The production of $W$ bosons in association with $D$ mesons
is a direct probe of the strange PDF~\cite{Stirling:2012vh}.
%
As shown in the left panel of Fig.~\ref{fig:wcharm}, at leading
order the $W$+charm process is proportional to the proton
strangeness, and therefore measurements of this cross-section
have long been advocated~\cite{Baur:1993zd} to provide direct information
on the strange content of the proton.
%
Moreover, by taking ratios or asymmetries
of the $W^++\bar{c}$ and
$W^-+c$ differential distributions, it is also possible
to extract some information on the strangeness asymmetry $s-\bar{s}$.

The ATLAS, CMS, and LHCb experiments have during Run I measured the production of $W$ bosons associated
with $D$ mesons using different final
states~\cite{Chatrchyan:2013uja,Aad:2014xca,Aaij:2015cha}.
%
In the case of the CMS measurements~\cite{Chatrchyan:2013uja},
the experimentally
accessible $W+D$ cross-sections were unfolded to the corresponding
$W+c$ parton level cross-sections, facilitating the comparisons
with theoretical calculations.
%
On the other hand, the ATLAS experiment has presented
their measurements~\cite{Aad:2014xca}
of this process in terms of final-state
quantities such as $D$ mesons and $c$-tagged jets.
%
While the two types of measurements provide in principle
comparable information, the interpretation of the latter
is somewhat more delicate since PDF fits are typically based
on parton-level (as opposed to hadron-level) calculations.
%
However, this is not a fundamental limitation anymore since,
as explained in Sect.~\ref{subsec:fast}, using
the {\tt aMCfast}~\cite{amcfast}
interface it is possible to use NLO+PS calculations
directly into a PDF analysis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{3-datatheory/plots/wcharm.pdf}
   \caption{\small Left: the LO diagrams for $W$+charm production
   in $pp$ collisions.
   %
   Center: the strangeness ratio $R_S=(s+\bar{s})/(\bar{u}+\bar{d})$
   at $Q=m_W$ comparing the results of the {\tt HERAfitter} analysis
   from~\cite{Chatrchyan:2013uja}, based on the CMS $W+c$ measurements,
   with the corresponding results based on the NNPDF reweighting
   method.
   %
   Right: the strangeness ratio $r_S=0.5(s+\bar{s})/\bar{d}$,
   comparing the ATLAS-epWZ12 fit (which included the ATLAS 2010
   $W,Z$ inclusive rapidity distributions) with a new fit including
   the ATLAS $W+D$ measurements~\cite{Aad:2014xca}.
   %
   See text for more details.
    \label{fig:wcharm}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In order to illustrate the impact of the LHC $W+D$ measurements
in the strange PDF, first of all we show in the central panel
of Fig.~\ref{fig:wcharm} the
strangeness ratio $R_S=(s+\bar{s})/(\bar{u}+\bar{d})$
   at $Q=m_W$, comparing the results of the {\tt HERAfitter} analysis
   from~\cite{Chatrchyan:2013uja}, based on the CMS $W+c$ measurements
   added to the HERA-I inclusive DIS data,
   with the corresponding results based on the NNPDF reweighting
   method.
   %
  From this comparison we find that the CMS $W+c$ data indeed allows
  to provide significant information on $s(x,Q^2)$ as compared
  to {\it i.e.} a HERA-only fit, and that the central value
  is consistent with the global fit results, which favors
  a suppressed strangeness as compared to the $\bar{u}$ and
  $\bar{s}$ light quark sea.
  %
  For instance, one sees that the CMS data prefers $R_S\simeq 0.6$
  at $x\simeq 0.1$, though $R_s$ tends to 1 as we move towards
  smaller values of $x$.

Next, in Fig.~\ref{fig:wcharm} we also show
the strangeness ratio $r_S=0.5(s+\bar{s})/\bar{d}$,
   comparing the ATLAS-epWZ12 fit, which included the ATLAS 2010
   $W,Z$ inclusive rapidity distributions~\cite{Aad:2011dm},
   with a new fit based on
   the ATLAS $W+D$ measurements~\cite{Aad:2014xca}.
   %
   As opposed to the CMS measurements, the ATLAS $W+D$ data favour
   a symmetric strange sea, namely $r_s\simeq 1$, which is also
   consistent with the constraints from the ATLAS 2010
   $W,Z$ inclusive rapidity
   distributions~\cite{Aad:2011dm,Aaboud:2016btc}
   (as well as with the updated analysis from the 2011 dataset).
   %
   While it is clear from Fig.~\ref{fig:wcharm} that the ATLAS
   and CMS $W$+charm measurements have opposite pulls
   on the strange PDF, the tension is only at the 1 or 2-$\sigma$ level.

   Further measurements of this process at 13 TeV should
   be able to shed more light on whether or not this tension
   between the ATLAS and CMS measurements
   persists.
   %
   Here, a complete fit including hadron-level
   $W$+$D$ measurements (and thus avoiding the unfolding
   to the parton level) would provide the cleanest possible
   interpretation of the PDF information from this process.
   %
   A more detailed discussion of the strange PDF, and of
   the constrains from the different strange-sensitive
   processes, is presented in Sect.~\ref{sec:structure.strange}.
   

   
