of $Q^2$ may mask the CT
effects at intermediate $Q^2\leq 2 GeV^2$.}
The analysis is done within the quantum diffusion model of CT for the range of
the expansion parameter $\Delta M^2$
($\Delta M^2=0.7$~GeV$^2$ in Fig.~\ref{Fig.trans12lim}(a) and $\Delta M^2=1.1$~GeV$^2$
in Fig.\ref{Fig.trans12lim}(b)) consistent with the EVA data.
In the case of the slower expansion rate, Fig.~\ref{Fig.trans12lim}(a), the transparency
is rather sensitive to $Q^2_0$ and the analysis yields a lower limit of $Q_0^2\approx 6$ GeV$^2$.
For a faster expansion rate, Fig.~\ref{Fig.trans12lim}(b), the nuclear transparency
is less sensitive to $Q^2_0$, since for intermediate range of $Q^2$ the
PLC expands
well before it escapes the nucleus.
The analysis in Fig.~\ref{Fig.trans12lim}(b) yields $Q^2_0\ge 4$~GeV$^2$.
Combining these two analyses one can set the lower limit for the formation of PLCs at
$Q^2_0 \approx 4$~GeV$^2$ (see Eqn.~(\ref{range}) for implication of this limit
in SRC studies).
The upgrade of Jefferson Lab would improve the situation, by pushing the
measurement of $T$ to a high enough $Q^2$ where the color
transparency predictions appreciably diverge from the predictions of
conventional calculations (see Figs.~\ref{Fig.transparent} and~\ref{Fig.trans12lim}).
Indeed, the EVA data have established in a
model independent way that at least for nucleon momenta $\geq 7.5$~GeV/c,
expansion effects are not large enough to mask the increase of the transparency.
Hence measurements at $Q^2 \geq 14$~GeV$^2$, corresponding to comparable momenta of
the ejectile nucleon, would unambiguously answer the question whether
nucleon form factors at these $Q^2$ are dominated by small or large
size configurations.
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.4in,file=trans12lim.eps}
\end{center}
\vspace{-0.9cm}
\caption{The $Q^2$-dependence of T. The solid line is the prediction of
the Glauber approximation. In (a) dashed curves correspond to the
CT prediction with $\Delta M^2=0.7$~GeV$^2$ and with $Q^2_0=1$(upper curve),
$2$, $4$, $6$ and $8$~GeV$^2$(lower curve).
In (b) dotted curves correspond to the
CT prediction with $\Delta M^2=1.1$~GeV$^2$ and with $Q^2_0=1$(upper curve),
$2$, $4$ and $6$~GeV$^2$(lower curve). All calculations are normalized to
the data at $Q^2=2$~GeV$^2$. The data are the same as in Fig.~\ref{Fig.transparent}. }
\label{Fig.trans12lim}
\end{figure}
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.8in,file=deep4.eps}
\end{center}
\vspace{-0.9cm}
\caption{The ratio of the cross section at $400$~MeV/c missing momentum to
the cross section at $200$~MeV/c as a function of $Q^2$.
Solid line corresponds to the GEA prediction. Dashed and dash-dotted lines
represent the quantum diffusion model of CT with $\Delta M^2=0.7$ and $1.1$~GeV$^2$
respectively. The drop with $Q^2$ in the color transparency models comes from a
reduction in the rescattering of the struck nucleon, which is the dominant
source of events with $P_m > k_F$.}
\label{Fig.CT_ratio}
\end{figure}
Although $(e,e'N)$ measurements will allow an unambiguous check of
the existence of color transparency, it will require a challengingly high
accuracy from experiments to investigate the details of the expansion
effects (Fig.~\ref{Fig.transparent}). Thus, although this is the simplest
reaction to measure, a much wider range of reactions would be necessary to
build a sufficiently complete picture of phenomenon and to scan the
expansion of the small size wave packets.
To obtain a more detailed knowledge of the interaction of PLCs with
nuclei we can also select a processes in which the ejectile could
interact a second time during its propagation through
the nucleus~\cite{FS91,double,FGMSS95} (double scattering reactions).
This can be done by studying recoil nucleons with perpendicular (vs $\vec q$)
momenta $p_{s,\perp} \geq 200$~MeV/c. At low $Q^2$, the majority of such high
momentum nucleons come from rescattering with the spectator nucleon in the
nucleus. Therefore, the number of such
nucleons should decrease substantially with the onset of CT which reduces
the probability of rescattering.
An important advantage of a double scattering reaction is that the
disappearance of the final state interactions can be studied using the
lightest nuclei (D,${^3}$He,${^4}$He), for which wave functions are known
much better and where one can use a generalized eikonal approximation, which
accounts for the nonzero values of the momenta of recoil nucleons
~\cite{FGMSS95,FSS97}.
Another advantage of double scattering reactions is that inter-nucleon distances
probed are not large, $1-2$~fm. These distances are comparable to the coherence
length for values of $Q^2$ as low as about $4-6$~GeV$^2$, and
may provide evidence for a number of color coherent phenomena in
the transitional $Q^2$ region. Ultimately, double scattering measurements
will allow us to determine whether the lack of the CT in $A(e,e'p)$
reactions at $Q^2\geq 8$~GeV$^2$ region is related to the large expansion
rate of PLCs, or if it is because PLCs are not produced at all for these
values of $Q^2$.
An appropriate measure for color transparency in double scattering reactions
is a ratio of cross sections, measured at kinematics for which
double scattering is dominant, to the cross section measured at
kinematics where the effect of Glauber screening is more
important. Theoretical investigations of these reactions~\cite{double,FGMSS95}
demonstrated that it is possible to separate these two kinematic regions
by choosing two momentum intervals for the recoil nucleon: ($300-500$~MeV/c)
for double scattering, and ($0-200$~MeV/c) for Glauber screening. To enhance
the effect of the final state interaction in both regions, the parameter $\alpha$,
characterizing the light cone momentum fraction of the nucleus carried by
the recoiling nucleon should be close to one ($\alpha = (E_s-p^z_s)/m \approx 1$,
where $E_s$ and ${\bf p_s}$ are the energy and momentum of recoil
nucleon in the final state).
Thus, the suggested experiment will measure the $Q^2$-dependence of the
following typical ratio at $\alpha=1$:
\begin{equation}
R = {\sigma(p_s = 400~MeV/c)\over \sigma(p_s=200~MeV/c)}
\label{rdouble}
\end{equation}
Figure~\ref{Fig.CT_ratio} shows this ratio, calculated within the generalized eikonal
approximation~(solid line),
and using the quantum diffusion model of CT with upper and lower values of the expansion
parameter $\Delta M^2$.
It is worth noting that in addition to the $d(e,e'pn)$ process, one can consider
excitation of baryon resonances produced in the spectator kinematics,
like $d(e,e'p)N^*$ and $d(e,e'N)\Delta$. The latter process is of special
interest for looking for the effects of so-called chiral transparency---the
disappearance of the pion field of the ejectile~\cite{fms,chiral}.
\subsubsection{Experimental Objectives}
The $A(e,e^\prime p)$ and $d(e,e^\prime p)$ experiments described in the
previous Section are rather straightforward: they require a high-luminosity
electron beam to access the very small cross sections at
high-$Q^2$ and a set of two medium-resolution magnetic spectrometers to
determine, with reasonable precision, the recoil nucleon's momentum and
the nucleon's binding energy.
In the case of the $A(e,e^\prime p)$ transparency measurements, the nuclear recoil
momentum is typically restricted to a momentum
smaller than the Fermi momentum $k_F$ ($\approx$ 250--300 MeV/c).
The missing energy (identical to the binding energy plus nuclear excitation
energy) is restricted to be well below pion production threshold
($\approx$ 100 MeV). SRCs within a nuclear system push a sizable amount
of the individual nucleons to large momenta and binding energies. As
precise, quantitative evidence of this effect remains elusive, it is
preferable to restrict oneself to the region of single particle strength
described by above cuts in recoil nucleon momentum and missing energy.
Using two medium-resolution magnetic spectrometers,
with momentum and angular resolutions of order 0.1\% and 1 mr, one can
easily make well-defined cuts in recoil nucleon momentum and missing energy.
In the case of the $d(e,e^\prime p)$ cross section ratio measurement, a
good determination of the recoil nucleon's momentum is a fundamental concern:
the recoil nucleon momentum distribution drops steeply with missing
momentum, even though the double rescattering mechanism partly counteracts
this effect. This is illustrated by the absolute value of the cross section
ratio in Fig.~\ref{Fig.CT_ratio}, $\approx$ 0.1, for recoil nucleon momenta of 400 MeV/c
with respect to 200 MeV/c. Moreover, at 200 MeV/c the cross section
varies with recoil nucleon momentum by about 30\% per 10 MeV/c.
Hence, an absolute comparison of the measured $d(e,e^\prime p)$ cross
section ratio with calculations requires determination of the recoil
nucleon momentum value with precision much smaller than 10 MeV/c.
The use of large non-magnetic devices, such as electromagnetic calorimeters,
may be possible in limited cases, but will seriously affect the required
missing momentum determination. Thus, for the $A(e,e^\prime p)$ and
$d(e,e^\prime p)$ examples shown in Figs.~\ref{Fig.transparent} and~\ref{Fig.CT_ratio},
two magnetic spectrometers
operating at a luminosity of 1$\times$10$^{38}$ electron-atoms/cm$^2$/s, with
3 and 6 msr solid angle, respectively, were assumed. The larger solid angle
magnetic spectrometer would detect the quasi-elastically scattered electrons,
the smaller solid-angle would map the (relativistically boosted) Fermi cone.
The latter magnetic spectrometer would need a momentum range between $4$~GeV/c
($Q^2 \approx 6$ (GeV/c)$^2$) and 10 GeV/c ($Q^2$ $\approx$ 17 (GeV/c)$^2$).
Missing energy and recoil nucleon momentum resolutions would still be better
than 10 MeV and 10 MeV/c, respectively.
Lastly, the estimated beam time for the projected uncertainties in
Figs.~\ref{Fig.transparent}
and~\ref{Fig.CT_ratio} would be less than one month of beam time for the
$^{12}$C$(e,e^\prime p)$ transparency measurements, and one month of beam time
for the d$(e,e^\prime p)$ cross section ratio measurements up to $Q^2$ =
12 (GeV/c)$^2$, with one additional month required to push these ratio
measurements to $Q^2$ = 14 (GeV/c)$^2$. For the cross section ratio
measurements, one would need to determine the cross section yields at
recoil nucleon's momenta of 200 and 400 MeV/s with two separate angle
settings of the magnetic spectrometer (included in the estimated beam times).
Note that the $d(e,e^\prime p)$ measurements will build on the
existing Jefferson Lab experiments\cite{KG,EGS} at $6$~GeV which plan to
study the ratio $R$~(Eq.(\ref{rdouble})) up to $Q^2$ of $6$~GeV$^2$.
\subsection{Color Coherent Effects with Coherent Vector Meson Production
off the Deuterium}
Although the main emphasis in color transparency studies is given to the
experiments with nucleon electroproduction, it is widely expected that one
should observe the onset of color coherence in meson electroproduction at lower
values of $Q^2$ than for the case of nucleon knockout. It should be easier
to find the quark and anti-quark of a meson close together to form a
point like configuration, than to find the three quarks of a
nucleon together.
The QCD factorization theorem\cite{CFS97} for the exclusive meson
production by the longitudinally polarized virtual photons demonstrates that
the $q\bar q$ PLCs dominate in the Bjorken limit and that
CT should occur both for the coherent and incoherent channels. In the
leading twist the exclusive meson production can proceed through the
quark-antiquark and, in case of vector mesons, also through the two gluon ladder
exchange in t-channel\cite{CFS97}.
At small $x$ (probed for example at HERA\cite{AC99}) the two gluon ladder exchange
dominates in the production of $\rho$ and $\omega$-mesons.
However at Jefferson Lab kinematics, production of $\rho$ and $\omega$-mesons is dominated by
the quark exchange \cite{CFS97,GV}. The latter is confirmed by the
analysis of HERMES data on $\gamma^* + p \rightarrow \rho+ p$ reaction\cite{GV}.
Additionally, the analyses of Ref.~\cite{GV,FKS96,Lech} indicate that the
leading twist approximation overestimates strongly (by a factor $\sim 4 $
for $Q^2\sim$ 4 GeV$^2$) the $\rho$-meson production cross
section. The suppression factor in leading approximation
contribution is explained in Refs.\cite{FKS96,GV} as a higher twist effect
due to the finite transverse size of the photon
wave function in the convolution integral involving the interaction block,
virtual photon, and meson wave functions.
However, the model analysis of \cite{FKS96} indicates that the higher twist
effects may not interfere with the transverse localization of the vector meson
wave function leading to a possibility of the sizable CT effects already at
$Q^2\ge $4 GeV$^2$. The suppression of the leading twist contribution was
observed by \cite{GV} for the quark exchange channel for
both vector meson and pion production. However the corresponding analysis of the
transverse interquark distances is not yet available.
Studies of the CT for meson production with the Jefferson Lab upgrade will be very
important for understanding of the onset of the leading twist contribution
and determining what transverse separations are important in the higher
twist contributions. An observation of CT for meson production
would allow us to use these processes at pre--asymptotic $Q^2$ for measuring
the ratios of different nucleon generalized parton distributions. In the
case of the vector meson production it is feasible to look for CT both
in coherent and incoherent scattering off
nuclei\cite{BM88,FS88,KNNZ,Brod94}.
Studies of incoherent reactions require a
very good energy resolution in the mass of the residual system to suppress
processes where a meson is produced in the elementary reaction,
processes like $\gamma_L +N \to M + \Delta$, as well as in a multi-step
processes like $\gamma_L +N \to \rho +N,
\rho+N^*\to \pi +N^{**}$. The first experiments looking for CT effects in the
incoherent production of pions and $\rho$-mesons were recently
approved at Jefferson Lab \cite{e01107,e02110}.
Here we will focus on the reactions of the
coherent meson production in which the background processes mentioned above
are suppressed. The first experiment dedicated to the studies of coherent
production of vector mesons from nuclei at $6$~GeV approved recently at
Jefferson Lab\cite{SKKpro}.
The proposed upgrade of Jefferson Lab,
which will provide high energy, high intensity,
and high duty factor beams makes systematic studies of
these reactions a very promising area of study.
\begin{figure}[th]
\vspace{-0.4cm}
\begin{center}
\epsfig{width=4.8in,file=edv.eps}
\end{center}
\vspace{-0.4cm}
\caption{Diagrams corresponding to single (a) and double scattering
contributions in coherent vector meson electroproduction.}
\label{Fig.edv}
\end{figure}
The most promising channel for studying color coherent effects with
meson electroproduction is the coherent production of vector mesons off
deuteron targets:
\begin{equation}
e+d \to e' + V + d'
\label{vmpro}
\end{equation}
where ``$V$'' is the $\rho$, $\omega$ or $\phi$ meson\footnote{The discussed process is
also of interest for measurements of the generalized parton distributions of the
deuteron which are different from the corresponding nucleon distributions
\cite{Pire}.}. This reaction is unique for the following reasons:
\begin{itemize}
\item Due to the large {\it photon-vector~meson} coupling the cross section of the
process is large, and at high energies and small $Q^2$($< 1$~GeV$^2$) is well
understood in the framework of the vector meson dominance (VMD) model;
\item The deuteron is the theoretically best understood nucleus. It has zero isospin
and as a result, in the coherent channel, $\rho-\omega$ mixing will be
strongly suppressed, since
the $\rho$ and $\omega$ have isospin one and zero respectively.
The technical advantage of using the deuteron in coherent reactions is the
possibility of detecting the recoil deuterons.
\item Coherent production of vector mesons off deuterium is characterized by two
contributions:
single and double scattering contributions (Fig.~\ref{Fig.edv}). Moreover, it is a
well known fact from the photo-production experiments~\cite{slac71,overman} that at
large $-t\geq 0.6$ GeV$^2$, the double scattering contribution can be unambiguously
isolated (Fig.~\ref{rhoexp}).
\end{itemize}
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.0in,file=dscrhoexp.eps}
\end{center}
\caption{Cross section of the coherent $\rho^0$ photo-production off
deuteron. Data are from~\protect{\cite{slac71,overman}}. Dashed, dash-dotted, dotted
and solid curves represent single scattering, double scattering, interference between
single and double scattering, and full contributions respectively. Theoretical
predictions are based on the vector meson dominance (VMD)
model~\protect{\cite{FPSS98}}.}
\label{rhoexp}
\end{figure}
The strategy of CT studies in the coherent reaction of Eqn.~(\ref{vmpro}) is somewhat
similar to the strategies of studying CT in double scattering $(e,e'NN)$ reactions.
First one has to
identify kinematics in which double scattering effects can be isolated from
Glauber type screening effects (corresponding to the interference term of single and
double scattering amplitudes). The availability of the $t$-dependence of the
differential cross section allows us to separate these kinematical regions.
As Fig.~\ref{rhoexp} demonstrates, at $-t\leq 0.6$~GeV$^2$ the cross section is
sensitive to the screening effects, while at $-t\geq 0.6$~GeV$^2$ it is
sensitive to the double scattering contribution. Afterwards one has to study the
$Q^2$-dependence of the cross sections in these kinematic regions.
To identify unambiguously the observed $Q^2$-dependence with the onset of
CT one should however impose additional kinematic constraints based on the fact that
in lepto--production processes, the longitudinal interaction length plays an
important role and has a characteristic $Q^2$-dependence(see e.g. \cite{Gribov}):
\begin{equation}
l_c~=~{{2\nu}\over{Q^2~+~m_V^2~-t_{min}}}.
\label{llenght}
\end{equation}
An important aspect of the measurements is the ability to separate the effects of
a changing longitudinal interaction length from those of color coherence with an
increase of $Q^2$. This can be achieved by keeping $l_c$ fixed in a $Q^2$ scan of
the coherent cross section at a wide range of momentum transfers $t$.
Based on the above discussions one can identify a CT observable as the ratio of
two differential cross sections at fixed $l_c$ but at different $t$:
one in the double scattering ($t_1$), and another in the screening ($t_2$) regions,
\begin{equation}
R = {d\sigma(Q^2,l_c,t_1)/dt\over d\sigma(Q^2,l_c,t_2)/dt}
\label{Rt}
\end{equation}
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.0in,file=dscrhoct.eps}
\end{center}
\caption{The ratio $R$ of the cross sections at transferred momenta
$-t~=~0.4$~GeV/c$^2$, and $-t~=~0.8$~GeV/c$^2$ as a function of $Q^2$.}
\label{tratio}
\end{figure}
Figure~\ref{tratio} presents model calculation of the $Q^2$-dependence of
ratio $R$ for $\rho$ production for $-t_1~=~0.8$ (GeV/c)$^2$
and $-t_2~=~0.4$ (GeV/c)$^2$.
The upper curve is calculated without CT effect, within VMD with a finite
longitudinal interaction length taken into account~\cite{FKMPSS97}. The lower
band corresponds to calculations within the quantum diffusion model of CT~\cite{FLFS88}
with different assumptions for CT~\cite{FPSS98} with respect to the expansion of the PLC and
its interaction with the spectator nucleon. The upper and the lower limits in
the band correspond to $\Delta M^2=$ $1.1$ and $0.7$~GeV$^2$ respectively
(see discussion in Sec.(3.3)).
It is worth noting that for a complete understanding of the coherent production
mechanism and the formation of the final mesonic states, these measurements should also
be carried out at $Q^2<1$~GeV$^2$ to allow us to match the theoretical calculations
with VMD.
\subsubsection {Experimental Objectives}
The experiment studying the reaction of Eqn.~(\ref{vmpro}) will be the part of a broad
effort to establish the existence of color transparency in QCD at intermediate
energies.
A large acceptance detector such as CLAS at Jefferson Lab is an ideal tool for
conducting such experiments. With a single setting it can simultaneously measure
the coherent production of all vector mesons in a broad kinematic range.
Figure~\ref{wq2} shows the accessible kinematical range for an
$11$~GeV electron beam energy with CLAS++. The lines show the $Q^2-W$ dependence at
fixed coherent length $l_c$. The plot shows that at this energy, the
shape of the $t$-dependence can be studied up to $Q^2=5$~(GeV/c)$^2$ at $l_c~\sim~$0.8.
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.0in,file=kinwq2_lc.eps}
\end{center}
\caption{
The accessible range of $Q^2$ and W at $11$~GeV beam energy
with new design of CLAS. Light shaded region is defined by
the detection of the scattered electrons in the forward region
of the CLAS. Dark shaded region is the preferred kinematical
region for the proposed experiment. Lines represent $Q^2-W$
dependence at a fixed longitudinal interaction length.}
\label{wq2}
\end{figure}
In the final state, the scattered electron and the recoil deuteron will
be detected together with the decay products of the produced vector meson
In the case of $\rho^0\to\pi^+\pi^-$, only one of the decay
pions will be detected, and missing mass technique will be used
to identify the second one. Additional suppression of the three pion final state
($\pi^+\pi^-\pi^0$) can be achieved by using a veto on a neutral hit in the CLAS
calorimeters. For identification of the $\omega$, a neutral
hit in the calorimeters will be used to suppress the $\rho^0$
background. $\phi$ mesons will be identified via their $K^+K^-$ decay,
detecting one of the kaons. Count rates are estimated with acceptance calculations using
CLAS++, assuming a luminosity ${\cal L}=10^{35}\times A/Z$~cm$^{-2}$
sec$^{-1}$.
\begin{figure}[ht]
\begin{center}
\epsfig{width=4.0in,file=rate11g_0804.eps}
\end{center}
\caption{Expected errors on the ratio of cross sections at transferred
momenta $0.4$~GeV/c$^2$ and $0.8$~GeV/c$^2$ for 2000 hours of running on CLAS
with $11$~GeV beam. The kinematics are fixed at $0.75\leq~l_c\leq~0.8$.
The solid curve is the calculation of the ratio assuming no color transparency
effects, the points are with color coherent effect. Events in each point are
integrated in the bins of $\Delta Q^2~=~0.4$~GeV/c$^2$ and $\Delta l_c~=~0.2$.
We assume a CLAS++ luminosity of ${\cal L} = 10^{35}\times A/Z$~cm$^{-2}$ sec$^{-1}$.}
\label{Fig.rates}
\end{figure}
In Fig.~\ref{Fig.rates}, the expected errors on the ratio of cross sections
of Eq.~(\ref{Rt}) is presented for the same kinematical conditions
as in Fig.~\ref{tratio}, with $l_c$ fixed at $0.8$~fm.
The cross sections are calculated according to Ref.~\cite{FPSS98}.
The statistical errors correspond to 30 days of beam time.
This figure shows that the accuracy of the
experiment will allow one to unambiguously verify
the onset of CT in this region of $Q^2$.
\section{Summary and Discussion}
Quantum Chromodynamics provokes a number of interesting questions related to nuclear
physics. This review has addresses two of these:
\begin{itemize}
\item What is the quark nature of nuclei at low temperature and high density?
\item What is the influence of color on hadron-nucleon interactions in nuclei?
\end{itemize}
Our central theme is that the use of Jefferson Laboratory, with electron energies
up to $11$~GeV, will lead to substantial progress in answering these questions.
New studies of deep inelastic scattering by nuclear targets will
focus on the first question. We discuss how the search for scaling in
deep inelastic scattering at values of Bjorken $x>1$ will focus on a microscopic
study of the nature of the quantum fluctuations which briefly transform
ordinary nuclear matter into a high density system. Furthermore,
the measurement of backward going nucleons in coincidence
with the outgoing electron (denoted as tagging the structure function) will
lead to disentangling the various models which have been proposed as explaining the
nuclear EMC effect and thereby establish a clear signature of quark degrees of freedom
in the nuclear structure
New studies of the knockout of one or two nucleons by electrons at high momentum
transfer offer the promise of revealing how color influences the interaction
between an ejected color singlet particle and the spectator nucleons.
The absence of significant final state interactions, known as color transparency,
would allow the discovery of a novel new phenomenon in baryon interactions.
New measurements of the electroproduction of vector mesons in coherent interactions
with a deuteron target will show how color influences meson-nucleon interactions.
The questions we discuss have been perplexing physicists for more than twenty years.
The use of Jefferson Laboratory, with its well known high resolution, high duty factor,
and high luminosity, at an energy of $\simeq 12$ GeV, will finally provide the long
desired answers.
\ack
This work is supported by DOE grants under contracts DE-FG02-01ER-41172,
DE-FG02-96ER-40960, DE-FG02-96ER40950, DE-FG03-97ER-41014 and W-31-109-ENG-38.
This work was supported also by the Israel Science Foundation funded
by the Israel Academy of Science and Humanities.
We gratefully acknowledge also the support from Jefferson Lab.
The Thomas Jefferson National Accelerator Facility (Jefferson Lab) is operated
by the Southeastern Universities Research Association (SURA) under DOE contract
DE-AC05-84ER-40150.
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\bibitem{UJ} Ulmer P and Jones M 1994 In-Plane Separations and High Momentum Structure
in d(e,e'p)n {\it Jefferson Lab Proposal E94-004}
\bibitem{KG} Kuhn S E and Griffioen K A (spokespersons) 1994
Electron Scattering from a High Momentum Nucleon in Deuterium
{\it Jefferson Lab Proposal E94-102}
\bibitem{EGS} Egiyan K Sh, Griffioen K A and Strikman M I (spokespersons) 1994
Measuring Nuclear Transparency in Double Rescattering Processes
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\bibitem{WJKUV}Boeglin W, Jones M, Klein A, Mitchell, Ulmer P and Voutier E
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\bibitem{BNL01} Leksanov A \etal 2001 \PRL {\bf 87} 212301
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Zakharov B G and Zoller V R 1994 \PR C {\bf 50} 1296
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\bibitem{double} Egiyan K S \etal 1994 \NP A {\bf 580} 365
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\bibitem{chiral} Frankfurt L L, Lee T S H, Miller G A and Strikman M 1997 \PR C {\bf 55} 909
\bibitem{GV}Vanderhaeghen M, Guichon P A and Guidal M 1999 \PR D {\bf 60} 094017
\bibitem{FKS96} Frankfurt L, Koepf W, and Strikman M 1996 \PR D {\bf 54} 3194
\bibitem{Lech} Mankiewicz L, Piller G and Weigl T 1998 \EJP {\bf 5} 119
\bibitem{KNNZ}Kopeliovich B Z, Nemchick J, Nikolaev N N and Zakharov B G 1993
\PL B {\bf 309} 179
\bibitem{e01107} Ent R, Garrow K (spokespersons) 2001 Measurement of Pion Transparency
in Nuclei {\it Jefferson Lab Proposal E01-107}
\bibitem{e02110} Hafidi K, Holtrop M, Mustapha B (spokespersons) 2002 $Q^2$-dependence of Nuclear
Transparency for Incoherent $\rho^0$ Production
{\it Jefferson Lab Proposal E02-110}
\bibitem{SKKpro}Stepanyan S, Kramer L and Klein F~(spokespersons) 2001
Coherent Vector Meson Production of Deuteron
{\it Jefferson Lab Proposal E02-012}
\bibitem{Pire}Berger E R, F.~Cano F, Diehl M and Pire B, 2001 \PRL {\bf 87} 142302
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\bibitem{FKMPSS97} Frankfurt L \etal 1997 \NP A {\bf 622} 511
\bibitem{FPSS98} Frankfurt L, Piller G, Sargsian M and Strikman M 1998 \EJP A {\bf 2} 301
\end{thebibliography}
\end{document}