\label{subsec:dglap}

As discussed above, the PDFs depend on two variables: the Bjorken
variable $x$, which at leading order can be identified with the momentum fraction carried by the considered parton,
and the scale $Q^2$, which in DIS corresponds to the virtuality of the exchanged gauge boson.
%
While the dependence of the PDFs on $x$ is determined by non--perturbative dynamics, and therefore
cannot be computed perturbatively, the situation is different for the $Q^2$ variable.
%
Here, the $Q^2$ dependence of the PDFs is introduced when higher--order initial--state collinear singularities
of the partonic cross section are regularised by means of
a PDF redefinition.
%
Such singularities arise from universal long-distance QCD dynamics,
and therefore are process-independent.

For this reason,  the $Q^2$ dependence of the PDFs can be computed in QCD perturbation theory up to any given order.
%
This dependence is determined by a series of integro--differential
equations, known as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations~\cite{Altarelli:1977zs,gl,dok,Lipatov:1974qm,dok}, which have the generic form
\be
  \label{eq:dglap}
  Q^2\frac{\partial}{\partial Q^2}f_i(x,Q^2) = \sum_j 
                                   P_{ij}(x,\as(Q^2))\otimes f_j(x,Q^2),\,
\ee
where $P_{ij}(x,\as(Q^2))$ are the Altarelli-Parisi splitting functions, which can be computed
in perturbation theory
\be
\label{eq:splittingfunctions}
P_{ij}(x,\as(Q^2)) = \sum_{n=0} \lp \frac{\alpha_s(Q^2)}{2\pi}\rp^{n+1}P_{ij}^{(n)}(x) \, ,
\ee
and where  $\otimes$ denotes the convolution
\be
\label{eq:conv}
f(x)\otimes g(x) \equiv \int_x^1 \frac{dy}{y} f(y) g\left(\frac{x}{y}\right),\,
\ee
which appears ubiquitously in QCD calculations.
%
The splitting functions Eq.~(\ref{eq:splittingfunctions}) depend on the type
of initial and final state parton that is  involved in the splitting.
%
At leading order, the DGLAP splitting functions are given by
\be
P_{qq}=\frac{4}{3}\lc \frac{1+x^2}{(1-x)_+}\rc \, ,
\ee
\be
P_{qg}=\frac{1}{2}\lc x^2+(1-x^2)\rc \, ,
\ee
\be
P_{qg}=\frac{4}{3}\lc \frac{1+(1-x)^2}{x}\rc \, ,
\ee
\be
\label{splittingfunctionsLO}
P_{gg}=6\lc \frac{1-x}{x}+x(x-1)+\frac{x}{(1-x)_+}\rc \, .
\ee
Note that both $P_{gg}$ and $P_{qg}$ have a singularity at $x=0$: this fact is responsible
for the rapid growth at small $x$ of the gluons and consequently of the sea quarks in this region.
%
The overall coefficients of the splitting functions are related
to the QCD color factors.
%
Some splitting functions exhibit a singularity at $x=1$, which
is regularized by means of the plus prescriptions, defined as
\be
\int_{0}^1 f(x)\lc \frac{1}{1-x}\rc_+\equiv \int_{0}^1 (f(x)-f(1))\lc \frac{1}{1-x}\rc_+ \, .
\ee

The structure of the DGLAP evolution equations is significantly simplified if we use specific
linear combinations of PDFs.
%
For instance, below the charm threshold, where there are only $n_f=3$ active quarks,
the following combination
\bea
\Sigma(x,Q^2) &\equiv& \sum_{i=1}^{n_f}\lp q_i+\bar{q}_i\rp (x,Q^2) \, , \nonumber \\
T_3(x,Q^2) &\equiv& \lp u+\bar{u} - d-\bar{d}\rp (x,Q^2) \, , \nonumber \\
T_8(x,Q^2) &\equiv& \lp u+\bar{u} + d+\bar{d}- 2(s+\bar{s})\rp (x,Q^2) \, ,  \\
V(x,Q^2) &\equiv& \sum_{i=1}^{n_f}\lp q_i-\bar{q}_i\rp (x,Q^2) \, , \nonumber \\
V_3(x,Q^2) &\equiv& \lp u-\bar{u} - d+\bar{d}\rp (x,Q^2) \, , \nonumber \\
V_8(x,Q^2) &\equiv& \lp u-\bar{u} + d-\bar{d}- 2(s-\bar{s})\rp (x,Q^2) \nonumber \, , 
\eea
has the important property that all the quark PDF combinations except for $\Sigma$, known as the total
quark singlet, evolve independently using their own specific splitting functions.
%
These combinations, known as non-singlet flavour combinations, therefore obey  a particularly simple evolution equation.
%
As the $g\to q\overline{q}$ splitting can only generate an overall $q+\overline{q}$ combination, only the singlet PDF evolution is explicitly coupled to the gluon.

The splitting functions Eq.~(\ref{eq:splittingfunctions}) are known up to $\mathcal{O}\lp \alpha_s^3\rp$
(NNLO)~\cite{mvvns,Vogt:2004mw},\footnote{Recently, the first
results towards N3LO splitting functions have been presented~\cite{Ablinger:2017tan,Moch:2017uml}.} and thus PDF evolution can be performed up to this order.
%
Several public codes implement the numerical solution of the DGLAP equations, with the {\tt HOPPET}~\cite{Salam:2008qg}, {\tt APFEL}~\cite{Bertone:2013vaa} and {\tt QCDNUM}~\cite{Botje:2010ay} codes using $x$-space methods, while the  {\tt PEGASUS}~\cite{pegasus} code performs the evolution in Mellin (moment) space.
%
These codes have undergone detailed benchmarking studies, with agreement at the
level of $\mathcal{O}\lp 10^{-5}\rp$ or better being found~\cite{Whalley:2005nh,Dittmar:2009ii}.

In order to illustrate the impact of the DGLAP evolution on the PDFs,
in Fig.~\ref{fig:pdf4lhcevolution} we show the PDF4LHC15 NNLO Hessian set with 100 eigenvectors,
comparing
   the PDFs at a low scale of $Q^2=10$ GeV$^2$ (left) with the same PDFs evolved
   up to a typical LHC scale of $Q^2=10^4$ GeV$^2$ (right plot).
   %
   In this plot, the PDFs are shown together with the corresponding
   one-sigma PDF uncertainty band.
   %
   From this comparison we see that the effects of the evolution are relatively mild in the
   non-singlet combinations $u_V=u-\bar{u}$ and $d_V=d-\bar{d}$, but they are dramatic on the gluon
   and the sea quarks, where they induce a very steep growth at small-$x$.
   %
   This steep growth is driven by the small-$x$ structure of the splitting functions, see
   Eq.~(\ref{splittingfunctionsLO}).
   %
   Another interesting aspect that can be observed from Fig.~\ref{fig:pdf4lhcevolution} is that
   the valence PDFs $xu_V$ and $xd_V$ are integrable, and the fact that they have a similar
   shape but with $u_V \simeq 2 d_V$.
   %
   This behaviour is
a consequence of the valence sum rules
   which fix the value of their integrals (see Sect.~\ref{sec:fitmeth}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
\includegraphics[scale=1.1]{2-pdffitting/plots/pdf4lhcq10.pdf}\qquad\quad
\includegraphics[scale=1.1]{2-pdffitting/plots/pdf4lhcq1000.pdf}
   \caption{\small The effect of the DGLAP evolution in the PDF4LHC15
   NNLO Hessian set (with 100 eigenvectors).
   %
   We compare
   the PDFs at a low scale of $Q^2=10$ GeV$^2$ (left) with the same PDFs evolved
   up to a typical LHC scale of $Q^2=10^4$ GeV$^2$ (right plot).
   %
   In this plot, the PDFs include the corresponding
   one-sigma uncertainty band.
    \label{fig:pdf4lhcevolution}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

