\subsection{Electroweak corrections}\label{sec:QED.photon.EW}

In addition to the QED photon--initiated corrections discussed above,
it can also be important to include other EW contributions, in particular those
arising from virtual EW bosons, in a PDF fit.
%
These corrections are most important at larger invariant masses
of the produced system, $Q\gg M_W$,
where virtual EW contributions receive logarithmic enhancements,
see Ref.~\cite{Mishra:2013una} for a review.
%
In particular, the virtual exchange of soft or collinear weak
bosons leads to Sudakov logarithms of the form $\alpha_W \ln^2 Q^2/M_W^2$,
where $\alpha_W=\alpha/\sin^2\theta_W$, which can lead to
large (negative) corrections for large values of $Q$.
%
Given that many of the LHC datasets that enter into the global
PDF are sensitive to the TeV region, from high-mass Drell-Yan
production
to the large $p_T$ tail of $Z$ production and inclusive jets
and dijets, the inclusion of such EW corrections is in general
required to achieve the best possible description of experimental
data in this region.

The state-of-the-art for EW corrections is NLO, that is,
$\mathcal{O}\lp \alpha_W^2\rp $, including in addition in some
cases mixed terms of the form $\mathcal{O}\lp \alpha_W\alpha_s\rp$
and related terms.
%
These corrections are available for most of the hadron
collider processes
that enter a typical global fit, including inclusive jet and dijet
production~\cite{Dittmaier:2012kx},
inclusive 
electroweak gauge boson production at high
$p_T$~\cite{Becher:2013zua,Campbell:2016lzl}
and high invariant mass $M_{ll(\nu)}$~\cite{Li:2012wna,Gavin:2012sy}
and differential top quark pair
production~\cite{Pagani:2016caq,Czakon:2017wor}.
%
Most of these calculations are implemented
in publicly available programs.
%
For instance, EW corrections to inclusive gauge boson
production  are available
in programs such as {\tt FEWZ}~\cite{Li:2012wna,Gavin:2012sy} and
{\tt HORACE}~\cite{CarloniCalame:2007cd}.
%
The latest version of {\tt MCFM}~\cite{Campbell:2016dks}
also includes the calculation of weak corrections to Drell-Yan, top-quark pair,
           and dijet production  at hadron colliders.
%
Recently, there has also been progress in the
automation of the calculation of these corrections,
both in the framework of {\tt MadGraph5\_aMC@NLO}~\cite{Frixione:2015zaa}
and of {\tt Sherpa}/{\tt OpenLoops}~\cite{Kallweit:2014xda}.

In Fig.~\ref{fig:Sect7-EWcorr} we show two
representative examples of NLO EW corrections
    for processes relevant for PDF determinations, computed with {\tt MCFM}
    at $\sqrt{s}=13$ TeV~\cite{Campbell:2016dks}.
    %
    In the left plot, we show the percentage NLO EW correction for
    high-mass dilepton
    production as a function of $M_{ll}$. The {\tt ZGRAD} calculation~\cite{Baur:2001ze} is also shown.
    %
    We see that these corrections are negligible for $M_{ll}\lsim 500$
    GeV, but that they can become significant as we increase $M_{ll}$,
    reaching $\delta_{\rm wk}\sim -20\%$ at 5 TeV.
    %
    In the right plot, we show the same quantity, now for dijet production
    as a function of the invariant mass of the dijet $M_{jj}$.
    %
    The two curves correspond to two possible ways to combine NLO
    QCD and EW corrections, known as additive ($\delta_{\rm add}$)
    and multiplicative ($\delta_{\rm prod}$).
    %
    Here the corrections are more moderate (since the Born
    is a pure QCD process) but they can still become up to  few
    percent in the region accessible at the LHC.
    %
    The results of Fig.~\ref{fig:Sect7-EWcorr} illustrate
    how a careful inclusion of NLO EW corrections is important
    for the description of the LHC data in the TeV region
    used for PDF determinations.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
  \begin{center}
 \includegraphics[width=0.99\textwidth]{7-QED/plots/EWcorr.pdf}
  \end{center}
  \vspace{-0.6cm}
  \caption{\small Two representative examples of NLO EW corrections
    for processes relevant for PDF determinations, computed with {\tt MCFM}
    at $\sqrt{s}=13$ TeV~\cite{Campbell:2016dks}.
    %
    In the left plot, we show the percentage NLO EW correction for high-mass dilepton
    production as a function of $M_{ll}$, comparing also with the
    corresponding {\tt ZGRAD} calculation.
    %
    In the right plot, we show the same quantity, now for dijet production
    as a function of the invariant mass of the dijet $M_{jj}$.
    %
    The two curves correspond to two possible ways to combine NLO
    QCD and EW corrections, known as additive ($\delta_{\rm add}$)
    and multiplicative ($\delta_{\rm prod}$).
    \label{fig:Sect7-EWcorr}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





