In the Standard Model, once the Higgs mass $m_h$ is measured,
all other parameters of the Higgs sector, such as the strength
of its coupling to fermions and vector bosons and its branching ratios, are uniquely
determined~\cite{deFlorian:2016spz,Dittmaier:2011ti,Dittmaier:2012vm}.
%
On the other hand, deviations of these Higgs couplings with respect
to the SM predictions are generically expected in BSM scenarios~\cite{Dawson:2013bba}.
%
Therefore, the precision measurements of the Higgs couplings represents a unique opportunity for BSM
searches, since any deviation with respect to the the
tightly fixed SM predictions would represent
a {\it smoking gun} for New Physics.
%
Crucially, realising this program requires not only
high precision experimental measurements of Higgs boson
production and its decay in various channels, but also the calculation
of the SM cross-sections and decay rates with matching theoretical precision.
%
It is therefore of particular importance that PDFs are one of the largest sources
of theoretical uncertainty affecting the predictions for Higgs boson production~\cite{deFlorian:2016spz}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.61]{8-pheno/plots/higgs-summary.pdf}
  \caption{\small The PDF dependence of the
  most important Higgs production inclusive
    cross-sections at the LHC 13 TeV.
    %
    The results are normalized to the central value of the NNPDF3.1 prediction,
    and only PDF uncertainties are shown.
    %
    In the case of ABMP16, we show results for the nominal $\alpha_s$ value
    as well as for the common $\alpha_s(m_Z)=0.118$ value.
    %
    See~\cite{Ball:2017nwa} for more details of the theoretical
    calculations.
    \label{fig:higgs}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Here we present a comparison of inclusive
Higgs production cross-sections at 13 TeV
with the latest releases of all PDF groups.
%
The settings of this comparison, and the codes
used for each process, are described in Ref.~\cite{nnpdf31}.
%
Specifically, 
we show the dominant Higgs boson production modes at hadron colliders:
gluon fusion, associated production with a $t\bar{t}$ pair,
$VZ$ associate production, and Higgs production in vector-boson
fusion.
%
In addition, we also show the results for double Higgs production
in the dominant gluon-fusion channel, which should become accessible
at the HL-LHC~\cite{Behr:2015oqq}.
%
Results are provided for NNPDF3.0 and 3.1, CT14, MMHT14 and for the
ABMP16 NNLO sets for $\alpha_s(m_Z)=0.118$, while in the latter
case we also indicate the result corresponding to
their best-fit value of $\alpha_s(m_Z)=0.1149$.
%
The theoretical settings for each cross-section calculation
are based on state-of-the-art
matrix element calculations, for instance the gluon-fusion
and VBF results are computed at N3LO using the
{\tt ggHiggs}~\cite{Ball:2013bra}
and {\tt vbf@n3lo}~\cite{Dreyer:2016oyx} codes respectively.
%
We only show here the PDF uncertainties, while
other sources of theoretical
errors
affecting these cross-sections
are listed in {\it e.g.} the latest Higgs Cross-Section Working
Group report~\cite{deFlorian:2016spz}.
%
Interestingly, we note that the uncertainty associated to the input value
of $\alpha_s(m_Z)$ can be comparable to the
PDF uncertainties in some channels.

There are a number of noteworthy features in the comparison
of  Fig.~\ref{fig:higgs}.
%
First of all, it shows that
in general there is
good agreement between the three
global fits, NNPDF3.1, CT14 and MMHT14 for all the
Higgs boson production modes.
%
The comparison between NNPDF3.1 and its predecessor
NNPDF3.0 highlights good agreement for the gluon
initiated channels, with a reduction of the PDF
uncertainties in the former case, while for quark-initiated
cross-sections such as $VH$ and VBF there is an upper shift
by around one sigma.
%
Another significant feature of this comparison is
that the recent ABMP16 set is also in reasonable agreement
with the rest of the groups, provided that
the same common value of the strong
coupling constant $\alpha_s(m_Z)=0.118$ is used.
%
On the other hand, if their best-fit value
$\alpha_s(m_Z)=0.1149$ is used in the calculation,
significant differences in the predicted cross-sections arise for the gluon--initiated channels.
%
In particular, the ABMP16 prediction is around 5--7\% (10--12\%) lower
than the other predictions for the gluon-fusion ($t\bar{t}$ associated
production) cross-section.

It is also worth mentioning here that PDF uncertainties
are relevant not only for the extraction
of Higgs couplings from inclusive cross-sections, but
also for the differential measurements~\cite{Aad:2016lvc,Khachatryan:2015yvw}
that will become
available thanks to the large statistics that will be accumulated
by the end of Run II as well as for the HL-LHC.
%
To illustrate this point, in
Fig.~\ref{fig:higgs_differential} we show the
PDF uncertainties in the
  $p_T^h$ distribution of Higgs bosons produced in the gluon-fusion
  mode at the LHC 13 TeV for $0 \le p_T^h \le 200$ GeV,
  computed using the PDF4LHC15 NNLO sets.
  %
  In this case we find that PDF uncertainties are at around the
  $\sim 2\%$ level.
  %
  However, these uncertainties will increase as
  the LHC becomes sensitive to higher $p_T$ values:
  as shown in the right plot of Fig.~\ref{fig:higgs_differential},
  at high invariant masses (high $p_T$ values) the
  PDF uncertainties in the gluon-gluon luminosity become
  rather larger, affecting the extraction of Higgs properties in that region.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{8-pheno/plots/higgs-differential.pdf}
  \caption{\small Left: the PDF uncertainties in the
  $p_T^h$ distribution of Higgs bosons produced in the gluon-fusion
  mode at the LHC 13 TeV for $0 \le p_T^h \le 200$ GeV,
  computed using the PDF4LHC15 NNLO sets.
  %
  Right: the gluon-gluon PDF luminosity with the same
  set now extending up to higher values of the invariant mass
  of the final state $M_X$.
    \label{fig:higgs_differential}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
