\section{Introduction}

In Ref.~\cite{Carrazza:2013wua} we have shown, using the latest NNPDF
technology~\cite{Ball:2012cx}, that the photon parton distribution
function (PDF) obtained from deep inelastic scattering (DIS) data has
large uncertainties. Furthermore, by computing with {\tt HORACE}
~\cite{CarloniCalame:2007cd} the $Z\rightarrow\mu^{+}\mu^{-}$
production in proton-proton collision, we have confirmed that the
photon PDF from DIS is not sufficiently precise for the level of
accuracy required by the LHC physics.

The large uncertainties of the photon PDF from DIS are due to the lack
of photon-induced contributions to DIS processes, when including QED
corrections up to leading order (LO) in $\alpha$ (i.e. $O(\alpha)$) to
PDFs. A possible cure for this issue is the inclusion of Drell-Yan
data where there are photon-induced contributions at $O(\alpha)$: for
example in $Z/\gamma^{*}$ production the $\gamma\gamma \rightarrow
l^{+} l^{-}$ process.

Here we present the determination of the photon PDF, for a PDF set
which is determined at NLO in QCD and LO in QED, by including a subset
of Drell-Yan data from LHC. Currently, fast interface codes for QCD
computations do not include contributions from photon PDFs, so we
decided to include these data through a reweighting procedure. In
short, the reweighting assigns weights to PDF replicas which satisfies
the new data by computing the $\chi^{2}$ between the new experimental
data and theoretical predictions obtained from each replica. Technical
details about the reweighting and unweighting procedure are presented
in Ref.~\cite{Ball:2010gb,Ball:2011gg} and will not be discussed here.

\section{Reweighting with ATLAS data}

We start the reweighting procedure by creating a prior PDF set with
500 replicas, combining the photon PDF from DIS with five copies of
the NNPDF2.3 NLO set~\cite{Ball:2012cx} at the initial scale
$Q^{2}_{0}=2$ GeV$^{2}$ and then evolving with the combined QCD+QED
DGLAP evolution equations. The prior PDF set thus defined preserves
the prediction accuracy of a global set for pure QCD computations:
indeed we have explicitly verified that PDF sets with and without QED
corrections are statistically equivalent which means that the
corrections to the quark and gluon due to their mixing with the photon
are indistinguishable from statistical fluctuations. Also, the
violation of the momentum sum rule due to the inclusion of
the photon PDF is less than 1\%.

We perform reweighting with two LHC data sets: ATLAS $W$,
$Z/\gamma^{*}$ rapidity distributions~\cite{Aad:2011dm} and ATLAS
Drell-Yan high mass differential
cross-section~\cite{Aad:2013iua}. These data sets provide constraints
to the photon PDF at medium and large-$x$ thanks to the presence of
photon-induced contributions in the $Z/\gamma^{*}$ and $W$
production. The QCD computation is performed with {\tt
  APPLGRID}~\cite{Carli:2010rw} for ATLAS $W$, $Z/\gamma^{*}$ rapidity
distributions data and {\tt DYNNLO}~\cite{Catani:2009sm} for ATLAS
Drell-Yan high mass data, while, in both cases we have computed the
Born and NLO photon-induced contributions with {\tt
  HORACE}~\cite{CarloniCalame:2007cd}, knowing that NLO contributions
have a very small effect.

Figure~\ref{fig:weights} shows the weights and $P(\alpha)$
distributions, defined in~\cite{Ball:2010gb}, for the present
reweighting. Recall that $\alpha$ is the rescaling of the covariance
matrix: $\alpha=1$ means that uncertainties are correctly
estimated. From these distributions we conclude that the procedure is
consistent and that the data is compatible with the reweighted PDF
set. After performing the reweighting we obtain a total number of
effective replicas of $N_{\rm eff}=266$ and the $\chi^{2}$ to the
ATLAS data is reduced from 1.98 to 1.14. In Table~\ref{table:chi2} we
collect the $\chi^{2}$ and $P(\alpha)$ values for the individual
reweighting procedure, looking at the $\chi^{2}$ values before and
after reweighting we confirm that data far from the $W$,
$Z/\gamma^{*}$ peaks, where the photon contribution is substantial,
contributes strongly to constrain the photon PDF.

\begin{table}
  \centering
  \begin{tabular}{c||c||c|c}
    \hline
    NLO &  ATLAS all  & ATLAS $W$, $Z/\gamma^{*}$  & ATLAS DY high mass  \\ 
    \hline
    \hline
    $\chi^2_{\rm in}$  & 1.98 & 1.20 & 3.78 \\
    \hline
    $\chi^2_{\rm rw}$  &  1.14 &  1.15 & 1.01 \\
    \hline
    $N_{\rm eff}$  & 266 & 364 & 326 \\
    \hline
    $\langle \alpha \rangle$  & 1.48 & 1.24 & 1.53 \\
    \hline
  \end{tabular}
  \caption{\label{table:chi2} Reweighting properties: $\chi^{2}_{\rm in}$ and $\chi^{2}_{\rm rw}$ are before and after reweighting, respectively; $N_{\rm eff}$ is the number of effective replicas; $\langle \alpha \rangle$ is the average rescaling of the covariance matrix.}
\end{table}

\begin{figure}
  \begin{centering}
    \includegraphics[scale=0.4]{whist-rw}\includegraphics[scale=0.4]{palpha-rw}
    \par\end{centering}
  \caption{\label{fig:weights} On the left, the distribution of
    weights after performing the reweighting with ATLAS $W$,
    $Z/\gamma^{*}$ rapidity data and ATLAS Drell-Yan high mass
    data. The prior set of $N_{\rm rep}=500$ replicas after the
    reweighting procedure is reduced to $N_{\rm eff}=266$ effective
    replicas. On the right, the associated $P(\alpha)$ distribution.}
\end{figure}

Figure~\ref{fig:pheno} shows a comparison between ATLAS data and
theoretical predictions computed with NNPDF2.3 NLO, with and without
photon-induced contributions using the photon PDF obtained after the
reweighting procedure. Again, the comparison shows that the
$Z/\gamma^{*}$ high mass production provides the most relevant
information to constrain the photon PDF.

\begin{figure}
  \begin{centering}
    \includegraphics[scale=0.39]{atlaswz-wpcomparison}\includegraphics[scale=0.39]{atlaswz-wmcomparison}
    \includegraphics[scale=0.39]{atlaswz-zcomparison}\includegraphics[scale=0.39]{atlashm-zcomparison}
    \par\end{centering}
  \caption{\label{fig:pheno} Comparison between experimental data and
    theoretical predictions computed with NNPDF2.3 NLO for each data
    set, with (blue lines) and without (green lines) photon-induced
    contributions.}
\end{figure}

\section{Unweighting the photon PDF}

Following the procedure presented in Ref.~\cite{Ball:2011gg} we
construct an unweighted photon PDF set. Figure~\ref{fig:rw} shows the
comparison between the photon PDF before and after performing the
reweighting with ATLAS data, in logarithmic scale (left) and linear
scale (right). We observe a significant reduction of the uncertainties
which essentially overlaps with the coverage in $x$ of the ATLAS data
used in the reweighting. It is interesting to observe that the central
value is also substantially reduced. This means that the central value
from the DIS fit is mostly determined by statistical fluctuations.

Figure~\ref{fig:photon} shows the details of the unweighted photon PDF
set with $N_{\rm rep}=100$ replicas. Also in this example, the current
reweighted photon PDF in comparison to the photon PDF from DIS has
small central values and smaller uncertainties at medium/small-$x$,
improving the quality of theoretical predictions. On the other hand,
we observe that at very small-$x$ the photon PDF has large
uncertainties, because the ATLAS data does not cover this region.

\begin{figure}
  \begin{centering}
    \includegraphics[scale=0.4]{photon_rw_log}\includegraphics[scale=0.4]{photon_rw}
   \par\end{centering}
 \caption{\label{fig:rw} Direct comparison between the photon PDF
   before (black lines) and after (green lines) the reweighting
   procedure. The reweighted photon PDF uncertainties are reduced at
   medium/small-$x$. The 68\% c.l. band is defined from the PDF mean value.}
\end{figure}

\begin{figure}
  \begin{centering}
    \includegraphics[scale=0.4]{qedcomparisonlog}\includegraphics[scale=0.4]{qedcomparison}
   \par\end{centering}
 \caption{\label{fig:photon} The unweighted photon PDF at
   $Q^{2}_{0}=2$ GeV$^{2}$ obtained after the reweighting procedure
   with ATLAS $W$, $Z/\gamma^{*}$ rapidity data and
   ATLAS Drell-Yan high mass data. The PDF set includes 100 replicas.}
\end{figure}

\section{Outlook}

In conclusion, the Drell-Yan LHC data from ATLAS reduces the photon
PDF uncertainties at medium/small-$x$, providing a much more
constrained and reliable photon PDF in comparison to the results
obtained by fitting DIS-only data.

Here we have shown only the impact of ATLAS data to the photon PDF. In
a the final photon PDF set~\cite{Carrazza:2013} we will also include
the LHCb low mass data, which reduces uncertainties at very
small-$x$. This will provide the first extraction of a modern photon
PDF with uncertainties and constrained by LHC data. Further
improvements of the photon PDF uncertainties will be possible when
more LHC data will be available, specifically, measurements of $WW$
production which at high energy scales constrains the photon PDF at
large-$x$.

