Supersymmetry (SUSY) is a compelling theory providing an 
extension of the Standard Model (SM) at high energies. 
In the SM, the Higgs boson mass suffers from large quantum loop 
corrections, as large as the cut-off scale of the theory, 
and therefore needs a high degree of 
fine-tuning of parameters to be cancelled.
With SUSY, this `naturalness problem' is solved by 
the addition of supersymmetric partner particles to the 
known fermions and 
bosons which 
cancels the largest of these loop effects
and permits the Higgs boson mass to 
lie naturally at the $\sim 100\,$GeV scale. 
Supersymmetric theories present other advantages, 
including the unification of running 
coupling constants at the Planck scale, 
see Sect.\,\ref{alfas}, and renormalization group 
equations that radiatively generate 
the scalar potential that leads to electroweak symmetry breaking. 
Yet, at the LHC there is so far no sign for SUSY particles.
It therefore is crucial to
enlarge the beam energy and luminosity
to extend these searches to the limit of phase space.

The possible (non)conservation of the discrete  $R$-parity, which relates 
spin (S), baryon and lepton numbers (B and L), is fundamental to determine the SUSY 
phenomenology.  In the framework of a 
generic $R$-parity conserving supersymmetric extension of the SM, 
SUSY particles are produced in pairs and the lightest supersymmetric 
particle (LSP) is stable.  In a large variety of models the LSP is the 
lightest neutralino, $\tilde{\chi}^{0}_{1}$, one 
of the SUSY partners of the gauge bosons 
together with its three heavier mass eigenstates 
($\tilde{\chi}^{0}_{2,3,4}$)  
and the charginos ($\tilde{\chi}^{\pm}_{1,2}$). 
The lightest neutralino only 
interacts weakly and provides a Dark Matter candidate with 
the appropriate 
relic density to possibly 
explain the cosmological Dark Matter. 


The possible appearance of $R$-parity-violating couplings, and hence the non-conservation 
of baryon and lepton numbers ($B$ and $L$) in supersymmetric reactions, imply an even 
richer  phenomenology. Although $R$-parity-violating interactions must be sufficiently 
small, a most dramatic implication of interactions from $R$-parity violations is 
the automatic  generation of neutrino masses and mixings. The possibility that 
the results of atmospheric 
and solar neutrino experiments may be explained by neutrino masses and 
mixings originating from $R$-parity-violating interactions has 
thus motivated a large number  of studies and models. 


The discovery (or exclusion) of supersymmetric particles remains a 
prime priority for the LHC experiments which
search for physics beyond the SM at the TeV scale. 
%The role of the LHeC is to accompany these
%searches with precision QCD dynamics and structure measurements,
%and to complement and possibly resolve the observation of new phenomena.  
At the LHeC, SUSY particles could be produced due to 
sizable lepton flavour violating 
terms or, in the framework of $R$-parity conserving models, via 
associated production of selectrons  and first and 
second generation squarks. The latter process 
would present a sizable 
cross section only if the sum of selectron and squark masses 
is below or around $1$\,TeV, i.e.  for relatively light squarks. 
Current exclusion limits set by the ATLAS and CMS 
experiments on first and second generation squarks 
extend  up to $\simeq 1.5$\,TeV, under the 
assumption that 
the first two squark families are degenerate. In models where 
this condition is relaxed, windows 
of discovery relevant for the LHeC 
might still be open at the time of start-up. 


Less stringent constraints exist in the context of $R$-parity violating scenarios. 
Processes of interest for the LHeC 
include leptoquark-like processes and associated 
production of quark-neutralinos via squark-quark-neutralino couplings, 
where squarks can be 
off-shell and thus beyond direct LHC exclusion limits. While stringent 
constraints exist for 
associated production of leptons and quarks possibly 
deriving from squark decay, processes 
of the kind $eu \rightarrow d\tilde{\chi}^{0}_{1}$ 
and subsequent decays of $\tilde{\chi}^{0}_{1}$ 
to leptons and quarks might be 
difficult to study at the LHC, due to the overwhelming SM multi-jet 
background, 
and be succesfully searched for at the LHeC. The LHeC will also provide 
indirect handles for the case of supersymmetry. 


The dominant SUSY production channels at the LHC are assumed to be 
squark-(anti)squark, 
squark-gluino, and gluino-gluino pair production. All gluon-initiated processes  
suffer from  very large uncertainties due to the highly limited knowledge
of the gluon density at high $x$.
If gluinos of $2-4$\,TeV mass exist, their discovery and kinematic characterisation 
will depend on the capability to predict their production cross section with 
good precision. 

\begin{figure}[t]
\unitlength=1cm
\begin{picture}(13,7)
\put(3,0){\includegraphics[clip=,angle=0.,width=.6\textwidth]{klees.pdf}}
%\put(9,7.){\includegraphics[clip=,angle=-90.,width=.43\textwidth]{alfbt-crop.pdf}}
\end{picture}
\caption{Calculation of gluino pair production in NLO SUSY-QCD using Prospino~\cite{prospino}
and assuming squark mass degeneracy and equality of squark and gluino masses for illustration.
The error bands are around central values (solid lines) and correspond to
the uncertainty quotations of the various PDF groups. The red band of uncertainty
for the LHeC corresponds to the statistical and systematic errors including their
correlations as treated in the NLO QCD fit described in the CDR.
}
\label{fig:klees}
\end{figure}
Fig.\,\ref{fig:klees} shows the amount of PDF 
uncertainties, current and expected, on the gluino-pair production 
cross section calculated at NLO SUSY-QCD expressed 
as the ratio to the MSTW08 predictions. 
The cross section calculation is based on \cite{Beenakker:1996ch} and assumes that first 
and second generation squarks are mass degenerate and equivalent to the gluino mass. 
The renormalization and factorization scale is set to the squark/gluino mass. 
Several of the current PDF fits are considered and suffer from very large uncertainties.   
They also differ considerably, by factors, in their central predictions,
which has to do with the smallness of the gluon distribution at large
Bjorken $x$ and the uncertainty of jet physics constraints
at the Tevatron and the LHC, related to scale and calibration
uncertainties and to the size of theory corrections
at high mass Drell-Yan scattering. On the other hand, predictions 
employing the LHeC-derived PDF fits exhibit
much smaller uncertainties, between 5\% and 20\% for gluino 
masses between 500 GeV and 5 TeV, thanks to the expected precision
LHeC measurements of the NC and CC cross sections, and the derived
quark and gluon densities, see above and \cite{AbelleiraFernandez:2012cc}. 
 
Such a greatly reduced level of uncertainty 
is comparable or below the experimental uncertainties 
expected for squark/gluino searches at the high-luminosity LHC, e.g. through 
enhanced multi-jet production rates (with or without missing transverse momentum) 
compared with SM predictions. It is difficult to predict the
fate of high mass searches for SUSY. However, it is a prime goal of
the GPD LHC experiments in the HL-LHC phase
to explore the phase space to the or close to the
kinematic limit. Fig.\,\ref{fig:klees} makes it clear that
the exploration of the 
region of above a few TeV gluino mass requires to
much improve the PDF, especially the gluon uncertainty
for which the LHeC is the most reliable and most precise
base, as has been argued above. 
Should deviations from SM predictions be observed, accurate predictions 
for inclusive squark and gluino production cross sections will be crucial to 
understand the nature of the new physics discovered and to determine 
SUSY particle masses and properties~\cite{Dreiner:2010gv}.
%Should new states indeed
%be observed, thier investigation will also require to
%accurately predict their production cross sections
%and further kinematic characteristics.
In this regard the
LHeC, as an ultra-precision QCD instrument, is a
necessary complement for the HL-LHC physics programme,
both for high mass searches as exemplified here and
for making the LHC a precision Higgs factory as
will be illustrated below.


The interest in $R$-parity violating SUSY translates directly 
into the striking potential of the LHeC to determine the
lepto-quark or lepto-gluon quantum numbers should such states
be discovered at the LHC. The $ep$ machine has a clean $s$-channel
single production mode, with variable input beam parameters
while the LHC produces them predominantly in pairs. 
This is discussed in detail in \cite{AbelleiraFernandez:2012cc}, 
as is also the reach in contact
interactions, excited leptons, anomalous lepton-quark interactions
and other BSM topics.
