%
\label{sec:higgslhec}
In the Standard Model, the breaking
of the electroweak $SU(2)_L \times U(1)_Y $ symmetry gives mass to the
electroweak gauge bosons via the Brout-Englert-Higgs mechanism,
while the fermions obtain their mass via Yukawa couplings with a scalar Higgs field. 
With the observation of a Higgs-like boson by the 
ATLAS~\cite{:2012gk} and CMS~\cite{:2012gu} 
collaborations with a mass around 126\,GeV, a new research field 
has opened in particle physics. The measurement of the couplings 
of the newly found boson to the known fundamental particles will 
be a crucial test of the SM and a window of opportunity to establish
physics beyond the SM. 

%
%\begin{figure}[h]
%\includegraphics[clip=,width=.47\textwidth]{higgs60a4.pdf}
%\caption{higgs
%}
%\label{figH}
%\end{figure}
%
At the LHeC, a light Higgs boson could be uniquely produced 
and cleanly reconstructed either 
via $HZZ$ coupling in neutral current  DIS or via $HWW$ coupling in 
charged current  DIS. Those vector boson fusion processes have 
sizeable cross sections, O(100)\,fb for $126$\,GeV mass,
and they can be easily distinguished, which is a unique advantage 
in comparison to  the VBF Higgs production in $pp$ scattering.
The observability of the Higgs boson signal at the LHeC was 
investigated in the CDR~\cite{AbelleiraFernandez:2012cc} using 
initially the dominant production and decay mode, i.e.
the CC reaction
 $e^- p \rightarrow H (\rightarrow b \bar{b}) + \nu + X$,
for the nominal $7$~TeV LHC proton beam  
and electron beam energies of $60$ and  
$150\,$GeV.
Simple and robust cuts are identified and found to reject 
effectively e.g. the dominant single-top background,
%as opposed to refined event weighting and network techniques
%presently utilised in complex final state (top and Higgs)
%physics at the LHC, 
providing an excellent S/B ratio of about~$1$ at the LHeC,
which may be further refined using sophisticated 
neural network techniques.
At the default electron beam energy of 
$60$\,GeV, for $80$\,\% $e^-$ polarisation and an 
integrated luminosity of 100 fb$^{-1}$, 
the $Hbb$ coupling is estimated to be measurable
with a statistical precision of about 4\,\%, which is
not far from the current theoretical uncertainty.
Typical coupling measurements,
as for $\gamma \gamma$ or $4l$, are of about $10$\,\%
precision with the HL-LHC, while the specific $b\overline{b}$
coupling will be particularly difficult to measure due
to high combinatorial backgrounds in $pp$.

The LHC is said to be inferior to a linear collider
in its coupling measurement prospects. Part of this
statement comes from large uncertainties, which
are related to the imperfect knowledge of the
PDFs and theory parameters. The LHeC, with its 
high precision PDF and QCD programme, will render many
of these uncertainties unimportant. Currently, for example,
an uncertainty of the $H \rightarrow \gamma \gamma$ 
cross section due to PDFs is quoted of nearly $10$\,\%
\cite{Baglio:2012et}, based on the variation of
the cross section predictions from different PDFs.
This will be very much reduced with the LHeC:
an iHix calculation of the NLO Higgs cross section
for MSTW08, NNPDF2.3 and HERAPDF1.5 leads to 
intrinsic uncertainties of 1.7, 1.2 and 2.2\,\%,
respectively, with a maximum deviation of 6.9\,\%.
The full experimental LHeC uncertainty, however, is 0.2\,\%.
The main advantage will be that the precision LHeC
data, possibly combined with HERA, will replace
essentially all previous data sets and thus
lead to a much better agreement between various
PDF determinations, besides the huge reduction
in uncertainty with the LHeC. 

A sizeable uncertainty
is also related to the strong coupling constant,
a difference in $\alpha_s$ of $\pm 0.005$
corresponding to an about  $10$\,\% cross
section uncertainty, see e.g. \cite{juan}
or \cite{Baglio:2010ae}. Obviously, the large
improvement in the determination of $\alpha_s$
with the LHeC will reduce this uncertainty
strongly too. Essentially with such a high
quality data set as the LHeC, one will simultaneously
determine the coupling and the PDFs, and
control their correlations at a very high level.
In \cite{Denner:2011mq} a systematic evaluation
has been presented of the effect of the
heavy quark masses and of $\alpha_s$ on
the uncertainties of the Higgs branching
fractions in various channels. One finds
partially large effects as $6$\,\% from $M_c$
on the $H \rightarrow c \overline{c}$ branching
or $5.6$\,\% from $\alpha_s$ on $H \rightarrow gg$.
These will certainly be much reduced. It is for
future studies to more systematically analyse
the striking potential of the LHeC to
remove or reduce the QCD uncertainties
of the Higgs cross sections and couplings.
There will also be improvements related to
QCD measurements at the LHC. Their level of
precision, however, especially for the
gluon, $\alpha_s$ and heavy quark QCD
cannot compete with the genuine DIS process.

It has also been observed~\cite{AbelleiraFernandez:2012cc},
 that the LHeC can
specifically explore well the CP structure 
of the $HWW$ coupling by separating it from the $HZZ$ 
coupling and the other signal production mechanisms. Any 
determination of an anomalous $HWW$ vertex will 
thus be free from possible contaminations of these.
A further advantage of the $ep$ collider kinematics stems 
from the ability to disentangle clearly the direction
 of the struck parton and the final state lepton (clear 
definition of forward and backward directions). Compared 
to the $pp$ situation, $ep$ lacks 
the complications due to underlying event and pile-up
driven backgrounds.

The few initial studies performed so far will be pursued further
given that the Higgs boson is now likely to indeed exist.
For the projected analyses, this first concerns using the
appropriate LHeC detector simulation, and optimising further
its design. For the accelerator design it is obvious that
a luminosity in excess of   $10^{33}$~cm$^{-2}$s$^{-1}$
is very desirable, see the discussion on machine
parameters in Section\,\ref{linac}. 
The then possible precision measurements
of rarer ($\tau$, $Z$, $W$, perhaps photon) 
decay channels, of the CP angular distributions, for
both the $HWW$ and $HZZ$ couplings, and
NC initiated production would make the LHeC, by design
a QCD machine, a collider to study the Higgs
boson and mechanism of very
remarkable and complementary potential. 
The assistance in removing a large part of the QCD 
and PDF related uncertainties of the Higgs measurements
in $pp$ will help making the LHC a genuine precision
Higgs facility.

