Many BSM physics scenarios predict
the existence of new heavy particles with masses around the
TeV scale.
%
For example, supersymmetry, composite Higgs,
and extra dimensions, are all classes of models, among many other others, that
motivate the search for new heavy resonances at the LHC
in the high-mass tail of various kinematic distributions.
%
Here, PDF uncertainties play an important
role in setting robust exclusion limits based on available
null results, and would become even more important
in the case of eventual discovery.
%
In particular, PDFs represent the dominant
theoretical uncertainty for the production of new
heavy particles in the TeV region, as such processes
are sensitive to the  large-$x$ behaviour of quarks and gluons.
%
The reason is that, as discussed in Sect.~\ref{sec:structure}, PDF
uncertainties are large in this region due to the
limited amount of experimental constraints.

In order to quantify the size
of the PDF uncertainties in the large invariant mass
region,
as well as the relative agreement between the PDF groups, it is useful
to compare the PDF luminosities for $M_X\ge 1$ TeV.
%
In this comparison we will restrict ourselves to ABMP16, CT14, MMHT14
and NNPDF3.1, in all cases using $\alpha_s(m_Z)=0.118$.
%
Results are shown in Fig.~\ref{fig:BSMlumi} for
$\sqrt{s}=13$, TeV normalized to the central value
of the MMHT14 calculation.
%
From the comparison in Fig.~\ref{fig:BSMlumi}, we find that
PDF uncertainties are small, at the few-percent level, up to
$M_X\simeq 5$
TeV for the quark-quark luminosities.
%
This is due to the fact that $\mathcal{L}_{qq}$ is dominated
by the rather accurately known up and down quark valence PDFs,
which are constrained by measurements of e.g. fixed-target
DIS structure functions.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.40]{8-pheno/plots/gg_lumi-tev.pdf}
  \includegraphics[scale=0.40]{8-pheno/plots/qg_lumi-tev.pdf}
  \includegraphics[scale=0.40]{8-pheno/plots/qq_lumi-tev.pdf}
  \includegraphics[scale=0.40]{8-pheno/plots/q2_lumi-tev.pdf}
  \caption{\small Comparison of PDF luminosities in the large
    invariant mass $M_X$ region between MMHT14, ABMP16, CT14 and
    NNPDF3.1.
    %
    From left to right and from top to bottom we show the results
    of the gluon-gluon, gluon-quark, quark-antiquark
    and quark-quark luminosities, normalized
    to the central value of MMHT14.
    %
    In this comparison, NNLO PDFs with $\alpha_s(m_Z)=0.118$
    sets are used.
    \label{fig:BSMlumi}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the gluon-gluon luminosity, $\mathcal{L}_{gg}$, we find a rather large spread
of the predictions between the different groups, with MMHT14 (ABMP16)
leading to the largest (smallest) central values.
%
For instance, at $M_X\sim 5$ TeV, which is close to the upper
limit of the kinematic coverage of the LHC, the envelope of the PDF uncertainty bands
spans $\sim 100\%$.
%
Even for more moderate invariant masses the spread is quite large, with the values of $\mathcal{L}_{gg}$
at $M_X\sim 2.5$ TeV varying between $\sim$ $+10\%$ and $-30\%$ in comparison to the central MMHT14 result.
%
It is thus clear that these uncertainties would represent
one of the limiting factors for BSM characterization
in the case of an eventual discovery.

For the quark-gluon luminosity, $\mathcal{L}_{qg}$, we
observe a similar trend to the gluon-gluon case, but
with reduced PDF uncertainties due to the contribution
of the well-constrained large-$x$ quark PDFs.
%
Interestingly, in the above two cases the results from all sets do not necessarily agree within the quoted 68\% confidence uncertainty bands.
%
As discussed further in Sect.~\ref{sec:structure.gluon}, it will be informative to see how this comparison changes when data sensitive to the high $x$ gluon, such as the $t\overline{t}$ differential distributions, currently only in the NNPDF3.1 set, are included in all the fits shown in this comparison.

PDF uncertainties, as well as the differences between groups,
are also large for the quark-antiquark PDF luminosity
$\mathcal{L}_{q\bar{q}}$, also shown in Fig.~\ref{fig:BSMlumi}.
%
The reason for this behaviour is two-fold.
%
On the one hand, the large-$x$ anti-quarks
are notoriously difficult to pin down, although recent
high-precision measurements from the Tevatron
and the LHC are improving the situation.
%
On the other hand, various groups parameterize the quark
sea content of the proton with rather different
assumptions~\cite{Ball:2016spl} (see Sect.~\ref{sec:pdfgroups}),
and this can have important
implications for the quark-antiquark luminosities.
%
We find that the spread of the different results
ranges between $+5\%$ and $-30\%$ for $M_X=3$ TeV,
with PDF uncertainties becoming $\mathcal{O}(100\%)$
for higher invariant masses.
%
Note here that the PDF uncertainties are the largest for
NNPDF3.1, despite this being the global analysis which
includes a largest amount of LHC electroweak data sensitive
to anti-quarks.
%
This highlights the fact that
methodological differences in the flavour assumptions
and parametrization of anti-quarks are one of the dominant
factors in explaining the difference between the various
groups for $\mathcal{L}_{q\bar{q}}$ at large invariant masses $M_X$.

In order to illustrate the phenomenological consequences of
these large PDF uncertainties at high $M_X$, in Fig.~\ref{fig:BSMhighmass}
we show the  PDF uncertainties for high-mass
  graviton production in the Randall-Sundrum
  scenario~\cite{Randall:1999vf,Randall:1999ee}
  induced by gluon-fusion at the LHC 8 TeV, computed at LO
  with {\tt MadGraph5}~\cite{Alwall:2007st}.
  %
   We compare the results of the NNPDF2.3 fit with those
  of the same fit including the constraints
  from top-quark production cross-sections~\cite{Czakon:2013tha}.
  %
  We observe that PDF uncertainties become $\mathcal{O}(100\%)$
  at large values of the graviton mass, consistent
  with the estimates from the gluon-gluon luminosity
  shown  in Fig.~\ref{fig:BSMlumi}.
  %
  We also see how these PDF uncertainties can be reduced
  by the inclusion of top quark pair production
  total cross-sections,
  highlighting the cross-talk between precision SM measurements
  and improving BSM searches.
  %
  Further constraints on the large-$x$ gluon can be obtained
  {\it e.g.} fitting top quark differential distributions,
  see Sect.~\ref{sec:datatheory.top}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{8-pheno/plots/BSMhighmass.pdf}
  \caption{\small Left: the PDF uncertainties for high-mass
  graviton production in the Randall-Sundrum scenario
  induced by gluon-fusion at the LHC 8 TeV, computed at LO
  with {\tt MadGraph5}.
  %
  We compare the results of the NNPDF2.3 fit with that
  of the same fit including the constraints
  from top-quark production cross-sections~\cite{Czakon:2013tha}.
  %
  Right: the $K$-factor for the NLO+NLL cross-section,
  including PDF uncertainties, for the production
  of a squark-anti-squark pair $\tilde{q}\tilde{q}^*$
  at the LHC 13 TeV with various PDF sets, from~\cite{Beenakker:2015rna}.
    \label{fig:BSMhighmass}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In Fig.~\ref{fig:BSMhighmass}
we also show the $K$-factor for the NLO+NLL cross-sections
for the production
  of a squark-anti-squark pair $\tilde{q}\tilde{q}^*$
  at the LHC 13 TeV.
%
The calculation has been performed using  various PDF sets,
  including PDF uncertainties, each set normalized to the corresponding NLO
  result~\cite{Beenakker:2015rna}.
  %
  This production channel is dominated by the quark-antiquark
  luminosity.
  %
  Specifically, we compare the predictions of NNPDF3.0,
  CTEQ6.6, and MSTW2008, all at NLO.
  %
  Note that by construction the central values of
  the three predictions are close since different trends
  cancel in this $K$-factor ratio, so the usefulness
  of this comparison arises mostly in estimating the PDF uncertainties
  in each case.
  %
  Here we also see that PDF uncertainties become very large
  at high mass, in particular in the case of NNPDF3.0,
  reflecting the underlying behaviour of the quark-antiquark
  luminosities (see Fig.~\ref{fig:BSMlumi}).
 
  The results shown in Fig.~\ref{fig:BSMhighmass} highlight that,
  in the case of an eventual discovery of novel high-mass
  particles at the LHC, it will be crucial to improve
  our knowledge of large-$x$ PDFs in order to be able
  to characterize the underlying BSM scenario.
%
To achieve this it will be essential to exploit as much as possible high-precision collider data, mostly from the LHC, in fits,
in order to pin down the large-$x$ gluons and
anti-quarks and thus reduce the PDF uncertainties
associated to high-mass BSM particle production.
%
For example, in Ref.~\cite{Czakon:2016olj} it was shown that
by including the $y_t$ and $y_{t\bar{t}}$ differential
distributions for top quark pair production
in a global PDF fit, it is possible to reduce the PDF
uncertainties that affect the high-mass tail of the $m_{t\bar{t}}$
distribution by up to a factor of two.
%
This distribution is widely use for searches, for instance
of new resonances that couple strongly to the top quark~\cite{Aaboud:2017hnm}.
%
In the future, it might also be feasible
to provide indirect constraints on BSM models,
for instance on the
coefficients of the SM-EFT~\cite{Brivio:2017vri} higher-order operators,
by including these in the global PDF fit, along the lines
of early studies aiming to constrain colored sparticles
from Tevatron jet production~\cite{Berger:2010rj}.
