Deep inelastic lepton-hadron scattering is the cleanest and most
precise probe of parton dynamics in protons and nuclei.
The LHeC is the only current proposal for TeV-scale lepton-hadron 
scattering and the only medium-term potential complement to the
LHC $pp$, $AA$ and $pA$ programme at the energy frontier. As such,
it has a rich and diverse physics programme of its own, as documented
extensively in the recent conceptual design report (CDR)
\cite{AbelleiraFernandez:2012cc}
and summarised in an initial submission by the LHeC Study Group
to the European Stratey of Particle Physics (ESPP) discussion
prior to the Cracow Symposium \cite{cracow1}.
The focus of this second submission to the ESPP is a 
further exploration of the relationship between the LHeC and the LHC. 
Specifically, by improving the understanding of the LHC initial state through 
tighter constraints on parton densities and providing information
on many other aspects of strong interactions, the LHeC 
extends the capabilities of the LHC programme substantially. 

%ORIGINAL PARAGRAPH FROM MAX
%
%The relation of the LHeC and the LHC is established with the
%prospect for synchronous operation of the new $ep$ (and $eA$)
%facility with the GPDs in the high luminosity phase of
%the Large Hadron Collider (HL-LHC). The physics prospects
%of the GPD programme, as illustrated in the ATLAS and
%CMS contributions to the ESPP process, are centered around
%two major questions: i) a desirably precise and complete
%exploration of the newly discovered particle, which most
%likely is the Higgs and ii) a continued search for
%new particles and symmetries or extra dimensions in the
%few TeV range of mass, or higher with indirect constraints.
%Seen from the perspective of this programme, one may
%ask, disregarding a genuine interest in QCD,
%in electroweak physics or in DIS at TeV
%energies and also disregarding all of the heavy ion
%physics, to which extent the precision LHeC results
%would be of interest for a) transforming the LHC into
%a precision Higgs facility of maximum Higgs coverage
%and b) for maximally exploiting and widening
%the search potential, especially at the limit of phase space,
%large masses which correspond to high Bjorken $x$.
%The present paper is written around these two questions,
%which evolve with time and which 
%deserve further study. It also recalls the
%importance of the deep inelastic electron-ion scattering
%for the completion of the LHC HI programme.
%The relations between the LHeC and the LHC
%are of course deeper and mutual but cannot be
%fully covered in a brief communication.

The LHeC offers the
prospect of synchronous operation of a new $ep$ (and $eA$)
facility with the GPDs in the high luminosity phase of
the LHC (HL-LHC). The physics programme 
of the GPD experiments on this timescale, as illustrated in the ATLAS and
CMS contributions to the ESPP process, is centered around
two major objectives: 
i) the most precise and complete
exploration possible of the newly discovered Higgs-like particle and 
ii) maximising the sensitivity of the continuing search for
new particles and symmetries or extra dimensions in the
few TeV range of mass.
%, and, with indirect constraints, beyond.
This document investigates the potential impact
%, disregarding a genuine interest in QCD,
%in electroweak physics or in DIS at TeV
%energies and also disregarding all of the heavy ion
%physics, 
of precision LHeC results on these objectives, as we understand them 
at present, recognising that the situation will 
evolve with time and 
deserves continuous further study. 
% transforming the LHC into
%a precision Higgs facility of maximum Higgs coverage
%and b) for maximally exploiting and widening
%the search potential, especially at the limit of phase space,
%large masses which correspond to high Bjorken $x$.
%The present paper is written primarily around these two questions,
It also recalls the
importance of the deep inelastic electron-ion scattering
for the completion of the LHC heavy ion programme.
The mutual relations between the LHeC and the LHC
are of course much deeper than can be covered in 
a brief communication such as this.

As documented in detail in \cite{AbelleiraFernandez:2012cc},
the parton density (PDF) determinations offered by the LHeC are substantially
superior to the possibilities using LHC data alone and, for the first
time, provide a full flavour decomposition essentially
free of assumptions. The LHeC also promises a broad and
unique programme of further strong interaction physics,
such as the exploration of a newly accessed
high density, low coupling regime at low $x$
and a new level of precision and hugely extended 
kinematic coverage on the partonic structure of nuclei.
Combined with competitive  sensitivity to new
physics in channels where initial state lepton quantum numbers are an advantage,
the LHeC represents a cost effective means of fully exploiting the LHC and
substantially extending its physics programme.

%As well as discovering 
%a new boson 
%with a mass of $126 \ {\rm GeV}$ \cite{:2012gk,:2012gu}
Following the discovery by 
ATLAS~\cite{:2012gk} and CMS~\cite{:2012gu} 
of a new boson
with a mass of about $126$\,GeV, 
the future  focus of the field will be firstly 
on determining whether this particle is the 
Standard Model Higgs boson or something more exotic and 
secondly on extending the sensitive
range for discovery of other new particles as far as possible. 
%Whilst new massive particles may be visible through single resonant production, 
%an equally likely signature is a non-localised excess, as 
%expected for example if SUSY is observable via 
%$gg \rightarrow \tilde{g} \tilde{g}$. 
ATLAS and CMS have reported exclusion limits for a wide range
of massive new particles in the $1 - 2 \ {\rm TeV}$ range. 
With no strong evidence for new effects so far,
the need to further extend such searches to the largest masses possible
is paramount. The increased beam energy
following LS1 will provide a first major step in this direction. 
Beyond that, further progress is limited by luminosity, due to the 
fast-falling cross sections as higher and higher $x$ PDFs are
involved (especially for gluon initiated processes), and by the 
uncertainities on those PDFs. To fully exploit the new particle
discovery range of the LHC,
both a luminosity upgrade and tighter external 
PDF constraints are therefore required.

The future exploration of the Higgs sector at the LHC,
for example by measuring
relative couplings and testing the CP structure, may similarly
become limited by theory uncertainties derived from PDF measurements
once the very high luminosities possible at HL-LHC have been 
accumulated \cite{ATLAS-LHLHC}. This is particularly true for the 
`working horse' channels in which the Higgs decays to $\gamma \gamma$
or four charged leptons. Whilst LHC inclusive $W$ and $Z$ production
data will somewhat improve constraints on the quark densities of the proton
at the electroweak scale, they
will have a limited impact on the gluon density, which is more pertinent
to Higgs physics, given the dominant $gg$ production mechanism. 

The present document is organised as follows. Section~2 presents a 
brief reminder of the Linac-Ring configuration of the LHeC,
which is being prepared via prototype and design
developments with a view to endorsement of the project in the
next few years. This includes a brief exploration of possible ways of
enhancing the luminosity to the level of $10^{34}$\,cm$^{-2}$s$^{-1}$.
Section~3 is devoted to PDF determinations, including the relation between
$pp$ and $ep$, the LHeC's potential to
unfold all quark flavours and to unfold the gluon
density to an unprecedented level of precision in a huge $x$ range. 
Section~4 provides a discussion of two key
features of the LHeC which may be 
crucial in the establishment of new theories or
particles: the prospect of measuring $\alpha_s$
to per mille accuracy and the precision calculation of
SUSY cross sections involving high $x$ partons at the limits of the 
accessible HL-LHC
mass range, 
gluino pair production being used as an example.
%for a maximum extension of the search range as required by
%the luminosity upgrade of the LHC. 
Section~5 discusses
the main contributions of the LHeC to Higgs physics;
its potential to find new physics at the cleanly accessible
$WWH$ (and $ZZH$) vertices and the corresponding
precision coupling measurements to vector bosons, 
as well as its reduction of the QCD-dominated
theory uncertainties on Higgs production in $pp$ collisions. 
Finally, in Section~6, the
importance of $eA$ DIS measurements for the LHC heavy ion programme is 
briefly reviewed.
