At proton-proton colliders, there are two main types of PDF-sensitive
measurements which contain charmed hadrons in the final state.
%
The first one is open $D$ meson production, which
provides important information on the small-$x$
gluon PDF beyond the kinematic coverage of the
HERA structure function data.
%
As such, it allows to constrain the gluon in a region
of great importance for high-energy astrophysics,
as well as for the tuning of Monte Carlo event
generators.
%
The second process is  $W$ production in association with charm
quarks, which is directly sensitive to the poorly
known strange PDF.
%
Here we focus on the first process, open charm
production, and we then briefly discuss $W$+charm production
in next subsection.

\subsubsection*{PDF sensitivity}\label{sec:datatheory.charm.sensitivity}

The production of charmed mesons at hadron colliders via
the $pp\to c\bar{c}$ reaction
is dominated by the gluon-gluon luminosity, and therefore
it provides a sensitive probe to the gluon PDF at small-$x$.
%
In particular, the forward measurements from the LHCb experiment
at 5, 7, and 13 TeV~\cite{Aaij:2013mga,Aaij:2016jht,Aaij:2015bpa}
provide information on the gluon at values of
$x$ as small as $x\simeq 10^{-6}$, well below the kinematic
reach of the HERA structure function data, and thus in a region
where PDF uncertainties are large due to the limited
amount of experimental information available.


This very small-$x$ region is not only important for the description of soft
and semi-hard QCD processes~\cite{Skands:2014pea}, it is also crucial for
the calculations of signal and
background processes~\cite{Gauld:2015kvh,
CooperSarkar:2011pa,Garzelli:2016xmx} for
ultra-high energy neutrino astrophysics.
%
In the former case, the small-$x$ gluon is relevant both for the calculation
of signal event rates, via the interaction cross-section between UHE
neutrinos and target nucleons (ice or water), as well as for the calculation
of the rates for the dominant background process, the production of
charm quarks in cosmic ray collisions in the atmosphere which then
decay into so--called `prompt' neutrinos and which dominate the
atmospheric neutrino flux at high energies.
%
With this motivation, various groups have studied the impact
of the LHCb charm measurements on the gluon
PDF using different theoretical
frameworks~\cite{Gauld:2016kpd,Cacciari:2015fta,Zenaiev:2015rfa,Zenaiev:2016kfl}, as we discuss in more detail now.

\subsubsection*{Experimental data}\label{sec:datatheory.charm.data}

The unique forward coverage of the LHCb experiment allows to
access the very small-$x$ region by means of the production of low-mass
final states such as $D$ mesons and $J/\Psi$ mesons.
%
LHCb has presented measurements
of $D$ meson production at three
center of mass energies, $\sqrt{s}=5,7$ and
13 TeV~\cite{Aaij:2013mga,Aaij:2016jht,Aaij:2015bpa}.
%
In particular, these datasets
are available double differentially in transverse momentum ($p_T^D$) and
rapidity ($y^D$) for a number of final states such as
$D^0, D^+, D_s^+$ and $D^{*+}$, which also
contain the contribution from charge-conjugate states.
%
The availability of the cross-sections for the same
process at different values of $\sqrt{s}$ is particularly
appealing for PDF studies, since many experimental and theoretical
systematic uncertainties cancel in their
ratios~\cite{Mangano:2012mh}.

The original LHCb measurements of charm production were affected
by an experimental problem, whose effects were particularly
marked in the central rapidity region.
%
These problems also affected the $B$ meson production cross-section,
where QCD calculations were far off the LHCb data for
$\eta_B\simeq 2.0$~\cite{Gauld:2017omh}.
%
This problem has already been corrected, and the LHCb collaboration
has presented revised versions of their measurements, which
exhibit now an improved agreement with the theoretical
predictions.
%
Note that this issue did not affect the PDF interpretation of the
LHCb $D$ meson data, for which the dominant sensitivity comes
from the region with forward rapidities (since it
matches to the small-$x$ PDF region).

\subsubsection*{Theoretical calculations and tools}\label{sec:datatheory.charm.theory}

Charm production is known at NLO in the perturbative QCD expansion
since almost three decades~\cite{Nason:1987xz,Nason:1989zy}.
%
This fixed-order calculation can be matched to NNLL
resummation~\cite{Cacciari:1998it} using a GM-VFN scheme, although
these corrections are numerically small in the kinematic region
accessible to the LHCb measurements.
%
The parton-level charm production cross-sections then
need to be hadronized into the experimentally observed
$D$ mesons, and this can be done either analytically, using
the heavy quark fragmentation functions extracted from LEP
data~\cite{Cacciari:2012ny}, or using hadronization models as implemented
in Monte Carlo event generators such as {\tt Pythia8}~\cite{Sjostrand:2007gs}
or {\tt POWHEG}~\cite{Alioli:2010xd}.
%
The recent calculation of NNLO corrections to differential
distributions on $t\bar{t}$ production in hadron collisions,
see Sect.~\ref{sec:datatheory.top},
makes conceivable that in the near future NNLO accuracy could be achieved
for charm production as well.

One important challenge in order to include the LHCb $D$ meson
measurements into a PDF fit is the large scale uncertainties
from the NLO calculation, which can be up to $\mathcal{O}(100\%)$.
%
In order to tackle this limitation, it is convenient to define
new observables as different cross-sections ratios in order
to cancel most of these theoretical errors.
%
In particular, the following two families of observables
have been advocated:
\begin{align}
  \label{eq:Obs}
N_X^{ij} &=  \frac{d^2\sigma({\rm X~TeV})}{dy_i^D d (p_T^D)_j} \bigg{/} \frac{d^2\sigma({\rm X~TeV})}{dy_{\rm ref}^D d (p_T^D)_j} \, , \\[1mm]
R_{13/X}^{ij} &= \frac{d^2\sigma({\rm 13~TeV})}{dy_i^D d (p_T^D)_j} \bigg{/} \frac{d^2\sigma({\rm X~TeV})}{dy_{\rm i}^D d (p_T^D)_j} \, , \nonumber
\end{align}
%
which benefit from the partial cancellation of the
residual scale dependence from missing higher-orders, while
retaining sensitivity to the gluon since different
regions of $x$ are probed in the numerator and denominator
of these observables.
%
In Eq.~(\ref{eq:Obs}), $y_{\rm ref}^D$ is a specific rapidity bin
chosen to normalize the absolute cross-section measurements.
%
As explained in~\cite{Gauld:2016kpd,Gauld:2015yia,
Zenaiev:2015rfa,Zenaiev:2016kfl}, its value
should be such that the results of the PDF fit are stable
with respect to small variations of this choice.
%
Thanks to these normalizations, scale variations are greatly
reduced, and in particular it can be shown~\cite{Gauld:2016kpd}
that the entire set of LHCb $D$ meson measurements at
5, 7 and 13 TeV can be satisfactorily
described $(\chi^2/n_{\rm dat})\simeq 1$
using NLO QCD theory.

\subsubsection*{Impact on PDFs}\label{sec:datatheory.charm.impact}

In order to illustrate the impact of the LHCb open charm
production data into the small-$x$ gluon,
in Fig.~\ref{fig:gPDF_rwgt} we show
the results of the gluon PDF from
   the PROSA fit, based on the HERA structure functions and
   LHCb $D$ meson 7 TeV data in the FFNS~\cite{Zenaiev:2015rfa}, compared
   with the corresponding fit based on the
   NNPDF3.0 framework~\cite{Gauld:2015yia}.
   %
   Comparing to the result of the HERA-only fit, we see that
   there is a marked reduction of the gluon PDF uncertainties
   down to $x\simeq 10^{-6}$.
   %
   We also find that the PROSA and NNPDF results, agree
   reasonably well with each other, both in terms
   of the central value and on the size of the gluon PDF
   uncertainties at small-$x$.
   %

Next, in Fig.~\ref{fig:gPDF_rwgt} we show
a comparison between the small-$x$ gluon
   at $Q^2=4$ GeV$^2$
   for NNPDF3.0 with the corresponding result after different
   combinations of the LHCb charm production data at 7, 8 and 13 TeV
    have been included in the fit~\cite{Gauld:2016kpd}.
   %
   We show the central value and one-sigma PDF uncertainty
   bands for the $N^7+N^{13/5}$ and the $N^5+N^7+N^{13}$
   combinations, as well as the central value for the
   $N^5+N^{13/7}$ case, see Eq.~(\ref{eq:Obs}) for the observable
   definitions.
   %
   We can see that the charm data reduces the uncertainty on the small-$x$ PDF uncertainties
   by up to almost an order of magnitude at $x\simeq 10^{-6}$.
   %
   One can also observe that the central value of the
   resultant small-$x$ gluon is robust with respect to
   the specific choice of LHCb $D$ meson observables
   used in the fit.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.46]{3-datatheory/plots/charmgluon.pdf}
   \caption{\small Left:
   the results of the gluon PDF from
   the PROSA fit of HERA structure functions and
   LHCb $D$ meson 7 TeV data in the FFNS~\cite{Zenaiev:2015rfa}, compared
   with the corresponding fit based on the
   NNPDF3.0 framework~\cite{Gauld:2015yia}.
   %
Right: comparison between the small-$x$ gluon
   at $Q^2=4$ GeV$^2$
   in NNPDF3.0 with the corresponding result after different
   combinations of the charm production data at
   LHCb have been included in the fit, from~\cite{Gauld:2016kpd}.
   %
   Specifically, we show here the central value and one-sigma PDF uncertainty
   bands for the $N^7+N^{13/5}$ and the $N^5+N^7+N^{13}$
   combinations, as well as the central value for the
   $N^5+N^{13/7}$ case, see text for more details.
    \label{fig:gPDF_rwgt}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The results of Fig.~\ref{fig:gPDF_rwgt} highlight that forward
$D$ meson production allows a precision determination
of the small-$x$ gluon PDF, with implications from UHE neutrino
astrophysics to future high-energy colliders (see also the
discussion of Sect.~\ref{sec:fcc}).
%
Furthermore, from the technical point of view, the inclusion
of these data into the next generation of global
PDF fits is facilitates by the availability of the
{\tt aMCfast} interface, which allows generating fast grids
for NLO calculations matched to parton showers, as required
for this process.
