text:
\section{Introduction}

Direct photons emerging from relativistic heavy ion collisions have long
been considered an important probe of the entire evolution of the
colliding system~\cite{Shuryak:1978ph}.  At almost all known or
conjectured stages of the collision there are processes producing photons.
Unlike hadronic observables that mostly encode the state of the medium at
 freeze-out, photons are emitted at all times throughout the rapid
evolution of the heavy ion collision and leave the interaction region
unmodified. Thus by measuring direct photons one has access to information
about the properties and dynamics of the medium integrated over space and
time. The measurement of direct photons is challenging due to a large
background of photons from the vacuum decay of final state hadrons (\mbox{$\pi^0$}\xspace,
$\eta$, $\omega$, etc.).

The PHENIX experiment at the Relativistic Heavy Ion Collider reported
large direct photon yields~\cite{ppg086} with strong
centrality dependence~\cite{ppg162} and significant azimuthal anisotropy
or ``elliptic flow''~\cite{ppg126}. Particularly surprising is the
discovery of large azimuthal anisotropy for direct
photons~\cite{ppg126}, which is comparable to that observed for
hadrons~\cite{ppg132}.  Preliminary results from the Large Hadron
Collider~\cite{wilde:2013phot,lohner:2013alicev2} indicate similar
direct photon yields and anisotropies.
The observation of large azimuthal anisotropy combined with observations
published earlier that the direct photon yields themselves are
large~\cite{ppg086,ppg162} contradicts several existing interpretations
where the large yields are provided at the very early production stage,
when the temperature of the system is highest but the collective flow
including azimuthal asymmetry is negligible. Conversely, the observed
large anisotropy suggests that photon production occurs at very late
stages of the collision when the collective flow of the system is fully
developed, while the temperature and the corresponding thermal photon
emission rates are already lower. Indeed, theoretical models have great
difficulty to simultaneously describe the observed yields and
anisotropy. This failure, colloquially called ``the direct photon
puzzle'', triggered a large amount of theoretical work, new models and
insights~\cite{turbide:2006flow,chatterjee:2006flow,chatterjee:2009flow,dusling:2010flow,hees:2011fb,linnyk:phsd2014,gale:2015semi,Muller:2013ila,shen:2013vja,shen:2015visc,hees:2015pce,monnai:2014scf,chatterjee:2013flow,dion:2011visc,fmliu:2014ft,vujanovic:2014early,mclerran:2014glas,tuchin:2013magn,mclerran:2015tail,basar:2012conf,gelis:2004noneq,biro:2014illu,pisarski:2015poly,campbell}.

In this paper we present new, more precise results on the azimuthal
anisotropy of direct photon emission from 200\,GeV \mbox{Au$+$Au}\xspace collisions
recorded in 2007 and 2010 by the PHENIX experiment. Results include second
and third order Fourier components of azimuthal distributions (\mbox{$v_2$}\xspace and
\mbox{$v_3$}\xspace, respectively) measured over a transverse momentum range extended
down to 0.4\,\mbox{GeV/$c$}\xspace. The new data, together with published results on
yields, are compared to some of the more recent model calculations.

The paper is organized as follows.  In Sec.~\ref{sec:expgeneral} we
describe the experiment, the data set, the way events are selected and
categorized, and the two methods by which photons are measured.  In
Sec.~\ref{sec:expv2} the steps needed to determine the direct photon
\mbox{$v_2$}\xspace, \mbox{$v_3$}\xspace, and their uncertainties are described, and the final results
are presented.  In Sec.~\ref{sec:modelcomp} the results are compared to a
few models treating yields and azimuthal asymmetries in a consistent
framework.  Sec.~\ref{sec:summary} summarizes our findings.

		\section{Experimental setup and photon measurements}
		\label{sec:expgeneral}

In PHENIX photons are detected by two substantially different techniques.
The first technique uses external conversion of photons as described in
detail in Ref.~\cite{ppg162}.  This method provides a high purity photon
sample with good momentum resolution, but requires large statistics due to
the few percent conversion probability and reduced acceptance. Therefore
the \mbox{$p_T$}\xspace range is limited. The second technique is a traditional
calorimetric measurement of photons similar to Ref.~\cite{ppg126}, but
with higher statistics. For photons identified by either technique, the
azimuthal anisotropy is extracted with the event plane (EP) method. Here
we give a brief summary of the PHENIX detector systems and a short
description of the two analyses.

	\subsection{Event selection and centrality determination}

Data from 200\,GeV \mbox{Au$+$Au}\xspace collisions were recorded with a minimum bias (MB)
trigger based on the signal in the beam-beam counters (BBC)~\cite{nimbbc},
which are located around the beampipe at $3.1<|\eta|<3.9$ and cover the
full azimuth. The minimum bias trigger requires at least two hits in each
of the two BBCs (north and south) as well as a reconstructed vertex from
the time-of-flight difference between the two sides.  The efficiency of
the MB trigger is $92.3\pm0.4(stat)\pm1.6(sys)$\%.

Collision centrality is calculated as percentiles of the total charge
distribution in the north and south BBC. The centrality determination is
based on percentiles of the total charge seen in the north and south BBC
and takes into account small shifts in $\eta$ coverage due to variations
of the collision $z$-vertex.

	\subsection{Inclusive photons via external conversion}
	\label{sec:Conv_photons}

External conversion photons are reconstructed from $2.6\times10^9$ MB \mbox{$\sqrt{s_{_{NN}}}$}\xspace = 200\,GeV \mbox{Au$+$Au}\xspace events recorded during the 2010 data taking
period. The event vertex in this dataset was $|z|<10$\,cm to ensure that
the magnetic field is sufficiently uniform. The same sample was previously
used in Ref.~\cite{ppg162} to determine direct photon yield and its
centrality dependence, where details of this analysis can be found. In the
rest of this paper this sample is referred to as ``conversion photons''.

Photons convert to $e^{+}e^{-}$ pairs in the readout plane of the Hadron
Blind Detector (HBD)~\cite{nimhbd}, which is located
at $\sim$60\,cm
radial distance from the collision vertex and corresponds to $\sim$3\%
$X_0$, where $X_{0}$ is the radiation length. The electron and positron
from the photon conversion are tracked through the PHENIX central tracking
detectors~\cite{nimtrk}. The azimuthal direction $\phi$ and the momentum
$p$ are reconstructed from the drift-chamber information, while the
polar angle of each track is determined by a point measurement in the
innermost pad-chamber (PC1) and the collision vertex. High efficiency
electron identification cuts are used to reduce the hadron contamination
in the sample. Light above a minimum threshold in the ring-imaging
\v{C}erenkov detector~\cite{nimrich} and a matching cluster of energy
$E$ in the electromagnetic calorimeter (EMCal)~\cite{nimemc} such that
$E>0.15$\,GeV and $E/p>0.5$, where $p$ is the momentum, are required. The
EMCal comprises two calorimeter types: 6 sectors of lead scintillator
sampling calorimeter (PbSc) and 2 sectors of lead glass \v{C}erenkov
calorimeter (PbGl). The typical energy resolution of the PbSc is $\delta
E/E=8.1\,\%/\sqrt{E(GeV)}\oplus2.1\,\%$, and that of the PbGl is $\delta
E/E=5.9\,\%/\sqrt{E(GeV)}\oplus0.8\,\%$.
The energy resolution, just as the photon identification efficiency,
depends on centrality and its (small) effect is corrected for using simulated
photon showers embedded into real events.

All remaining tracks with \mbox{$p_T$}\xspace$>0.2$\,\mbox{GeV/$c$}\xspace, are combined to pairs.
Conversion photons are identified by analyzing the invariant mass of the
pairs. The default tracking in PHENIX assumes that each track originates
at the collision vertex. Thus, if the $e^{+}e^{-}$ pair comes from a
conversion of a real photon in the HBD readout plane, the momenta will be
mis-measured and a finite mass, in this case about
$m_{ee}\sim$12\,MeV/$c^2$, is reconstructed. Conversely, if the momenta
are re-calculated assuming the HBD readout plane as origin, the invariant
mass is close to zero. Through a simultaneous cut on both mass
calculations a sample of photon conversions with a purity of 99\% is
obtained down to \mbox{$p_T$}\xspace = 0.4\,\mbox{GeV/$c$}\xspace~\cite{ppg162}. The remaining 1\% of
pairs are mostly from the \mbox{$\pi^0$}\xspace Dalitz decays. The effect on the inclusive photon $v_n$ is estimated to be smaller than 1\%.

	\subsection{Inclusive photons and \mbox{$\pi^0$}s\xspace via the calorimeter}
	\label{sec:EMCAL_photons}

The PHENIX EMCal is the principal detector in the calorimetric analysis,
which is performed in a similar way as in Ref.~\cite{ppg126}.  The \mbox{$v_2$}\xspace
and \mbox{$v_3$}\xspace are measured simultaneously for inclusive photons and \mbox{$\pi^0$}s\xspace.  A
total of $4.4\times10^{9}$ MB \mbox{Au$+$Au}\xspace events from the 2007 data taking
period are analyzed.  The event vertex in this sample was $|z|<30$\,cm.

Photon candidates in the EMCal are clusters above a threshold energy of
0.2\,GeV that pass a shower shape cut as well as a charged particle veto
cut by the pad chamber PC3 immediately in front of the EMCal.
However,
photon candidates with less than 1\, GeV energy are only used to reconstruct
\mbox{$\pi^0$}\xspace, but are not included in the inclusive photon sample of the calorimeter.
As described in Ref.~\cite{Afanasiev:2012dg}, the remaining hadron
contamination was estimated by comparing {\sc geant} simulations,
verified with actual data.
The \mbox{$\pi^0$}\xspace is
measured via the 2$\gamma$ decay channel, with a cut on the energy
asymmetry of the two photons $\alpha = \frac{|E_1-E_2|}{E_1+E_2} <$ 0.8.
For each \mbox{$p_T$}\xspace bin the number of reconstructed \mbox{$\pi^0$}s\xspace is taken as the
integral of the two-photon invariant mass distribution, with the
combinatorial background subtracted by the mixed event
method~\cite{ppg080}.  The signal to background ratio at
$1.0<p_{T}<1.5$\,\mbox{GeV/$c$}\xspace is 0.1, rapidly improving with increasing \mbox{$p_T$}\xspace.

For the inclusive photon measurement it is important to restrict the
measurement to a region where the residual contamination from
misidentified hadrons is small.  Therefore, in the inclusive photon
sample only clusters with $E>1$\,GeV are considered. On the other
hand the inclusive (and direct) photon results presented here have
an upper range of 4~GeV/$c$, which is far from the threshold where
two decay photons from a $\pi^0$ can merge in the calorimeter.
Within this \mbox{$p_T$}\xspace range a purity of larger than 95\% is achieved.
The largest contamination of the photon sample results from
antineutrons, which are not removed by the charge particle veto
but deposit significant energy through annihilation.
The systematic uncertainty from
particle identification (PID) of photons is estimated by varying
both the shower shape cut (five different settings) and,
independently, applying or omitting the charged particle veto cut.
Results from all cut variations are then fully corrected. The deviation
between results is 3-4\%, which is quoted as systematic uncertainty on the
inclusive photon yield.

	\subsection{Event plane determination}
	\label{sec:EP}

\begin{figure}[htbp]
\includegraphics[width=0.998\linewidth]{c_resolution_PPG176_for_final}
  \caption{Event plane resolution as a function of centrality for the RxN(I+O)
detector (a) used for the final results in this paper, and (b) for the MPC
detector used to cross-check the results.
  }
    \label{fig:eventplanereso}
\end{figure}

PHENIX has different detector systems to establish the EP, which cover
different pseudorapidity ($\eta$) ranges:  the outer and inner
reaction plane
detector (RxNO, 1$<|\eta|<$1.5, RxNI, 1.5$<|\eta|<$2.8), the muon piston
calorimeters (MPCS, -3.7$<\eta<$-3.1, MPCN, 3.1$<\eta<$3.9), and the BBC
(3.1$<|\eta|<$3.9).  All these detectors cover the full $2\pi$ azimuth
and are sufficiently separated in $\eta$ such that we do not
expect auto correlations between the event plane determination and the
photon production asymmetry measured.  The RxNI and RXNO are
scintillation counter systems with a 2\,cm Pb converter that makes them
sensitive to photons in addition to charged particles.
While these photons contribute to the determination of the event
plane, note that they are separated at least $\Delta\eta$=0.7
from the central region, which is where the
photon $v_2$ and $v_3$ are measured.

The results in this paper are obtained using the event planes measured by
the combination of the RxNI and RxNO~\cite{nimrxn}. Due to the large
rapidity coverage this combination has the best resolution. The resolution
$Res(\Psi_n)$ is measured with the 2-subevent
method~\cite{poskanzer:1998sube}.  The resolution for RxN and MPC is shown
in Figure~\ref{fig:eventplanereso}. The final results are cross-checked by
using the other detectors for the event plane determination. Despite the
significant difference in resolution the measured direct photon
anisotropies are consistent, within the systematic uncertainties.

\begin{figure*}[htbp]
  \includegraphics[width=1.0\linewidth]{c_Figure2_160418}
  \caption{Inclusive photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace at midrapidity ($|\eta| <$ 0.35) for
\mbox{Au$+$Au}\xspace collisions at \mbox{$\sqrt{s_{_{NN}}}$}\xspace = 200 GeV in different centrality bins 0\%--20\%
(a,d), 20\%--40\% (b,e), and 40\%--60\% (c,f) with the event plane
estimated with the reaction plane detector (1$<|\eta|<$2.8). The data from
the external conversion method are shown as solid circles and from the
calorimeter method as solid squares. The error bars (boxes) around the
data points are statistical (systematic) uncertainties. Also shown are the
calculated decay photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace (thick solid line) along with the
statistical (dotted line) and systematic (light solid line) uncertainties
resulting from uncertainties on the input data. An additional systematic
uncertainty due to the finite event plane resolution is not shown (see
Table~\ref{tab:syserr}), because it is common to all $v_n$ measurements.
  }
    \label{fig:vincl}
\end{figure*}

		\section{Direct photon {\mbox{$v_2$}\xspace} and {\mbox{$v_3$}\xspace}}
		\label{sec:expv2}

The photon anisotropy is measured via the coefficients of a Fourier
decomposition of the azimuthal distributions of photons with respect to
the event plane~\cite{poskanzer:1998sube}

\begin{equation}
\frac{dN}{d(\phi-\Psi_k)} \propto 1+
\sum\limits_{n} [v_{kn} \cos{ \{ n(\phi-\Psi_k)\} }],
\label{eq:flow}
\end{equation}

\noindent
where $\phi$ is the azimuthal angle of the photon, $\Psi_k$ is the
orientation of the $k^{th}$ event plane for a given event, and $v_{kn}$
are the $n^{th}$ coefficients with respect to the $k^{th}$ event plane. In
our analysis we made and explicitly tested the assumption that the 2nd
and 3rd order event planes are uncorrelated, which allows us to ignore the
$k\neq n$ terms and to introduce the notation \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace for the case
$k=n$, i.e. in the rest of the paper we use $v_2 \equiv v_{22}$ and $v_3
\equiv v_{33}$.

The determination of the direct photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace proceeds in three
steps: (i) \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace are determined for the conversion photon sample
(Section~\ref{sec:Conv_photons}) and for the calorimeter photon sample
(Section~\ref{sec:EMCAL_photons}) with respect to the event plane
(Section~\ref{sec:EP}). We refer to these coefficients as inclusive photon
$v^{\rm inc}_{2}$ and $v^{\rm inc}_3$. In the second step (ii), the decay photon
$v^{\rm dec}_2$ and $v^{\rm dec}_3$ are estimated, i.e. the anisotropy resulting
from the decays of hadrons to photons. It is calculated based on \mbox{$v_2$}\xspace,
\mbox{$v_3$}\xspace, and yields measured for charged and neutral pions; contributions
from heavier mesons are taken into account using proper scaling (see
Sec.~\ref{sec:decv2v3}). As a final step (iii) the direct photon \mbox{$v_2$}\xspace and
\mbox{$v_3$}\xspace are calculated statistically through a subtraction of the results
from step (i) and (ii) weighted by the ratio $R_\gamma$, the ratio of the
yields of direct photons to the yield of photons from hadron decays (see
Eq.~\ref{eq:vdirect}).

	\subsection{Inclusive photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace}
	\label{sec:incv2v3}

The inclusive photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace are measured with respect to the
event plane.  We employ two methods to determine these coefficients. For
each photon the azimuthal angular difference $(\phi-\Psi_k)$, with
$k=2,3$, is calculated. In the first method the coefficients are
determined as the event ensemble average for individual bins in photon \mbox{$p_T$}\xspace
and centrality:

\begin{equation}
v_{n} = \langle \cos{\{ n (\phi-\Psi_n)}\} \rangle/Res(\Psi_n).
\label{eq:avgcos}
\end{equation}

\noindent
Here $Res(\Psi_n)$ is the resolution function that accounts for the finite
event plane resolution (see Figure~\ref{fig:eventplanereso}).

In the second method the azimuthal distribution of photons in a given \mbox{$p_T$}\xspace
and centrality bin is fitted as:

\begin{equation}
\frac{dN}{d(\phi-\Psi_n)} = N_0 [1+ 2 v'_{n} \cos{\{ n (\phi-\Psi_n)\} }],
\label{eq:flowfit}
\end{equation}
\begin{equation}
v_n = v'_{n}/Res(\Psi_n).
\end{equation}

This is Eq.~\ref{eq:flow} for the case $k=n$ and neglecting all
$k\neq n$ terms. The measured values of \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace ($v'_2, v'_3$)
need to be corrected for the event plane resolution.

In the conversion photon method the quoted $v_n$ values come from the
average cosine method, while in the calorimeter analysis the quoted $v_n$
values are the average of the results obtained with the two methods. The
difference between the two methods is less than 1\%. The results
for the inclusive photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace are shown in
Figure~\ref{fig:vincl}. Both measurements agree in the region where they
overlap.

	\subsection{Decay photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace}
	\label{sec:decv2v3}

\begin{figure}[htbp]
\includegraphics[width=0.998\linewidth]{c_Figure3_PPG176_for_final1_160828}
\caption{Top panels: charged and neutral pion \mbox{$v_2$}\xspace (a) and \mbox{$v_3$}\xspace (c) for the
20\%--40\% centrality class, including previously published results. The
averaged values used in our analysis are shown as a thick solid line
together with the estimated statistical (dotted line) and systematic
(light solid line) uncertainties. Bottom panels: ratio of the measured
\mbox{$v_2$}\xspace (b) and \mbox{$v_3$}\xspace (d) values to the averaged values. }
\label{fig:vpion}
\end{figure}

About 80\%--90\% of the inclusive photons come from decays of neutral mesons
and exhibit an anisotropy with respect to the event plane that results
from the anisotropy of the parent mesons~\cite{ppg126}. To estimate this
contribution we use measured yields and anisotropy for charged and neutral
pions; $v_n$ for heavier mesons is obtained by $KE_T$ scaling as described
below. The yields of mesons used here are the same as are used for the
measurement of $R_{\gamma}$ in Ref.~\cite{ppg162}.

The \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace for pions are determined by combining data from
different measurements of charged and neutral pion \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace. The
\mbox{$\pi^0$}\xspace \mbox{$v_2$}\xspace has been published in Ref.~\cite{ppg129} but the measurement
has been repeated in this analysis to check the consistency of the
results.  The method to count the number of \mbox{$\pi^0$}s\xspace in any \mbox{$p_T$}\xspace bin is
briefly described in Sec.~\ref{sec:EMCAL_photons}. To obtain
\mbox{$v_2$}\xspace (\mbox{$v_3$}\xspace) for each \mbox{$p_T$}\xspace the number of reconstructed \mbox{$\pi^0$}s\xspace is extracted
in six 15 (10) degree wide bins of the azimuthal angle $\Delta\Phi = \Phi
- \Psi_n$ where $\Phi$ is the azimuth of the \mbox{$\pi^0$}\xspace and $\Psi_n$ is the
second (third) order event plane (see Sec.~\ref{sec:EP}).  These
distributions of the raw \mbox{$\pi^0$}\xspace counts vs.~$\Delta\Phi$ are then fitted as
described in Sec.~\ref{sec:incv2v3} to obtain \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace for \mbox{$\pi^0$}\xspace.
Note that because the individual \mbox{$\pi^0$}s\xspace are not identified, the average
cosine method~\cite{poskanzer:1998sube} is not applicable.

These data are combined with $\pi^{\pm}$ data in the \mbox{$p_T$}\xspace range 0.5 to 4
\mbox{GeV/$c$}\xspace~\cite{ppg147}. For \mbox{$v_2$}\xspace we also use $\pi^{\pm}$ data from
Ref.~\cite{ppg123}. For the centrality class 20\%--40\% these data are
compiled in Figure~\ref{fig:vpion}.  We interpolate the data, weighted by
their statistical and systematic uncertainties, to obtain an average value
$v_n$ for pions as a function of \mbox{$p_T$}\xspace. The result of this averaging
procedure, including our estimate of the systematic uncertainties, is also
shown in Figure~\ref{fig:vpion}.

For the heavier mesons, $\eta, \omega, \rho, \eta^{'}$, the $v_n$ is
derived from the $v_n$ of the pions by scaling with the kinetic energy
\cite{ppg062,ppg147}.

\begin{equation}
v^{meson}_{n} (KE_{T}) = v^{\pi}_n (KE_T),
\label{eq:ketscaling}
\end{equation}
\noindent where
\begin{equation}
KE_T = m_T - m = \sqrt{p_T^2+m^2} - m,
\end{equation}
where $m$ is the mass of the corresponding meson.

The yields of the heavier mesons are determined from the \mbox{$\pi^0$}\xspace yields at
\mbox{$p_T$}\xspace = 5\,\mbox{GeV/$c$}\xspace using the following ratios: $\eta/\pi^0 = 0.46\pm0.060$,
$\omega/\pi^0 = 0.83\pm0.12$, $\rho/\pi^0 = 1.00\pm0.300$ and
$\eta^{'}/\pi^0 = 0.25\pm0.075$.
Below \mbox{$p_T$}\xspace = 2\,\mbox{GeV/$c$}\xspace $KE_{T}$-scaling is only an extrapolation
for the $\eta$ yields.  Therefore, we also applied a
blast-wave fit, and the difference is included in the systematic
uncertainties. Note that the blast-wave fit results in lower $\eta$
yield at small \mbox{$p_T$}\xspace, increasing the direct photon yield and its \mbox{$v_2$}\xspace,\mbox{$v_3$}\xspace.
The meson yields, momentum spectra and
$v_n$ are used to simulate mesons that are then decayed to all decay
chains including photons. From the simulation we calculate the decay
photon $v_n^{\rm dec}$ using Eq.~\ref{eq:avgcos} with $Res(\Psi_n)=1$,
because the event plane is known in the simulation. The only source of
systematic uncertainty on $v_n^{\rm dec}$ is the uncertainty of the measured
\mbox{$\pi^0$}\xspace \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace, and the resulting decay photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace, derived from it.
The resulting $v_n^{\rm dec}$ is compared to the inclusive photon $v_n$ in
Figure~\ref{fig:vincl}. We find that the decay photon and inclusive photon
$v_n$ are similar. This was already observed for \mbox{$v_2$}\xspace in
Ref.~\cite{ppg126}, but is now also found for \mbox{$v_3$}\xspace. Given that a finite
direct photon yield has already been established~\cite{ppg086,ppg162}, the
similarity of $v_3^{\rm inc}$ and $v_3^{\rm dec}$ implies a large direct photon
\mbox{$v_3$}\xspace, as will be shown in the next section.

	\subsection{Direct photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace}
	\label{sec:dirv2v3}

The \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace for direct photons are extracted from the measured
inclusive photon $v^{\rm inc}_n$, the decay photon $v_n^{\rm dec}$, discussed in
the previous sections, and the ratio of the inclusive to decay photon
yield $R_\gamma$ measured in Ref.~\cite{ppg162}. The procedure was
introduced in Ref.~\cite{ppg126}:
\begin{equation}
v_n^{\rm dir} = \frac{R_{\gamma}v_n^{\rm inc} - v_n^{\rm dec}}{R_{\gamma}-1}.
\label{eq:vdirect}
\end{equation}
We reproduce $R_\gamma$ from Ref.~\cite{ppg162} with statistical and
systematic uncertainties in Figure~\ref{fig:rgamma}.

\begin{figure}[htbp]
\includegraphics[width=0.998\linewidth]{fig6}
\caption{The inclusive over decay photon ratio $R_{\gamma}$
used in the current analysis.  Present data means the results
published in Ref.~\cite{ppg162}.}
\label{fig:rgamma}
\end{figure}

All systematic uncertainties on the individual contributions on
$v_n^{\rm dir}$ are summarized in Table~\ref{tab:syserr}. Uncertainties that
are uncorrelated between data points are called Type A, those that are
correlated are Type B and uncertainties that change all points by a common
multiplicative factor are called Type C. Uncertainties on $R_\gamma$ are
common for \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace and for the conversion and calorimeter method.
For photon and pion $v_n$ measurements with PHENIX the orientation of the
event planes, i.e. $\Psi_n$, is determined with the same detectors using
the same algorithms. Thus the systematic uncertainty on the event plane
determination is common for all \mbox{$v_2$}\xspace (\mbox{$v_3$}\xspace) measurements. The
uncertainties on the decay photon $v_n$ are common to the conversion and
calorimeter method. The systematic uncertainty on $v_n^{\rm inc}$ is
independent for the two methods and mostly reflects the different purity
of $>$95\% compared to $>$99\% for the calorimeter and conversion method,
respectively.

\begin{table*}[htb]
\caption{
Summary of systematic uncertainties on the input to the measurement of
$v_n^{\rm dir}$, where the $R_\gamma$ is from Ref.~\cite{ppg162}, and the
$v_n^{\rm inc}$ and $v_n^{\rm dec}$ indicate ``inclusive"
and ``decay" photons, respectively.
The values are quoted for \mbox{$p_T$}\xspace $<$ 3 \mbox{GeV/$c$}\xspace, although most do not
vary with \mbox{$p_T$}\xspace, as can be seen from Figures~\ref{fig:vincl} and
\ref{fig:vpion}. The uncertainties on the $v_n^{\rm dec}$ due to the
statistical uncertainty of the input data are uncorrelated between data
points (type A); they are included in the statistical errors on the final
results. Type B uncertainties are correlated in \mbox{$p_T$}\xspace, i.e. they can vary
with \mbox{$p_T$}\xspace but only smoothly in the quoted range. Type C uncertainties
change $v_n^{\rm dir}$ for all \mbox{$p_T$}\xspace by a constant multiplicative factor.
The systematic uncertainties on \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace are typical values.
}
\label{tab:syserr}
\begin{ruledtabular} \begin{tabular}{clcccc}
 &        & \multicolumn{3}{c}{centralities} & \\
Input & Source & 0\%--20\% & 20\%--40\% & 40\%--60\% & Type \\
\hline
$R_\gamma$ && 5.5\% & 5.5\% & 5.5\% & B \\
\\
$v^{\rm inc}_2$
& conversion method   & $<$1\% & $<$1\% & $<$1\% & B \\
& calorimeter method  & 4\%    & 3\%    & 4\%    & B \\
\\
$v^{\rm dec}_2$
& meson $v_2$ (stat)  & $<$1\% & $<$1\% & $<$1\% & A \\
& $\pi^0$ $v_2$ (sys) & 5\%    & 3\%    & 2\%    & B \\
& $\eta,\omega$ $v_2$ (sys)  & $<$1\% & $<$1\% & $<$1\%  & B \\
& \bf{Event plane}         & 3\%    & 3\%    & 3\%    & C \\
\\
$v^{\rm inc}_3$
&conversion method   & $<$1\% & $<$1\% & $<$1\% & B \\
&calorimeter method  & 5\%    & 7\%    & 10\%   & B \\
\\
$v^{\rm dec}_3$
&meson $v_3$ (stat)  & 1\%    & 2\%    & 4\%    & A \\
&$\pi^0$ $v_3$ (sys)   & 11\%   & 11\%   & 11\%   & B \\
&$\eta,\omega$ $v_3$ (sys)  & $\sim$ 1\% & $\sim$1\% & $\sim$1\%  & B \\
& \bf{Event plane}         & 6\%    & 7\%   & 18\%   & C \\
\end{tabular} \end{ruledtabular}
\end{table*}

Using Gaussian error propagation, the statistical and systematic
uncertainties would be calculated as:
\begin{eqnarray}
\sigma_{v_n^{\rm dir}}^2
&=& \Big( \frac{R_\gamma}{R_\gamma-1} \Big)^2
\times \sigma_{v_n^{\rm inc}}^{2}+ \Big( \frac{1} {R_{\gamma}-1} \Big)^2 \nonumber \\
&\times& \sigma_{v_n^{\rm dec}}^{2}+ \Big( \frac{v_n^{\rm dec}-v_n^{\rm inc}}{R_\gamma-1} \Big)^2
\times \sigma_{R_\gamma}^{2}+ \sigma_{EP}^2.
\label{eq:syserr}
\end{eqnarray}
Except for the case $v_n^{\rm inc} = v_n^{\rm dec}$, there is a nonlinear
dependence on $R_{\gamma}$ that, combined with uncertainties of 20\%--30\% on
$(R_\gamma-1)$, results in asymmetric uncertainties, which are not
described by Eq.~\ref{eq:syserr}. In particular, for the case
$v_n^{\rm dec}>v_n^{\rm inc}$ the uncertainties on $v_n^{\rm dec}$ and
$v_n^{\rm inc}$ are amplified if $R_{\gamma}$ is small.

We estimate these asymmetric uncertainties by modeling a probability
distribution for possible values of $v_n^{\rm dir}$ using the statistical and
systematic uncertainties on $v_n^{\rm inc},v_n^{\rm dec},R_{\gamma}$, and the
event plane resolution. We assume that the individual statistical and
systematic uncertainties follow Gaussian probability distributions. The
probability distribution for $v_n^{\rm dir}$ is then determined by generating
many combinations of $v_n^{\rm inc}$, $v_n^{\rm dec}$, and $R_{\gamma}$.
Figure~\ref{fig:syserrmc} shows one example of a probability distribution
based on the systematic uncertainties on the calorimeter measurement for
0\%--20\% centrality and $1<\mbox{$p_T$}\xspace<1.5$\,\mbox{GeV/$c$}\xspace. In Figure~\ref{fig:syserrmc}
the effect of the
uncertainty of only $v_n^{\rm inc}, v_n^{\rm dec}$ or $R_{\gamma}$, are plotted
separately. The asymmetry due to the uncertainty of $R_\gamma$ is clearly
visible.

Probability distributions based on statistical (including type A
systematics) and systematic uncertainties are determined for each
$v_n^{\rm dir}$ data point in \mbox{$p_T$}\xspace and centrality and for both analyses. The
central value for each data point was calculated using
Eq.~\ref{eq:vdirect}.  We note that the peak or median of the
probability distributions used to determine the statistical and systematic
uncertainties agrees with the calculated central value to better than the
symbol size. From each distribution we calculate the lower and upper bound
on the uncertainty by integrating from $\pm\infty$ to a $v_n$ for which
the integrated probability reaches 15.9\%. These values bracket a 68\%
probability range for $v_n$ and are quoted as upper and lower statistical
and systematic uncertainties on the final result.

\begin{figure}[htbp]
 \includegraphics[width=0.998\linewidth]{c_Figure5_160418}
  \caption{
This example shows the direct photon $v_{2}^{\rm dir}$ measured via the
calorimeter method with the event plane estimated by the reaction plane
detector (1 $< |\eta| <$ 2.8) in the 0\%--20\% centrality bin. Each of the
various dashed curves indicate the probability distribution of the
$v_{2}^{\rm dir}$ result due to the variation of a single term in
Eq.~\ref{eq:vdirect}. While varying $v_2^{\rm inc}$ and $v_2^{\rm dec}$
alone
leaves the uncertainty on $v_2^{\rm dir}$ Gaussian, varying $R_{\gamma}$
results in strongly asymmetric shapes.  The black solid curve shows the
result when all uncertainties are taken into account simultaneously.
  }
    \label{fig:syserrmc}
\end{figure}

\begin{figure*}[htbp]
  \includegraphics[width=1.0\linewidth]{c_Figure6_160418}
  \caption{
Direct photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace at midrapidity ($|\eta| <$ 0.35), for
different centralities, measured with the conversion method (solid
circles, green) and calorimeter method (solid squares, black).  The event
plane was determined with the reaction plane detector (1 $< |\eta| <$
2.8). The error bars (boxes) around the data points are statistical
(systematic) uncertainties.}
   \label{fig:vdir}
\end{figure*}

The final results for the direct photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace, including
statistical and systematic uncertainties as outlined above, are shown in
Figure~\ref{fig:vdir} for three centralities and separately for the two
analysis methods.  For the conversion method \mbox{$v_3$}\xspace is shown only for the
highest centrality bin; the statistical fluctuations preclude any
meaningful measurement in the more peripheral bins. The data and their
uncertainties are shown in Tables~\ref{tab:photon_v2}
and~\ref{tab:photon_v3}.

\begingroup \squeezetable
\begin{table*}[htb]
  \caption{\label{tab:photon_v2}
Direct photon \mbox{$v_2$}\xspace for the indicated centrality
bins for the two methods used.  Uncertainties are shown
separately as upper and lower.
}
\begin{ruledtabular}
   \begin{tabular}{cccccc}
Centrality & Method &   $<\mbox{$p_T$}\xspace>$ [\mbox{GeV/$c$}\xspace] & \mbox{$v_2$}\xspace & Statistical uncert.
& Systematic uncert. \\
\hline
0\%--20\% & conversion photon &
     0.50 &    0.0531 &    +0.0084,  -0.0076 &    +0.0200,  -0.0187 \\
&&   0.70 &    0.0387 &    +0.0070,  -0.0087 &    +0.0252,  -0.0291 \\
&&   0.90 &    0.0357 &    +0.0080,  -0.0104 &    +0.0185,  -0.0246 \\
&&   1.10 &    0.0456 &    +0.0105,  -0.0135 &    +0.0208,  -0.0277 \\
&&   1.30 &    0.0713 &    +0.0116,  -0.0128 &    +0.0185,  -0.0207 \\
&&   1.50 &    0.0979 &    +0.0162,  -0.0153 &    +0.0227,  -0.0214 \\
&&   1.70 &    0.0735 &    +0.0148,  -0.0160 &    +0.0157,  -0.0173 \\
&&   1.90 &    0.1560 &    +0.0291,  -0.0229 &    +0.0254,  -0.0192 \\
&&   2.25 &    0.1034 &    +0.0247,  -0.0243 &    +0.0223,  -0.0215 \\
&&   3.00 &    0.0699 &    +0.0316,  -0.0338 &    +0.0140,  -0.0155 \\
&&   4.25 &   -0.3534 &    +0.8077,  -0.1197 &    +0.1149,  -0.1831 \\
& calorimeter &
     1.19 &    0.0591 &    +0.0038,  -0.0058 &    +0.0225,  -0.0266 \\
&&   1.69 &    0.0852 &    +0.0029,  -0.0035 &    +0.0163,  -0.0170 \\
&&   2.20 &    0.0957 &    +0.0046,  -0.0050 &    +0.0214,  -0.0218 \\
&&   2.70 &    0.0903 &    +0.0074,  -0.0078 &    +0.0186,  -0.0190 \\
&&   3.20 &    0.0747 &    +0.0098,  -0.0122 &    +0.0177,  -0.0189 \\
&&   3.85 &    0.0339 &    +0.0282,  -0.0430 &    +0.0218,  -0.0298 \\
20\%--40\% & conversion photon &
     0.50 &    0.0964 &    +0.0125,  -0.0113 &    +0.0133,  -0.0113 \\
&&   0.70 &    0.0668 &    +0.0173,  -0.0289 &    +0.0336,  -0.0485 \\
&&   0.90 &    0.0640 &    +0.0178,  -0.0281 &    +0.0308,  -0.0555 \\
&&   1.10 &    0.0866 &    +0.0155,  -0.0217 &    +0.0240,  -0.0403 \\
&&   1.30 &    0.1251 &    +0.0146,  -0.0170 &    +0.0178,  -0.0240 \\
&&   1.50 &    0.1405 &    +0.0182,  -0.0202 &    +0.0185,  -0.0227 \\
&&   1.70 &    0.2074 &    +0.0316,  -0.0269 &    +0.0291,  -0.0212 \\
&&   1.90 &    0.1511 &    +0.0314,  -0.0342 &    +0.0207,  -0.0245 \\
&&   2.25 &    0.1846 &    +0.0279,  -0.0273 &    +0.0186,  -0.0174 \\
&&   3.00 &    0.1412 &    +0.0407,  -0.0431 &    +0.0137,  -0.0160 \\
&&   4.25 &    0.1561 &    +0.1048,  -0.0992 &    +0.0133,  -0.0121 \\
& calorimeter &
     1.19 &    0.0902 &    +0.0097,  -0.0151 &    +0.0236,  -0.0377 \\
&&   1.69 &    0.1403 &    +0.0066,  -0.0104 &    +0.0185,  -0.0248 \\
&&   2.20 &    0.1649 &    +0.0046,  -0.0056 &    +0.0188,  -0.0202 \\
&&   2.70 &    0.1592 &    +0.0071,  -0.0083 &    +0.0189,  -0.0200 \\
&&   3.20 &    0.1327 &    +0.0098,  -0.0136 &    +0.0190,  -0.0216 \\
&&   3.85 &    0.0972 &    +0.0155,  -0.0277 &    +0.0153,  -0.0192 \\
40\%--60\% & conversion photon &
     0.50 &    0.1173 &    +0.0272,  -0.0252 &    +0.0117,  -0.0086 \\
&&   0.70 &    0.0905 &    +0.0214,  -0.0266 &    +0.0149,  -0.0280 \\
&&   0.90 &    0.1128 &    +0.0261,  -0.0327 &    +0.0192,  -0.0349 \\
&&   1.10 &    0.1101 &    +0.0338,  -0.0444 &    +0.0243,  -0.0473 \\
&&   1.30 &    0.1978 &    +0.0325,  -0.0313 &    +0.0163,  -0.0138 \\
&&   1.50 &    0.1608 &    +0.0465,  -0.0508 &    +0.0168,  -0.0244 \\
&&   1.70 &    0.3154 &    +0.0943,  -0.0687 &    +0.0771,  -0.0366 \\
&&   1.90 &    0.1848 &    +0.0943,  -0.0969 &    +0.0184,  -0.0224 \\
&&   2.25 &    0.0173 &    +0.1036,  -0.1478 &    +0.0584,  -0.1188 \\
&&   3.00 &    0.2305 &    +0.2262,  -0.1954 &    +0.0473,  -0.0310 \\
&&   4.25 &   -0.0043 &    +0.4198,  -0.2826 &    +0.0466,  -0.0920 \\
& calorimeter &
     1.19 &    0.0960 &    +0.0147,  -0.0247 &    +0.0226,  -0.0462 \\
&&   1.69 &    0.1412 &    +0.0139,  -0.0255 &    +0.0162,  -0.0334 \\
&&   2.20 &    0.1687 &    +0.0172,  -0.0258 &    +0.0212,  -0.0313 \\
&&   2.70 &    0.1624 &    +0.0323,  -0.0427 &    +0.0302,  -0.0405 \\
&&   3.20 &    0.1388 &    +0.0539,  -0.0657 &    +0.0319,  -0.0487 \\
&&   3.85 &    0.0999 &    +0.0670,  -0.0788 &    +0.0290,  -0.0533 \\
   \end{tabular}
   \end{ruledtabular}
\end{table*}
\endgroup

\begingroup \squeezetable
\begin{table*}[!ht]
  \caption{
Direct photon \mbox{$v_3$}\xspace for the indicated centrality
bins for the two methods used.  Uncertainties are shown
separately as upper and lower.
}
\label{tab:photon_v3}
\begin{ruledtabular}
   \begin{tabular}{cccccc}
Centrality & Method &   $<\mbox{$p_T$}\xspace>$ [\mbox{GeV/$c$}\xspace] & \mbox{$v_3$}\xspace & Statistical uncert.
& Systematic uncert. \\
\hline
0\%--20\% & conversion photon &
     0.50 &    0.0094 &    +0.0155,  -0.0163 &    +0.0039,  -0.0052 \\
& &  0.70 &    0.0237 &    +0.0142,  -0.0146 &    +0.0099,  -0.0111 \\
& &  0.90 &    0.0094 &    +0.0143,  -0.0163 &    +0.0119,  -0.0173 \\
& &  1.10 &    0.0333 &    +0.0204,  -0.0218 &    +0.0193,  -0.0223 \\
& &  1.30 &    0.0558 &    +0.0247,  -0.0247 &    +0.0233,  -0.0233 \\
& &  1.50 &    0.0299 &    +0.0314,  -0.0346 &    +0.0246,  -0.0301 \\
& &  1.70 &    0.0476 &    +0.0305,  -0.0317 &    +0.0161,  -0.0177 \\
& &  1.90 &   -0.0006 &    +0.0461,  -0.0535 &    +0.0189,  -0.0265 \\
& &  2.25 &    0.2094 &    +0.0657,  -0.0516 &    +0.0461,  -0.0299 \\
& &  3.00 &    0.0637 &    +0.0672,  -0.0672 &    +0.0172,  -0.0174 \\
& &  4.25 &    0.2753 &    +0.4140,  -0.4118 &    +0.1492,  -0.0765 \\
& calorimeter &
     1.19 &    0.0298 &    +0.0055,  -0.0073 &    +0.0214,  -0.0256 \\
&&   1.69 &    0.0461 &    +0.0040,  -0.0053 &    +0.0166,  -0.0182 \\
&&   2.20 &    0.0587 &    +0.0096,  -0.0110 &    +0.0170,  -0.0185 \\
&&   2.70 &    0.0696 &    +0.0129,  -0.0129 &    +0.0180,  -0.0180 \\
&&   3.20 &    0.0726 &    +0.0191,  -0.0175 &    +0.0231,  -0.0221 \\
&&   3.85 &    0.0677 &    +0.0380,  -0.0332 &    +0.0408,  -0.0378 \\
20\%--40\% & calorimeter &
     1.19 &    0.0178 &    +0.0085,  -0.0127 &    +0.0240,  -0.0343 \\
&&   1.69 &    0.0415 &    +0.0108,  -0.0154 &    +0.0304,  -0.0381 \\
&&   2.20 &    0.0619 &    +0.0128,  -0.0146 &    +0.0339,  -0.0365 \\
&&   2.70 &    0.0703 &    +0.0198,  -0.0206 &    +0.0326,  -0.0336 \\
&&   3.20 &    0.0637 &    +0.0244,  -0.0256 &    +0.0274,  -0.0284 \\
&&   3.85 &    0.0308 &    +0.0265,  -0.0331 &    +0.0228,  -0.0250 \\
40\%--60\% & calorimeter &
     1.19 &    0.0346 &    +0.0131,  -0.0157 &    +0.0362,  -0.0422 \\
&&   1.69 &    0.0638 &    +0.0271,  -0.0273 &    +0.0497,  -0.0494 \\
&&   2.20 &    0.0920 &    +0.0651,  -0.0567 &    +0.0780,  -0.0676 \\
&&   2.70 &    0.1011 &    +0.1224,  -0.1028 &    +0.0973,  -0.0793 \\
&&   3.20 &    0.0187 &    +0.1580,  -0.1476 &    +0.0823,  -0.0877 \\
&&   3.85 &   -0.0430 &    +0.1421,  -0.1289 &    +0.0751,  -0.0938 \\
   \end{tabular}
   \end{ruledtabular}
\end{table*}
\endgroup

The two analysis techniques are very different but the results agree well
in the overlap region, and they are also consistent with the results
published earlier~\cite{ppg126}. The direct photon \mbox{$v_2$}\xspace centrality
dependence, both in trend and magnitude, is quite similar to the observed
pion \mbox{$v_2$}\xspace. The third order coefficients \mbox{$v_3$}\xspace are consistent with no
centrality dependence.

		\section{Comparisons to models}
		\label{sec:modelcomp}

As already mentioned, the essence of the ``direct photon puzzle'' is that
current theoretical scenarios have difficulties explaining the large
direct photon yield and azimuthal asymmetries at the same time. This is
illustrated by a recent state-of-the-art calculation of viscous hydrodynamic calculation of photon emission with fluctuating initial density profiles and standard thermal
rates \cite{shen:2015visc}, which falls significantly short in describing
yield and \mbox{$v_2$}\xspace. Over the past few years many new ideas have been proposed
to resolve this puzzle, including non equilibrium effects
\cite{monnai:2014scf,
gelis:2004noneq,mclerran:2015tail,mclerran:2014glas}, enhanced early
emission due to large magnetic fields
\cite{Muller:2013ila,basar:2012conf,tuchin:2013magn}, enhanced emission at
hadronization \cite{campbell}, as well as modifications of the formation
time and initial conditions
\cite{chatterjee:2013flow,fmliu:2014ft,vujanovic:2014early}.

In this subsection we compare our results to a subset of the models which
(i) consider thermal radiation from the QGP and HG (hadron gas) plus
additional proposed sources, (ii) have a complete model for the space-time
evolution, and (iii) calculate absolute yields and \mbox{$v_2$}\xspace. For the
comparison we use the data for the 20\%--40\% centrality class, and note that
the comparison leads to similar conclusions for the other centrality bins.
While none of the models describes all aspects of the available data, they
are representative of how different theories are trying to cope with the
challenge.

\begin{figure}[htbp]
  \includegraphics[width=0.998\linewidth]{PHENIXAndTheory_Opt1}
  \caption{Comparison of the direct photon yields~\cite{ppg162}
and \mbox{$v_2$}\xspace with the fireball model~\cite{hees:2015pce}.
The two curves for \mbox{$v_2$}\xspace correspond to two different
parametrizations of the prompt photon component. See text for details.
  }
    \label{fig:TheoryOpt1}
\end{figure}

First, we compare the data to the ``fireball'' scenario originally
calculated in Ref.~\cite{hees:2011fb}. The model includes pQCD, QGP and HG
contributions, with the instantaneous rates convoluted with a fireball
expansion profile. The basic parameter is the initial transverse
acceleration of the fireball, $a_T$. The prompt photon component is
estimated in two ways. The first variant is a parametrization of the
photon yields measured in \mbox{$p$$+$$p$}\xspace by the PHENIX experiment~\cite{ppg060}
(labeled as ``primordial 1"), the second is an $x_t$-scaling motivated
parametrization (labeled as ``primordial 2"), modified with an empirical
factor $K=2.5$ to match the measured data at high \mbox{$p_T$}\xspace (above 4\,\mbox{GeV/$c$}\xspace).
The yield calculation includes thermal yields from the QGP with
$T_0=350$\,MeV and from the hadronic phase.  Different from an earlier
version of the model, chemical equilibrium prior to kinetic freeze-out is
no longer assumed.  This results in a large enhancement in photon
production in the later hadronic stages via processes like meson
annihilation (for instance $\pi+\rho\rightarrow\pi+\gamma$). With an
initial transverse acceleration $a_T=0.12$\,$c^2$/fm and
$\tau\approx15$\,fm/$c$ fireball lifetime, 100\,MeV freeze-out temperature
and $\beta_s=0.77$ surface velocity, the observed low \mbox{$p_T$}\xspace photon yields
are recovered within systematic uncertainties, but underpredict the
data~\cite{hees:2011fb}. In Figure~\ref{fig:TheoryOpt1} the data are
compared to the most recent updated ``fireball'' scenario shown in
Ref.~\cite{hees:2015pce}, which includes a calculation with ideal
hydrodynamics with finite initial flow at thermalization and enhanced
yields around chemical freeze-out temperature $T_c$ that improves the
description of the data. The direct photon \mbox{$v_2$}\xspace has its maximum at about
the same \mbox{$p_T$}\xspace in both theory and data. The \mbox{$v_2$}\xspace calculated in the original
fireball scenario~\cite{hees:2011fb} under predicts the measured one. The
radial boost hardens the photons from the hadronic gas (HG) and in this
way increases \mbox{$v_2$}\xspace as well, but the calculation still falls short of the
measurement. \mbox{$v_3$}\xspace is currently not calculated in this model.

\begin{figure}[htbp]
  \includegraphics[width=0.998\linewidth]{PHENIXAndTheory_Opt0}
    \caption{
Comparison of the direct photon yields and \mbox{$v_2$}\xspace, \mbox{$v_3$}\xspace with a
hydrodynamical model~\cite{Ryu:2015vwa,Paquet:2015lta} calculated under
three different assumptions including the ``semi-QGP'' scenario~\cite{gale:2015semi}.
      }
    \label{fig:TheoryOpt0}
\end{figure}

Second, in Figure~\ref{fig:TheoryOpt0} the data are compared to three
calculations evaluated with the hydrodynamical background as described in
Ref.~\cite{Ryu:2015vwa,Paquet:2015lta}. The first calculation, labeled
"QGP w/ viscous", was evaluated using the AMY photon emission rate in the high-temperature
(QGP) region, and included viscous corrections to the photon emission
rates~\cite{Shen:2014nfa,dion:2011visc} due to both bulk and shear
viscosities. The same calculation without the viscous corrections
corresponds to the curve labeled "QGP, w/o viscous".  Once viscous
corrections are included, \mbox{$v_2$}\xspace drops by more than 50\% at 3\,\mbox{GeV/$c$}\xspace, while
the yield decreases just by $\sim$10\%. The third curve, labelled
"semi-QGP, w/o viscous", shows the consequence of including the effect of
confinement on the photon emission rate, as computed in the semi-QGP
approach~\cite{gale:2015semi}. The utilization of the semi-QGP photon
rates at high temperatures suppresses the spectrum, but does not change
the $v_n$ significantly. This is a consequence of the small contribution
of QGP photons to the thermal photon \mbox{$v_2$}\xspace, which is dominantly produced
at temperatures around and smaller than the confinement temperature. The
prompt photon contributions in all three calculations are evaluated within
the perturbative QCD framework.

\begin{figure}[htbp]
  \includegraphics[width=0.998\linewidth]{PHENIXAndTheory_Opt2}
    \caption{
Comparison of the direct photon yields and \mbox{$v_2$}\xspace with the PHSD
model~\cite{linnyk:phsd2014,linnyk:phsd2015}.
      }
    \label{fig:TheoryOpt2}
\end{figure}

Third, we compare the data with PHSD (parton-hadron-string dynamics), a
microscopic transport model~\cite{linnyk:phsd2014}. In addition to the
traditional QGP and HG sources (resonance decays) this model includes late
stage meson-meson and meson-baryon Bremsstrahlung, which enhances the
yield at the lowest \mbox{$p_T$}\xspace substantially and increases \mbox{$v_2$}\xspace by almost 50\%
in the \mbox{$p_T$}\xspace$<3$\,\mbox{GeV/$c$}\xspace region (see Figure 2 in
Ref.~\cite{linnyk:phsd2014}).  Contributions from photonic decays of
$\phi$ and $a_1$ are also included, because these are not subtracted in the
measurement.  After all other sources are added, the direct photon
spectrum is very well reproduced below 3\,\mbox{GeV/$c$}\xspace, but \mbox{$v_2$}\xspace under predicts
the measured values. Also, the \mbox{$p_T$}\xspace where \mbox{$v_2$}\xspace reaches its maximum is
under predicted. In Figure~\ref{fig:TheoryOpt2} the data are compared to
the latest PHSD model calculation~\cite{linnyk:phsd2015} that included
additional photon production channels in the hadronic phase and improved
the Bremsstrahlung calculation.  The model also provides \mbox{$v_3$}\xspace. It is
positive and consistent with the data within uncertainties.

Explaining the large yield and strong flow simultaneously requires
significant improvements in quantifying the contributions from the late
stage QGP and hadron-gas interactions. Even deeper insight on both the
photon sources and the time profile of the system may be necessary to
further improve the models. Future measurements of more differential
quantities will help to distinguish and quantify the individual photon
sources.

		\section{Summary and conclusions}
		\label{sec:summary}

The PHENIX experiment at the Relativistic Heavy Ion Collider measured 2nd
and 3rd order Fourier coefficients of the azimuthal distributions of
direct photons emitted at midrapidity in \mbox{$\sqrt{s_{_{NN}}}$}\xspace = 200\,GeV \mbox{Au$+$Au}\xspace
collisions, for various collision centralities.  Two different and
independent analyses are used to determine the inclusive photon yield. The
external conversion photon measurement allows one to extend the \mbox{$p_T$}\xspace range
down to 0.4 \mbox{GeV/$c$}\xspace compared to 1.0 \mbox{GeV/$c$}\xspace for the calorimetric measurement.
In the overlap region the two results are consistent.  The \mbox{$v_2$}\xspace
measurements are also consistent with earlier published results, while
\mbox{$v_3$}\xspace is published for the first time.

Both the direct photon \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace are found to be large. The \mbox{$v_2$}\xspace
exhibits a clear centrality-dependence, while \mbox{$v_3$}\xspace is consistent with no
centrality dependence. At all centralities, the direct photon \mbox{$v_2$}\xspace is
similar in magnitude to the hadron \mbox{$v_2$}\xspace for \mbox{$p_T$}\xspace $<$ 3\,\mbox{GeV/$c$}\xspace, The direct
photon \mbox{$v_3$}\xspace is consistent with that for hadrons over the entire \mbox{$p_T$}\xspace
range.

We compare the data to several recent calculations, which treat the direct
photon yields and the azimuthal asymmetries in a consistent production and
evolution framework. None of them describe the full systematics of the
data adequately, but there has been progress in the last few years. The
general trend of the models appears to be including sources from the
earliest (pre-equilibrium, see for instance Ref.~\cite{Muller:2013ila})
or very late times in the evolution of the system, while giving less
emphasis to photon production at intermediate times, when most of the
expansion occurs. PHSD includes new sources from the hadron gas and photon
production even after the hadrons are decoupled from each other, which
improves description of the yields but still under predicts \mbox{$v_2$}\xspace.  The
model that best approximates the measured \mbox{$v_2$}\xspace, including the \mbox{$p_T$}\xspace region
where \mbox{$v_2$}\xspace reaches its maximum value, starts the evolution with a large
initial boost even before thermalization~\cite{hees:2011fb}.  It is also
worth noting that the microscopic transport model~\cite{linnyk:phsd2014}
is able to describe the anisotropies as well as the full-scale viscous
hydrodynamics~\cite{gale:2015semi}.

While the data are getting more differential and more accurate, and model
calculations improve, the ``direct photon puzzle'' remains unresolved.
High quality data of yields and \mbox{$v_2$}\xspace and \mbox{$v_3$}\xspace for different collision
systems, including very asymmetric ones, and energies would help to
further improve our understanding of direct photon production because robust
models must be able to describe the data over a wide range of experimental
conditions.

		\section*{ACKNOWLEDGMENTS}

We thank the staff of the Collider-Accelerator and Physics
Departments at Brookhaven National Laboratory and the staff of
the other PHENIX participating institutions for their vital
contributions.  We acknowledge support from the
Office of Nuclear Physics in the
Office of Science of the Department of Energy,
the National Science Foundation,
Abilene Christian University Research Council,
Research Foundation of SUNY, and
Dean of the College of Arts and Sciences, Vanderbilt University
(U.S.A),
Ministry of Education, Culture, Sports, Science, and Technology
and the Japan Society for the Promotion of Science (Japan),
Conselho Nacional de Desenvolvimento Cient\'{\i}fico e
Tecnol{\'o}gico and Funda\c c{\~a}o de Amparo {\`a} Pesquisa do
Estado de S{\~a}o Paulo (Brazil),
Natural Science Foundation of China (Peoples' Republic of~China),
Croatian Science Foundation and
Ministry of Science, Education, and Sports (Croatia),
Ministry of Education, Youth and Sports (Czech Republic),
Centre National de la Recherche Scientifique, Commissariat
{\`a} l'{\'E}nergie Atomique, and Institut National de Physique
Nucl{\'e}aire et de Physique des Particules (France),
Bundesministerium f\"ur Bildung und Forschung, Deutscher
Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany),
National Science Fund, OTKA, K\'aroly R\'obert University College,
and the Ch. Simonyi Fund (Hungary),
Department of Atomic Energy and Department of Science and Technology (India),
Israel Science Foundation (Israel),
Basic Science Research Program through NRF of the Ministry of Education (Korea),
Physics Department, Lahore University of Management Sciences (Pakistan),
Ministry of Education and Science, Russian Academy of Sciences,
Federal Agency of Atomic Energy (Russia),
VR and Wallenberg Foundation (Sweden),
the U.S. Civilian Research and Development Foundation for the
Independent States of the Former Soviet Union,
the Hungarian American Enterprise Scholarship Fund,
and the US-Israel Binational Science Foundation.

redpajama_set_name: RedPajamaArXiv


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