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\journal{Nuclear Physics A} 

\begin{document}

\begin{frontmatter} 

% Your Title - please insert
\title{Exploring Gluonic Matter with Electron-Ion Collisions}


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\author{J.H. Lee}
\address{Brookhaven National Laboratory, Upton, NY11973, USA}


\begin{abstract}
Light and heavy nuclei probed in deep inelastic scattering and diffractive processes in the 
high-energy regime open a new precision window into fundamental 
questions in QCD.   The proposed Electron-Ion Collider (EIC) is 
a new high-energy and high-luminosity electron-ion machine.
The design offers unprecedented access to explore the nature of QCD matter 
and strong color fields. In particular, the new collider will allow us to reach and explore the regime 
where the gluon density saturates, one of the fundamental outstanding questions in QCD, and test 
the validity of the Color Glass Condensate approach. 
Selected key measurements in electron-ion collisions with an emphasis on probing and 
characterizing the gluonic matter are discussed.
\end{abstract}

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\section{Introduction}
Quantum Chromodynamics has established itself as a successful theory of strong interactions, yet
the details of the underlying mechanism  of the theory have not been fully understood. 
Despite gluons being the mediator of the strong force and mostly responsible for the visible mass of the universe, 
their dynamics are far from being completely understood.
The difficulty of the theoretical understanding lies mainly in the nature of the self-coupling of gluons,
which is conjectured to lead to the phenomenon of parton saturation at small parton momentum fraction ($x$) 
theorized as the Color Glass Condensate (CGC)~\cite{cgc}.  
The non-linear saturated regime, where the increase of the number of gluons through 
gluon radiation and parton splitting is balanced by multi-gluon fusion between self-interacting gluons, is
the state with the maximum occupancy allowed in QCD. There have been strong hints of  gluon saturation 
in high-energy $ep$, $dA$, and $pp$ collisions~\cite{sat1,sat2,sat3,sat4}.
The dynamic scale $Q_s^2$ characterizes the onset of saturation for a high density of gluons in the target, 
 and increases as 
 $x$ gets smaller,  as shown in Fig.~\ref{fig1}.
The proposed electron-ion collider (EIC)~\cite{eic1,eic2} is designed to explore the nature of QCD matter 
and strong color fields
with a unprecedentedly wide kinematic reach accessing deep into the saturation regime
by utilizing the nuclear enhanced  saturation scale $(Q_s^A)^2\approx A^{\frac{1}{3}}(Q_s^p)^2$. 
The kinematic coverages of the EIC for the planned maximum electron and nucleon energy ranges 
from 5($e$)+100($N$) GeV (stage-I) to 30+100 GeV (stage-II)
with predicted saturation scales are  
shown for $ep$ and $eA$ collisions in Fig.~\ref{fig1}.
Selected key measurements in $eA$ collisions with the EIC,
probing and characterizing the gluonic matter,  
will be discussed. A more detailed description of the physics capabilities and 
 proposed machine designs of the EIC can be found in~\cite{white,int,erhic}.
\section{Key Measurements with the EIC}
\begin{wrapfigure}{r}{0.4\textwidth}
  \includegraphics[width=0.4\textwidth]{Qs_eic_acceptance.pdf} 
  \caption{Kinematic coverage in $x$ and $Q^2$ of the EIC for different beam energies,
  compared with predictions of the saturation scale, $Q_s^2$, in $p$, $Ca$, and $Au$.
  }
  \label{fig1}
\end{wrapfigure}
\subsection{Nuclear Structure Functions}
The nuclear structure functions $F_2^A(x,Q^2)$ and $F_L^A(x,Q^2)$, which characterize the partonic structure of nuclei,
are the most basic observables in
 $eA$,  and they will be one of the first observables at the EIC. 
 $F_2$  describes the quark and anti-quark momentum distribution, and is sensitive to gluons via scaling violations. 
The longitudinal structure function $F_L$ is directly proportional to the gluon momentum distribution.
The extraction of nuclear $F_L$ requires running at multiple beam energies, which can be accommodated by the flexibility of
the EIC.
Higher-twist effects~\cite{twist} in non-linear evolution in $x$ can be realized as variations of the nuclear modification factors of 
the structure functions defined as 
$R_{2,L}(x,Q^2)={F_{2,L}^A(x,Q^2)}/({AF_{2,L}^p(x,Q^2)})$.
Various models with different treatments of conventional QCD effects and the expected 
phenomenology of saturation, 
have a wide range of predictions for $R_{2,L}(x,Q^2)$.
Within the expected precision of the measurements at the EIC, differentiation between the different models is clearly possible 
in the region $10^{-4} \lesssim x \lesssim 1$.
\begin{wrapfigure}{r}{0.6\textwidth}
%  \centering
  \includegraphics[width=0.6\textwidth]{dihadrons_dphi.pdf}
  \caption{Saturation model~\cite{cyrille} prediction of azimuthal angle difference ($\Delta\varphi$) between two hadrons 
  in $ep$, $eCa$, and $eAu$ (left), 
   and comparisons with calculations from conventional non-saturated model  for $eAu$ (right).} 
  \label{fig2}
\end{wrapfigure}
In addition to inclusive DIS measurements for structure functions, 
nuclear diffractive structure functions $F^D_{2,L}$, which have never been measured in $e+A$, can be also accessed at the EIC.
At the EIC energy regime, diffractive processes are expected to share a large ($>$ 30\%) fraction of the total cross-section, and 
nuclear $F^D_{2,L}$ will be a sensitive measurement of saturation~\cite{diff}.  
\subsection{Di-hadron Correlations}
The nonlinear QCD evolution of multi-gluon distributions are expected to be 
different from that of the single-gluon distribution, and it can be measured through modification 
of di-hadron correlations~\cite{dima} in the semi-inclusive DIS process $e+A \rightarrow e^\prime+h_1+h_2+X$ dominantly through
a photon-gluon fusion process.
Saturation physics models allow us to compute the functional form of the multi-gluon distribution as functions 
of gluon transverse momentum $k_\perp$ and saturation momentum $Q^2_s$($x$).
Figure~\ref{fig2} shows the difference in azimuthal angle between a trigger- ($p_T^{trig}>2$ GeV/$c$) 
and associate- ( $p_T^{trig}$ $>p_T>1$ GeV/$c$) hadrons
for theoretical predictions and expected experimental measurements at the EIC.  
 Precise measurements of these di-hadron correlations at the EIC would allow one to extract the spatial multi-gluon 
 correlations in the transverse plane and study their non-linear evolution.  Saturation effects in this channel correspond 
 to a progressive disappearance of the back-to-back correlations of hadrons with increasing atomic number $A$.
\subsection{Exclusive Diffractive Vector Meson Production}
For precise transverse imaging of gluon distributions and how the small-$x$
evolution modifies the transverse distributions, exclusive vector meson production, $e+A \rightarrow e^\prime + V + A^\prime$ where 
$V = \rho, \phi, J/\psi$ in coherent and incoherent diffractive processes will be studied.  The vector mesons are formed from the interaction of a 
$q$$\bar{q}$ color dipole fluctuation of the virtual photon with the nucleus.
\begin{wrapfigure}{r}{0.7\textwidth}
%  \centering
  \includegraphics[width=0.7\textwidth]{dsigma_dt_jpsi_phi.pdf}
  \caption{$d\sigma/dt$ distributions for exclusive $J/\psi$ (left) and $\phi$ (right) production in coherent and incoherent events in 
  diffractive $eAu$ collisions. Predictions from saturation and non-saturation models are shown.}
  \label{fig3}
\end{wrapfigure}
Coherent diffraction, where the nucleus stays intact, probes the space-time distribution of the partons in the nucleus.
Incoherent diffractive processes with the excited nucleus breaking up is sensitive to fluctuations of high parton densities at small impact 
parameter with large momentum transfer in the collision. 
The impact parameter  dependence of the gluon distribution is obtained from the dependence of 
the squared momentum transfer ($t$) on the cross-section by Fourier transformation.
The EIC allows one to study how this changes with the non-linear QCD evolution towards small $x$ in exclusive diffractive processes.
Experimentally coherent processes can be tagged by vetoing on the neutrons emitted by the nuclear breakup of the nucleus 
in the incoherent diffractive $eA$ collision. Figure~\ref{fig3} shows the $d\sigma/dt$ distribution for $J/\psi$ on the left and 
$\phi$ mesons on the right, and they are compared with predictions of saturation and non-saturation models. The curves 
were generated with the Sartre~\cite{int} event generator. 
\subsection{Energy loss, Fragmentation at Large $x$}
The EIC can reveal the nuclear structure throughout the ($x,Q^2$) plane, from
gluon saturation at low-$x$ to the gluon EMC effect and its $Q^2$ evolution at high-$x$,
which allows the estimation of the  nuclear quark and gluon distributions and their uncertainties. 
The EIC will provide a wide $\nu$ $( = \frac{Q^2}{2Mx})$ range, which greatly extend the existing
fixed-target measurements~\cite{accardi}. 
At small $\nu$, 
studying in-medium hadronization can provide information on
the dynamics of confinement: the stages of hadronization 
and their time scales.
At large $\nu$,  parton propagation through the medium will allow us to study
the energy loss and $p_T$-broadening of leading partons as well as jet-shape modifications.
For the first time, the in-medium hadronization and propagation of heavy quarks can be also studied, and the pQCD 
description of cold nuclear matter can be tested.  The EIC can provide the insight into how colorless 
hadrons emerge from the quarks and gluons and the dynamics governs color 
neutralization and hadron formation. 
\\\\
\noindent Table~\ref{table} shows a summary of the key measurements at the EIC. 
\begin{table}[htbp]
   \centering
   \begin{tabular}{@{} lcccc @{}} % Column formatting, @{} suppresses leading/trailing space
      \toprule
      to study    & measurements & access to & stage-I & stage-II\\
      \midrule
      nuclear wave function;     &  $F_{2,L}$ & integrated gluon                &   gluons at                                             &  exploration of \\ 
      saturation, $Q_s$              &                     &      momentum                   &     $10^{-3}$$\lesssim$$x$$\lesssim 1$      &  saturation regime\\
                                                    &                     &         distribution                                                             &                    &  saturation\\
      \midrule
         nonlinear QCD                &  di-hadron                    &        $k_T$-dependent     &    onset of                &  nonlinear\\
        evolution/universality      &     correlations              &         gluons;      gluon                &    saturation;                &  small-x\\
                                                    &                                        &         correlations               &     $Q_s$               &  evolution\\
       \midrule
         nonlinear small-$x$                         &  diffractive process;                     &        spatial gluon     &    moderate $x$              &  saturation\\
        evolution;  saturation                        &    $\sigma_{diff}$, exclusive                              &        distributions                     &    with nuclei                &  regime; \\
        dynamics                                            &        vector mesons      &                                    &     $Q_s$               &  evolution\\
       \midrule
         parton energy loss,                        &   large-$x$ SIDIS;                     &        transport              &    light flavors              &  rare probes and\\
        shower mechanism                        &     jets                                                 &      coefficients in         &    and charm;                &  bottom; \\
        and evolution                                   &                                             &     cold matter               &   jets                         & large-$x$ gluons  \\         
      \bottomrule
   \end{tabular}
   \caption{Key measurements in $eA$ collisions at the EIC (stage-I and stage-II) addressing the physics of high gluon densities and dynamics of quarks and gluons in a nucleus.}
   \label{table}
\end{table}
\section{Conclusion}
The proposed EIC will be the world's first electron ion collider with high luminosity at high-energy.
The versatility of the machine will 
allow systematic exploration of strong gluon fields in nucleons and nuclei
with unprecedented precision and kinematic range and reach/establish a new QCD phase - gluon saturation in $eA$ collisions. 
The physics program of the EIC is geared
towards a unified understanding of strongly interacting matter. 
\section*{References}
{\raggedright
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\begin{thebibliography}{00}
%------- replace following references ;-)
\bibitem{cgc} F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan. Ann. Rev. Nucl. Part. Sci., {\bf 60} (2010) 463.
\bibitem{sat1} J.L. Albacete {\it et al.} arXiv:1203.1043.
\bibitem{sat2} A. Stasto, B. Xiao, and F. Yuen, arXiv:1109.1817.
\bibitem{sat3} P. Tribedy and R. Venugopalan, arXiv:1112.2445.
\bibitem{sat4} K. Dusling and R. Venugopalan, arXiv:1201.2658.
\bibitem{eic1} A. Deshpande, these proceedings.
\bibitem{eic2} C. Marquet, these proceedings.
\bibitem{white} C. Aidala {\it et al.}  ``A Hign Luminosity, High Energy Electron Ion Collider", A white paper prepared for the NSAC LRP 2007. 
\bibitem{int} D. Boer  {\it et~al.}, ``Gluons  and the quark sea at high energies: distributions, polarization, tomography'', A report on the joint BNL/INT/Jlab program on the science case for an Electron-Ion Collider, Sept. 13 to Nov. 19 2010, Institute for Nuclear Theory, Seattle, arXiv:1108.1713, 2011.
\bibitem{erhic} V.N. Litvinenko {\it et~al.}, ``High-energy high-luminosity electron-ion collider eRHIC'', arXiv:1109.2819, 2011.
\bibitem{twist} J. Bartels, K. Golec-Biernat, and L. Motyka.  Phys. Rev., {\bf D81} (2010) 054017.
\bibitem{diff} M.S. Kugeratski {\it et~al.}, Eur. Phy. J., {\bf C46} (2006) 413.
\bibitem{dima} D. Kharzeev, E. Levin, and L. McLerran.  Nucl. Phys., {\bf A748} (2005) 627.
\bibitem{cyrille} C. Marquet, B. Xiao, and F. Yuan. Phys. Lett., {\bf B682} (2009) 207.
\bibitem{accardi} A. Accardi  {\it et~al.}, Rivista del Nuovo Cimento, {\bf 32} (2009) 439.
\end{thebibliography}


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