As discussed in \cite{AbelleiraFernandez:2012cc}, the study of $e$A collisions at the LHeC will have strong implications on physics of ultrarelativistic heavy-ion collisions presently studied at RHIC and at the LHC. This applies both to the initial state which determines the subsequent behaviour of the dense medium produced  in the collisions, and to the ability of hard probes to characterize such medium.  Different physical effects need to be cleanly separated, that are not due to the presence of a medium, like nuclear modifications of parton densities or details of the mechanism of particle production. Finally, it is  also crucial to understand how the medium affects the probe, for example the mechanism of QCD radiation and parton energy loss in a medium. %

%\subsection{Nuclear parton distributions and initial conditions for collective behaviour}
\subsection{Aspects of heavy-ion physics that can be addressed in $e$A at the LHeC}

Currently, nuclear parton distribution functions (nPDFs) suffer from large
 uncertainties  due to the very limited coverage in the kinematics obtained
 by the measurements in low-energy fixed-target experiments. As a result, 
gluon and quark distribution functions are  presently almost unconstrained 
for values of $x$ below $10^{-2}$, which translates into uncertainties in 
the precise characterization of the medium in heavy-ion collisions through
 hard probes e.g. $J/\Psi$ production \cite{Abelev:2012rv}. 

The LHeC will 
provide an unprecedented precision for the measurement of the parton content of nuclei. The kinematic coverage for $e$A collisions at the LHeC  will allow to extend the $x$ and $Q^2$ ranges by 3 to 4 orders of magnitude down to $x\sim 10^{-5}-10^{-6}$ and up to $Q^2 \sim 10^6 \; {\rm GeV}^2$ respectively, see \cite{AbelleiraFernandez:2012cc,cracow1}. Flavor decomposition of the nuclear structure functions will be performed for the first time, including the measurement of the previously unknown charm and beauty  components of nPDFs. An example of the constraining power of the LHeC is illustrated in Fig.~\ref{fig:rpdf} (left) where the nuclear modification factor is shown for the gluon distribution  as a function of $x$ for a fixed value of $Q^2$.  Clearly, the LHeC offers  huge possibilities for distinguishing between different models of nuclear shadowing and for  constraining the nuclear parton dynamics, particularly at small values of $x$. Note also that diffraction in $ep$ and nuclear shadowing are theoretically related through Gribov relation that can be cleanly tested at the LHeC. Finally, access to large values of $x > 0.1$ will also be possible at the LHeC, thus providing additional information about the antishadowing and EMC effects for nuclear ratios.

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\begin{center}
\centerline{\includegraphics[clip=,width=.32\textwidth]{Pb_allsets2}\hskip -0.3cm
\includegraphics[width=0.30\textwidth]{ds0w-skew}\hskip -0.4cm
\includegraphics[width=0.45\textwidth,angle=0]{dihadron.pdf}
}
\vskip -0.5cm
\caption{Left: Ratio of gluon density for protons bound in Pb to those in a free proton at $Q^2$ = 1.69 GeV$^2$. The red
 band corresponds to the uncertainty   in the original EPS09 analysis, while the brown band corresponds to the uncertainty
 obtained after including nuclear LHeC pseudodata on the total reduced cross sections (Fit 1).  The light blue band corresponds 
to the uncertainty after including the information on charm  and beauty cross sections (Fit 2).
Center: Energy ($W$) dependence of the coherent photoproduction of the $J/\Psi$ on a proton and different nuclei in the forward case $t=0$
according to the b-Sat model.  Right: Di-hadron correlation function  for the case of the scattering off the proton 
(red-dashed and black-solid lines) compared to the $e$A case (blue-dotted line). The energy of the electron is assumed 
to be $E_e=50 \  {\rm GeV}$. The observed hadrons are pions.
See \cite{AbelleiraFernandez:2012cc} for details.}
\label{fig:rpdf}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Initial condition for collective behaviour}

Ultrarelativistic heavy-ion collisions at RHIC and the LHC allow to create and characterize partonic matter under extreme conditions. The description of the collective behaviour of such matter involves several stages, which include modeling the initial conditions prior to isotropization, the subsequent evolution of the system through relativistic hydrodynamics and, finally, hadronisation. The initial conditions for the heavy ion collisions are currently being parametrized and fitted to the data. They involve large uncertainties which  translate into the uncertainties of the extracted bulk properties of the medium such as shear viscosity \cite{Shen:2011zc}.
The LHeC offers unique possibilities for pinning down these initial conditions. The initial state can be determined through the details of nPDFs and of particle production, both measured precisely in $e$A collisions.
Furthermore, there are sound theoretical indications that at low $x$ and for large nuclei, a novel regime of QCD appears that is characterized by high parton densities. In this regime, the standard collinear framework that was developed for a dilute parton system must break down and should be superseded by a more complicated, non-linear evolution setup that leads to the saturation of parton densities. 
The existence and properties of this partonic dense regime will be tested at the LHeC  through several measurements ranging from those of structure functions, with subsequent determination of parton distribution functions in nuclei (see Fig.~\ref{fig:rpdf} (left)) and inclusive diffraction to more exclusive ones such as elastic vector meson production (Fig.~\ref{fig:rpdf} (center)). Thus it will be possible to locate the onset of the saturation regime as a function of $x$, mass number and impact parameter of the collision.
Remarkably, and unlike in lower energy facilities, at the LHeC this phenomenon can be investigated in the DIS region where the coupling is small and perturbative techniques are applicable to the non-linear regime. Furthermore, saturation can be observed both in $ep$ and $e$A, which gives a unique possibility for disentangling saturation from other nuclear effects.


Finally, one of the standard tools for characterizing the medium created in heavy-ion collisions is the modification of the yield of high-energy particles - jet quenching - due to the changes in QCD radiation and hadronization induced by the presence of a dense medium.
Such characterization demands a detailed understanding of these phenomena that can be achieved by studying particle and jet yields and correlations in $e$A collisions at the LHeC. In this respect, the kinematics of the parton whose radiation and hadronization undergoes medium modifications is much more precisely constrained in DIS than in hadronic collisions. As an example, the large kinematic range of LHeC allows to investigate the dynamics  of partons traveling through the nucleus
with energies from moderate to very large.
% from low energies, for which hadronization is expected to occur inside the nucleus, to high energies
% with hadronization outside the nucleus.
 Besides, the study of particle angular correlations (e.g. back-to-back ones, Fig.~\ref{fig:rpdf} (right)) will offer insight  both on these changes and on the possible breakdown of collinear factorization and  the existence of a dense partonic regime. 


\subsection{Prospects in $p$Pb collisions at the LHC compared to $e$Pb at the LHeC}

A first $p$Pb run at the LHC is scheduled for early 2013 (a pilot run providing ${\cal O}(10^6)$ collisions per experiment happened last September 13th 2012 and one publication already appeared \cite{alicehndeta}). Tentative plans exists for additional runs in the future \cite{jowettcrac}. They offer information \cite{Salgado:2011wc} about similar aspects relevant for heavy-ion collisions to those discussed here for $e$A at the LHeC. 
%By comparing Fig.~\ref{fig:rpdf} left with the corresponding one for $p$Pb in \cite{AbelleiraFernandez:2012cc} or Fig. 9 in \cite{Salgado:2011wc}, one sees that the kinematical region explored at the LHC with forward instrumentation, and at the LHeC, will be similar.
The $x-Q^2$ region explored at the LHC with forward instrumentation both in hadronic and ultraperipheral collisions, and at the LHeC, will be comparable~\cite{AbelleiraFernandez:2012cc,Salgado:2011wc}

Nevertheless, even assuming that corrections to collinear factorization are small - though they are expected to be larger in $p$A than in $pp$, the nuclear modifications of partonic densities and of fragmentation and hadronization, come intrinsically mixed for most observables in $p$A collisions, see e.g. the discussion in \cite{Eskola:2012rg}. Therefore, they will be far more difficult to constrain in $p$Pb at the LHC than in $e$Pb at the LHeC. Besides, the accuracy - of a few percent - for measurements of cross sections achievable at the LHeC cannot be matched in a $p$A collider.
% The systematic uncertainties for several observables like jets, photons,$\dots$, are bound to be larger in $p$Pb than in $e$Pb due to the larger background for reconstruction, contribution from multiple hard collisions, etc.
Finally, the determination of the kinematic variables relevant for parton densities, fragmentation functions and other quantities of interest, is much more direct in DIS than in hadronic collisions. Thus, the information that can be obtained in $e$Pb collisions at the LHeC will be substantially more precise, and the possibilities for discovery of the dense regime of QCD at small $x$ through quantitative studies as those proposed in \cite{AbelleiraFernandez:2012cc} significantly larger, than in $p$Pb collisions at the LHC.

