\label{sec:global.heavyq}

The contribution of the charm structure function $F_2^c(x,Q^2)$ to the total inclusive
neutral current structure function
$F_2^p(x,Q^2)$ at HERA can be as high as 25\% at small-$x$ and $Q^2$, and therefore
it is crucial to compute it with high accuracy.
%
In particular, the finite value of heavy quark mass must be taken into account.
%
There are various theoretical schemes that have been proposed for the computation of heavy quark production
in DIS structure functions:
\begin{itemize}

\item The Zero-Mass Variable Flavour Number scheme (ZM-VFNS), where all heavy quark
mass effects are ignored but potentially large logarithms of $\ln Q/m$ are resummed
into the heavy quark parton distribution.
%
This is also known as the massless scheme.

\item The Fixed-Flavor Number Scheme (FFNS), where the heavy quark is always treated
as a massive particle and never as a massless parton irrespective of the value of the scale $Q$.
%
In this scheme, the heavy quark PDF does not exist and the number of active flavours is always
kept fixed.
%
This scheme takes into account heavy quark mass effects in the coefficient functions,
but does not resum logarithmically enhanced terms of the form $\ln Q/m$ that
become numerically relevant at high scales.
%
This is also known as the massive scheme

\item The General-Mass Variable Flavor Number scheme (GM-VFNS) combines the advantage
of the massive and massless calculations by means of an interpolated scheme which is valid
for any value of the scale $Q$, and that matches the FFN and ZM-VFN schemes at small and large
values of $Q$, respectively.

\end{itemize}

Here we review the basic steps that enter into the construction of the GM-VFNS calculation
of heavy quark DIS structure functions.
%
Although for illustration purposes we will focus on the FONLL derivation from Ref.~\cite{Forte:2010ta},
the construction of related GM-VFNS such as ACOT~\cite{acot2}, S-ACOT~\cite{Kramer:2000hn} and
TR~\cite{Thorne:2012az} has most
of these steps in common.
%
We start by the expression of a generic DIS structure function $F(x,Q^2)$,
in a kinematic regime where one has $n_l$ light flavours and
 a single heavy
flavour of
mass $m$.
%
In the massless scheme, accurate when  $W \gg 4 m^2$,
the expression of $F$ in terms of PDFs and coefficient functions is the following
\begin{equation}
  F^{(n_l+1)} (x, Q^2) = x \int_x^1 \frac{dy}{y} \sum_{i = q, \bar{q}, h,
  \bar{h}, g} C_i^{(n_l+1)} \left( \frac{x}{y}, \alpha_s^{(n_l+1)} (Q^2) \right)
  f_i^{(n_l+1)} (y, Q^2), 
\end{equation}
where $q$ are the light quarks and $h$ is the heavy quark.
%
As indicated from the sum, in this scheme the heavy quark is treated as a massless
parton, with all finite mass effects therefore neglected.
%

Now, in the massive (or decoupling~\cite{Collins:1978wz}) scheme, which is most suitable
when $W \approx 4 m^2$ and thus heavy quark mass effects must be accounted for,
this structure function reads
\begin{equation}
  F^{(n_l)} (x, Q^2) = x \int_x^1 \frac{dy}{y} \sum_{i = q, \bar{q}, g}
  C_i^{(n_l)} \left( \frac{x}{y}, \frac{Q^2}{m^2}, \alpha_s^{(n_l)} (Q^2) \right)
  f_i^{(n_l)} (y, Q^2). \label{eq:Fnl}
\end{equation}
where now the massive coefficient functions $C_i^{(n_l)}$ includes the full mass dependence,
and the heavy quark is on longer treated as a massless initial--state parton.
%
In this scheme, the PDFs and $\alpha_s$ satisfy evolution equations with $n_l$ active quarks.
%
The construction of the GM-VFNS structure functions is based on two steps.
%
First of all, one needs
to express PDFs and $\alpha_s$ in the massless scheme by means of the matching conditions
\begin{eqnarray}
  \alpha_s^{(n_l+1)} (Q^2) & = & \alpha_s^{(n_l)} (Q^2) + \sum^{\infty}_{i = 2} c_i (L)
  \times \left( \alpha_s^{(n_l)} (m^2) \right)^i \ ,  \label{eq:matchingalpha}\\
  f_i^{(n_l+1)} (x, Q^2) & = & \int_x^1 \frac{dy}{y} \sum_{j = q, \bar{q}, g} K_{ij} \left( \frac{x}{y}, L,
  \alpha^{(n_l)}_s (Q^2) \right) f_j^{(n_l)} (y, Q^2) \ ,  
  \label{eq:matchingpdf}
\end{eqnarray}
where $L \equiv \log Q^2 / m^2$, and then using these transformed expressions to write down $F^{(n_l)}$
in terms of PDFs and $\alpha_s$ in the massless scheme,
\begin{equation}
  F^{(n_l)} (x, Q^2) = x \int_x^1 \frac{dy}{y} \sum_{i = q, \bar{q}, g}
  B_i \left( \frac{x}{y}, \frac{Q^2}{m^2}, \alpha_s^{(n_l+1)} (Q^2) \right)
  f_i^{(n_l+1)} (y, Q^2), \label{eq:Fnlbar}
\end{equation}
Once we have expressed both $F^{(n_l)}$ and $F^{(n_l+1)}$ in terms of PDFs and $\alpha_s$ in the massless scheme,
the second step is to match the two expressions while removing any double counting.
%
This way we will maintain the main advantages of the two schemes (heavy quark mass effects in $F^{(n_l)}$,
resummation of large $\ln Q^2/m^2$ logarithms in $F^{(n_l+1)}$) within a single scheme that is valid
for any scale $Q$.
%
To achieve this, one defines the massless limit of the massive scheme structure function
as follows
\begin{equation}
  F^{(n_l,\,0)} (x, Q^2) = x \int_x^1 \frac{dy}{y} \sum_{i = q, \bar{q}, g}
  B^{(0)}_i \left( \frac{x}{y}, \frac{Q^2}{m^2}, \alpha_s^{(n_l+1)} (Q^2) \right)
  f_i^{(n_l+1)} (y, Q^2), \label{eq:Fnlbarzero}
\end{equation}
where in the coefficient functions $B^{(0)}_i$ all the terms which are power suppressed
of the form $m/Q$ are neglected, and the only dependence on the heavy quark mass $m$
is on logarithms of the form $\ln Q/m$,

The FONLL approximation for $F$ is then given by
\begin{eqnarray}
&&  F^{\rm{FONLL}} (x, Q^2) =  F^{(d)} (x, Q^2)  + F^{(n_l)} (x,
  Q^2), \label{eq:FONLL}
\\
 &&\quad F^{(d)} (x, Q^2)\equiv
\left[ F^{(n_l+1)} (x, Q^2) - {F}^{(n_l,\,0)} (x,
  Q^2) \right]
\label{eq:fdiff}
\end{eqnarray}
where Eq.~(\ref{eq:fdiff}) is constructed out of the
massless-scheme expression $F^{(n_l+1)}$,
and 
the massless limit  $F^{(n_l,\,0)}$ of the massive-scheme
expression as in Eq.~(\ref{eq:Fnlbarzero}).
%
It is thus easy to see that in the limit where $Q \gg m$, the FONLL
structure function reduces to the massless calculation, while for $Q \sim m$
the FONLL result coincides with the massive calculation up to subleading (higher order) terms.
%
Since close to threshold the difference term $F^{(d)}$ is formally subleading but numerically
non-negligible, it is customary to further suppress it using for instance a damping factor~\cite{Cacciari:1998it}
or the $\chi$ rescaling~\cite{Nadolsky:2009ge}.
%
The GM-VFN scheme formalism that be generalized as well to include mass effects
for heavy-quark initiated contribution~\cite{Ball:2015tna,Ball:2015dpa}, such as those required {\it i.e.} in the
presence of a non-perturbative charm content in the proton.

To illustrate the numerical impact the heavy quark mass effects
have in deep-inelastic structure functions, in
Fig.~\ref{fig:f2c} we show the
inclusive proton structure
   function $F_2(x,Q^2)$ at NNLO~\cite{Thorne:2014toa}
   as a function of $Q^2$ for two different
   values of $x$ in the TR' GM-VFNS~\cite{Thorne:2006qt} as compared to the FFNS calculation.
   %
   We see that differences can be as large as few percent, comparable
  or larger with the precision of available DIS data.
  %
  In the same figure (right) we show the NNLO charm structure function $F_2^c(x,Q)$
   as a function of $Q$ for $x=0.01$ comparing the S-ACOT-$\chi$~\cite{Guzzi:2011ew}
   GM-VFNS with the corresponding ZM and FFN scheme calculations,
   where we can observe how the S-ACOT-$\chi$ calculation
   smoothly interpolates between the FFN scheme at low values of
   $Q$ and the massless result at high $Q$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.45]{2-pdffitting/plots/f2c.pdf}
   \caption{\small Left plot: the inclusive proton structure
   function $F_2(x,Q^2)$ at NNLO as a function of $Q^2$ for two different
   values of $x$ in the TR' GM-VFNS as compared to the FFNS calculation~\cite{Thorne:2014toa}.
   %
   Right plot: the NNLO charm structure function $F_2^c(x,Q)$
   as a function of $Q$ for $x=0.01$ comparing the S-ACOT-$\chi$
   GM-VFNS with the corresponding ZM and FFN scheme calculations~\cite{Guzzi:2011ew}.
    \label{fig:f2c}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
