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

Before the establishment of QCD as the renormalizable quantum field theory of the strong interaction, the results of DIS
experiments were interpreted in the context of the so--called quark
parton model.
%
In this model, the proton was composed by non-interacting, co-moving quarks
each of them carrying a given fraction $x$ of its total momentum, and then the
DIS structure functions have particularly simple
expressions in terms of the PDFs.
%
Moreover, in this model the PDFs have a simple probabilistic interpretation,  with $q_i(x)\Delta x$ 
giving the probability of finding a quark of flavour $i$ inside the proton carrying out
a momentum fraction in the range $\lc x, x+\Delta x\rc$.
%
The expressions of the DIS structure functions in the quark parton model
therefore provide a useful way to illustrate the PDF sensitivity of this process.

For the NC DIS structure functions $F_2$ and
$F_3$, as defined in (\ref{eq:ncxsect}), the quark parton model expressions
are given by
\be
\label{eq:partonmodel1}
\lc  F_2^{\gamma},F_2^{\gamma Z}, F_2^Z\rc = x\sum_{i=1}^{n_f}\lc e^2_i, 2e_ig_V^i, g_V^{i2}+g_A^{i2}\rc \lp q_i+\bar{q}_i\rp \, ,
\ee
\be
\label{eq:partonmodel2}
\lc  F_3^{\gamma},F_3^{\gamma Z}, F_3^Z\rc = x\sum_{i=1}^{n_f}\lc 0, 2e_ig_A^i, 2 g_V^i g_A^i\rc \lp q_i-\bar{q}_i\rp \, ,
\ee
while the longitudinal structure function vanishes in this model, $F_L=0$, and the superscripts on the LHS
indicate the gauge boson which is being interchanged, as well as the contribution from the $\gamma Z$
interference term.

In Eqns.~(\ref{eq:partonmodel1}) and~(\ref{eq:partonmodel2}), $e_i$ is the electric charge of the quark of
flavour $i$ and the weak couplings are given by $g_V^i=\pm \frac{1}{2}-2e_i\sin^2\theta^2_W$
and $g_A^i=\pm \frac{1}{2}$, where the $\pm$ corresponds to a $u$ or $d$ type quark.
%
The sum runs over all the $n_f$ quarks that are active for the specific scale at which the scattering
takes place.
%
From Eqns.~(\ref{eq:partonmodel1}) and~(\ref{eq:partonmodel2}) we see that the main limitation
of the NC structure functions is that they provide limited access to quark flavour
separation and in particular they cannot separate quarks from antiquarks, unless one goes to very high $Q^2$
values where the suppression induced by the $Z$ boson propagator can be ignored.

In the case of CC DIS, the corresponding expressions for the structure functions
in the parton model, assuming that we are above the charm threshold but below the top quark
threshold, and the CKM suppressed transitions can be neglected, are given by
\bea
F_2^{W^-}&=&2x\lp u+\bar{d}+\bar{s}+c \rp \, , \nonumber
\\
F_3^{W^-}&=&2x\lp u-\bar{d}-\bar{s}+c \rp \, , 
\\
F_2^{W^+}&=&2x\lp d+\bar{u}+\bar{c}+s \rp \, , \nonumber
\\
F_3^{W^+}&=&2x\lp d-\bar{u}-\bar{c}+s \rp \, , \nonumber
\eea
where again the longitudinal structure function $F_L^{W^\pm}=0$ vanishes
in this model.
%
By comparing the NC and CC expressions, we can see that the main difference between
them is that in the latter case the $F_3^W$ structure function, which provides information
on the difference between quark flavours, is not suppressed with respect to $F_2^W$.
%
For this reason, CC structure functions, both from HERA and from neutrino
fixed-target experiments,
are generally included in global fits in order to improve the discrimination between quarks and anti--quarks.

These quark parton model expressions are also valid at LO in perturbative QCD, once
the effects of the DGLAP evolution are accounted for as described in Sect.~\ref{subsec:dglap}.
%
It is only at NLO that the contribution from the gluon PDF must also be included, and therefore the inclusive DIS structure functions will only be weakly sensitive to the gluon PDF,
either through scaling violations (that is, the effect on the quark DGLAP evolution) or via the small $\mathcal{O}(\alpha_s)$ contribution to the coefficient functions.
%
The exception is the longitudinal structure function $F_L$, which vanishes at LO, while at
NLO it is non-zero and directly sensitive to the gluon PDF.
%
Indeed, it can be shown that this structure function is given by
\be
F_L(x,Q^2)=\frac{\alpha_s(Q^2)}{\pi}\lc \frac{4}{3}\int_0^1 \frac{dy}{y}\lp \frac{x}{y}\rp^2F_2(y,Q^2)
+2\sum_i e_i^2 \int_x^1 \frac{dy}{y}\lp \frac{x}{y}\rp^2 \lp 1-\frac{x}{y}\rp g(x,Q^2)\rc \, ,
\ee
which is known as the Altarelli-Martinelli relation~\cite{Altarelli:1978tq}.
%
For this reason, $F_L$ measurements can, in principle, provide direct constraints
on the gluon in particular at low-$x$.

Finally, in addition to the inclusive structure functions, it is also possible to determine the heavy quark structure functions experimentally, by selecting DIS events with charm or bottom mesons in the final state. The LO process proceeds via the photon-gluon fusion mechanism $\gamma g \to q\overline{q}$, see Fig.~\ref{fig:dis-diagrams} (right), and therefore heavy quark structure functions
offer direct information on the gluon PDF, as well as on the treatment of heavy quark mass effects in the theoretical calculation.
%
Charm structure functions in addition are an
important ingredient for the determination of the charm
mass $m_c$ together with the PDFs~\cite{Bertone:2016ywq,Gao:2013wwa,Alekhin:2012vu}.
%
On the other hand, while data on $F_2^b$ is known to have a small impact in the global
 fit, it is relevant for specific applications, for instance
 the determination of the bottom quark mass $m_b$ from
 the PDF fit.

\subsubsection*{Experimental data}\label{sec:datatheory.data}
%
Since the pioneering DIS experiments at SLAC in the late 60s
and early 70s, there have been many other
measurements of the DIS structure functions.
%
These have been performed using either electrons,
positrons or muons as projectile, and scattering  off protons, deuterons
and neutrons, either for fixed--target or for collider kinematics.
%
We now discuss this various measurements in turn.

To begin with, the fixed-target DIS measurements available for PDF fits
can be divided into neutral current and charged
current datasets. In the NC case this includes:

\begin{itemize}
\item Proton and deuteron structure function data
by the BCDMS collaboration~\cite{bcdms1,bcdms2}, using muons
as projectiles.
\item Proton and deuteron structure function data
by the NMC collaboration~\cite{Arneodo:1996kd,Arneodo:1996qe},
as well as of the ratio between deuteron
to proton structure functions, $F_2^d/F_2^p$.
\item  SLAC measurements of the proton and deuteron
NC structure functions~\cite{Whitlow:1991uw}.
\item Proton, neutron
and deuteron structure function data at high-$x$ and low--$Q^2$ performed by 
JLAB experiments such as CLAS~\cite{Tkachenko:2014byy}.
%
While these are excluded from most PDF fits by the typical DIS cuts
in $x$ and $Q^2$, these are included in the CJ fits.
\item Older structure function data from the EMC collaboration~\cite{Aubert:1982tt}.
%
Despite their age, the EMC measurements of $F_2^c$ have never
been repeated and thus provide unique information on the charm
content of the proton at high--$x$.
\end{itemize}

In the CC case we have:

\begin{itemize}

\item Inclusive structure function measurements due to neutrino beams on nuclear targets, by the CDHSW, CCFR~\cite{Seligman:1997mc,Yang:2000ju} and CHORUS~\cite{Onengut:2005kv} and NuTeV~\cite{Zeller:2001hh} collaborations.

\item Charm production in neutrino--induced DIS.
%
This process is often referred to as dimuon production, since the charm quark hadronizes into a $D$ meson
which then decays semi-leptonically, see Fig.~\ref{fig:dis-diagrams}.
%
Data has been taken by the CCFR and NuTeV~\cite{Goncharov:2001qe,MasonPhD} and
and CHORUS~\cite{KayisTopaksu:2008aa} collaborations on the same nuclear
targets as the corresponding inclusive measurements, and also by the NOMAD collaboration~\cite{Samoylov:2013xoa}.

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.42]{3-datatheory/plots/dis-diagrams.pdf}
   \caption{\small Left plot: $D$ meson production
   in charged-current neutrino-induced DIS.
   %
   This is known as the ``dimuon'' process, since events are tagged
   when the $D$ meson decays semi-leptonically, with
   the pair of oppositely-charged muons providing a clean signature.
   %
   Right plot:
   %
   charm production in neutral current DIS at leading order proceeds
   via the photon-gluon fusion process, 
   highlighting its sensitivity to the gluon PDF.
    \label{fig:dis-diagrams}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the DIS measurements from the HERA lepton-proton collider we have:

\begin{itemize}
\item The final measurements of the NC and CC differential cross-sections
using electron and positron projectiles from the the combination
of the Run I and Run II data-taking periods~\cite{Abramowicz:2015mha}.

\item The latest heavy flavour measurements from HERA include
the combined NC cross-sections of charm production
in DIS, $\widetilde{\sigma}_{c}$~\cite{Abramowicz:1900rp}  and
the H1 and ZEUS data on the bottom structure function
 $F_2^b(x,Q^2)$~\cite{Aaron:2009af,Abramowicz:2014zub}.

\end{itemize}

This HERA legacy combination of DIS inclusive structure
functions supersedes all previous
inclusive measurements from H1 and ZEUS, including the Run I
combined dataset~\cite{Aaron:2009aa} as well as the separate
measurements by the two experiments
from Run II~\cite{Aaron:2012qi,Collaboration:2010ry,
ZEUS:2012bx,Collaboration:2010xc}.
%
The impact of replacing these individual datasets by the final
HERA combination of inclusive structure functions has been studied
by different
groups~\cite{Rojo:2015nxa,Harland-Lang:2016yfn,Hou:2016nqm},
and is found to be quite moderate in general.
%

We also note that previous measurements of the longitudinal structure
function $F_L$ by the H1 and ZEUS collaborations~\cite{Collaboration:2010ry,Andreev:2013vha,Abramowicz:2014jak}
are now superseded by the final inclusive HERA combination.

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

The coefficient functions
of the DIS structure functions in the neutral current case
 are available up to  $\mathcal{O}\lp \alpha_s^3\rp$ in the massless
limit~\cite{Moch:2004xu,Vermaseren:2005qc}, and up to  $\mathcal{O}\lp \alpha_s^2\rp$ taking into account
heavy quark mass effects~\cite{Laenen:1992zk,Laenen:1992xs}, though there has
been considerable recent progress towards the completion of the $\mathcal{O}\lp \alpha_s^3\rp$ calculation
of massive DIS structure functions~\cite{Ablinger:2014nga,Ablinger:2016eyd},
in particular of the terms that dominate in the $Q^2\gg m^2$ limit.
%
For charged current structure functions, massless
coefficients are available up to $\mathcal{O}\lp \alpha_s^3\rp$ and
massive coefficient functions
up to $\mathcal{O}\lp \alpha_s^2\rp$~\cite{Berger:2016inr}.
%
For heavy-quark initiated processes, massive coefficient functions are available only
up to $\mathcal{O}\lp \alpha_s\rp$~\cite{Kretzer:1998ju}.

These coefficient functions have been implemented in a number of private and public
codes, which allow the efficient calculation of DIS structure functions using
state of the art theoretical information, such as {\tt QCDNUM}~\cite{Botje:2010ay},
{\tt APFEL}~\cite{Bertone:2013vaa},\footnote{The {\tt APFEL}
program is currently being rewritten into
C++~\cite{Bertone:2017gds}.}
and {\tt OpenQCDrad}~\cite{Alekhin:2010sv}.
%
This efficient evaluation of DIS cross-sections is possible
because the lengthy expressions for the NNLO DIS coefficient functions
are also available in the form of more compact interpolated expressions, which
reduce the computational burden of their evaluation.
%
Moreover, DIS structure functions can be evaluated either in terms of the heavy pole mass
or in terms of the running $\overline{\rm MS}$ mass, as discussed in~\cite{Alekhin:2010sv}.
%
This statement is valid both in the FFNS scheme as well as in any GM-VFNS, see for instance
the discussion of the FONLL case in~\cite{Bertone:2016ywq}. 

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

From purely theoretical arguments, we know that DIS structure functions will be constraining for some specific
PDF combinations, in particular the total quark singlet $\Sigma$, the quark
isotriplet $T_3$ (thanks to the deuteron measurements), the strange PDF
(from the neutrino-induced dimuon cross-sections), as well as the gluon PDF from scaling
violations at HERA.
%
The same arguments indicate that other PDF combinations would be poorly constrained
in a DIS-only fit, such as the separation between quarks and antiquarks and the gluon
at large-$x$.
%
On the other hand,
given that DIS constitutes the backbone of any PDF fit, it is not straightforward to gauge its impact by adding
say DIS data on top of a non-DIS dataset, as in the latter case the PDFs will be poorly constrained.
%
Here, in order to illustrate the role that DIS structure functions have in a PDF analysis, we have chosen
instead two other related examples.

First of all, in Fig.~\ref{fig:DISonlyImpact} we show
the relative PDF uncertainties in the HERAPDF2.0NNLO analysis, separated
   into experimental, model, and parametrization uncertainties, at
   $Q^2=10$ GeV$^2$, for the various antiquarks.
   %
   This PDF fit is based only on the inclusive HERA structure function data, and therefore
   it can be used to illustrate the information from quark flavour separation that
   can (and that cannot) be obtained from DIS measurements.
   %
   We observe that while at medium and small-$x$ the experimental PDF uncertainties
   are small, the contribution from model choices is larger, adding up
   to between 5\% and 10\%.
   %
   The exception is the strange PDF, which is only poorly constrained by the HERA charged current
   cross-sections, and where errors are at least $\simeq 20\%$, if not bigger.
   %
   Moreover, at large $x$ the PDF uncertainties for all the antiquarks increase rapidly,
   reflecting the fact that the HERA data has no sensitivity to the quark
   flavour separation in this region.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.50]{3-datatheory/plots/DISonly.pdf}
   \caption{\small
   Left: the relative PDF uncertainties in the HERAPDF2.0NNLO fit, separated
   into experimental, model, and parametrization uncertainties, at
   $Q^2=10$ GeV$^2$, for the various antiquark PDFs.
   %
   Right: the gluon PDF in the NNPDF2.0 analysis at small-$x$ (top) and large-$x$ (bottom),
   comparing the results of a DIS-only fit with those of a DIS+jet fit.
   %
   See text for more details.
    \label{fig:DISonlyImpact}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

While DIS data provide only loose information on quark flavour separation,
it gives us instead some of the most constraining information on the gluon PDF
in the medium and small-$x$ region, in particular in the HERA region
due to scaling violations.
%
To quantify this, in the right plot of Fig.~\ref{fig:DISonlyImpact}  we show
 the gluon in the NNPDF2.0 analysis at small-$x$ (top) and large-$x$ (bottom),
   comparing the results of a DIS-only fit with those of a DIS+jet fit.
   %
   At small and medium $x$, the impact of jet data is minimal, showing that
   in a global fit the DIS data provides the most important constraints
   on the gluon in this kinematic region.
   %
   On the other hand, at large-$x$ we observe that the information on the gluon from DIS data is only partial,
   and adding jet production to the fit helps to significantly reduce the PDF uncertainties there.

We also emphasize that neutrino-induced charged-current structure functions, in
particular the dimuon cross-sections, provide valuable constraints on the strange
content of the proton~\cite{Alekhin:2017olj,Alekhin:2014sya,Ball:2009mk,Lai:2007dq,Olness:2003wz}.
%
The impact of various datasets into the strange PDF will be discussed in
more detail in Sect.~\ref{sec:structure.strange}.




