The factorization theorems of perturbative QCD (see
Ref.~\cite{Collins:2011zzd} and references therein) express physical
observables for hard processes characterized by a large momentum scale
as the convolution of a perturbatively computable
partonic cross-section, determined 
in terms of the degrees of freedom of the QCD Lagrangian
--- quarks and gluons --- and universal, process--independent parton
distributions, up to corrections suppressed by powers of the
ratio of a characteristic QCD scale $\Lambda\sim100$~MeV to the large scale.

Factorization is most rigorously established for deep inelastic
lepton-hadron scattering, where the expression for the
cross-section has schematically the form
\begin{equation}\label{eq:disfact}
\sigma_{DIS}(x,Q^2)=\int_x^1\frac{dz}{z}\,\hat\sigma_{DIS}(z,\alpha_s(Q^2))
f\left(Q^2,\frac{x}{z}\right),
\end{equation}
where $\sigma_{DIS}(x,Q^2)$ denotes generically a deep inelastic
 structure function, $Q^2$ is the hard scale of the process, namely,
 the virtuality of the gauge boson which mediates the scattering
 process, and the scaling variable is
 $x=Q^2/2p\cdot q$ in terms of the incoming hadron momentum $p$ and
 momentum transfer $q=k^\prime-k$ between the incoming ($k$) and
 outgoing ($k^\prime$) lepton momenta. In Eq.~(\ref{eq:disfact}), the
 partonic cross section $\hat\sigma_{DIS}(z,\alpha_s(Q^2))$ can be
 computed as a perturbative expansion in $\alpha_s(Q^2)$, while
 $f\left(Q^2,x\right)$ is a parton distribution; a sum over different
 kinds of partons (individual quark flavours and gluons) is
 understood but omitted for simplicity.
Equation~(\ref{eq:disfact}) directly follows from the Operator-Product
 Expansion, though it can also be derived from the computation of
 parton Feynman diagrams. 

Factorization for hadronic cross-sections, such as for example the
production of an electroweak final state such as a Higgs or a $W$ or
$Z$ takes the form
\begin{equation}\label{eq:hadfact}
\sigma_{DY}(\tau,M^2)=\int_\tau^1\frac{dz}{z}\,\hat\sigma_{DY}(z,\alpha_s(M^2))
{\cal L}\left(\frac{\tau}{z}\right),
\end{equation}
where now $\sigma_{DY}(\tau,M^2)$ and
$\hat\sigma_{DY}(z,\alpha_s(M^2))$ are respectively hadronic and
partonic 
cross sections
for production of the electroweak final state of mass $M$: with the hadronic
$\sigma$ measurable, and the partonic $\hat \sigma$ computable in
perturbation theory, and  the scaling variable is
$\tau=\frac{M^2}{s}$.
The dependence on parton distributions now goes through the parton luminosity
\begin{equation}\label{eq:lumi}
{\cal L}(\tau)=\sum_{a,b} \int_{\tau}^1 \frac{dx}{x}  f_{a/h_1}(x) 
f_{b/h_2}(\tau/x),
\end{equation}
which depends on the parton distribution from parton $a$ and $b$,
respectively, extracted from the two incoming hadrons $h_1$ and $h_2$. Again, we
have neglected a sum over partonic channels for simplicity.

Because taking a Mellin transform turns the convolutions in
Eqs.(\ref{eq:disfact}-\ref{eq:hadfact}) into ordinary products, the
universality of PDFs implies that one can form a PDF-independent
measurable ratio:
\begin{equation}\label{eq:pdfindep}
\frac{\sigma_{DY}(N,M^2)}{\sigma^2_{DIS}(N,M^2)}=\frac{\hat\sigma_{DY}(N,M^2)}{\hat\sigma_{DIS}^2(N,M^2)},
\end{equation}
where  $\sigma_{DY}(N,M^2)=\int_0^1d\tau\,\tau^{N-1}\sigma(\tau,M^2)$,
and analogously for the other cross sections. It is the possibility of
constructing such a ratio, which only depends on perturbatively
computable partonic cross sections, which guarantees predictivity
of Drell-Yan processes for example.

Factorization for hadronic processes in which a colorless electroweak
final state is produced, such as Higgs, or a real or virtual  $W$, $Z$
or $\gamma$ is firmly established on the basis of an all-order
analysis of the relevant Feynman diagrams. Furthermore, such processes
are currently available up to NNLO in perturbation theory at the
differential level and partly at N$^3$LO for total cross sections: no
counterexample to factorization has been found up to this order.
Factorization is also well-established for sufficiently inclusive
colored final states, such as the one-jet and dijet cross section. In
this case no fully rigorous all-order argument is really available,
but no counterexample to factorization has ever been found either.

Recent global determinations of parton distributions, such as CTEQ~\cite{Lai:2010vv},
MSTW~\cite{Martin:2009iq} or NNPDF~\cite{Ball:2011uy} combine both
deep inelastic scattering data with a variety of beams (electrons,
muons, neutrinos) and targets (protons, deuterons and heavier nuclei)
as well as hadron-hadron data such as virtual photon (Drell-Yan)
production, $W$ and $Z$ production, and single jet production. 
A typical such analysis includes over 2000 deep inelastic data points,
and over 1000 hadronic data points. It thereby provides both a test of
the factorization framework which guarantees the mutual consistency of
these data, and a possibility of assessing their relative impact.

%
\begin{figure}[htb]
\begin{center}
\vspace{-5cm}
\includegraphics[clip=,width=.49\linewidth]{xg_Q_2_log}
\includegraphics[clip=,width=.49\linewidth]{xg_Q_2_lin}
\caption{The gluon distribution determined using deep inelastic
scattering data only (blue) or also including jet data (red), plotted
on a logarithmic (left) and linear (right) scale vs $x$ at
$Q^2=2$~GeV$^2$ (from~\cite{Forte:2010dt}).
}
\label{fig:jetdisconsist}
\end{center}
\end{figure}
As an example of both, Figure~\ref{fig:jetdisconsist} compares the
gluon distribution extracted by only fitting deep inelastic scattering
data, and
by also adding jet data to the DIS dataset. Such a comparison provides an
interesting test of factorization for at least three reasons. First,
because one is comparing a lepton-hadron and a hadron-hadron
process. Second, because the gluon is determined through rather
different physical mechanisms: in deep inelastic scattering, where it
does not couple to the leading-order perturbative process because it
carries no electric or weak charge, the gluon is determined
by the $Q^2$ evolution. In jet production, instead, it
directly contributes both to the initial and final state. Finally,
because the jet data are taken at scales which are typically one or
two orders of magnitude larger than that of the deep inelastic data,
and thus the parton distribution extracted from one process has to be
evolved through renormalization group equations in order to be  used
in the other process. Nevertheless, it is apparent from    
Fig.~\ref{fig:jetdisconsist}  that the gluon determined by
deep inelastic scattering and jets is consistent: when
adding jet to deep inelastic 
data, the central value moves very little, while the
uncertainty is somewhat reduced. It is also apparent from the
figure that the bulk of the information on the gluon distribution is
coming from deep inelastic scattering, with the jet data only
providing some useful but no determinant extra 
information.
% in a more limited kinematic region.

Both these situations are generic, and likely to persist in the
future. On the one hand, perturbative factorization is now firmly
extablished for a variety of processes, and electron-proton data can thus
reliably provide input for hadron collider processes. In fact, because
factorization is most reliably established in deep inelastic
scattering, it is the availability of precise parton distributions
from lepton-hadron scattering which may allow detailed tests of the
validity of factorization for  processes for which it is less 
established. On the other hand, because lepton-hadron data are
generally subject
to smaller power-suppressed corrections, perturbatively more stable,
easier to compute than most hadronic processes so results to the
highest pertiurbative orders are available, and finally free of many
complications which arise when dealing with hadronic initial and final
states (such as jet definitions, or underlying event), 
lepton-proton data always provide comparatively  more
competitive and theoretically reliable
determination of parton distributions than hadron-hadron data. The
natural scenario is one in which lepton-proton data are used to
determine parton distributions, and the latter are then used for
hadron collider processes, and there are strong reasons of principle
why this is the case.
 
