The charm content of the proton is a topic
that has recently received quite a lot of attention
(see~\cite{Brodsky:2015fna} for a recent review).
%
As discussed in Sect.~\ref{sec:fitmeth.PDFpara}, there are two different
approaches to treat the charm PDF within the global QCD analysis.
%
On the one hand, one can assume that the charm PDF is generated
entirely
from perturbative evolution, and thus compute the charm
PDF from the gluon and light quark PDFs starting from
the charm threshold $\mu_c\simeq m_c$ by means of
the DGLAP evolution equations.
%
On the other hand, it is also possible to release this assumption
and treat the charm PDF on an equal footing to the light quark
PDFs, namely introducing a functional form for $c(x,Q_0)$
with parameters to be determined by
experimental data.

Until recently, in most global fits the charm PDF was generated
using the perturbative evolution {\it ansatz}, and then separately,
in dedicated intrinsic charm studies, variants of these global fits
were performed with specific 
models for the charm PDF.
%
In these studies, the parameters
of the model charm PDF, typically
its overall normalization, where constrained by experimental data, see
for instance Refs.~\cite{Pumplin:2007wg,
  CT14IC,Hou:2017khm,Jimenez-Delgado:2014zga}.
%
An alternative approach is taken
by the NNPDF3.1 global analysis, which fits the charm PDF
 using  the same parametrization as for
the light quarks.
%
When the charm PDF is fitted, the dominant constraints
arise from processes sensitive to initial-state
charm, such as the charm structure
functions of the EMC experiment~\cite{Aubert:1982tt},
other fixed-target DIS datasets, and collider electroweak
gauge boson production.

For the first approach, the
 CT14IC analysis~\cite{Hou:2017khm} provides a recent and representative example.
%
Here, the charm PDF
is parametrized according to two theoretical
scenarios.
%
The first scenario assumes either the exact
or the approximate BHPS model~\cite{Brodsky:1980pb},
which predicts a valence-like charm PDF at the input scale.
%
In the case of the approximate solution of the model,
we have
\be
c(x,Q_0)=\frac{1}{2}Ax^2 \lc \frac{1}{3}(1-x)(1+10x+x^2)
-2x(1+x)\ln(1/x)\rc \, ,
\ee
while  a more complicated, non-analytic expression is used
for the exact BHPS solution.
%
The other scenario explored in the CT14IC study
is the SEA model, where the charm
PDF is parametrized by a ``sea-like'' function taken
to be proportional to the light quark sea PDFs, namely
\be
c(x,Q_0)=A\lp \bar{d}(x,Q_0)+\bar{u}(x,Q_0)\rp \, .
\ee
In these three scenarios, the overall normalization $A$ of
the fitted charm PDF is a free parameter to be determined
from the experimental data.

To illustrate the main results of this study,
In Fig.~\ref{fig:ct14ic} we plot
the deviation of the $\chi^2$ in the CT14IC fits,
computed with respect to the
     best-fit value of the CT14 fit with perturbative charm,
     as a function of the charm
     momentum fraction $\la x\ra_{\rm IC}=C(Q_0)$, where
     we have defined
     \be
     \label{eq:charmmomfrac}
C(Q^2) \equiv \int_0^1dx~x~\lp c(c,Q^2)+\bar{c}(x,Q^2)\rp \, .
\ee
     Results are shown for the BHPS and SEA models, with the
    `1' points labeling the preferred value of  $\la x\ra_{\rm IC}$,
    with those labelled with `2' indicating the largest
    values of the charm momentum fraction allowed by the
    fit tolerance criteria.
    %
    We observe that the BHPS model is preferred, leading
    to a best-fit value of $\la x\ra_{\rm IC}\simeq 0.6\% $.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.44]{6-structure/plots/ct14ic.pdf}
     \caption{\small 
     Left: the deviation of the $\chi^2$ in the CT14IC fits,
with respect to the
     best-fit value of the CT14 fit with perturbative charm,
     as a function of $\la x\ra_{\rm IC}$
     %
     Results are shown for the BHPS and SEA models, with the
    ``1'' points labeling the preferred value of  $\la x\ra_{\rm IC}$,
    with those labelled with ``2'' indicate the largest
    values of the charm momentum fraction allowed by the
    fit tolerance criteria.
    %
    Right: the transverse momentum distribution of $Z$ bosons
    in the $pp \to Z+c$ process at 13 TeV, comparing the CT14 NNLO
    result with various of the CT14 IC models, as a function
    of $p_T^Z$.
      \label{fig:ct14ic}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As mentioned above, a different approach to fitted charm is adopted by the
NNPDF collaboration~\cite{Ball:2016neh,Ball:2017nwa}.
%
In this case, the charm PDF is treated on an equal footing
to the  light quark PDFs, and therefore it is parametrized
with a 37-parameter artificial neural network with  2-5-3-1
architecture,
\be
c^{+}(x,Q_0) = c(x,Q_0)+\bar{c}(x,Q_0)=
x^{a_{c^+}}(1-x)^{b_{c^+}}\,{\rm NN}_{c^+}(x) \, ,
\ee
where $a_{c^+},b_{c^+}$ are the corresponding preprocessing
exponents, whose range is determined by an iterative
procedure.
%
The charm asymmetry, on the other hand, is assumed to vanish
$c^{-}(x,Q_0) = 0$.
%
The charm PDF is then determined at the input evolution
scale $Q_0=1.64$ GeV from the experimental data,
finding that the recent LHC high-precision electroweak
gauge boson production measurements provide the
best constraints~\cite{Ball:2017nwa}.
%
 DIS structure functions are treated
      with the FONLL general-mass VFN scheme, modified
      to account for initial-state massive
      contributions~\cite{Ball:2015tna,Ball:2015dpa}.

One of the potential benefits of this model-independent approach is
that it improves the stability of the
fitted PDFs with respect to the value of the charm mass $m_c$,
since its variations can be re-absorbed into the fitted charm
boundary condition.
%
To illustrate this point, in Fig.~\ref{fig:nnpdf31ic}
we show the dependence of the quark-antiquark
      PDF luminosity at the LHC 13 TeV in the NNPDF3IC fits
      with the value of the charm mass $m_c$ used in the fit.
      %
      We find that even for a relatively large variation
      of $\delta m_c =\pm 0.14$ GeV, the $q\bar{q}$ luminosity
      is very stable in most of the $M_X$ range.
       
The amount of charm present inside the proton is most
usefully quantified by its
momentum fraction, defined in
Eq.~(\ref{eq:charmmomfrac}).
%
In the case of perturbative charm, by construction we have $C(Q^2< \mu_c^2)=0$,
while if there is a non-perturbative charm component in the proton
in general we have $C(Q^2)\ne 0$ at all values of the scale $Q^2$.
%
In Fig.~\ref{fig:charmfrac} we show the momentum fraction carried by charm quarks
both at a low scale $Q=1.51$ GeV and at a high scale
$Q=M_Z$, comparing the results from  NNPDF3.0, based on
perturbative charm,  with those from NNPDF3.1, based on
       fitted charm, with and without the inclusion of the EMC charm
       data, as well as with the BHPS and SEA scenarios of the
       CT14IC fits.

In the comparison of Fig.~\ref{fig:charmfrac},
   the uncertainty bands of the CT14IC results
   indicate the
       range between no intrinsic charm
       and the maximum amount of IC allowed within the CT14 tolerance,
       with the central
       value corresponding to the best-fit.
%
Note that the standard CT14IC fits do not include the EMC data, with the rationale
that  the experimental
systematics are not well documented.
%
It was found in alternative fits that
neither the BHPS or SEA models can describe
well the EMC data, and the same applied to the fit with perturbative charms.
%
Thus, within the CT framework, the EMC charm
structure function data is found to provide
no definitive discrimination between the purely
perturbative and intrinsic charm models.
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40]{6-structure/plots/Cplot.pdf}
\includegraphics[scale=0.40]{6-structure/plots/Cplot-highQ.pdf}
     \caption{\small 
       The momentum fraction carried by charm quarks, $C(Q)$
       Eq.~(\ref{eq:charmmomfrac}),
       both at a low scale $Q=1.51$ GeV (left) and at a high scale
       $Q=M_Z$ (right plot).
       %
       We compare NNPDF3.0 (perturbative charm) with NNPDF3.1 (based on
       fitted charm) with and without the inclusion of the EMC charm
       data, as well as with the BHPS and SEA scenarios of the
       CT14IC fits.
       %
       See text for more details.
      \label{fig:charmfrac}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The comparisons of Fig.~\ref{fig:charmfrac} highlight, first of all, that 
when the charm PDF is generated perturbatively
its uncertainties are very small, but that
this is not necessarily the case once charm is fitted.
%
Reassuringly, once charm is fitted (NNPDF3.1),
the results with perturbative charm (NNPDF3.0)
are consistent within uncertainties.
%
The NNPDF3.1 analysis also finds that adding the EMC charm
data helps in reducing the PDF uncertainties on $\la x\ra_{\rm IC}$
by around a factor 3, but that even without it one can achieve a quite
competitive determination of $c(x,Q)$, due to the precision
collider electroweak data.
%
The CT14IC results are consistent within PDF errors
with the NNPDF3.1IC, although the highest values within
the CT14IC are disfavored by the NNPDF fit with latest LHC data.
%
The rapid growth of $C(Q)$
from $Q=1.51$ GeV to $Q=M_Z$, driven by the perturbative
component, is also apparent from this comparison.

In Fig.~\ref{fig:nnpdf31ic} we show
a comparison of the fitted charm PDF at $Q=1.65$ GeV
      between the NNPDF3IC set and the different model scenarios
      considered in 
      the CT14IC analysis.
      %
      We see that NNPDF3IC prefers a valence-like shape,
      along the lines of the BHPS model, though uncertainties
      are still large.
      %
      The CT14IC BHPS results tend to have the maximum
      at slightly lower values of $x$; note also that
      they develop a perturbative tail since the plot
      is performed at a value $Q > Q_0$.
      %
      The CT14IC SEA models predict that the enhancement of
      the charm PDF is localized at medium and small-$x$,
      while in the valence region the fitted charm agrees
      with the perturbative {\it ansatz}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.40]{6-structure/plots/nnpdf31ic.pdf}
     \caption{\small 
      Left: comparison of the fitted charm PDF at $Q=1.65$ GeV
      between the NNPDF3IC set and the different models
      of the CT14IC analysis.
      %
      Right: the dependence of the quark-antiquark
      PDF luminosity at the LHC 13 TeV in the NNPDF3IC fits
      with the value of the charm mass $m_c$ used in the fit.
      \label{fig:nnpdf31ic}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If the charm content of the nucleon is indeed different
from the one predicted by the perturbative ansatz, there
are a number of phenomenological consequences that
could be studied at the LHC.
%
To begin with, it would modify the kinematic distributions
of the $Z$ bosons
    in the $pp \to Z+c$
    process~\cite{Boettcher:2015sqn,Bailas:2015jlc,Beauchemin:2014rya}, leading to an enhancement
    of the cross-section which grows with the value
    of $p_T^Z$.
    %
    To illustrate this point, in Fig.~\ref{fig:ct14ic}
we show the transverse momentum distribution of $Z$ bosons
    in the $pp \to Z+c$ process at $\sqrt{s}=13$ TeV, comparing the CT14 NNLO
    result with the CT14 IC models, as a function
    of $p_T^Z$.
    %
    Depending on the specific model considered, enhancements
    of up to 50\% are predicted.

    A closely related process is photon production
    in association with charm
    mesons~\cite{Stavreva:2009vi,Bednyakov:2013zta},
    which is however theoretically less clean as it
    is affected by the poorly-understood parton-to-photon
    fragmentation component.
    %
    Another important process where non-perturbative charm
    could make a difference is open $D$ meson production~\cite{Vogt:1994zf,Kniehl:2012ti},
    in particular at large $p_T$ and at forward rapidities,
    which enhance the sensitivity to the large-$x$ region.
    %
    By exploiting the information from these various LHC processes using
    data from the present and future runs,
    we can hope in the future to shed more light
    on the charm content of the proton.
