\label{subsec:photons}

In this section we discuss the PDF constraints that can be derived
from  isolated prompt photon production at hadron colliders.

\subsubsection*{PDF sensitivity}\label{sec:datatheory.photon.sensitivity}

The leading
parton--level processes for direct (also known as `prompt') photon production,
that is,  where the photon is produced by point--like emission from a quark,
are given by
\begin{align}
{\rm QCD~Compton}&:\qquad qg \to q \gamma \;,\\
{\rm Annihilation}&:\qquad q\overline{q}\to g\gamma\;.
\end{align}
The QCD Compton process gives the dominant contribution, in particular at the LHC,
and is directly sensitive to the gluon PDF.
%
For LO kinematics, the momentum fraction carried by the incoming gluon is directly proportional to the transverse energy $E_\perp^\gamma$ of the produced photon, and thus for higher $E_\perp^\gamma$ this process provides a direct probe of the gluon PDF at high $x$.
%
Moreover, this represents the highest rate electroweak process at the LHC, while the produced photon directly reflects the production kinematics, without for example requiring any additional hadronization corrections, as in the case of jet production. This can therefore provide a valuable tool with which to constrain the gluon.

However, the PDF interpretation of prompt photon
production is not without its complications.
%
In particular, the `direct' process described above is not the only way in which high $E_\perp$ photons can be produced in hadronic collisions.
%
We must also include the `fragmentation' contribution, whereby a standard $2\to 2$ QCD scatter involving a final--state quark (or anti--quark) produces a photon through a collinear $q\to q\gamma$ emission.
%
While the parton--level process for this fragmentation component carries an extra power of $\alpha_s$ compared to the direct production mechanism, the collinearly enhanced photon emission is effectively of order $\alpha/\alpha_s$, and thus fragmentation
enters at the same order.

Technically speaking, this fragmentation emerges from the higher order corrections to the direct process. These correspond to multiple collinear splittings of a high $p_\perp$ parton which end up with a photon, and that can be absorbed into universal `fragmentation functions'.
%
These cannot be calculated perturbatively, but rather must be fit to data, for example in $e^+e^-$ annihilation to hadrons. This introduces a potentially significant additional source of uncertainty, since our understanding of the non-perturbative quark-to-photon fragmentation
mechanism is very poor.

In fact, the situation is greatly improved by noting that physically this fragmentation process corresponds to same multiple emission process that generates final--state jets, and indeed such fragmentation photons are typically accompanied by significant additional hadronic activity in the vicinity of the detected photon.
%
This is to be contrasted with direct emission, where at LO the produced photon and outgoing quark are produced completely back--to--back.
%
The direct mechanism may therefore be greatly enhanced by introducing 'isolation' criteria whereby the total sum of the transverse energy of the hadrons present in some cone $R$ centred on the photon is less than a given value.
%
These also reduce the additional `non--prompt' background due to the electromagnetic decay of hadrons. The impact of such a cut is shown in Fig.~\ref{fig:isol}, where it seen that the contribution from the less well known fragmentation contribution is small.
%
This also demonstrates the dominance of the direct Compton production process for prompt photon production at the LHC.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
  \begin{center}
 \includegraphics[width=0.45\textwidth]{3-datatheory/plots/subproc_ppgamma_jetphox_lhc_inc_stack}
\includegraphics[width=0.45\textwidth]{3-datatheory/plots/subproc_ppgamma_jetphox_lhc_is_stack.pdf}
  \end{center}
\caption{\small
Relative contributions from Compton ($qg$), annihilation ($q\overline{q}$) and fragmentation to prompt photon production at central rapidities at the 14 TeV LHC, before (left) and after (right) the application of isolation cuts. Figures taken from~\cite{d'Enterria:2012yj}.}\label{fig:isol}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Historically, isolated photon production represented one of the first
non-DIS processes used to constrain PDFs in the global analysis,
and was used in such early fits as those of
Refs.~\cite{Aurenche:1988vi,Harriman:1990hi,Morfin:1990ck}.
%
However, the  difficulties in describing the fixed target photon
production measurements from
the E706 experiment~\cite{Apanasevich:1997hm,Apanasevich:2004dr}
raised questions about the reliability of this process for PDF fits within
the perturbative QCD framework, as well as about
potential sensitivities to non--perturbative effects.
%
Combined with the increasing availability of high precision jet data from the Tevatron, which also constrain the high $x$ gluon, this lead to the process falling out of favour in the PDF fitting community. 
%
The last PDF set to include any such data is the MRST99~\cite{Martin:1999ww} fit. 

However, the subsequent studies of~\cite{Ichou:2010wc,d'Enterria:2012yj} (see also~\cite{Carminati:2012mm} for a study of the related $\gamma$ + jet process) have shown that by increasing the
center-of-mass energy $\sqrt{s}$ from fixed target to collider energies and, as discussed above, imposing a suitable isolation condition on the produced photon, the process may be brought under reasonable theoretical control.
%
Moreover, a comparison of the NLO perturbative QCD predictions to the ATLAS
Run I measurements~\cite{Aad:2016xcr,Aaboud:2017cbm}
exhibits
an adequate description of the data, albeit with fairly large $\sim 10-15\%$ scale variation uncertainties.
%
These largish theoretical uncertainties have been recently reduced down
to a few percent thanks to  the availability of the NNLO calculation, discussed below.
%
Therefore, isolated photon production may well provide a useful tool for PDF constraints
for the next generation of global fits.

\subsubsection*{Experimental data}\label{sec:datatheory.photon.data}

The hadron available collider data on isolated photon production is summarised below:

\begin{itemize}

\item The most recent data at 1.96 TeV from the CDF~\cite{Aaltonen:2017swx} and D0~\cite{Abazov:2005wc}
collaborations extends up to $E_\perp^\gamma<0.5$ TeV and 0.3 TeV, respectively, while the photon pseudorapidity is restricted to have $|\eta^\gamma|\lesssim 1$.
%
In the CDF case this corresponds to the full Run II $9.5\,{\rm fb}^{-1}$ data set, and so represents the final legacy measurement.

\item The ATLAS 7 TeV measurement~\cite{Aad:2013zba} covers up to $E_\perp^\gamma<1$ TeV,
while those at 8~\cite{Aad:2016xcr} and 13~\cite{Aaboud:2017cbm} TeV extend up to $E_\perp^\gamma<1.5$ TeV.
%
These datasets correspond to the full available integrated luminosities of $4.6\,{\rm fb}^{-1}$ and  $20.2\,{\rm fb}^{-1}$ at 7 and 8 TeV, respectively, while the 13 TeV measurement uses a $3.2\,{\rm fb}^{-1}$ data set.

\item The most precise CMS data at 7 TeV~\cite{Chatrchyan:2011ue}, corresponding to $36\,{\rm pb}^{-1}$ of integrated luminosity and extending to $E_\perp^\gamma<0.4$ TeV.

\item Data from a smaller sample at $2.76$ TeV have also been taken by ATLAS~\cite{Aad:2015lcb} and CMS~\cite{Chatrchyan:2012vq}, to be used as baseline for heavy ion collision studies..

\item For the ATLAS and CMS measurements,
the photon pseudorapidity is restricted to satisfy $|\eta^\gamma|\lesssim 2.4$.

\end{itemize}

In addition to these collider datasets, there exist a number of older measurements taken from
fixed-target experiments (see~\cite{Ichou:2010wc,d'Enterria:2012yj} for a summary) but
these are characterized by large experimental uncertainties, as well as by an inadequate
treatment of photon isolation in some cases, and thus they should not be considered
for PDF studies.

\subsubsection*{Theoretical calculations and tools}\label{sec:datatheory.photon.theory}
\label{isolgam:theory}

For the past 15 years, the theoretical state--of--the--art was provided by the~\texttt{JETPHOX}~\cite{Catani:2002ny} MC generator, which implements both the direct and the fragmentation contributions consistently at NLO. The NLO EW corrections have also been calculated in~\cite{Kuhn:2005gv}.
%
Recently, the first NNLO calculation of direct photon production has been reported~\cite{Campbell:2016lzl}, although the (numerically small) fragmentation component still remains at NLO.
%
The NNLO prediction for the ATLAS 8 TeV data~\cite{Aad:2016xcr} is compared to the
corresponding NLO calculation in~\cite{Campbell:2016lzl} and found to lie consistently within the NLO scale uncertainty band, with the central value being $\sim 5\%$ higher.
%
Moreover, the NNLO scale uncertainty is found to be greatly reduced, leading now
to scale variations at the 3\% level at most.

In order to achieve the best possible description of the LHC measurements, it is crucial
to include the leading-log EW Sudakov corrections of~\cite{Becher:2013zua}, as well as
to evaluate the coupling $\alpha$ at the scale $m_Z$, as recommended in~\cite{Becher:2013zua}.
%
In particular, the EW corrections are found to reduce the cross section by as much as $10\%$, that is, significantly outside the QCD scale variation band, at the highest $E_\perp$, improving the shape description.
%
The results of the \texttt{PeTeR} code~\cite{Schwartz:2016olw}, which combines the NLO calculation with ${\rm N}^3$LL threshold resummation in addition to these EW corrections, is found to lie close to the combined NNLO+EW prediction, but with a larger uncertainty band, indicating that the data may not be too sensitive to such additional resummation effects.
%
This is not completely unexpected, since direct photons are produced far from the
kinematic threshold, where the effects of resummation are expected to be more important.

Therefore, while the NNLO calculation is a very encouraging step towards including isolated photon data in high precision PDF fits, there remain still some further theoretical issues to be investigated, related to the impact of EW corrections and, as discussed in~\cite{Campbell:2016lzl} the choice of photon isolation, which can also affect the comparison with the experimental data.

\subsubsection*{Impact on PDFs}\label{sec:datatheory.photon.impact}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
  \begin{center}
 \includegraphics[width=0.4\textwidth]{3-datatheory/plots/g_PDFerr_red_Qs}
  \end{center}
\caption{\small Relative reduction in the NNPDF2.1 NLO gluon PDF uncertainty when including a 36 ${\rm pb}^{-1}$ LHC 7 TeV isolated photon data set via Bayesian reweighting.
%
Taken from~\cite{d'Enterria:2012yj}.}\label{fig:isolPDF}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Currently, no up to date studies of the impact of isolated photon data on the PDFs have been performed, in particular taking into account the new NNLO calculation and the high precision LHC data. However, in~\cite{d'Enterria:2012yj} (see also~\cite{Ichou:2010wc}) the impact of a range of data, including the earlier 36 ${\rm pb}^{-1}$ ATLAS and CMS measurements at 7 TeV, on the PDFs has been studied in detail through a reweighting of the NNPDF2.1 set.
%
In Fig.~\ref{fig:isolPDF} the impact of the this LHC data on the gluon PDF is shown.
%
A moderate reduction in the uncertainty, of up to 20\%, is found in the intermediate $x$ region. Interestingly, this overlaps with the kinematically relevant region for Higgs boson production via gluon fusion at the LHC, and indeed a $\sim 20$\% reduction in the Higgs production cross section is found.
%
Given these results correspond to a reasonably limited LHC data set, it would be interesting to see the impact of the latest LHC data, as well as that of the NNLO corrections.




