In this section we explore a topic that have received a lot
of attention in PDF fits in the recent years, namely
the role of QED and weak corrections, in particular concerning
photon-initiated processes.
%
Here first of all we discuss the role of QED corrections
and the photon PDFs, and then we review pure weak corrections
to hard scattering matrix elements
arising from virtual massive weak boson exchange.

\subsection{Photon-induced processes}\label{sec:QED.photon}

It has been over a decade since the calculation of the splitting functions at NNLO in $\alpha_s$~\cite{mvvns,Vogt:2004mw} provided the necessary tools to
be able to carry out NNLO PDF fits.
%
Moreover, we have seen in Sect.~\ref{sec:datatheory} that for the majority of PDF sensitive observables, the perturbative calculation is available at this NNLO order. Given that the data from the LHC are available at increasing precision, to below the percent level, NNLO PDF fits are essential to match
this unprecedented  precision and have naturally become the standard. However, a simple counting of powers of $\alpha_S$ indicates that
\be
\alpha^2_s(M_Z)\sim \frac{1}{70}\;,\qquad \alpha_{\rm EM}(M_Z^2)\sim \frac{1}{130}\;.
\ee
That is, we may roughly expect the NNLO QCD  and NLO EW corrections to be of the same order of magnitude. While such an argument clearly neglects the non--trivial differences in the structure of the QCD and EW corrections, this nonetheless serves as a warning that we must at least consider the impact of going to NLO EW if we are to claim percent-precision to LHC cross-sections.

A specific type of EW correction of particular relevance to PDF studies is the contribution from photon--initiated processes, such as those
shown schematically in Fig.~\ref{fig:photfeyn}.
%
As this involves a photon in the initial state, this requires the introduction of a PDF for the photon in the proton\footnote{For brevity, we will refer to this throughout as the photon PDF, but this should not be confused with the partonic content of the photon itself, which often receives a similar label, see e.g.~\cite{Gluck:1999ub}.}. This is included in complete analogy to the QCD partons, and moreover as it involves a massless boson in the initial--state, higher order QED $q\to q\gamma$ and $\gamma\to q\overline{q}$ splitting will generate collinear singularities that must be absorbed into the corresponding PDFs.
%
In other words, this will produce QED corrections to the DGLAP evolution of the PDFs.
%
Another important type of EW corrections relevant for PDF fits, namely
those associated to virtual massive weak boson exchange, are
discussed in Sect.~\ref{sec:QED.photon.EW}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
  \begin{center}
    \includegraphics[width=0.28\textwidth]{7-QED/plots/photon}\qquad
    \includegraphics[width=0.45\textwidth]{7-QED/plots/lumi13r.pdf}
  \end{center}
  \caption{\small Left: schematic diagram for the photon--initiated production of a system $X$, and the corresponding photon PDFs.
    %
    Right: the $\gamma\gamma$ luminosity as a function of the invariant mass, $M_X$, of the produced final-state.
The ratio to results with $O(\alpha\alpha_S)$ and $O(\alpha^2)$ to the leading $O(\alpha)$ DGLAP evolution shown. Calculated using the approach described in~\cite{Harland-Lang:2017dzr}.
  }\label{fig:photfeyn} \label{fig:qedevol}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsubsection*{QED corrections to DGLAP evolution}\label{sec:QED.photon.dglap}

The introduction of the photon PDF requires the following straightforward extension of the DGLAP evolution equations,
\begin{align}\nonumber
 Q^2\frac{\partial}{\partial Q^2}g(x,Q^2) &=\sum_{q,\overline{q},g}P_{g a}(x,\as(Q^2))\otimes f_a(x,Q^2)+P_{g\gamma}(x,\as(Q^2))\otimes \gamma(x,Q^2),\,\\ \nonumber
 Q^2\frac{\partial}{\partial Q^2}q(x,Q^2) &=\sum_{q,\overline{q},g}P_{q a}(x,\as(Q^2))\otimes f_a(x,Q^2)+P_{q\gamma}(x,\as(Q^2))\otimes \gamma(x,Q^2),\,\\ \label{eq:photdglap}
   Q^2\frac{\partial}{\partial Q^2}\gamma(x,Q^2) &=P_{\gamma\gamma}\otimes \gamma(x,Q^2)+\sum_{q,\overline{q},g}P_{\gamma a}(x,\as(Q^2))\otimes f_a(x,Q^2).\,
\end{align}
The splitting functions can then be expanded in powers of both the QCD and QED coupling
\be
P_{ij}=\sum_{m,n}\left(\frac{\alpha_S}{2\pi}\right)^m\left(\frac{\alpha}{2\pi}\right)^n P_{ij}^{(m,n)}\;.
\ee
The lowest order QED splitting function $P_{\gamma q}^{(0,1)}$ is due to the same type of Feynman diagram as in the LO QCD case for $P_{g q}^{(1,0)}$, with the gluon simply replaced by a photon, and similarly for $P_{qq}$ and $P_{q\gamma}$. Thus these are trivially related by suitable adjustments of the colour factors and inclusion of the electric charges $e_q$ of the quark, with
\be
P_{qq}^{(0,1)}=\frac{e_q^2}{C_F}P_{qq}^{(1,0)}\;,\qquad
P_{q\gamma}^{(0,1)}=\frac{e_q^2}{T_R}P_{q g}^{(1,0)}\;,\qquad
P_{\gamma q}^{(0,1)}=\frac{e_q^2}{C_F}P_{g q}^{(1,0)}\;,\qquad
P_{\gamma\gamma}^{(0,1)}=-\frac{2}{3}\sum_f e_f^2 \,\delta(1-x)\;,
\ee
where for the $P_{\gamma\gamma}$ case only the Abelian contribution is present and the sum is over all fermions in the loop, that is quarks and leptons\footnote{To be consistent, and in particular to preserve momentum fully, this requires the introduction of lepton PDFs in the proton. However as discussed in~\cite{Bertone:2015lqa} the contribution from these is generally of limited phenomenological relevance, and can be safely neglected. Note in any case that including leptons
in the running of the QED coupling $\alpha(Q)$ is still required.}. The calculation of the $O(\alpha_s \alpha)$ ($m,n=1$) terms, where the $P_{g\gamma}$ and $P_{\gamma g}$ splittings enter for the first time, is given in~\cite{deFlorian:2015ujt}, while the $O(\alpha^2)$ ($m=0$, $n=2$) terms are given in~\cite{deFlorian:2016gvk}. Publicly available implementations of the DGLAP evolution including these QED corrections are provided by the \texttt{APFEL}~\cite{Bertone:2013vaa} and \texttt{QEDEVOL}~\cite{Sadykov:2014aua} packages.

The impact of the $O(\alpha \alpha_S)$ and $O(\alpha^2)$ corrections on the $\gamma\gamma$ luminosity, defined as
\be
\mathcal{L}_{\gamma\gamma}=\frac{1}{s}\int_{M_X^2/s}^1 \frac{{\rm d}x}{x}\,\gamma(x,M_X^2)\,\gamma\left(\frac{M_X^2}{xs},M_X^2\right)\;,
\ee
is shown in Fig.~\ref{fig:qedevol}. The $O(\alpha\alpha_S)$ corrections have a fairly small but clearly non--negligible impact on the luminosity, giving up to a $\sim 5\%$ negative correction. As we would expect, the $O(\alpha^2)$ corrections are significantly smaller, but can reach the percent level. 
%In both cases, the impact of these corrections decreases with increasing $M_X$, due to the smaller relative contribution from the PDF evolution in comparison to the input photon, see (\ref{eq:photsol}) below.
%
%Note that in general these corrections  depend quite a bit on PDF set used,
%for instance the results of Fig.~\ref{fig:qedevol} have been computed
%using NNPDF3.0QED, while the ratio to the $\mathcal{O}(\alpha)$
%result for $M_X=200$ GeV is $R\simeq 0.96$
%is the HKR16 photon PDF is used instead.
%Note that in general these corrections will depend on the PDF set used. 
The results of Fig.~\ref{fig:qedevol} have been computed
using the MMHT framework~\cite{Harland-Lang:2017dzr}, which is closely based on the LUXqed formalism described below.  
%Using the NNPDF3.0QED instead, the $\mathcal{O}(\alpha)$
%result for $M_X=200$ GeV is about 10\% lower.

 As discussed in detail in~\cite{Harland-Lang:2016lhw}, the $P_{\gamma\gamma}$ self--energy contribution to the DGLAP evolution of the photon PDF is intimately connected to the choice of renormalization scale for the initial--state photon coupling to the hard process. 
It is well known in QED that
 for on--shell external photons the coupling receives no renormalization, and is completely determined to be $\alpha(0)$.
However, the $P_{\gamma\gamma}$ term breaks this simple picture, and we should instead use $\alpha(\mu_F)$ in the calculation.
%
 Physically, the photon substructure is being resolved by the introduction of a photon PDF and the contribution from $\gamma\to q\overline{q}$ splittings in the evolution, such that a purely on--shell renormalization scheme is no longer appropriate. This has been confirmed at NLO EW order in~\cite{Kallweit:2017khh}, where it is shown that using the on--shell scheme leads to uncanceled fermion--mass singularities in the hard cross section. 
 
\subsubsection*{The photon PDF}\label{sec:QED.photon.PDF}

The first attempts at describing the photon PDF can be divided into two distinct categories, either being phenomenological approaches that model the photon PDF, as in the MRST2004QED~\cite{Martin:2004dh} and more recent CT14QED~\cite{Schmidt:2015zda} sets, or the data-driven
approach of the NNPDF2.3/3.0QED~\cite{Ball:2013hta,Bertone:2016ume} sets.
%
The first attempt to include the photon in a PDF set was provided by MRST2004QED. Here, a simple model for the photon PDF at input scale $Q_0$ due to one--photon emission off the valence quarks was
assumed.
%
In other words, the quark valence distributions were frozen at $Q_0$ and the LO QED DGLAP evolution for the photon is integrated between the light quark mass $m_q$ and $Q_0$, so that
\be\label{eq:photinel}
\gamma(x,Q_0^2)=\frac{\alpha}{2\pi}\left[\frac{4}{9}\log\left(\frac{Q_0^2}{m_u^2}\right) u(x,Q_0)+\frac{1}{9}\log\left(\frac{Q_0^2}{m_d^2}\right)d(x,Q_0)\right]\otimes\frac{1+(1-x)^2}{x}\;.
\ee
The CT14QED set generalised this approach, allowing additional freedom in the normalization of the photon, which was fit to ZEUS data~\cite{Chekanov:2009dq} on isolated photon production\footnote{In fact, as we will discuss below, this has been supplemented with the elastic component to give the inclusive set CT14QEDinc.}. Thus, within such approaches the photon PDF is completely {\it predicted} within the specific model, up to any freedom in the model parameters, such as the choice of quark masses for MRST2004QED or the overall normalization for CT14QED. 

On the other hand, the NNPDF2.3QED~\cite{Ball:2013hta} set (subsequently updated to NNPDF3.0QED~\cite{Bertone:2016ume}), freely parameterises the photon PDF at input. In other words, the photon is treated on exactly the same footing as the QCD partons. This is then extracted from a fit (or more precisely, a Bayesian reweighting) to DIS and LHC $W,Z$ data; in the former case the constraint comes purely from the effect on the PDF evolution, with no explicit photon--initiated contribution included. 
%
However, in general the contribution of photon--initiated process are small, leading to significant
uncertainties on the extracted photon PDF. 

More recently, there has been a great deal of progress in our understanding of the photon PDF. One crucial point that was missed in the above approaches is the long range nature of QED.
%
That is, the proton is itself an electrically charged object which can coherently emit a photon, with the proton remaining intact afterwards. The possibility for such elastic photon emission is of course very well established. Elastic $ep$ scattering is an extremely well measured process, providing for example the first measurement of the proton charge radius~\cite{Hofstadter:1955ae,Mcallister:1956ng} in the 1950s, with further precise measurements of this process~\cite{Bernauer:2013tpr} continuing to this day.
%
Theoretically, the well known equivalent photon approximation (EPA)~\cite{Budnev:1974de} provides a precise foundation for describing the elastic scattering process in terms of a flux of coherently emitted photons from the proton.

The connection of this fact to the photon PDF was discussed in~\cite{Gluck:2002fi} and more recently in~\cite{Martin:2014nqa,Harland-Lang:2016qjy}. Following the equivalent photon approximation, it is straightforward to show that elastic photon emission leads to a contribution to the photon PDF at a scale $Q_0\sim $1 GeV given by
\be \label{eq:photcoh}
\gamma_{\rm el}(x,Q_0^2)=\frac{1}{x}\frac{\alpha}{\pi}\int_{\frac{x^2 m_p^2}{1-x}}^{Q_0^2}\!\!\frac{{\rm d}Q^2 }{Q^2}\left[\left(1-x-\frac{x^2 m_p^2}{Q^2}\right)F_E(Q^2)+\frac{x^2}{2}F_M(Q^2)\right]\;,
\ee
where $F_E$ and $F_M$ are the elastic and magnetic form factors of proton, which are related to the electric and magnetic charge distributions in the proton. These are steeply falling functions of $Q^2$ that are probed very precisely in a range of elastic $ep$ scattering experiments, see e.g.~\cite{Bernauer:2013tpr}.

To demonstrate the connection of this elastic
component to the inclusive photon PDF, 
if we for simplicity omit the small backreaction
that the photon has on the quark and gluon PDFs via the evolution
equations, then we can solve Eq.~(\ref{eq:photdglap}) to get~\cite{Harland-Lang:2016apc}
\begin{align}\label{eq:photsol}
\gamma(x,\mu^2)&=\frac{1}{\alpha(\mu^2)}\left(\alpha(Q_0^2)\,\gamma(x,Q_0^2)+\int_{Q_0^2}^{\mu^2}\frac{{\rm d}Q^2}{Q^2}\alpha(Q^2)\sum_{q,\overline{q},g}P_{\gamma a}(x,\as(Q^2))\otimes f_a(x,Q^2)\right)\;,\\ \label{eq:photsol1}
&\equiv\gamma_{\rm input}(x,\mu^2)+\gamma_{\rm evol}(x,\mu^2)\;.
\end{align}
Thus the photon is given separately in terms of an input at low scale $Q_0$ and an evolution component due to the usual DGLAP $q\to q\gamma$ emission for $Q^2>Q_0$.
%
The latter is completely determined in terms of the quark and gluon PDFs, leaving the input photon at $Q_0$, which is dominantly due to elastic emission.  Thus this already provides quite a strong constraint on the photon PDF; as we will see below, the impact in comparison to the model-independent
NNPDF approach can be dramatic.

However, even for relatively low photon virtualities, $Q^2<Q_0$, the emission may also be inelastic, such that the proton no longer remains intact afterwards. In other words we have
\be
\gamma(x,Q_0^2)=\gamma_{\rm el}(x,Q_0^2)+\gamma_{\rm inel}(x,Q_0^2)\;,
\ee
In~\cite{Martin:2014nqa,Harland-Lang:2016qjy} fairly simple phenomenological models for this inelastic component, given by suitable generalizations of (\ref{eq:photinel}), were taken, while the CT14QED set allowed an additionally free normalization to be fitted to ZEUS data on isolated photon production, as described above. 

Given that the elastic component is directly determined from the form factors $F_E$ and $F_M$, that are themselves measured from elastic $ep$ scattering, it is natural to ask whether the inelastic component can be similarly determined. In other words, rather than relying on a phenomenological model, can $\gamma_{\rm inel}$ instead be calculated directly from the form factors for inelastic $ep$ scattering, that is, from the proton structure functions?
%
In the analysis of~\cite{Manohar:2016nzj} it was shown that this is indeed the case, with the corresponding {\tt LUXqed} PDF set made publicly available. In particular, they find that the photon PDF can be expressed as\footnote{Following the publication of~\cite{Manohar:2016nzj} it was discovered that this expression had been derived in the earlier papers of~\cite{Anlauf:1991wr,Mukherjee:2003yh}, but without the correct limits on the $Q^2$ integral.}
%  \begin{align}
%\nonumber
%  x\gamma(x,\mu^2) &=  \frac{1}{2\pi \alpha(\mu^2)} \! \int_x^1 \frac{dz}{z} \Bigg\{ \int^{\mu^2}_{\frac{x^2 m_p^2}{1-z}}   \frac{dQ^2}{Q^2} \alpha^2(Q^2) \Bigg[\! \left(  zp_{\gamma q}(z)+ \frac{2 x ^2 m_p^2}{Q^2}\right)\! F_2\left(\frac{x}{z},Q^2\right)-z ^2 F_L\!\left(\frac{x}{z},Q^2\right)\Bigg] \\  \label{eq:luxqed}
% & - \alpha^2(\mu^2) \left(z^2+zp_{\gamma q}\ln(1-z)\right) F_2\left(\frac{x}{z},\mu^2\right)\Bigg\}\,.
%  \end{align}
   \begin{align}
\nonumber
  x\gamma(x,\mu^2) &=  \frac{1}{2\pi \alpha(\mu^2)} \! \int_x^1 \frac{dz}{z} \Bigg\{ \int^{\frac{\mu^2}{1-z}}_{\frac{x^2 m_p^2}{1-z}}   \frac{dQ^2}{Q^2} \alpha^2(Q^2) \Bigg[\! \left(  zp_{\gamma q}(z)+ \frac{2 x ^2 m_p^2}{Q^2}\right)\! F_2\left(\frac{x}{z},Q^2\right)-z ^2 F_L\!\left(\frac{x}{z},Q^2\right)\Bigg] \\  \label{eq:luxqed}
 & - \alpha^2(\mu^2) z^2F_2\left(\frac{x}{z},\mu^2\right)\Bigg\}\,.
  \end{align}
  Thus in this formalism the photon PDF is a derived quantity, which can be written purely in terms of the inclusive DIS structure functions, which are known quite precisely from
  the experimental point of view. More recently, the detailed study of~\cite{Manohar:2017eqh} has demonstrated how this expression may be derived in a process independent way by using the operator definition of the photon PDF, as well as generalising this expression to the case of the polarized and transverse momentum dependent PDFs. This approach is also shown to provide quite simple derivation of the known $O(\alpha\alpha_s)$ and $O(\alpha^2)$ splitting functions, $P_{\gamma i}$.

  While the connection of Eq.~(\ref{eq:luxqed}) to the considerations above is not immediately clear, some similarity in the overall form with Eq.~(\ref{eq:photsol}) is apparent.
%
  Indeed, by substituting for $F_{2,L}$ in terms of the quark and gluon PDFs, at high $Q^2$  this readily reduces to  $\gamma_{\rm evol}$ in (\ref{eq:photsol}); indeed this is how the \texttt{LUXqed}
  photon PDF is calculated in this region.
%
  In addition, using the known expression for the elastic contributions to $F_{2,L}$ reproduces Eq~(\ref{eq:photcoh}) when combined with Eq.~(\ref{eq:photsol}); this elastic contribution is also included. By using fits to the experimentally determined inelastic structure functions at low $Q^2$, including in the resonance region, it is shown in~\cite{Manohar:2016nzj} that the remaining inelastic contribution, and therefore the photon PDF in its entirely, is very precisely determined.
  %
  In Fig.~\ref{fig:LUXqed} we show an overview of the various contributions to the photon
    PDF $\gamma(x,Q^2)$ in the LUXqed approach as a function of $x$
    at $Q=100$ GeV.
    %
    We see that at small-$x$ it is dominated by the PDF contribution, while
    at large-$x$ the elastic contribution accounts for up to half
    of the size of $\gamma(x,Q)$.
  
    It is worth emphasizing that the expression Eq.~(\ref{eq:luxqed}) does not rely on any explicit separation between an input and evolution component to the photon as in (\ref{eq:photsol}), and corresponds to the exact result for the photon within the quoted accuracy of~\cite{Manohar:2016nzj,Manohar:2017eqh}, valid to all orders in QED and QCD, and including non--perturbative corrections.
   %
    Indeed, applying standard DGLAP above the starting scale $Q_0$ terms the power--like $\sim m_p^2/Q^2$ correction would be missed, while for $Q^2>Q_0^2$ the contribution from the elastic component would be omitted and the inelastic resonance component, which contributes at higher $x$ in this region, would not be correctly modelled.
  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
  \begin{center}
    \includegraphics[width=0.99\textwidth]{7-QED/plots/LUXqed.pdf}
  \end{center}
  \vspace{-0.6cm}
  \caption{\small Left: overview of the various contributions to the photon
    PDF $\gamma(x,Q^2)$ in the LUXqed approach as a function of $x$
    at $Q=100$ GeV.
    %
    Right: comparison of the photon PDFs from CT14qed\_inc, MRST2004, NNPDF2.3/3.0
    and LUXqed, normalized to the central value of the latter.
  }\label{fig:LUXqed}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

However, from the point of view of a global PDF set it may be preferable to use Eq.~(\ref{eq:luxqed}) in a form that can be more directly implemented within the standard fitting framework. That is, applying this approach after suitable modification to calculate the input photon, which can then be included as part of the default input parton set for any future fits and studies, see~\cite{Harland-Lang:2017dzr} for initial discussion. An alternative iterative approach is proposed in~\cite{Manohar:2017eqh}.

To illustrate the differences and similarities between these various determinations
of $\gamma(x,Q)$, in Fig.~\ref{fig:LUXqed} we show the comparison of the photon PDFs from CT14qed\_inc, MRST2004, NNPDF2.3/3.0
    and LUXqed, normalized to the central value of the latter.
    %
    It is clear from this comparison that the theoretical uncertainties associated
    with the LUXqed determination are much smaller than in any other of the
    previous approaches.
    %
    Interestingly, the LUXqed PDF is in good agreement within uncertainties
    with the model-independent NNPDF3.0QED fit in the entire range of $x$, though
    the PDF uncertainties are much larger in the latter case specially at large $x$.
   
\subsection*{Phenomenology}\label{sec:QED.photon.pheno}

In Fig.~\ref{fig:photlumi} (left) we show the NNPDF3.0QED $\gamma\gamma$ luminosity at $\sqrt{s}=13$ TeV, including the 68\% C.L. error bands. A large PDF uncertainty is evident, in particular at higher system mass $M_X$. As discussed above, the input component in Eq.~(\ref{eq:photsol1}) is poorly determined within this approach, due to limited constraints placed by the available experimental
data.
%
It is therefore unsurprising that the PDF errors should be most significant at higher mass,  as here the relative contribution from this input component is larger, due the reduced phase space for PDF evolution. In addition, the central value of the luminosity is seen to lie towards the upper end of the uncertainty band. As discussed in~\cite{Mangano:2016jyj,Harland-Lang:2016qjy}, this exhibits a much gentler decrease with $M_X$ in comparison to the QCD parton luminosities. However, also plotted is the LUXqed result, and the difference is dramatic. The central value lies towards the lower end of the NNPDF band at higher mass, with a PDF uncertainty that is smaller than the line width of the plot. We also show the result of~\cite{Harland-Lang:2016qjy}, labelled HKR16, which demonstrates a similar trend. Thus, simply applying basic physical conditions on the form of the photon PDF, and including the dominant coherent input
Eq.~(\ref{eq:photcoh}) gives a qualitatively similar result.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
  \begin{center}
 \includegraphics[width=0.45\textwidth]{7-QED/plots/lumi13m}
  \includegraphics[width=0.45\textwidth]{7-QED/plots/lumi13r1}
  \end{center}
\caption{$\gamma\gamma$ luminosities at $\sqrt{s}=13$ TeV. (Left) Absolute values for the HKR16,  NNPDF3.0QED and LUXqed sets. (Right) Ratios of the CT14QED, HKR16 and {\tt xFitter\_HMDYep} sets to the LUXqed prediction. 68\% uncertainty bands are shown, with the exception of the HKR16 set, where the error is due to model variation in the inelastic input (lower edge corresponds to elastic only).}\label{fig:photlumi}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5

Taking a closer look, in Fig.~\ref{fig:photlumi} (right) we show the ratio of the HKR16, CT14QED and xFitter\_HMDYep results to the LUXqed luminosity. The {\tt xFitter\_HMDYep} set is extracted
in~\cite{Giuli:2017oii} by applying a similar agnostic methodology to NNPDF, but including the more constraining ATLAS high mass Drell--Yan data~\cite{Aad:2016zzw} in the fit; this therefore represents the most up to date set within such an approach. Again, the LUXqed uncertainty band is barely visible on the curve, varying from $1-2$\% over the considered mass interval.  The CT14QEDinc prediction, which includes an elastic component, is consistent, but with larger $\sim 20-40\%$ uncertainties, due to the more limited constraints placed by the ZEUS isolated photon production data on the inelastic input\footnote{In addition, the ZEUS data are selected by requiring that at least one track associated with the proton side is reconstructed within the detector acceptance. While this will remove the elastic component, which will produce no extra tracks, entirely it is also possible for the proton dissociation products in the inelastic case to fall outside the acceptance and therefore not pass such a cut. Thus while CT14QED use these data to  extract the total inelastic component, at least part of this will also be removed by this cut.}. The HKR prediction lies somewhat below the LUXqed result, outside of the quoted model variation band, in particular at larger $M_X$. This is due in large part to the lack of any explicit resonant contribution in the inelastic input, which becomes more important at higher $x$ and hence $M_X$. Finally, the xFitter\_HMDYep set is seen to have a sizeable uncertainty band (albeit smaller than the NNPDF3.0 set~\cite{Giuli:2017oii}), and interestingly appears to lie somewhat above the LUXqed result. From this it is clear that any attempt to extract the photon PDF within such an approach will almost certainly not be competitive. More generally, we can see that the LUXqed set exhibits by far the smallest PDF uncertainties. 

Prior to these most recent developments, a range of phenomenological studies pointed out similar trends in the NNPDFQED predictions for the photon--initiated contributions to lepton and $W$ pair~\cite{Mangano:2016jyj,Bourilkov:2016qum,Accomando:2016tah} and $t\overline{t}$~\cite{Pagani:2016caq} production. At high system invariant mass these could be significant, and even dominant over the standard channels, with a large PDF uncertainty. From Fig.~\ref{fig:photlumi} the reason for this is clear, being driven by the large PDF uncertainty in the $\gamma\gamma$ luminosity at high mass, and the relatively gentle decrease with mass in the central value. However, from the discussion above we know that using the NNPDF set will dramatically overestimate the uncertainty on the photon--initiated contribution. In Fig.~\ref{fig:lepphot} we show the lepton pair production cross section at high mass, at the $\sqrt{s}=13$ TeV LHC and a $\sqrt{s}=100$ TeV FCC. We can see that indeed at the LHC, the NNPDF prediction for the photon--initiated contribution can even be larger than the standard Drell--Yan contribution. However, the up--to--date LUXqed prediction exhibits no such behaviour. The prediction is under good theoretical control, and gives a small, though not negligible, contribution.

\begin{figure}[t]
  \begin{center}
 \includegraphics[width=0.45\textwidth]{7-QED/plots/crosslux.pdf}
  \includegraphics[width=0.45\textwidth]{7-QED/plots/crossluxfcc.pdf}
  \end{center}
\caption{The differential lepton pair production cross sections at $\sqrt{s}=$ 13 TeV and 100 TeV with respect to the invariant mass of the pair $M_{ll}$, for lepton $|\eta| <2.5$ and $p_\perp  > 20$ GeV. The photon--initiated contributions predicted using the LUXqed and NNPDF3.0QED sets, including the 68\% C.L. uncertainty bands. The NLO Drell--Yan cross section, calculated with \texttt{MCFM}~\cite{Campbell:2011bn}, is also shown.}\label{fig:lepphot}
\end{figure}

Thus, by considering the physics that generates the photon PDF, and recognising the dominance of the elastic emission process, we already achieve a significant reduction in PDF uncertainty in comparison to the model-independent approach, even when accounting for the most sensitive
data available in the latter case.
%
Moreover, the additional information provided by Eq.~(\ref{eq:luxqed}) in combination with the high precision data on the inelastic (and elastic) proton structure functions provides extremely tight constraints on the photon PDF, resulting in a $\sim 1\%$ level PDF uncertainty. It is worth emphasising that while consistency tests are of course to be encouraged, this is not the result of a particular theoretical model, to be treated on the same footing as older PDF sets. The LUXqed set is a fundamentally experimental determination of photon PDF; it is simply that by doing this directly in terms of the measured structure functions the tightest constraints can be achieved. Such information must be included in any future photon PDF set, and we have therefore moved beyond the era of large photon PDF uncertainties.
%
Indeed, the photon PDF, which used to be the poorest known of all the proton PDFs, now has the smallest uncertainty.

% Subsection on electroweak corrections
\input{7-QED/subsec-EW.tex}
