To begin with, in Fig.~\ref{fig:pdfcomp-xg} we show the gluon PDF $xg(x,Q^2)$ at
a low hadronic scale $Q=1.7$ GeV (left plot)
     and at a typical LHC scale of $Q=100$ GeV (right plot).
     %
     We find that
in general there is reasonable agreement between the four sets 
considered within PDF uncertainties.
%
This remains true at small-$x$, where the PDF uncertainties become rather
large, due to the limited amount of experimental constraints.
%
While the central value of the MMHT14 gluon becomes negative for $x\lsim 5\times 10^{-5}$,
the CT14 result exhibits a flat behaviour, and the NNPDF3.1 gluon increases rapidly.
%
We also note that the agreement between ABMP16 and the other three sets
becomes rather worse
if the PDF set corresponding to their best-fit $\alpha_s(m_Z)=0.1149$ value is used.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.40]{6-structure/plots/xg-abs-pdfcomp-lowQ.pdf}
  \includegraphics[scale=0.40]{6-structure/plots/xg-pdfcomp-highQ.pdf}
   \caption{\small 
     The gluon PDF $xg(x,Q^2)$ at $Q=1.7$ GeV (left)
     and $Q=100$ GeV (right) comparing the ABMP16, CT14, MMHT14,
     and NNPDF3.1 NNLO sets with $\alpha_{S}(m_Z)=0.118$.
     %
     In the right plots, results are normalized to the central value
     of NNPDF3.1.
    \label{fig:pdfcomp-xg}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Perhaps the most important discrepancy between the gluon PDFs from the four
groups arises in the large-$x$ region, where the NNPDF3.1 result
(and even more markedly ABMP16) is rather softer in comparison to CT14 and MMHT14.
%
For example, at $x\simeq 0.2$ the differences between the NNPDF3.1 and
CT14 central values are at the 2-$\sigma$ level.
%
One of the reasons for these differences could be related to the use
of different datasets to constrain the large-$x$ gluon, and specifically
by the use of top-quark differential distributions in NNPDF3.1,
which have been shown to lead to a softer large-$x$ gluon as compared to the
same fit without any $t\bar{t}$ data included~\cite{Czakon:2016olj},
see also Sect.~\ref{sec:datatheory.top}.
%
Indeed,  the CT14 and previous NNPDF3.0 set (which do not fit such data)
 are in better agreement within uncertainties.
%
We also note that in the large-$x$ region PDF uncertainties
are quite large, leading to siginificant theoretical
uncertainties for the production of new BSM heavy particles,
as will be discussed in Sect.~\ref{sec:LHCpheno.BSM}.
%
In this respect, it will be interesting to compare the large-$x$
gluon PDF once other groups include also the $t\bar{t}$
differential measurements.

It is worth emphasising that one of the most important differences
of the current realization of global PDF fits as compared to previous versions is the
fact that now several datasets provide independent constraints on the large-$x$
gluon.
%
Until recently, the gluon at large-$x$ was only constrained in the
PDF fit by inclusive jet cross-sections, and to a lesser extent by
DIS data via scaling violations.
%
However, we now have at least three datasets that can constrain
the large-$x$ gluon, namely inclusive jets, the $p_T$ distribution
of $Z$ bosons, and top quark differential distributions. In all cases, NNLO calculations are now available.
%
To illustrate the robustness of the resulting gluon, in 
Fig.~\ref{fig:largexgluon} we show a comparison of the NNPDF3.1 NNLO global
fit at $Q=100$ GeV with
     the corresponding fits where the $Z$ $p_T$, top quark, or inclusive
     jet data have been removed.
     %
     We can observe that the four fits agree within
     PDF uncertainties,
    highlighting that these three families of physical
     processes have statistically consistent pulls on the large-$x$ gluon.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.44]{6-structure/plots/largexgluon.pdf}
   \caption{\small 
     Left: comparison of the NNPDF3.1 NNLO global fit at $Q=100$ GeV with
     the corresponding fits where the $Z$ $p_T$, top quark, or inclusive
     jet data have been removed.
     %
     Right: same, now comparing with the NNPDF3.1 NNLO fit
     where the ATLAS and CMS 7 TeV inclusive jet data have been treated
     using exact NNLO theory, from~\cite{Ball:2017nwa}.
    \label{fig:largexgluon}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Another consideration that is relevant for the determination of the large-$x$ gluon in
a PDF analysis
are the settings for the theoretical calculations used for the inclusive jet
cross-sections.
%
Until 2016, only the NLO calculation was available, and different groups
treated jet data in the global fit in different ways, from adding
the NLO scale errors as additional systematic uncertainties
as in CT14 and NNPDF3.1, to using the threshold approximation
to
the full NNLO result as in MMHT14, to excluding jet data altogether as advocated
by ABMP16 (see Sect.~\ref{sec:pdfgroups}).
%
The availability of the complete NNLO calculation, discussed
in Sect.~\ref{sec:datatheory.jets}, makes now possible that
future installments of these PDF analyses will be able to fit to inclusive jet
data using the exact NNLO theory.

This said, there are some indications that, at least for specific settings of the NLO
calculation, the inclusion of jet data with exact NNLO theory leads to a moderate
phenomenological impact.
%
This is because, if the jet $p_T^{\rm jet}$ scale is adopted as the central renormalization
and factorization scale, and a not too small value of $R$ is used, the
NNLO/NLO $K$-factor is a few percent at
most~\cite{Currie:2016bfm,Currie:2017ctp}.
%
To illustrate this, in Fig.~\ref{fig:largexgluon} we compare the baseline
NNPDF3.1 NNLO fit (where jet data are treated at NLO using $p_T^{\rm jet}$
as central scale, with scale variations added
as additional systematic error) to the same fit where exact NNLO theory
has been used for the ATLAS and CMS 7 TeV data.
%
We see that the resulting differences are small at the PDF level, and at the
$\chi^2$ level one finds~\cite{Ball:2017nwa} a small but non-negligible improvement
once NNLO theory is used.
