The size and shape of the strange PDF has recently attracted a lot
of debate (see~\cite{Alekhin:2017olj,Alekhin:2014sya} for recent summaries).
%
On the one hand,
most PDF analyses find a suppressed strangeness as compared to
the non-strange light quark sea, a pull driven mostly
by the deep-inelastic neutrino inclusive and  charm production
(``dimuon'') data (see Sect.~\ref{sec:datatheory.DIS}).
%
On the other hand, high-precision collider data from the LHC
has instead exhibited the opposite trend, with a recent
QCD analysis from ATLAS based on the $W,Z$ 7 TeV rapidity distributions
from the 2011 dataset finding 
a strange sea that is in fact larger than the non-strange sea.
%
Given the importance of strangeness for many phenomenological
applications, for instance the measurement of the $W$ mass,
it will be important to resolve this issue in the future.
%
In this respect, alternative probes of $s(x,Q)$ are most useful,
such as the recent suggestion of using semi-inclusive DIS measurements with kaons
in the final state~\cite{Borsa:2017vwy}.

In Fig.~\ref{fig:pdfcomp-xsp} we show the
total strange PDF $xs^+(x,Q^2)$ at $Q=100$ GeV, 
     in the same format as that of Fig.~\ref{fig:pdfcomp-xg}.
     %
     The strangeness-sensitive datasets included in
     the four analysis are rather different,
     both
     in terms of the neutrino fixed-target data and 
      the LHC collider data.
     %
     For example, only the ABMP16 fit includes the NOMAD dimuon
     data~\cite{Samoylov:2013xoa}, while only
     NNPDF3.1 includes the ATLAS $W,Z$ 2011 rapidity distributions.
     %
     We can see that there is reasonable
     agreement within uncertainties between the four groups
     except for ABMP16 for $x\lsim 10^{-3}$, which has a much harder
     strangeness than the other fits.
     %
     We also note that the differences in the size of the
     strange PDF uncertainty can vary by up to a factor $\sim 5$,
     with ABMP16 exhibiting the smallest uncertainties. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40]{6-structure/plots/xsp-abs-pdfcomp-lowQ.pdf}
\includegraphics[scale=0.40]{6-structure/plots//xsp-pdfcomp-highQ.pdf}
     \caption{\small 
      Same as Fig.~\ref{fig:pdfcomp-xg} for the total strangeness
      $xs^+(x,Q^2)$.
      \label{fig:pdfcomp-xsp}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A more physically transparent method to quantify the strange
content of the proton is given by the ratio of the strange to the
non-strange sea quark PDFs, defined as
\be
\label{eq:rs}
R_s (x,Q^2) \equiv
\frac{ s(x,Q^2)+\bar{s}(x,Q^2) }{ \bar{u}(x,Q^2)+\bar{d}(x,Q^2) } \, .
\ee
In this ratio, a symmetric strange sea would correspond to $R_S\simeq 1$,
while a suppressed strangeness instead leads to $R_S\ll 1$.
%
Traditionally, the constraints from neutrino dimuon production
have suggested a value $R_S\sim 0.5$ in most global fits.
%
In Fig.~\ref{fig:Rsx} we show the ratio of strange to non-strange sea quarks
     $R_s (x,Q^2)$, Eq.~(\ref{eq:rs}),
     for $Q^2=1.9$ GeV$^2$ and $x=0.023$.
     %
     We compare the results of the various global PDF fits with those
     of the ATLAS/{\tt xFitter} study~\cite{Aaboud:2016btc}, which includes the recent ATLAS $W,Z$ high precision data, as well as with
     those of a NNPDF3.1 fit based on the same dataset as the ATLAS
     study.
     %
     The vertical lines indicate two possible scenarios for the strange
     PDFs, namely a suppression of size $R_{s}\simeq 0.5$ and 
     a strange sea which is symmetric with the non-strange one,
     $R_{S}\simeq 1$.

From this comparison,
     we can observe that the four PDF sets considered here exhibit a preference for
     a suppressed strangeness.
     %
     On the other hand, this comparison also shows that
     if only the HERA and ATLASWZ11 data are considered, the
     NNPDF3.1 analysis yields an unsuppressed strangeness,
     although with PDF uncertainties rather larger than
     those from the {\tt xFitter} analysis.
     %
     This comparison demonstrates that the opposite pull between the low-energy
     neutrino data and the high-energy collider data is genuine,
     although the tension is only at the 1- or 2-$\sigma$ level, as indicated
     by the fact that the NNPDF3.1 global and HERA+ATLASWZ11 results
     agree within PDF uncertainties.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.45]{6-structure/plots/strangeratio-summary.pdf}
     \caption{\small 
     The ratio of strange to non-strange sea quarks
     $R_s (x,Q^2)$, Eq.~(\ref{eq:rs}) for $x=0.023$
     and $Q^2=1.9$ GeV$^2$.
     %
     We compare the results of various global PDF fits with those
     of the ATLAS/{\tt xFitter} interpretation study as well as with
     those of a NNPDF3.1 fit based on the same dataset as the ATLAS
     study.
     %
     The vertical lines indicate two possible scenarios for the strange
     PDFs, namely a suppression of size $R_{s}\simeq 0.5$ and then
     a strange sea which is symmetric with the non-strange light quark sea,
     $R_{S}\simeq 1$.
      \label{fig:Rsx}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One restriction of the comparison summarised
in Fig.~\ref{fig:Rsx}
is that it is restricted to a specific point $x\simeq 0.023$.
%
To bypass this limitation, in
Fig.~\ref{fig:Rsxplot} we therefore show the $R_s (x,Q^2)$ ratio
as a function of $x$ both at low and at high scales.
%
There are a number of interesting features that can be observed from
this comparison.
%
First, we observe that DGLAP evolution automatically increases
the value of $R_S$ as we go to higher scales, specially at medium and
small $x$,
since as we go to higher values of $Q$ the sea component
dominates over the valence components.
%
Second, we find  a consistent strangeness content for the four groups in most of the range
of $x$, although the size of corresponding uncertainties in each case
can vary by a significant amount.

Another important
point from this comparison is that
clearly any statement about whether or not strangeness is
suppressed depends on the region of $x$ that is being considered.
%
For instance, in the MMHT14 analysis for $Q=1.38$ GeV the value
of $R_S$ changes from around 0.4 at $x\simeq 0.1$ to around 0.8
for $x\simeq 0.007$.
%
So different $x$ regions exhibit different amounts of suppression
with respect to the light sea quarks, and therefore the question
of the suppression (or lack thereof) of the strange PDF
is a more nuanced issue than what is sometimes stated.
%
In any case, is clear from the comparison of
Fig.~\ref{fig:Rsxplot} that a symmetric strange sea in the entire
range of $x$ is not favored by any of the four fits shown here,
in particular in the region around $x\simeq 0.1$ and above.
%
In this respect, future data from the LHC and elsewhere will help
to shed some light on this important issue.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40]{6-structure/plots/xRs-abs-RslowQ.pdf}
\includegraphics[scale=0.40]{6-structure/plots/xRs-abs-RshighQ.pdf}
     \caption{\small 
     The ratio of strange to non-strange sea quarks
     $R_s (x,Q^2)$, Eq.~(\ref{eq:rs}), as a function of $x$
     for $Q=1.38$ GeV (left plot) and for $Q=100$ GeV (right plot).
               \label{fig:Rsxplot}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
