In a similar way to the DIS structure functions for electron--proton collisions, the production cross sections in proton--proton collisions can be factorized in terms of the convolution between
two universal PDFs and a process-dependent partonic cross section.
%
For example, the Drell-Yan production cross section, $\sigma^{\rm DY}(pp\to l^+l^-+X)$,
can be expressed~\cite{Collins:1987pm,Collins:1989gx} as 
\be
\label{eq:qcdfact1}
\frac{d^2\sigma^{\rm DY}}{dydQ^2}(y,Q^2,\mu^2_R,\mu_F^2)=\sum_{a,b=q,\bar q, g}
\int^1_{\tau_1} dx_1 f_{a}(x_1,\mu^2_F)\int^1_{\tau_2} dx_2 f_{b}(x_2,\mu_F^2
)\,\frac{d^2\hat \sigma^{\rm DY}_{ab}}{dydQ^2}(x_1,x_2,y,Q^2,\mu^2_R,\mu^2_F),
\ee
where $y$ and $Q^2$ are the rapidity and invariant mass square of the lepton pair,
and $s$ is the centre--of--mass energy of the two
incoming protons, while $\mu_F$ ($\mu_R$) are the factorization (renormalization) scales.
The lower integration limits  are $\tau_{1,2}=\sqrt{Q^2/s}\,e^{\pm y}$.
%
The partonic cross sections that appear in Eq.~(\ref{eq:qcdfact1}) can be computed as a perturbative expansion in $\alpha_S$:
\begin{equation}
\frac{d^2\hat \sigma^{\rm DY}_{ab}}{dydQ^2}(x_1,x_2,y,Q^2,\mu^2_R,\mu_F^2)=\sum_{n=0}^{\infty}
\left(\frac{\alpha_s(\mu^2_R)}{2\pi}\right)^n
\frac{d^2\hat \sigma^{(n)\,{\rm DY}}_{ab}}{dydQ^2}.
\end{equation}

From Eq.~(\ref{eq:qcdfact1}) we observe that the definition of the PDFs, once
perturbative QCD corrections are accounted for, requires the introduction of a factorization scale $\mu_F$,
below which additional collinear emissions are absorbed into a PDF redefinition.
%
To all orders, the physical
cross section, as a product of the PDFs and partonic cross section,
is independent of the choice of the factorization scale.
%
However, at any fixed order in the perturbative series, there will be some  sensitivity to the missing
higher order dependence, which can be minimised by choosing a suitable value of $\mu_F$
so as to maintain a better convergence of the series.
%
In Drell-Yan production, the conventional scale choice is $\mu^2_F=Q^2$, namely the invariant mass of the
dilepton pair.

In full generality, for
the case of the total inclusive cross section for a narrow resonance
production with mass $M$, the cross section can be factorized as
\begin{equation}
\label{eq:qcdfact2}
\sigma=\sum_{a,b=q,\bar q, g}\int_{M^2}^{s}{d\hat s\over \hat s}
\mathcal{L}_{ab}(\hat s,\mu_F^2)\,
\hat s \hat \sigma_{ab}(\hat s, M^2, \mu^2_R,\mu_F^2),
\end{equation}
where $\hat s$ is the center of mass energy of the two incoming partons,
the parton-parton luminosity can be defined as~\cite{Campbell:2006wx}
\begin{equation}
\mathcal{L}_{ab}(\tau,\mu_F^2)={1\over s}\int_{\tau/s}^1{dx\over x}f_{a}(
\tau/sx,\mu_F^2)\,f_{b}(x,\mu_F^2) \, .
\end{equation}
The partonic cross section depends only on the kinematic variable $z\equiv
M^2/\hat s$ and $\mu_{F,R}$
\begin{equation}
\hat s \hat \sigma_{ab}(\hat s, M^2, \mu^2_R,\mu^2_F)=\sum_{n=0}^{\infty}\left(
\frac{\alpha_s(\mu_R^2)}{2\pi}\right)^nC_{ab}^{(n)}(z,\mu^2_R,\mu_F^2) \, .
\end{equation}
The usefulness of this factorized form Eq.~(\ref{eq:qcdfact2}) is that the complete
PDF dependence of the hadronic cross-section is now encoded in the partonic luminosities $\mathcal{L}_{ab}$.
%
As will be discussed in Sect~\ref{sec:datatheory}, there has been significant
recent progress in the higher-order calculation of partonic cross-sections.
%
In the case of inclusive processes, for instance,
the coefficient functions $C^{(n)}(z,\mu_R^2,\mu_F^2)$ are known to NNLO for Drell-Yan
production~\cite{Hamberg:1990np} and top-quark
pair production~\cite{Czakon:2013goa}, and to N$^3$LO for Higgs boson production via
gluon fusion in the limit of infinite top quark mass~\cite{Anastasiou:2015ema}.  


