%
Deep inelastic scattering is an ideal process for the determination
of the strong coupling constant, which determines the scaling
violations of the parton distributions. Despite a major effort for
the past more than $30$ years, there is no precise determination
of $\alpha_s$ available, of a precision competing with QED or weak
couplings, and a number of severe questions remains to be solved as
is discussed in~\cite{AbelleiraFernandez:2012cc} and has recently 
been summarised in~\cite{job1012}. Questions regard the (in)consistency 
of previous DIS data, the (in)consistency of inclusive DIS and jet based data, the
true uncertainty of the world average on $\alpha_s$ including the
role of various lattice QCD determinations etc. It is for these
reasons and because of the importance of $\alpha_s$ for the
grand unification of gauge theories as for a plethora of predictions
of cross sections, as of Higgs production at the LHC, 
that an experimental determination of the strong coupling
with an order of magnitude improved precision is crucial.
It is also time to challenge the lattice QCD $\alpha_s$ results, which
seem to be most accurate but which
exhibit variations which are non-negligible~\cite{job1012}.

Two independent simulations and fit approaches have been undertaken
in order to verify the potential of the LHeC to determine $\alpha_s$,
see the CDR~\cite{AbelleiraFernandez:2012cc} for details. 
Table~\ref{tab:alfa} summarises the main results. It can be seen 
that the total experimental uncertainty on $\alpha_s$ is $0.2$\,\%
from the LHeC and $0.1$\,\% when combined with HERA. This determination
is free of higher twist, hadronic and nuclear corrections relying 
solely on inclusive DIS $ep$ data at high $Q^2$. 
%
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
case & cut [$Q^2$ (GeV$^2$)] & uncertainty & relative precision (\%)\\
\hline
HERA only & $Q^2>3.5$ & 0.00224 &  1.94  \\
HERA+jets & $Q^2>3.5$  & 0.00099 & 0.82 \\
\hline
%LHeC only (14p)& $Q^2>3.5$  & 0.000180  & 0.15  \\
LHeC only & $Q^2>3.5$  & 0.00020   & 0.17 \\
%LHeC only (14p)& $Q^2>20.$  & 0.00029 &  0.25 \\
%\hline
LHeC+HERA & $Q^2>3.5$  & 0.00013 & 0.11 \\
%LHeC+HERA (10p)& $x>10^{-5}$  & 0.11777 & 0.000298 & 0.25   \\
%LHeC+HERA (10p)& $x>10^{-4}$  & 0.11809 & 0.000319  & 0.27  \\
%\hline
LHeC+HERA  & $Q^2>7.0$   & 0.00024   & 0.20 \\
LHeC+HERA  & $Q^2>10.$  & 0.00030 & 0.26  \\
\hline
\end{tabular}
\caption{Results of NLO QCD fits to HERA data (top, without and with jets)
to the simulated LHeC data alone and to their combination,
for details of the fit see ~\cite{AbelleiraFernandez:2012cc}.
The resulting uncertainty includes all the statistical and experimental
systematic error sources taking their correlations into account. 
}
\label{tab:alfa}
\end{center}
\end{table}
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There are known further parametric uncertainties in DIS determinations
of $\alpha_s$. These can safely be expected to be much reduced
also by the LHeC, which promises to determine the charm mass, for
example, by $3$\,MeV, as compared to $30$\,MeV at HERA, corresponding
to an $\alpha_s$ uncertainty of $0.04$\,\%. Matching the
experimental uncertainty requires
that when the LHeC operates such analyses were performed in N$^3$LO
pQCD in order to reduce the scale uncertainty. The ambition to
measure $\alpha_s$ to per mille accuracy therefore represents
a vision for a renaissance of the physics of deep inelastic scattering
which is a major goal of the whole LHeC enterprise.
Due to the huge range
in $Q^2$ and the high precision of the data new, decisive tests will
also become available for answering the question whether the
strong coupling determined with jets and in inclusive DIS
are the same. If confirmed, as is demonstrated in Table\,\ref{tab:alfa}
with the HERA data, a joint inclusive and jet analysis has the 
potential to even further reduce the uncertainty of $\alpha_s$
as simulated here.

Numerous tests of the running of $\alpha_s$ have been performed.
At even higher scales, the law which governs this behaviour would 
be affected, possible strongly as is illustrated in 
Figure\,\ref{alfa} (left), if extra dimensions showed up in the
kinematic range accessible to the LHeC. Besides effects on
$\alpha_s$ one would expect to see changes of the NC cross section,
as in contact interaction patterns, for which the LHeC
provides a range up to about $50$\,TeV, which is discussed in the CDR.

