The importance of deep--inelastic scattering
for PDF fits cannot be overemphasised.
%
As discussed in the previous section, this process was instrumental in the discovery of quarks, and has since then represented the backbone of global PDF fits. 
%
The DIS mechanism is schematically represented
in Fig.~\ref{fig:DIS-main}.
%
Here, an energetic lepton, which can be either charged
(electron or muon) or neutral (a neutrino) scatters off a proton
(or some other hadron) by means of the interchange of a virtual photon $\gamma^*$ or
a $W^{\pm}$ or $Z$ boson.
%
 The large virtuality $Q$ of the gauge boson, $Q\gg \Lambda_{\rm QCD}$,
   ensures that the process can be described within the perturbative
   QCD
   factorization framework in terms of coefficient functions
   and parton distributions, as we show below.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.41]{2-pdffitting/plots/DIS-main.pdf}
   \caption{\small Schematic representation of
  the deep--inelastic scattering process.
   %
   An energetic lepton (electron, muon or neutrino) scatters
   off one of the quarks
   in the proton by means of the interchange of an electroweak gauge boson
   ($\gamma, W^{\pm}$ or $Z$).
   %
   The large virtuality carried by the gauge boson, $Q\gg \Lambda_{\rm QCD}$,
   ensures that the process can be described within the perturbative QCD
   factorization framework in terms of coefficient functions
   and PDFs.
    \label{fig:DIS-main}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The kinematics of the DIS process can be described in terms
of a few invariant quantities, namely
%
\be
\label{eq:disvars}
x\equiv \frac{Q^2}{2 P\cdot q} \, , \quad Q^2 \equiv q^2 \, ,
\quad y\equiv \frac{q\cdot P}{k\cdot P} \, ,
\ee
where $k$ and $k'$ are the four--momenta of the incoming and outgoing
leptons, $q$ is the four--momentum of the exchanged gauge boson, and
$P$ is incoming proton's momentum.
Here $x$ is known as the Bjorken variable, and although it is defined purely
in terms of the kinematics of the initial and final--state particles,
it can be shown that in the parton model it corresponds
to the proton momentum fraction carried by the struck parton in the centre--of--mass frame of the high--energy lepton--proton collision.
%
Recall that by momentum conservation $q=k'-k$, and thus
all the variables in Eq.~(\ref{eq:disvars}) can be determined
by the knowledge of the incoming momenta of the lepton $k$ and
of the proton $P$ as well as the outgoing momentum of the lepton $k'$
without any reference to the final hadronic state $X$.
%
In Eq.~(\ref{eq:disvars}), $Q^2=-q^2$ is the virtuality of the electroweak
gauge boson and the variable $y$ is known as the inelasticity.
%
The centre of mass energy $W$ of the proton--photon collision is given by
\be
W^2 = (P+q)^2 = Q^2 \frac{1-x}{x} + m_p^2\, .
\ee
The value $x=1$ corresponds
to the elastic limit, where the proton remains intact after the collision.

It can be shown that only two of the three variables in Eq.~(\ref{eq:disvars}) are independent, and therefore differential cross sections in DIS are measured for instance as a function of $(x,Q^2)$ or $(x,y)$.
%
Using Lorentz invariance and kinematic
arguments, it can be shown that the DIS cross sections can be expressed
in terms of a series of independent ``structure functions'' that
describe the dynamics of the interaction between the gauge boson
and the incoming hadron.
%
In the neutral current (NC) case, that is, where
either a virtual photon $\gamma^*$ or a $Z$ boson
is exchanged, the DIS
differential cross section 
for a charged lepton $\ell^\pm$ scattering off a proton
can be decomposed in terms of structure functions as follows:
\be
\frac{d^2\sigma^{\rm NC,\ell^{\pm}}}{dxdQ^2} (x,y,Q^2)=\frac{2\pi \alpha^2}{ x Q^4}
 \lc
Y_+ F_2^{NC}(x,Q^2) \mp Y_- x F_3^{NC}(x,Q^2)-y^2 F_L^{NC}(x,Q^2)\rc \ ,
\label{eq:ncxsect}
\ee
where we have defined 
\begin{equation}
 Y_{\pm}=1\pm (1-y)^2.
\label{eq:ypmdef}
\end{equation}
In most cases, experimental measurements are given
in terms of a reduced cross section, defined as
\begin{equation}
  \label{eq:rednc}
  \widetilde{\sigma}^{\rm NC,\ell^{\pm}}(x,y,Q^2)=\lc
  \frac{2\pi \alpha^2}{ x Q^4} Y_+\rc^{-1}\frac{d^2\sigma^{\rm NC,\ell^{\pm}}}{dxdQ^2}(x,y,Q^2) \,,
\end{equation}
which is more closely related to the dominant structure function $F_2(x,Q^2)$, and thus to the underlying PDFs of the proton.

In the case of charged current (CC) DIS, when neutrinos
are used as projectiles or when the incoming charged leptons
interact with the proton by means of the exchange of a charged
weak gauge boson $W^{\pm}$,
the differential cross sections are given by:
\begin{eqnarray}
\label{eq:ccxsect}
  \frac{d^2\sigma^{\rm CC,\ell^{\pm}}}{dxdQ^2}(x,y,Q^2) &=&\frac{G_F^2}{4\pi x}
  \lp \frac{M_W^2}{M_W^2+Q^2}\rp^2 \\\nonumber
&\times&\frac{1}{2}\lc
  Y_+ F_2^{CC,\ell^{\pm}}(x,Q^2)\mp Y_- x F_3^{CC,\ell^{\pm}}(x,Q^2)
  -y^2 F_L^{CC,\ell^{\pm}}(x,Q^2)\rc\,.
\end{eqnarray}
which is generally rescaled
to define a reduced cross section 
\begin{eqnarray}
  \label{eq:redcc}
  \widetilde{\sigma}^{\rm CC,\ell^{\pm}}(x,y,Q^2)=\lc
  \frac{G_F^2}{4\pi x}
  \lp \frac{M_W^2}{M_W^2+Q^2}\rp^2 \rc^{-1}
  \frac{d^2\sigma^{\rm CC,\ell^{\pm}}}{dxdQ^2}(x,y,Q^2)\,,
\end{eqnarray}
similarly to the NC case.
%
In Eqns.~(\ref{eq:ccxsect}) and~(\ref{eq:redcc}), $\ell^\pm$ labels
either the incoming or outgoing charged lepton.

According to the QCD factorization theorem, the
general expression for the DIS structure functions can be written
schematically as
\begin{equation}
\label{eq:QCDfactorization}
  F (x, Q^2) = x \int_x^1 \frac{dy}{y} \sum_{i}
  C_i \left( \frac{x}{y}, \alpha_s (Q^2) \right)f_i (y, Q^2), 
\end{equation}
where $C_i \left( x/y, \alpha_s (Q^2) \right)$ are known as the
coefficient functions and $f_i (y, Q^2)$ are the PDFs.
%
The coefficient functions represent the cross section for the
partonic scattering process $q_i+\gamma^*\to X$, and can be computed in
perturbation theory as a series expansion in the strong coupling
$\alpha_s$ (as well as in the electroweak coupling $\alpha_W$ if these
corrections are included).

While the coefficient functions
encode the short distance dynamics of the parton--boson collision,
the PDFs are instead determined 
by long distance non--perturbative QCD dynamics, and 
so cannot be computed using perturbative methods.
%
On the other hand, the crucical factorization property of Eq.~(\ref{eq:QCDfactorization}) is that while the coefficient
functions (or in general the partonic cross sections) are process
dependent, the PDFs themselves are universal. 
%Therefore, they need to the parametrised and extracted from a global
%analysis of hard scattering measurements.
This allows us to parameterise and extract the PDFs from a global
analysis of hard scattering measurements. These can then be used to make predictions
 for other PDF--dependent processes.
%
%This is possible due to the crucial factorization property of Eq.~(\ref{eq:QCDfactorization}); while the coefficient
%functions (or in general the partonic cross sections) are process
%dependent, the PDFs instead are universal. 
%The PDFs extracted in such a global fit can therefore be used to make predictions
% for other PDF--dependent processes.
