\section{Introduction}

Collision of heavy nuclei at relativistic energies is expected
to lead to formation of a deconfined state of matter known as
Quark-Gluon Plasma (QGP)~\cite{muller}, where quarks and gluons
are the effective degrees of freedom rather than nucleons or
hadrons~\cite{wong, hwa}. It is now well accepted that a few
microseconds after the `Big Bang' the whole universe was in
the state of QGP~\cite{uli}.

Several experiments performed at the Super Proton Synchrotron
(SPS) at CERN and Relativistic Heavy Ion Collider (RHIC) at
Brookhaven National Laboratory, New York, have provided a
significant evidence of the formation of this novel state
of matter. A giant accelerator known as Large Hadron Collider
(LHC) at CERN will be in operation very soon and will provide
many new insights about the properties of QGP and the theory of
strong interactions.

Heavy ion collisions at relativistic energies produce extremely high
temperatures and energy densities within a very small volume. As a
result, quarks and gluons (also known as partons) no longer remain
confined within the nucleonic volume and create a deconfined state
of partons due to multiple scatterings  and production of secondaries
due to gluon multiplications. The system (or the fireball) may reach
a state of local thermal equilibrium. It cools  by expansion and
below a certain critical temperature ($T_c \sim 180$ MeV) or energy
density, the partons are confined to form hadrons and the system
reaches a hadronic state. It may undergo a further expansion and
cooling before the freeze-out takes place.

Radiation of photons and dileptons has been proposed as the most
promising and efficient tool to characterize the initial state of
heavy ion collisions. Unlike hadrons, which are emitted from the
freeze-out surface  after undergoing  intense re-scatterings,
photons come out from each and every phase of the expanding fireball.
Being electromagnetic in nature, they interact only weakly and their
mean free path is larger than the typical system size ($\sim 10 $ fm).
As a result once produced, they do not suffer further interaction
with the medium ($\alpha \ll \alpha_s$) and carry undistorted
information about the circumstances of their production to the
detector~\cite{feinberg}.

Initially, photons (real as well as virtual), were studied in
order to get only the temperature of the plasma. Several other
possibilities, e.g., (i) evolution of the system size by intensity
interferometry~\cite{prl_inter}, (ii) momentum anisotropy of the
initial partons~\cite{CFHS} as well as formation time of quark-gluon
plasma~\cite{cs} using  elliptic flow of thermal photons, (iii) an
accurate check on jet quenching and other aspects of the collision
dynamics by photons due to passage of high energy jets through
plasma~\cite{fms_phot} etc. have come to the fore. Of-course
dileptons are considered as the most reliable messengers of the
medium modification of vector mesons~\cite{KKMM}.
\begin{figure}
\centering
\includegraphics[height=4.8cm, width=5.50cm]{old_phot_dn.eps}
\caption{Schematic diagram of different sources of photons and
their relative $p_T$ spectra.}
\label{phot_mass}
\end{figure}
\section{Sources of photons}

In order to proceed, it is useful to identify various sources of
photons from relativistic heavy ion collisions. Their production
is a result of convolution of the emissions from the entire history
of the nuclear collision. Photons are emitted from the
pre-equilibrium stage, from QGP phase, from hadronic phase and also
from the decay of hadrons produced at the time of freeze-out. An
ideal situation would ensue if the contributions from different
stages dominate different parts of the $p_T$ spectrum. A schematic
diagram of the different sources of photons and their slopes is
shown in Fig.~\ref{phot_mass}. We, thus need rates and models to
study the evolution from different sources. Hydrodynamics, Cascade,
Fire-ball, Cascade+Hydrodynamics are the vastly used models for
this purpose.

\subsection{Direct photons}

The term `direct photons' stands for the photons which emerge
directly from a particle collision. In a heavy ion collision
experiment, the detector captures all the emitted photons
including those from decay of final state hadrons. The resultant
spectrum is the inclusive photon spectrum. However, more than 90\%
of the photons in this spectrum are from hadron decay. One can
subdivide this broad category of `direct photons' into `prompt',
pre-equilibrium, `thermal' (from QGP as well as hadronic phase)
and `jet conversion' depending on their origin. Before we come
to the different sub-categories of direct photons, we start our
discussion with decay photons and their subtraction from the
inclusive photon spectrum.

\subsection{Decay Photons}

As mentioned earlier, most of the decay  photons are from
2-$\gamma$ decay of $\pi^0$ and $\eta$ mesons. $\omega$,
$\eta'$ etc. also contribute to the decay photon spectrum,
marginally. Subtraction of the decay background from inclusive
photon spectrum is a very challenging task. WA98 Collaboration
~\cite{subtraction} used the subtraction method using invariant
mass analysis for decay background and later PHENIX Collaboration
~\cite{rhic} has developed this method to a much higher level
of sophistication.

\subsubsection{Invariant mass analysis}
\begin{figure}
\centering
\includegraphics[height=7.2cm, width=10.0cm]{subtract_mass_pt10.eps}
\caption{ Invariant mass distributions of pairs of electromagnetic
clusters passing photon selection cuts for pair transverse momenta
satisfying $3.0 < 3.5 $ GeV. Top panels: $m_{\gamma \gamma}$
distributions in $Au+Au$ events compared to a normalized mixed-event
sample representing the combinatoric background. Bottom panels:
The $m_{\gamma \gamma}$ distributions after subtraction of the
combinatoric background for 10-20\% and 60-70\% centrality
bins~\cite{pion_supp}.}
\label{decay_subtraction}
\end{figure}
Extraction of direct photon spectrum from the inclusive photon spectrum
is done using invariant mass analysis as the primary step. First of all,
all the detected photons are listed on an event by event basis. Then by
selecting two photons randomly from an event, invariant mass of the
 pair is calculated. If $E_1$, $E_2$ are the energies and $\bf{p_1}$,
$\bf{p_2}$ are the three momentum of the photons respectively, then
the invariant mass of the pair is,
\begin{equation}
M_{\gamma\gamma} = [ (E_1 + E_2)^2 - ({\bf{p_1}} +{\bf{p_2}})^2 ]^{1/2}.
\end{equation}
If the value of $M_{\gamma\gamma}$ is close to $m_{\pi^0}$,
it is assumed that they are the decay products from the same
pion and a $\pi^0$ spectrum is obtained. Similarly the $\eta$
spectrum is obtained. These two spectra are then used to
determine the decay photon spectrum using kinematics and the
subtraction of this spectrum from  the inclusive spectrum gives
the direct photon spectrum.

However, in an event having N number of photons, the total number
of photon pairs that can be formed is $^NC_2$. For large values
of N, this number gets very large. Thus, there is a very high
probability of getting a pair of photons having an invariant mass
$m_{\pi^0}$ or $m_\eta$, which are not decay product of the same pion
or eta meson.

Several other issues also need to be accounted for: (i) the
detector resolution is finite, (ii) the opening angle between
the decay photons can get very small, specially if the energy
of the pion is large, (iii) the photons may not deposit all their
energy in the detector, and (iv) one of the photons may be
outside the coverage of the detector.

\subsubsection{Mixed event analysis}

Thus, the major problem of invariant mass analysis is that, the
accidental (false) photon pairs can also give rise to pion mass
and it is not possible to distinguish them from the correlated
pairs. To overcome this problem, a mixed event analysis
~\cite{pion_supp} procedure has been used successfully. The
basic idea of mixed event technique is to compare particle
spectrum from one event to the result for particle combinations
from different events, which are {\it a priori} not correlated.
As a first step,  properly normalized mixed events are constructed
by randomly sampling photons from different events. The difference
of the invariant mass spectra (see Fig.~\ref{decay_subtraction})
of the real events and the mixed events then gives the pion and $\eta$
distributions. Once again, the decay photon spectrum is subtracted
from the inclusive photon spectrum to get the direct photons.

\subsubsection{Internal conversion and tagging of decay photons}

An alternative approach of separating direct photons from decay
background is by measuring the `quasi-real' virtual photons which
appear as low mass electron positron pair. It is assumed that, any
source of real photons also produces low mass virtual photons which
decay  into  $e^+e^-$ pair. This method is known as internal
conversion method~\cite{nucl_ex_0804.4168} and is based on two
assumptions. The first assumption is that the ratio of direct to
inclusive photons is the same for real as well as virtual photons
having $m_{\gamma} < 30$ MeV, i.e, $ \gamma^*_{\rm dir}/\gamma^*_{\rm incl}
= \gamma_{\rm {dir}}/\gamma_{\rm {incl}}$. Secondly, the mass
distribution follows the Kroll-Wada formula~\cite{kroll_wada}:
\begin{equation}
\frac{d^2n_{ee}}{dm_{ee}} = \frac {2 \alpha} {3 \pi} \frac {1} {m_{ee}}
\sqrt{ 1- \frac {4m_e^2}{m_{ee}^2}} \left ( 1+ \frac {2m_e^2}{m_{ee}^2}
\right ) S dn_{\gamma}.
\label{kroll}
\end{equation}
Here, $m_{e}$ and $m_{ee}$ are the masses of electron and $e^+e^-$
pair respectively and $\alpha$ is the fine structure constant. This
method is used for Compton scattering ($ q + g \rightarrow q + \gamma^*
\rightarrow q + e^+ + e^- $), Dalitz decay ( $\pi^0, \eta \rightarrow
e^+e^- \gamma, \ \omega \rightarrow e^+ e^- \pi^0$) and also for two
$\gamma$ decay of several other hadrons. The factor S in Eqn.~(\ref{kroll})
is process dependent and  for $\pi^0$ ($\rightarrow \gamma \gamma^*
\rightarrow \gamma e^+e^-$) decay it is expressed as~\cite{S_decay}:
\begin{eqnarray}
S= | F(m_{ee}^2)|^2 \left (1- \frac {m_{ee}^2} {M_h^2} \right )^3,
\end{eqnarray}
where, $M_h$ is the hadron mass and $F(m_{ee}^2)$ is the form factor.
The factor S is 0 for $m_{ee} > M_h$ and goes to 1 as $m_{ee}
\rightarrow 0$ or $m_{ee} \ll p_T$.  The key advantage of this method
is the greatly improved signal to background ratio which is achieved
by elimination of the contribution of Dalitz ($\pi^0$) decay. The
experimentally measured quantity is the ratio of $e^+ e^-$ pairs
in a particular invariant mass bin and the direct photon spectrum
is obtained by multiplying $ \gamma^*_{\rm{dir}}/\gamma^*_{\rm incl}$
to the measured inclusive photon spectrum (left panel of
Fig~\ref{decay_phot_tag}).

Due to the excellent resolution of the PHENIX detector to measure
charged particles at low momenta, another powerful  technique
known as `tagging of decay photons'~\cite{gong} is used to eliminate
the $\pi^0$ decay background. This method is very useful in the low
and intermediate $p_T$ range ( $1 < p_T < 5$ GeV) as the systematic
uncertainties introduced by detector efficiency, acceptance etc.
cancel out for measuring  $\gamma/\gamma_{\pi^0}$ directly in-place
of conventional double ratio ($R= (\gamma/\pi^0)_{\rm measured}/(\gamma/
\pi^0)_{\rm decay}$) technique. In this method, the invariant mass
($m_{\gamma e^+ e^-}$) distribution  of the $\pi^0$ ( $\rightarrow
\gamma \gamma^* \rightarrow \gamma e^+ e^- $) decay products is
constructed and then mixed event analysis is used for the final
subtraction (middle and right panel of Fig~\ref{decay_phot_tag}).
\begin{figure}
\centering
\includegraphics[height=3.2cm, width=3.20cm]{bathe.eps}
\includegraphics[height=3.2cm, width=7.60cm]{gong.eps}
\caption{Left Panel: Invariant mass distribution of virtual
photons from $\pi^0$ and $\eta$ Dalitz decay as well as from
direct photons~\cite{nucl_ex_0804.4168}. Middle and right panels:
Invariant mass spectrum in `tagging' of decay photon
method~\cite{gong}.}
\label{decay_phot_tag}
\end{figure}
\subsection{Sources of direct photons}

Direct photons can be classified into different categories depending
on their origin from different stages of the expanding fireball formed
after the collision. These are: (1) prompt photons, which originate
from initial hard scatterings, (2) pre-equilibrium photons, produced
before the medium gets thermalized, (3) thermal photons from quark-gluon
plasma as well as by hadronic reactions in the hadronic phase, and (4)
photons from passage of jets through plasma. It is not possible
experimentally to distinguish between the different sources. Thus,
theory can be used with a great advantage to identify these sources
of direct photons and their relative importance in the
spectrum~\cite{dks_qm}.

\subsubsection{Partonic processes for production of prompt photons}

\begin{figure}
\centering
\includegraphics[height=4.cm, width=12 cm]{qgp_phot.eps}
\caption{Partonic processes for production of photons from (a) quark gluon
Compton scattering, quark anti-quark annihilation process and (b) quark
fragmentation.}
\label{partonic}
\end{figure}

In relativistic heavy ion collisions, prompt photons are produced
due to quark gluon  Compton scattering ($ q + g \rightarrow g +
\gamma$), quark anti-quark annihilation process ($q + \bar q \rightarrow
g + \gamma$), and quark fragmentation ( $ q \rightarrow q + \gamma$)
following scattering of partons of the nucleons in the colliding nuclei
(see Fig~\ref{partonic}).
At lowest order in $\alpha \alpha_s$, quark gluon Compton scattering
and quark anti-quark annihilation processes dominate the photon
production. In Next to Leading Order (NLO) calculation, many more
complicated scattering processes appear
in the photon production cross-section and the total contribution
can be written as addition of two different terms as~\cite{Aurenche},
\begin{equation}
\frac {d \sigma} {d \overrightarrow p_T d \eta} = {\frac {d \sigma^{(D)}}
{d \overrightarrow p_T d \eta}} + {\frac {d \sigma^{(F)}} {d
\overrightarrow p_T d \eta}}
\label{prompt}
\end{equation}
In the above equation `D' stands for `Direct' or the total Compton
scattering and annihilation contribution while the photons from
fragmentation are denoted by `F'. It is clear that the `D' photons
are well separated from hadrons. The `F' photons on the other hand
have their origin in collinear fragmentation of coloured high $p_T$
partons and are accompanied by hadrons. This can be used with
advantage as the produced photons can be separated out by employing
useful isolation cuts. The two terms in Eqn.~(\ref{prompt}) can be
written explicitly
as~\cite{old_aurenche}:
\begin{eqnarray}
\frac {d \sigma^{(D)}} {d \overrightarrow p_T d \eta} &=& \sum_{i,j=q,
\bar{q},g} \int dx_{1} dx_{2} \ F_{i/h_1}(x_{1},M)\ F_{j/h_2}(x_{2},M)
{\frac { \alpha_s (\mu_R)}{2 \pi}}
\nonumber\\
& \times& \left
({\frac {d \hat{\sigma}_{ij}}{d \overrightarrow p_T d \eta}} +{\frac
{ \alpha_s (\mu_R)} {2 \pi}} K_{ij}^{(D)}  (\mu_{R},M,M_{_F})\right )
\end{eqnarray}
and
\begin{eqnarray}
\frac {d \sigma^{(F)}} {d \overrightarrow p_T d \eta} &=& \sum_{i,j,k=q,
\bar{q},g} \int dx_{1} dx_{2} {\frac{dz}{z^2}}\ F_{i/h_1}(x_{1},M)
\ F_{j/h_2}(x_{2},M) \ D_{\gamma/k}(z, M_{_F})
\nonumber \\
&\times &({\frac { \alpha_s (\mu_R)}{2 \pi}})^2 \left( {\frac {d
\hat\sigma_{ij}^k}{d \overrightarrow p_T d \eta}} +{\frac { \alpha_s
(\mu_R)}{2 \pi}} K_{ij,k}^{(F)} (\mu_{R},M,M_{_F}) \right).
\end{eqnarray}
Here $F_{i/h_{1,2}}(x,M)$ are the parton distribution functions
and $\alpha_s(\mu_R)$ is the strong coupling defined in the
$\overline{MS}$ renormalization scheme at the renormalization
scale $\mu_R$. For details see Ref.~\cite{Aurenche}. As mentioned
earlier, in a complete and consistent NLO pQCD $[O (\alpha
\alpha_s^2)]$ calculation important contribution to prompt
photon result arises from various possible $ 2 \rightarrow
3 $ process like $ ab \rightarrow \gamma cd$ for the Direct
as well as for the Fragmentation processes~\cite{old_aurenche,
Vogel}.
\begin{figure}[b]
\centering
\includegraphics[height=7.5cm, width=8.cm]{aurenche1.eps}
\caption{World's inclusive and isolated direct photon
production cross-sections measured in $pp$ and $p\bar p$
collisions compared to JETPHOX NLO predictions, using BFG
II(CTEQ6M) for fragmentation (structure) functions and a
common scale $p_T/2$~\cite{Aurenche}.}
\label{all_phot}
\end{figure}

In case of photons from fragmentation, the higher order correction
is very important at low $x_T\,( =\, 2 p_T/\sqrt{s})$. By choosing a
scale for factorization, renormalization, and fragmentation, all
equal to $p_T/2$, a very good quantitative description is obtained
for all the available $pp$ and $p \overline{p}$ data without
introduction of any intrinsic $k_T$. Results spanning over two
orders of magnitude in energy and over nine orders of magnitude
in cross sections, are shown in Fig.~\ref{all_phot}.

It should be mentioned though, that a similar exercise for pions
requires a scale of $p_T/3$~\cite{pi_aurenche}.

In the early calculations, the results for nucleus-nucleus scatterings
were often obtained by multiplying the $pp$ results for some $\sqrt{s}$
with corresponding scaling factor $N_{\rm {coll}}$ ($ = \sigma_{NN}
T_{AB}$; where, $\sigma_{NN}$ is the nucleon-nucleon cross-section
and $T_{AB}$ is the nuclear overlapping function for nuclei A and B)
or number of binary collisions.  In actual practice, often enough,
$\sigma_{NN}$ was replaced by $\sigma_{pp}$. However, as the valence
quark structures of protons $(uud)$ and neutrons $(udd)$ are different,
one needs to correctly account for the iso-spin of the nucleons to
calculate the prompt contribution. This correction will strongly
affect the results in the $p_T$ range, where the valence quark
contribution is significant. Also, remember that there is no direct
measurement for $pn$ and $nn$ cross-sections, though they can be
estimated by comparing results of scatterings involving deuterons.
The effect of shadowing on structure function and energy loss of
final state quarks before they fragment into hadrons are two other
corrections~\cite{JMS} which need to be accounted for.

\begin{figure}[t]
\centering
\includegraphics[height=4.0cm]{1.eps}
\includegraphics[height=4.00cm]{2.eps}
\caption{Left panel: Prompt photons for $pn$ and $nn$ collisions
normalized to those for $pp$ collisions at $\sqrt{s}=200$ GeV.
Right panel: Iso-spin, shadowing and energy loss corrected $R_{AA}$
for prompt photons at $200A$ GeV $Au+Au$ collisions using NLO pQCD.}
\label{ifs1}
\end{figure}

In order to clearly see this point we show the results for prompt
photons for $nn$ and $pn$ collisions normalized to those for $pp$
collisions at $\sqrt{s}= 200$ GeV in the left panel of Fig.~\ref{ifs1}.
It is clear that at low $x_T$ (where processes involving gluons
dominate) the effect is not very significant, however with larger
values of $x_T$ (where the processes involving valence quarks
dominate), the production of photons decreases by a large factor
for the case of $pn$ and $nn$ collisions.

The transverse momentum dependent  nuclear modification factor
$R_{AA}$, defined as,

\begin{equation}
R_{AA}(p_T)= \frac {1} {N_{\rm {coll}}} \frac{d \sigma^{AA}_{\gamma}
(p_T)/dy d^2{p_T}} {d \sigma^{pp}_{\gamma}(p_T)/dy d^2{p_T}}
\end{equation}
for prompt photons using NLO pQCD and considering iso-spin, shadowing
and energy-loss effects for $200A$ GeV $Au+Au$ collisions at RHIC  is
shown in the right panel of Fig.~\ref{ifs1}. We note that, for $p_T
< 10$ GeV, the iso-spin and shadowing corrected result shows an
enhancement in the prompt photon production compared to the situation
when only the iso-spin correction is incorporated~\cite{DKS_ifs}. This
is due to anti-shadowing for larger values of Bjorken $x$ or large $x_T$.
The inclusion of the energy-loss pushes down the value of $R_{AA}$ to less
than one for all $p_T$. We also give a comparison with the
PHENIX~\cite{Phenix} experimental data in the bottom lower panel of
Fig.~\ref{ifs4}.

\begin{figure}[t]
\centering
\vspace{0.3cm}
\includegraphics[height=3.80cm]{4.eps}
\includegraphics[height=3.80cm]{5.eps}
\caption{Iso-spin, shadowing and energy-loss corrected $R_{AA}$ for
prompt photons at LHC (left panel) and SPS (right panel) energies
using NLO pQCD.
}
\label{ifs2}
\end{figure}
NLO results at LHC ($Pb+Pb@5.5A$ TeV) and SPS ($Pb+Pb@17.4A$ GeV) are
shown in Fig.~\ref{ifs2}. To clearly demonstrate the relative features
of iso-spin, shadowing and energy-loss corrected prompt photons at
different collider energies, results for $R_{AA}$ as function of
$x_T$ are shown in Fig.~\ref{ifs4}.

\begin{figure}[ht]
\centering
\includegraphics[height=3.8cm]{6.eps}
\includegraphics[height=3.8cm]{7.eps}

\vspace{1cm}
\includegraphics[height=3.8cm]{8.eps}
\includegraphics[height=3.8cm]{3.eps}
\caption{ Iso-spin, shadowing and fragmentation energy-loss
corrected NLO pQCD results at different collider energies with
different target projectile combination and comparison of the
RHIC results with PHENIX~\cite{Phenix} data.}
\label{ifs4}
\end{figure}

\subsection{Photon production from Quark-Gluon Plasma}

The thermal emission rate of photons with energy $E$ and momentum
$p$ from a small system (compared to the photon mean free path)
is related to the imaginary part of photon self energy by the
following relation
\begin{eqnarray}
E\frac{dR}{d^3p}=\frac{-2}{(2\pi)^3}Im{\Pi_{\mu}}^{R,\mu}\frac{1}
{e^{E/T}-1}
\label{phorate}
\end{eqnarray}
where, ${\Pi_{\mu}}^{ R,\mu}$ is the retarded photon self-energy
at a finite temperature $T$. This relation is valid in the
perturbative~\cite{Mc_Toi_&} as well as non-perturbative~\cite{gale_kapu}
limits. It is also valid to all orders in the strong interactions
and to order $e^2$ in the electromagnetic interactions. If the photon
self-energy is approximated by carrying out a loop expansion to some
finite order, then the formulation of Eqn.~(\ref{phorate}) is equivalent
to relativistic kinetic theory. In order to illustrate this, we closely
follow the treatment of Ref.~\cite{KLS}.

Thus using relativistic kinetic theory formulation, the contribution
of these processes to the rate can be written as~\cite{KLS,BNNR}:
\begin{eqnarray}
{\mathcal R}_i &=&{\mathcal N}\int \frac{d^3p_1}{2E_1(2\pi)^3}
\frac{d^3p_2}{2E_2(2\pi)^3}f_1(E_1)f_2(E_2)(2\pi)^4
\delta({p_1}^{\mu}+{p_2}^{\mu}-{p_3}^{\mu}-p^{\mu}) \nonumber\\
&\times& |{\mathcal M}_i|^2\frac{d^3p_3}{2E_3(2\pi)^3}
\frac{d^3p}{2E(2\pi)^3}[1 \pm f_3(E_3)]
\label{rate1}
\end{eqnarray}
where ${\mathcal M}_i$ represents the amplitude for one of these
processes and the $f's$ are the Fermi-Dirac or Bose-Einstein
distribution functions as appropriate. Positive and negative signs
in the last part of Eqn.~(\ref{rate1}) correspond to Bose enhancement
and Pauli suppression respectively.

The integral above can be simplified by introducing Mandelstam
variables $s=(p_1+p_2)^2$, $t=(p_1-p_3)^2$ and $u=(p_1-p)^2$. Now
the differential
photon rate can be written as:
\begin{eqnarray}
E\frac{d{\mathcal R}_i}{d^3p}&=&\frac{{\mathcal N}}{(2\pi)^7}
\frac{1}{16E}\int ds dt ~|{\mathcal M}_i(s,t)|^2\int
dE_1 dE_2 f_1(E_1)f_2(E_2)\nonumber\\ & \times &
[1\pm f_3(E_1+E_2-E)]\theta(E_1+E_2-E)
(a{E_1}^2+bE_1+c)^{-1/2}
\end{eqnarray}
where,
\begin{eqnarray}
a &=&-(s+t)^2, \nonumber\\
b &=&2(s+t)(Es-E_2t),\nonumber\\
c &=&st(s+t)-(Es+E_2t)^2.
\end{eqnarray}
Considering the photon energy to be large, one can consider,
 $f_1(E_1)f_2(E_2) \approx e^{-(E_1+E_2)/T}$ and simplify the
above as
\begin{eqnarray}
E\frac{d{\mathcal R}_i}{d^3p}=\frac{{\mathcal
N}}{(2\pi)^6}\frac{T}{32E}e^{-E/T} \int \frac{ds}{s}~\ln(1\pm
e^{-s/4ET})^{\pm 1}\int dt |{\mathcal M}_i(s,t)|^2
\end{eqnarray}
In the above equation, positive and negative signs stand for
fermions ($q$) and bosons ($g$) in the final state respectively.

The relation between the amplitude and differential cross-section for
massless particles can be written as,
\begin{eqnarray}
\frac{d\sigma}{dt}=\frac{|{\mathcal M}|^2}{16\pi s^2}
\end{eqnarray}
And thus, the differential cross-sections for annihilation process and
Compton scattering are:
\begin{eqnarray}
\frac{d\sigma^{\rm {annihilation}}}{dt}=\frac{8\pi \alpha \alpha_s}
{9s^2}\frac{u^2+t^2}{ut}
\end{eqnarray}
and
\begin{eqnarray}
\frac{d\sigma^{\rm {Compton}}}{dt}=\frac{-\pi \alpha \alpha_s}{3s^2}
\frac{u^2+s^2}{us}
\end{eqnarray}
For annihilation process, ${\mathcal N}=20$ when summing over
$u$ and $d$ quarks and for Compton scattering ${\mathcal N}=320/3$.
The total cross-section can be obtained after integrating over $t$.
These differential cross sections have a singularity at $t$ and/or
$u=0$ and the total cross-section is infinite as the processes
involve exchange of mass-less particle.

To screen this divergence many-body effects are necessary. This
approach will be discussed later. As a first step let us isolate the
region of phase space causing the divergences. The integration is
done over
\begin{eqnarray}
&&-s+{k_c}^2 \le t\le -{k_c}^2, \hspace{0.2cm} 2{k_c}^2\le s \le \infty,
\end{eqnarray}
where $T^2 \gg {k_c}^2>0$ is an infrared cut-off.

This treats $u$ and $t$ symmetrically and maintains the
identity $s+t+u=0$ appropriate for all massless particles.

In the limit that ${k_c}^2\to 0$,
\begin{eqnarray}
E\frac{d{\mathcal R}^{\rm {Compton}}}{d^3p}= \frac{5}{9}\frac{\alpha
  \alpha_s}{6{\pi} ^2} T^2 e^{-E/T}[\ln (4ET/{k_c}^2)+C_F]\\
E\frac{d{\mathcal R}^{\rm {annihilation}}}{d^3p}= \frac{5}{9}\frac{\alpha
  \alpha_s}{3{\pi} ^2} T^2 e^{-E/T}[\ln (4ET/{k_c}^2)+C_B]
\end{eqnarray}
where
\begin{eqnarray}
C_F=\frac{1}{2}-C_{\rm {Euler}}+\frac{12}{{\pi}^2}\sum_{n=2}^{\infty}
\frac{(-1)^n}{n^2}\ln~ n= 0.0460.........,
\label{cf}
\end{eqnarray}
\begin{eqnarray}
C_B=-1-C_{\rm{ Euler}}-\frac{6}{{\pi}^2}\sum_{n=2}^{\infty}
\frac{1}{n^2} \ln~n= -2.1472.......  .
\label{bf}
\end{eqnarray}
These expressions use the full Fermi-Dirac or Bose-Einstein
distribution functions in the final state.

These results have a very interesting structure. Thus, the
factor $5/9$ arises from the sum of the squares of the electric
charges of the $u$ and $d$ quarks, the factor $\alpha \alpha_s$
comes from the topological structure of the diagrams, a factor $T^2$
comes from phase space which gives the overall dimension of the rate,
and we have the Boltzmann factor $e^{-E/T}$ for photons of energy $E$.
The logarithm arises due to the infrared behavior.

\subsubsection{Infrared contribution}

The infrared divergence in the photon production rate \cite{KLS}
discussed above is caused by a diverging differential cross-section
when the momentum transfer goes to zero. Often-times long-ranged
forces can be screened by many-body effects at finite temperatures.
Braaten and Pisarski have analyzed problems such as this one in QCD
\cite{wel13}. They have argued that a cure can be found in reordering
perturbation theory by expanding correlation functions in terms
of effective propagators and vertices instead of bare ones. These
effective propagators and vertices are just the bare ones plus
one-loop corrections, with the caveat that the one-loop corrections
are evaluated in the high temperature limit. This makes them relatively
simple functions.

The analysis of Braaten and Pisarski shows that a propagator must be
dressed if the momentum flowing through it is soft (small
compared to $T$). This is because propagation of soft momenta is
connected with infrared divergences in loops. Dressing of propagators
are necessary, otherwise corrections due to these are also infinite.

Using the one loop corrected propagators and vertices and the
contribution to the rate coming from the infrared sensitive part
of phase space can be written as,
\begin{eqnarray}
E\frac{d{\mathcal R}^{\rm BP}}{d^3p}=\frac{5}{9}\frac{\alpha \alpha_s}
{2 {\pi}^2}T^2e^{-E/T}\ln(\frac{{k_c}^2}{2{m_q}^2})
\end{eqnarray}
where, $2{m_q}^2=\frac{1}{3}g^2 T^2$.

Adding this contribution to those given by Eqns.~(\ref{cf}) and
~(\ref{bf}), the final result can be written as;

\begin{eqnarray}
E\frac{d{\mathcal R}}{d^3p}=\frac{5}{9}\frac{\alpha \alpha_s}{2
  {\pi}^2}T^2e^{-E/T}\ln \left (\frac{2.912}{{g}^2} {\frac{E}{T}} \right).
\end{eqnarray}

This is independent of cut-off $k_c$. Thus, the Braaten-Pisarski
method has worked beautifully to shield the singularity encountered
above. We also note that in kinetic theory calculation, the effective
infrared cutoff is ${k_c}^2=2{m_q}^2$. In earlier works an infrared
cut-off was often imposed by giving the exchanged quark an effective
temperature-dependent mass.

These early results have been brought to a high degree of
sophistication and results complete to leading order in $\alpha_s$
with inclusion of LPM effects are now available, which should be
used for detailed calculation~\cite{AMY}.

\subsubsection{Photons from passage of jets through QGP}

The relativistic heavy ion collisions at RHIC (and LHC) energies
are marked by a large production of high energy quark and gluon
jets which lose energy while passing through the QGP due to
collision and radiation of gluons. This is the celebrated phenomenon
of jet-quenching. A quark jet having a transverse momentum $p_T$
would be formed within $\tau \sim 1/p_T$ which can be much smaller
than $\tau_0$, when QGP is formed, for large $p_T$. This quark (or
anti-quark) jet, while passing through QGP may annihilate with a
thermal anti-quark (or quark) or undergo a Compton scattering with
a thermal gluon and lead to production of a high energy photon. This
process is called the jet-photon conversion \cite{fms_phot}.

We have seen that the  kinematics of the annihilation of a quark
anti-quark pair ($q + {\bar q}\to \gamma+g$) is expressed in terms
of the Mandelstam variables $s=(p_q+p_{\bar q})^2$, $t=(p_q-p_{\gamma})^2$
and $u=(p_{\bar q}-p_{\gamma})^2$. We also recall that the largest
contribution to the production of photons arises from small values
of $t$ or $u$, corresponding to $p_{\gamma}\sim p_q$ or $p_{\gamma}
\sim p_{\bar q}$~\cite{wong,fms_phot,nad}.

The phase-space distribution of the quarks and gluons produced in a
nuclear collision can be approximately decomposed into two components,
a thermal component $f_{\rm th}$ characterized by a temperature $T$ and a
hard component $f_{\rm jet}$ given by hard scattering of the partons and
limited to transverse momenta $p_T \gg 1$ GeV: $f({\bf p})=f_{\rm th}
({\bf p})+f_{\rm jet}({\bf p})$. $f_{\rm jet}$ dominates for large momenta,
while at small momenta $f$ is completely given by the thermal part.

The phase space distribution for the quark jets propagating through
the QGP is given by the perturbative QCD result for the jet yield
\cite{fms_phot}:
\begin{eqnarray}
f_{\rm jet}({\bf p})= \frac{1}{g_q}\frac{(2\pi)^3}{\pi R_{\perp}^2
\tau p_T}\frac{dN_{\rm jet}}{d^2p_T dy}R(r)\delta(\eta-y)
\Theta(\tau_{\rm max}-\tau_i)\Theta(R_{\perp}-r)
\label{dks31}
\end{eqnarray}
where $g_q=2\times 3$ is the spin and colour degeneracy of the quarks,
$R_{\perp}$ is the transverse dimension of the system,
and the $\eta$ is the
space-time rapidity. $R(r)$ is a transverse profile function.
$\tau_{\rm max}$ is the smaller of the lifetime $\tau_f$ of the QGP
and the time $\tau_d$ taken by the jet produced at position {\bf r} to
reach the surface of the plasma.

One can approximate the invariant
photon differential cross sections for the annihilation process and
Compton scattering as~\cite{wong,fms_phot},
\begin{eqnarray}
E_{\gamma}\frac{d\sigma^{(a)}}{d^3p_{\gamma}}\sim \sigma^{(a)}(s)\frac{1}
{2}E_{\gamma}[\delta({\bf p}_{\gamma}-{\bf p}_q)+\delta({\bf p}_{\gamma}-
{\bf p}_{\bar q})],
\label{dks27}
\end{eqnarray}
and
\begin{eqnarray}
E_{\gamma}\frac{d\sigma^{(C)}}{d^3p_{\gamma}}\sim \sigma^{(C)}
(s)E_{\gamma}\delta({\bf p}_{\gamma}-{\bf p}_q)
\label{dks28}
\end{eqnarray}
Here $\sigma^{(a)}(s)$ and $\sigma^{(C)}(s)$ are the corresponding total
cross sections.

Using Eqn.~(\ref{dks27}) and Eqn.~(\ref{dks28}), the rate of production
of photons due to annihilation and Compton scattering are
given by~\cite{wong}:
\begin{eqnarray}
E_{\gamma}\frac{dN^{(a)}}{d^4xd^3p_{\gamma}}&=&\frac{16E_{\gamma}}
{2(2\pi)^6}\sum_{q=1}^{N_f}f_q({\bf p_{\gamma}})\int d^3pf_{\bar q}
({\bf p})[1+f_g({\bf p})]\sigma^{(a)}(s) \nonumber\\
&\times& \frac{\sqrt{s(s-4m^2)}}{2E_{\gamma}E}+(q\leftrightarrow{\bar q})
\label{dks29}
\end{eqnarray}
\begin{eqnarray}
E_{\gamma}\frac{dN^{(C)}}{d^4xd^3p_{\gamma}}&=&\frac{16E_{\gamma}}
{(2\pi)^6}\sum_{q=1}^{N_f}f_q({\bf p_{\gamma}})\int d^3pf_g({\bf p})
[1-f_q({\bf p})]\sigma^{(C)}(s)\nonumber\\
&\times& \frac{(s-m^2)}{2EE_{\gamma}}+(q\to{\bar q})
\label{dks30}
\end{eqnarray}
The $f's$ are the distribution functions for the quarks, anti-quarks
and gluons. Inserting thermal distributions for the gluons and quarks
one can obtain an analytical expression for these emission rates for
an equilibrated medium~\cite{wong,KLS,BNNR}.

In order to estimate the jet photon conversion contribution, we first
note that, the integrals over {\bf p} in Eqns.~(\ref{dks29}) and
(\ref{dks30}) are dominated by small momenta. Therefore dropping
the jet part in the distributions, $f({\bf p})$ in the integrands is
approximated by the thermal part. Now performing the integrals and
identifying the quark and anti-quark distributions outside the
integrals with the jet distributions, results for Compton and
annihilation scatterings due to jet-conversions are given as,
\begin{eqnarray}
E_{\gamma}\frac{d{N_{\gamma}}^{(a)}}{d^3p_{\gamma}d^4x}&=&E_{\gamma}
\frac{d{N_{\gamma}}^{(C)}}{d^3p_{\gamma}d^4x}\nonumber\\
&=&\frac{\alpha \alpha_s}{8{\pi}^2}{\sum^{{\mathcal N}_f}_{f=1}}
(\frac{e_{q_f}}{e})^2 [f_q({\bf p}_{\gamma})+f_{\bar q}({\bf p}_{\gamma})
]T^2 [\ln(\frac{4E_{\gamma}T}{m^2})+C]
\label{phjet}
\end{eqnarray}
Here, $C=-1.916$. If we include  three lightest quark flavours,
then $\sum_f {e_{q_f}}^2/e^2=2/3$. We also assume that the mass $m$
introduced here to shield the infrared divergence can be identified
 with the thermal quark mass $m_{th}$.

These pioneering works have now been corrected for energy loss and flavour
change suffered by the jets, as they pass through the plasma~\cite{tur}
as well as bremsstrahlung induced by the passage of the jets through the
plasma~\cite{zakha}.

\subsubsection{Equilibration time for photons}

It is of interest to get an idea about the equilibration
time of photons in the medium. Following the treatment of
Ref.~\cite{KLS},  we note that the six-dimensional phase
space distribution $dn/d^3p$($=dN/d^3xd^3p$), for the photons
satisfies the rate equation:
\begin{equation}
\frac {d} {dt} \left ( {\frac {dn}{d^3p}} \right ) = {\frac {dR}
{d^3p}} \left ( 1 - {\frac {dn/d^3p}{dn_{\rm eq}/d^3p}} \right ).
\label{eq}
\end{equation}
In the above equation, $dn_{\rm eq}/d^3p$ is the equilibrium
distribution and is expressed as Planck's distribution,
\begin{equation}
\frac {dn^{\rm eq}} {d^3 p} = {\frac {2} {(2 \pi)^3}} {\frac {1}
{e^{E/T}-1} }.
\end{equation}
Considering $\tau_{\rm eq}$ ($ = \frac
{dn_{\gamma}^{\rm eq}} {d^3p}/ \frac {dR} {d^3p}$) as the time for
equilibration and assuming zero photons at the beginning we can
write,
\begin{equation}
\frac {dn} {d{^3}p} = \frac {dn^{\rm eq}} {d^3p}
\left(1-e^{t/\tau_{\rm eq}}\right).
\end{equation}
Using the rate equation, the thermalization time can be expressed
in a simplified form (considering $ E > 2T $) as:
\begin{equation}
\tau_{\rm eq} = {\frac {9E} {10 \pi \alpha \alpha_s T^2 }} {\frac {1}
{\ln( {\frac {2.9} {g^2}} {\frac {E} {T}}+1)}}.
\end{equation}
For energy values E =0.5, 1, 2, 3 GeV, corresponding values for
$\tau_{\rm eq}$  will be about 270, 356, 505, 639  fm/$c$ respectively,
when the temperature is 200 MeV. As the life time of the system is
very short, of the order of few tens of fm/$c$, it is very clear that
the high energy photons will never reach the equilibrium state in heavy
ion collisions. This is an important confirmation for the validity of the
assumption made in all such studies that photons, once produced in the
collision, leave the system with out any further re-interaction.

\begin{figure}[t]
\centering
\includegraphics[height=5.50cm]{had_phot.eps}
\caption{ Typical hadronic reactions for photon production.}
\label{phot_fms+had_yield}
\end{figure}

\subsection{Photons from hot hadronic matter}

Hot hadronic matter produced after the hadronization of the quark-gluon
plasma will also lead to production of photons due to hadronic reactions.
These photons will dominate the spectrum at lower $p_T$ ( $< 1 $ GeV ).
The first ever calculation of production of thermal photons from
hadronic matter was performed by Kapusta {\it et al.}~\cite{KLS}.

In a hot hadronic gas (having temperature of the order of pion mass),
the most important hadronic constituents for photon production are
$\pi$ and $\rho$ mesons (see Fig.~\ref{phot_fms+had_yield}).
The low mass of pions and the large spin iso-spin degeneracy of
$\rho$ mesons, make them the most easily accessible particles in
the medium. In order to illustrate the photon production from these
two mesons, we closely follow the treatment of KLS~\cite{KLS}. In a
hadronic reaction involving charged $\pi$ and  $\rho$ meson, the
typical Lagrangian describing the interaction can be written as:
\begin{eqnarray}
\mathcal L &=& \mid D_\mu \Phi \mid^2 -m_\pi^2 \mid \Phi \mid^2 -
\frac {1} {4} \rho_{\mu \nu} \rho^{\mu\nu} + \frac {1} {2} m_\rho^2
\rho_\mu \rho^\mu - \frac {1} {4} F_{\mu\nu}F^{\mu\nu}
\end{eqnarray}
where,
\begin{eqnarray}
 D_\mu &=&\partial_\mu -ieA_\mu - ig_\rho \rho_\mu, \nonumber
\end{eqnarray}
$\Phi$ is the complex pion field, $\rho_{\mu\nu}$ is the $\rho$ field
strength and $F_{\mu\nu}$ is the photon field tensor. The differential
cross-sections for the dominating photon producing processes ($\pi \rho
\rightarrow \pi \gamma$) in the hadronic phase is expressed as~\cite{KLS},
\begin{figure}[th]
\centering
\includegraphics[height=9.0cm, width=7.0cm]{kls.eps}
\caption{Left Panel: Yield of photons from different  hadronic
channels (taken from~\cite{KLS}).}
\label{check}
\end{figure}
\begin{eqnarray}
\frac{d \sigma} {dt} (\pi^+ \rho^0 \longrightarrow \pi^+ \gamma)&=&\frac {d
\sigma} {dt} (\pi^- \rho^0 \longrightarrow \pi^- \gamma)\nonumber\\
& =& {\frac {\alpha g_\rho^2} {12sp_{\rm c.m.}^{2}}}
 \left [ 2 - {\frac {(m_\rho^2-4m_\pi^2)s}{(s-m_\pi^2)^2}} -
(m_\rho^2-4m_\pi^2) \right.\nonumber\\ &\times & \left.
 \left (\frac {s-m_\rho^2 + m_\pi^2} {s-m_\pi^2} \frac{1} {u-m_\pi^2} +
\frac {m_\pi^2}{(u-m_\pi^2)^2} \right ) \right ].
\end{eqnarray}
Similarly for
\begin{eqnarray}
\frac{d \sigma} {dt} (\pi^- \rho^+ \longrightarrow \pi^0 \gamma)&=&\frac {d
\sigma} {dt} (\pi^+ \rho^- \longrightarrow \pi^0 \gamma)\nonumber\\
&=& - {\frac {\alpha g_\rho^2} {48sp_{\rm c.m.}^2}}\left [ 4(m_\rho^2-4m_\pi^2)
 \left [ {\frac {u} {(u-m_\pi^2)^2}} + {\frac {t} {(t-m_\rho^2)^2}}
\right.\right.\nonumber\\ &-& \left.{\frac {m_\rho^2}{s-m_\pi^2}}
\left( {\frac {1} {u-m_\pi^2}} + {\frac {1}{t-m_\rho^2}} \right)\right ]
+ \left [\left(3+ {\frac {s-m_\pi^2}{m_\rho^2}}\right) \right.
\nonumber\\ &\times & \left. {\frac {s-m_\pi^2} {t-m_\rho^2}}\right] -
\left. {\frac {1} {2}} + {\frac {s} {m_\rho^2}} - \left( {\frac
{s-m_\pi^2} {t-m_\rho^2}} \right)^2 \right ]
\end{eqnarray}
and also,
\begin{eqnarray}
\frac{d \sigma} {dt} (\pi^0 \rho^+ \longrightarrow \pi^+ \gamma)&=&\frac {d
\sigma} {dt} (\pi^0 \rho^- \longrightarrow \pi^- \gamma)\nonumber\\
&=& \frac {\alpha g_\rho^2} {48sp_{\rm c.m.}^2} \left [ \frac {9} {2} -
\frac {s}{m_\rho^2} - \frac {4(m_\rho^2-4m_\pi^2)s} {(s-m_\pi^2)^2}
\right. \nonumber\\ &+& \frac {(s-m_\pi^2)^2 - 4m_\rho^2(m_\rho^2-4m_\pi^2)}
{(t-m_\rho^2)^2} \nonumber\\
&+& \frac {1} {t-m_\rho^2} \left (5(s-m_\pi^2)- \frac {(s-m_\pi^2)^2}
{m_\rho^2} \right.\nonumber\\
 & - & \left. \left. {\frac {4(m_\rho^2 - 4 m_\pi^2)}{s-m_\pi^2} }
(s-m_\pi^2+m_\rho^2) \right ) \right ]
\end{eqnarray}
In the above set of equations, $s, \ t, \ u$ are the Mandelstam variables
and $p_{\rm c.m.}$ is the three momentum of the interacting particles
in their centre of mass frame. Typical results are shown in
Fig.~\ref{check}.

Xiong {\it et al.}~\cite{XIONG} and Song~\cite{SONG} first introduced
the $\pi \rho \rightarrow a_1 \rightarrow \pi \gamma$ channel for
photon production in hadronic phase, whereas baryonic processes and
medium modification were included by Alam {\it et al.}~\cite{alam1,alam2}.
Several refinements, e.g., inclusion of strange sector, use of massive
Yang-Mills theory and t-channel exchange of $\omega$ mesons, were
incorporated by Turbide {\it et al.}~\cite{TRG}. The last calculation
is essentially the state of art result at the moment. Photon spectra
considering a complete leading rate from QGP~\cite{AMY} and exhaustive
reactions in hadronic matter~\cite{KLS,TRG} are shown in Fig.~\ref{phot_rate}.

\begin{figure}[t]
\centering
\includegraphics[height=5.0cm, width=6.0cm]{rate.eps}
\caption{Complete leading order rates from QGP and
exhaustive reactions in hadronic matter~\cite{KLS,AMY,TRG}.}
\label{phot_rate}
\end{figure}
\section{Photons from Pb+Pb@SPS to Au+Au@RHIC}

\subsection{SPS}

In order to get an idea of greatly increased insights  provided by
single photon production, let us briefly recall some of the important
results from the SPS era, many of which preceded the large strides
made in our theoretical understanding mentioned above. The first hint
of single photon production, which later turned out to be the upper
limit of their production, came from the $S+Au$ collisions studied at the
SPS energies~\cite{wa80}.

These results were analyzed in two different scenarios by authors
of Ref.~\cite{ss}. In the first scenario, a thermally and chemically
equilibrated quark-gluon plasma was assumed to be formed at some
initial time ($\tau_0\approx$ 1 fm/$c$), which expanded~\cite{assb},
cooled, and converted into a mixed phase of hadrons and QGP at a
phase-transition temperature, $T_C \approx$ 160 MeV. When all the
quark-matter was converted into a hadronic matter, the hot hadronic
gas continued to cool and expand, and underwent a freeze-out at a
temperature of about 140 MeV. The hadronic gas was assumed to consist
of $\pi$, $\rho$, $\omega$, and $\eta$ mesons, again in a thermal and
chemical equilibrium. This was motivated by the fact that the included
hadronic reactions involved~\cite{KLS} these mesons. This was already
a considerable improvement over a gas of mass-less pions used in the
literature at that time. In the second scenario, the collision was
assumed to lead to a hot hadronic gas of the same composition. The
initial temperature was determined by demanding that the entropy of
the system be determined from the measured particle rapidity
density~\cite{HK}. It was found that the scenario which did not
involve a formation of QGP led to a much larger initial temperature
and a production of photons which was considerably larger than the
upper limit of the photon production, and could be ruled out. The
calculation assuming a quark-hadron phase transition yielded results
which were consistent with the upper limit of the photon production.
These results were confirmed~\cite{wa80_ana} by several calculations
exploring different models of expansion (see left panel of
Fig.\ref{fig_wa98}).

\begin{figure}[t]
\centering
\includegraphics[height=4.50cm, width=5.50cm]{s_au.eps}
\includegraphics[height=4.20cm, width=5.50cm]{wa98_dks.eps}
\caption{ Left panel: Upper limits at the 90 \% confidence level on
the invariant excess photon yield for the 7.4 \% $\sigma_{mb}$ most
central collisions of $200A$ GeV $^{32}S+Au$~\cite{wa80}. The solid
and the dashed curves give the thermal photon production expected
from hot hadron gas calculation~\cite{ss}, while the dotted curve
is the calculated thermal photon production expected in the case
of a QGP formation. Right panel: Single photon production in $Pb+Pb$
collision at CERN SPS. Prompt photons are estimated using pQCD
(with a K-factor estimated using NLO calculation) and intrinsic
$k_T$ of partons~\cite{DKS_WA98}.}
\label{fig_wa98}
\end{figure}

It was soon realized that one may not limit the hadronic gas to contain
just $\pi$, $\rho$, $\omega$, and $\eta$ mesons, as there was increasing
evidence that perhaps all the mesons and baryons were being produced in
a thermal and chemical equilibrium in such collisions. Thus authors of
Ref.~\cite{Clay} explored the consequences of using a hadronic gas
consisting of essentially all the hadrons in the Particle Data Book,
in a thermal and chemical equilibrium. This led to an interesting
result for the $Pb+Pb$ collision at SPS energies, for which experiments
were in progress. It was found that with the rich hadronic gas, the
results for the production of photons in the phase-transition and no
phase transition models discussed above were quite similar, suggesting
that measurement of photons at the SPS energy could perhaps not
distinguish between the two cases. However, in a very important
observation, it was also noted that the calculations involving hot
hadronic gas at the initial time would lead to hadronic densities of
several hadrons/fm$^3$, and while those involving a quark gluon plasma
in the initial state would be free from this malady. Thus, it was
concluded that the calculations involving a phase transition to
QGP offered a more natural description.

The WA98 experiment~\cite{subtraction}, reported the first observation
of direct photons in central $158A$ GeV $Pb+Pb$ collisions studied at the
CERN SPS. This was explained~\cite{ss1} in terms of formation of quark
gluon plasma in the initial state (at $\tau_0\approx$ 0.2 fm/$c$), which
expanded, cooled and hadronized as in Ref.~\cite{Clay} (see right panel
of Fig.~\ref{fig_wa98}). An independent confirmation of this approach was
provided by an accurate description~\cite{dks_na50} of excess dilepton
spectrum measured by the NA60 experiment for the same system.

Once again the results for single photons  were analyzed by several
authors using varying models of expansion as well as rates for production
of photons; viz., with or with-out medium modification of hadronic properties
(see, e.g., Ref.~\cite{TRG,sps1,sps2}). The outcome of all these efforts can
be summarized as follows: the single photon production in $Pb+Pb$ collisions
at SPS energies can be described either by assuming a formation of QGP in
the initial state or by assuming the formation of a hot hadronic gas whose
constituents have massively modified properties. The later description,
however, involved a hadronic density of several hadrons/fm$^3$, which
raises doubts about the applicability of a description in terms of hadrons,
as suggested by Ref.~\cite{Clay}.

\subsection{RHIC}
\begin{figure}[b]
\centering
\includegraphics[height=6.2cm]{rhic1.eps}
\caption{Direct photon results for $Au+Au$ collisions (left panel) and
$p+p$ collisions (right panel) at $\sqrt{s_{NN}} = 200A$ GeV~\cite{rhic}.}
\label{rhic_fig}
\end{figure}

The first experimental photon result obtained from the Relativistic
Heavy Ion Collider was the centrality dependent single photon data
for $200A$ GeV $Au+Au$ collisions by PHENIX collaboration~\cite{rhic},
where single photons were identified clearly for  $p_T > 4$ GeV. For
central collisions, the lower bound on photon transverse momenta was
even much lower, upto 2 GeV (Fig.~\ref{rhic_fig}). This data was in
good agreement with NLO pQCD results for $pp$ collisions, scaled by
number of binary collisions ( without considering the iso-spin effect).

Thermal radiation dominates the direct photon spectrum at lower values
of $p_T$ ($ \le 3$ GeV) and the measured slope of the thermal photon
spectrum can be related to the temperature of the system.

In Fig.~\ref{ther_phot_rhic} results from several theoretical models
based on hydrodynamics for thermal photon production at RHIC are shown.
The initial temperature and thermalization time for $Au+Au$ collision at
$200A$ GeV quoted by several theoretical groups~\cite{TRG, ss1, sps1, sps2,
anrds, Ente_Press} are in the range of 450 - 650 MeV  and about 0.2
fm/$c$ respectively.

All these calculations are comparable to the experimental data and
with each others within a factor of 2 and also confirm the dominance
of thermal radiation in the direct photon spectrum in low and
intermediate $p_T$ range.
\begin{figure}
\centering
\includegraphics[height=4.5cm, width=5.5cm]{peressounko.eps}
\caption{Thermal photon spectra for central $200A$ GeV $Au+Au$ collision
computed within different models ~\cite{Ente_Press} compared to the
expected pQCD prompt photon yield and to the experimental total direct
photon spectrum measured by PHENIX~\cite{Phenix}.}
\label{ther_phot_rhic}
\end{figure}

\subsubsection{Indications for jet conversion photons}

We have already discussed the procedure for calculating the high
energy photons due to passage of jets through the quark gluon plasma.

The  parametrized $p_T$ distribution of jets (quarks, anti-quarks, and
gluons) obtained by using CTEQ5L parton distribution function
and EKS98 nuclear modification factor is given by:
\begin{eqnarray}
\frac{dN^{\rm jet}}{d^2p_Tdy}|_{y=0}=
T_{AA}\frac{d\sigma^{\rm jet}}{d^2p_Tdy}|_{y=0}=K \frac{a}
{(1+p_T/b)^c}
\end{eqnarray}
Here $T_{\rm AA}=9A^2/8\pi {R_{\perp}}^2$ is the nuclear thickness
for a head-on collision and to include the higher order effects
a $K$ factor of $2.5$ is introduced. Numerical values for the
parameters for quarks, anti-quarks and gluons
can be found in the Ref.~\cite{fms_phot}.

Now as a first step, let us ignore the transverse expansion of the
plasma and assume that a thermally and chemically equilibrated plasma
is produced in the collision at an initial time $\tau_0$ at temperature
$T_0$. In an isentropic longitudinal expansion, ${T_0}$ and ${\tau_0}$
are related by the observed particle rapidity density ($dN/dy$) by:
\begin{eqnarray}
{T_0}^3{\tau_0}= \frac{2 {\pi}^4}{ 45 \zeta(3)}\frac{1}{ 4 a \pi
R_\perp^2}\frac{dN}{dy}
\end{eqnarray}
where $a=42.25 \pi^2/90$, for QGP consisting of $u, d$ and $s$ quarks and
gluons. $dN/dy$ can be taken as $\simeq 1260$, based on the charge particle
pseudo-rapidity density measured by PHOBOS experiment~\cite{Phobos_coll}
for central collision of $Au$ nuclei at ${\sqrt s_{NN}}=200$ GeV. For
central collision of $Pb$ nuclei at LHC energies $dN/dy \simeq 5625$
was used by Fries {\it et al.}~\cite{fms_phot} as in Ref.~\cite{prl25}.

Further assuming a rapid thermalization, initial conditions can then
be estimated as~\cite{prl25} $T_0=446$ MeV and $\tau_0=0.147$ fm/$c$
for the RHIC and $T_0=897$ MeV and $\tau_0= 0.073$ fm/$c$ for the LHC.
Now, taking the nuclei to be  uniform spheres, the  transverse profile
for initial temperature is given by $T(r)=T_0[2(1-r^2/{R_{\perp}}^2)
]^{1/4}$ where $R_{\perp}=1.2A^{1/3}$ fm. The same profile
$R(r)=2(1-r^2/{R_{\perp}}^2)$ is used for the jet
production.

The jet traverses the QGP medium until it reaches the surface or
until the temperature drops to the transition temperature $T_c$
($\sim 160$ MeV).
\begin{figure}
\centering
\includegraphics[height=4.6cm, width=5.50cm]{fms_rhic.eps}
\includegraphics[height=4.6cm, width=5.50cm]{fms_lhc.eps}
\caption{Jet conversion photons at RHIC [left panel] and LHC
[right panel] energies (taken from Ref.~\cite{fms_phot}).}
\label{FMS_rhic_lhc}
\end{figure}

First estimates for the jet conversion photons at RHIC and LHC energies
along with other sources of photons having large transverse momentum are
given in Fig.~\ref{FMS_rhic_lhc}. We see that the jet conversion photons
make a fairly large contribution in the $p_T$ range of 4-10 GeV.

\begin{figure}[t]
\centering
\includegraphics[height=4.7cm]{fms.eps}
\caption{Photon yield as a function of $p_T$ in central (0-10\%)
$Au+Au$ collisions at $200A$ GeV. Results of jet-photon conversion
(solid and labeled), primary hard photons (dashed) and the sum of
the two (upper-most solid curve) along with thermal photons are
shown separately. Data are from the PHENIX collaboration~\cite{Phenix}.}
\label{FMS_phot}
\end{figure}

The centrality dependence of jet conversion photons was also studied by
Fries {\it et al.}~\cite{fms_phot2} (see figure~\ref{FMS_phot}), which
indicates a small but clear contribution of photons due to passages of
jets through the plasma. These early calculations have been brought to
a high degree of sophistication by the McGill group~\cite{TGFH} (see
figure~\ref{gale_evan_1_2}), where jet quenching and jet-photon conversion
along with bremsstrahlung is treated in a single frame work. The details
can be seen in Ref.~\cite{TGFH}.

In Fig.{\ref{gale_evan_1_2}} the results for thermal photons, direct
photons due to primary processes, bremsstrahlung photons and the
photons coming from jets passing through the QGP in central collision
of gold nuclei at RHIC energies is plotted. The quark jets passing
through the QGP give rise to a large yield of high-energy photons. For
RHIC this contribution is dominant source of photons up to $p_T \simeq
6$ GeV. Due to multiple scattering suffered by the fragmenting
partons a suppression of the bremsstrahlung contribution is found. This
will further enhance the importance of the jet-photon conversion process.

\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{gale1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{gale2.eps}
\caption{Yield of photons in $Au+Au$ collisions at RHIC, for centrality
classes $0-10\%$ (left panel) and $0-10\%$ (right panel). See~\cite{TGFH}
for details. The data sets are from Refs.~\cite{HB} and ~\cite{Phenix}
respectively.}
\label{gale_evan_1_2}
\end{figure}

Obviously a high statistics data at several centralities and energies
will go a long way in clearly establishing the presence of jet-conversion
photons at RHIC and LHC energies. This is also of importance, as there are
indications that these photons measure the initial spatial anisotropy of
the system~\cite{TGFH}.

\section{Dileptons}

\begin{figure}
\centering
\includegraphics[height=3.50cm ]{dilepton_prod.eps}
\caption{ Dilepton production from (a) quark anti-quark annihilation and
(b) $\pi^+ \pi^-$ annihilation through $\rho$ channel.}
\label{dil_dia}
\end{figure}

Virtual photons or dileptons are also very powerful and efficient
probes like the real photons, to study the dynamics of heavy ion
collisions and the properties of the medium created in the
collisions. Real photons are massless, whereas dileptons are massive.
Thus, invariant mass $M$ and the transverse momentum $p_T$ are the
two parameters available for dileptons, which can be tuned
to investigate the different stages of the expanding fireball. Dileptons
having large invariant mass and high $p_T$, are emitted very early,
soon after the collision when the temperature of the system is very
high. On the other hand, those having lower invariant masses come out
later from a relatively cooler stages and at low temperatures.
\begin{figure}
\centering
\includegraphics[height=5.50cm, width= 6.0cm]{dilepton.eps}
\caption{Expected sources of dilepton production as a function of
invariant mass in relativistic heavy ion collisions (schematic).}
\label{dil_mass}
\end{figure}

Similar to the real photons, dileptons  are also emitted from every
stage of heavy ion collisions~\cite{wong}. In the QGP phase, a quark
can interact with an anti-quark to form a virtual photon $\gamma^*$,
which subsequently decays in to a lepton ($l^-$) and an anti-lepton
($l^+$) pair, together known as dilepton. In the hadronic
phase, dileptons are produced from interactions of charged hadrons
with their anti-particles by processes like ($\pi^+ + \pi^- \rightarrow
l^+ + l^-$) (see Fig.~\ref{dil_dia}). They are also produced from decay
of hadronic resonances
like $\rho$, $\omega$, $\phi$, and $J/\Psi$ as well as from Drell-Yan
process. In the Drell-Yan process, a valence quark from a nucleon in
the projectile nucleus interacts with a sea anti-quark from a nucleon
in the target nucleus to form a virtual photon, which decays into a
lepton pair. The dilepton
emissions can be classified into three distinct regimes in a rough
estimation, depending on the invariant mass $M$ of the emitted lepton
pairs. These are:

a) Low Mass Region (LMR): $M \le M_{\phi}$(= 1.024 MeV): In this
mass range, vector meson decays are the dominating source of dilepton
production and medium modified spectral density is one of the key issues
which needs to be addressed.

b) Intermediate Mass Region (IMR): $ M_{\phi}<  M < M_{J/\Psi}$(=
3.1 GeV): In intermediate mass region, continuum radiation from QGP
dominates the dilepton mass spectrum and thus this region is important
for getting a pure QGP signature.

c) High Mass Region (HMR): $M \ge M_{J/\Psi}$: In the HMR, the most
interesting phenomenon are the primordial emission and heavy quarkonia
like $J/\Psi$  and $\Upsilon$ suppression.

A schematic diagram of continuous dilepton mass spectrum is shown in
Fig.~\ref{dil_mass}.

\subsection{Dileptons from QGP and hadronic phase}

Dilepton production by quark anti-quark annihilation  process or by
$\pi^+$ $\pi^-$ annihilation process can be expressed in a general
form as,
\begin{equation}
{a^+ + a^- \rightarrow  l^+ + l^-}
\end{equation}
where, the particle `a' can be either a quark or a $\pi$ meson. We
closely follow the treatment of Ref.~\cite{KKMM} for calculation of
dilepton spectra in this section. From Quantum Electrodynamics (QED),
the cross section for $e^+ e^- \rightarrow \mu^+ \mu^-$ can be
written as,
\begin{eqnarray}
\tilde\sigma(M)= \frac {4 \pi } {3} \frac {\alpha^2} {M^2} \left [1+ \frac
{2m_l^2}{M^2} \right] \left [1- \frac {4m_l^2} {M^2} \right]^{1/2}
\label{cross}
\end{eqnarray}
In the above equation, $\alpha$ is the fine structure constant
($\approx 1/137$), $m_l$ is the mass of $\mu$, and M is the dilepton invariant
mass. For the quark anti-quark annihilation process, this cross-section
is modified by inclusion of a colour factor $N_c (= 3)$, and the fractional
charges of the up(u) and down(d) quarks. Thus, the new cross-section takes
the form as,
\begin{eqnarray}
\sigma_q(M) & = & F_q \tilde \sigma(M),\\
 F_q  & = & N_c(2s+1)^2 {\sum_{f} e_f^2}
\end{eqnarray}
where s is the spin of quarks, $e_f$ is the fractional charge, and the
sum is over different quark flavour `f'. If we consider only u and d
quarks having fractional charges 1/3 and 2/3 respectively, then the factor
$F_q$ is 20/3.

In the hadronic phase, the $\pi^+\pi^-$ annihilation from vector
meson dominance model can be expressed as,
\begin{equation}
\pi^+ + \pi^- \rightarrow \rho \rightarrow l^+ + l^-
\end{equation}
For this case, the QED cross-section is multiplied by form factor
$F_\pi(M)$ which is of Breit-Wigner form as;
\begin{equation}
F_\pi(M)= {\frac {m_\rho^4} {(m_\rho^2-M^2)^2+m_\rho^2\Gamma_\rho^2}}
\end{equation}
Here, $M_\rho$ is the mass of the $\rho$ meson ($\sim 770$ MeV) and
$\Gamma_\rho$ ( $\sim 155$ MeV ) is the decay width. Thus, the total pion
cross-section as a function of M becomes
\begin{equation}
\sigma_\pi(M)=F_\pi \tilde \sigma(M)(1-4m_\pi^2/M^2)^{1/2},
\end{equation}
The reaction rate `R' can be obtained from the kinetic theory:
\begin{equation}
R(a^+a^- \rightarrow l^+ l^-) = \int \frac {d^3p_1} {(2\pi)^3} f({\bf p_1})
\int \frac {d^3p_2} {(2\pi)^3} f({\bf p_2})\sigma(a^+a^- \rightarrow
l^+l^-;{\bf p_1p_2})v_{\rm rel}
\label{rate}
\end{equation}
where,
\begin{equation}
v_{rel}= {\frac {[(p_1.p_2)^2 - m_a^4]^{1/2}} {E_1E_2}}
\nonumber\\
\end{equation}
$f({\bf p})$, is the occupation probability at momentum ${\bf p}$ and
energy $E=\sqrt{{\bf p}^2 + m_a^2}$. Now, using the distribution
function $f({\bf p})\sim e^{-E/T}$, and integrating over five out
of the six variables, the reaction rate takes a simplified form as,
\begin{equation}
R(T)= \frac {T^6} {(2\pi)^4} \int_{z_0}^{\infty} \sigma(z)z^2(z^2-4z_a^2)
K_1(z)dz
\end{equation}
Here, $z=M/T$, $z_a=m_a/T$, and $K_1$ is the modified Bessel function of
the first kind. The value of the parameter $z_0$ is taken as the larger
of $2m_a/T$ and $2m_l/T$. Now, for massless $u$ and $d$ quarks ( as $m_u$,
$m_d$ $\ll T$), the $e^+e^-$ emission rate takes a simple form of $T^4$
law,
\begin{equation}
R= \frac {10} {9\pi^3} \alpha^2T^4 .
\end{equation}
One can also estimate the relaxation time for lepton pairs to come
to equilibrium with the QGP:
\begin{equation}
{t_{\rm rel}= {\frac {n_{\rm eq}^l} {2R}} = {\frac {9 \pi}
{10 \alpha^2 T}}}  .
\end{equation}
For a temperature range of 200 to 500 MeV, the value of $t_{\rm rel}$
varies from $20 - 60 \times 10^3$ fm/$c$. We know that the life time of
the QGP phase is only $\sim 10$ fm/$c$, which is more than three order
of magnitude smaller than $t_{\rm rel}$. Thus, like the real photons,
the produced lepton pairs also escape the system without suffering
significant absorption in the medium.

Now, the dilepton emission rate $R$ is actually defined as the total number
of lepton pairs emitted from a 4-volume element $\ d^4x \ (= \ d^2x_T \
d\eta \ \tau \ d\tau)$ at a particular temperature $T$ is given by:
\begin{equation}
R=dN/d^4x .
\end{equation}
Thus, the rate of production of dileptons having invariant mass $M$
can be expressed using Eqn.(\ref{rate}) as,
\begin{eqnarray}
\frac{dN} {d^4xdM^2}= \frac {\sigma(M)}{2(2\pi)^4}M^3TK_1(M/T) \left[1-
\frac {4m_a^2}{M^2} \right].
\end{eqnarray}

From the last equation and using properties of the modified Bessel's
functions, the production rate per unit 4-volume for total energy
$E$, momentum $p$ and invariant mass $M$ (where, $E = \sqrt{ p^2 +
M^2}$ ) can be written as,
\begin{eqnarray}
E \frac {dN}{d^4xdM^2d^3p} = \frac {\sigma(M)} {4(2 \pi)^5} M^2 e^{-E/T}
\left[1- \frac{4m_a^2}{M^2}\right] .
\end{eqnarray}
Integrating the rate of emission over the entire 4-volume from QGP and
hadronic phase, one can obtain the $p_T$ spectra at a particular $M$ as,
\begin{eqnarray}
\frac {dN} {dM^2 d^2p_T dy} & = & \int \ \tau \ d\tau \ r \ dr \ d\eta \
d\phi  \left [ \left [E \frac {dR} {dM^2 d^3p} \right ]_{\rm QGP}
\it{f}_{\rm QGP} (r, \tau)  \right. \nonumber\\ & + & \left. \left [E
\frac {dR} {dM^2 d^3p} \right ]_{\rm HM}  \it{f}_{\rm HM}(r, \tau) \right]
\label{dil_spec}
\end{eqnarray}
where, $\eta $ is the space time rapidity. In the above equation,
the temperature $T$, QGP and hadronic matter (HM) distribution
functions ($\it{f}_{\rm QGP}, \it{f}_{\rm HM}$), all are functions
of space $\bf{r}$ (x, y), transverse velocity $v_T$, and proper time
$\tau$. Eqn.~(\ref{dil_spec}) can be solved
numerically by using any appropriate model [e.g., Ref.~\cite{KKMR}]
with a proper equation
of state (EOS). One can also get the invariant mass spectrum by
integrating out the variable $p_T$ from Eqn.~(\ref{dil_spec}).
The invariant mass spectrum of thermal dilepton is dominated
by QGP radiation above $\phi$ mass and hadronic radiation
outshines the QGP contribution for $M \le M_{\phi}$.

\subsection{Medium modification}

As mentioned earlier, in the low mass region, dilepton emission is
largely mediated by $\rho(770)$, a broad vector meson, as a result
of its strong coupling to the $\pi \pi$ channel and a short lifetime,
which is about 1.3 fm/$c$. In-medium properties of vector mesons, like
change in medium mass and width have long been considered as prime signatures
of a hot and dense hadronic medium. CERES/NA45~\cite{ceres, NA45} started the
pioneering experiment on dilepton measurement during the period 1989-1992 at
CERN SPS.
For $p+Be$ and $p+Au$ collisions, the theoretical
estimates, considering only hadron decay (as source of dilepton
production), were in excellent agreement with the experimental data
points. However, the CERES/NA45 experimental data for $S+Au$ and $Pb+Au$
at $200A$ GeV and $158A$ GeV respectively exhibit excess radiation
beyond the electromagnetic final state decay of produced hadrons below
the $\phi$ mass. From these observations, it was concluded that, the
theoretical models based on $\pi \pi$ annihilation can reproduce the
experimental data, only if the properties of the intermediate $\rho$
meson is modified in the medium. This was an exciting discovery and
a genuine consequence of many body physics. However, the resolution and
statistical accuracy of the data was insufficient to distinguish between
models suggesting a drop in the mass of $\rho$ mesons~\cite{BR} and those
which suggest an increase in their decay width~\cite{RW} and thus,
determine the in-medium spectral properties of the $\rho$ meson.

Shortly after this, an excess dimuon (over the sources expected from
$pA$  measurement)  was identified experimentally by Helios/3~\cite{heli}
(measured both $e^+ e^-$ and $\mu^+ \mu^-$) and NA38/NA50~\cite{na50}
collaborations.
\begin{figure}[ht]
\centering
\includegraphics[height=4.2cm, width=5.50cm]{ceres1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{ceres2.eps}
\caption{ Left panel: Invariant mass spectrum of $e^+ e^-$ pairs
emitted in $158 A$ GeV $Pb+Au$ collisions from the combined analysis
of 1995 and 1996 data by CERES/NA45~\cite{ceres,NA45}. The solid line
shows expected yield from hadron decay and dashed lines indicate the
individual contribution to the total yield. Right panel: Comparison
of the experimental data to i) free hadron decays without $\rho$
decays (thin solid line), ii) model calculations with a vacuum $\rho$
spectral function (thick dashed line), iii) with dropping in-medium
$\rho$ mass (thick dash-dotted line), iv) with a medium modified
$\rho$ spectral function (thick solid line).}
\label{ceres_1_2}
\end{figure}

Rapp and Shuryak~\cite{rapp_sur} and Kvasnikova {\it et al.}~\cite{dks_na50}
showed that, the excess dimuon observed by NA50 in the mass region $ 1.5
< M  < 2.5$ GeV can be explained by thermal signal without invoking any
anomalous enhancement in the charm production. Kvasnikova {\it et al.}
studied the NA50 intermediate mass dimuon result using hydrodynamic
model and a detailed analysis of the rates of dilepton production from
hadronic phase~\cite{dks_na50}. Detector resolution and acceptance were
accurately modeled in their calculation and the normalization was
determined by a fit to the Drell-Yan data
using the MRSA parton distribution function as done in the experimental
NA50 analysis. The results are shown in Fig.~\ref{DKS_na50}. They also
studied the centrality dependence of the NA50 data using hydrodynamics
by incorporating azimuthal anisotropy in the calculation. The results
were in fairly good agreement with the measured excess dilepton data,
as can be seen in the third panel of Fig.~\ref{DKS_na50}.
\begin{figure}
\centering
\includegraphics[height=3.2cm, width=3.60cm]{dks_11.eps}
\includegraphics[height=3.2cm, width=3.60cm]{dks_12.eps}
\includegraphics[height=3.6cm, width=3.20cm, angle=90]{dks_13.eps}
\caption{(a) The dimuon invariant mass spectrum after correcting for
detector acceptance and resolution and NA50 data. The Drell-Yan and
thermal contributions are shown separately along with correlated charm
decay and direct decays of the $J/\Psi$ and $\bar{J}/\Psi$. (b) The
dimuon transverse momentum spectrum. (c) Centrality dependent
results from Ref.~\cite{dks_na50} and NA50 data~\cite{na50}.}
\label{DKS_na50}
\end{figure}

\subsubsection{Dropping $m_{\rho}$ vs. increasing $\Gamma_{\rho}$}

\begin{figure}
\centering
\includegraphics[height=4.25cm, width=5.0cm]{rapp_1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{na60-med.eps}
\caption{ Left panel: In medium vector meson spectral functions
from hadronic many body theory. The $\rho$ in cold nuclear matter
at several densities proportional to the saturation density $\rho_0
=0.16$ fm$^{-3}$. Right panel: NA60~\cite{NA60} excess dimuons in
central $In+In$ collisions at SPS compared to thermal dimuon radiation
using in-medium electromagnetic rates~\cite{Hees_Rapp}.}
\label{rapp_1}
\end{figure}

The CERES data~\cite{ceres,NA45} was unable to distinguish between
the models suggesting drop in the mass of $\rho$ meson and those
suggesting an increase in its width, as seen in Fig.~\ref{ceres_1_2}.
The high statistics data obtained by the NA60 Collaboration for $In+In$
collisions at SPS energies for the
dimuon excess rather clearly ruled out the models advocating the
dropping mass scenarios. They rather firmly establish the models
advocating the substantial broadening of the decay width of the
$\rho$ meson due to many body effects (see Fig~\ref{rapp_1}).

\section{Elliptic Flow}

Elliptic flow is one of the key observables in relativistic collisions
of heavy nuclei, which confirms the collectivity and early thermalization
in the created hot and dense matter. For a non-central collision (impact
parameter $b \ne 0$ ) of two spherical nuclei, the overlapping zone
between the nuclei no longer remains circular in shape, rather it takes
an almond shape. This initial spatial anisotropy of the overlapping zone
is converted into momentum space anisotropy of particle distribution via
the action of azimuthally anisotropic pressure gradient, which gives rise
to elliptic flow~\cite{olli}. Note that, this anisotropic flow or elliptic
flow can also be produced in central collisions of deformed nuclei, such
as $U+U$ collisions~\cite{uli_khul}. The driving force for momentum
anisotropy is the initial spatial eccentricity $\epsilon_x$ ($ = <y^2 -
x^2>/<y^2 + x^2>$) and the momentum anisotropy can grow as long as
$\epsilon_x > 0$~\cite{QGP3}. Elliptic flow coefficient $v_2$ is quantified
as the second Fourier co-efficient of particle distribution in the $p_T$
space, which is of the form:
\begin{eqnarray}
\frac{dN(b)}{p_T dp_T \, dy \,d\phi} & = & \frac{dN(b)}{ 2 \pi p_T \,
dp_T \, dy} (b)\times [1+ 2 v_1(p_T,b) \cos( \phi)
\nonumber\\
& + & 2 v_2(p_T,b) \cos(2 \phi)+ 2 v_3(p_T,b) \cos(3 \phi)+ \dots ].
\end{eqnarray}
At mid-rapidity ($y=0$) and for collisions of identical nuclei, only
the even cosine terms survive in the above Fourier series and $v_2$
is the lowest non-vanishing anisotropic flow co-efficient. Now, the
value of $v_2$ depends on the impact parameter $b$, transverse
momentum $p_T$ as well as on the particle species through their
rest masses $m$. For massless real photons, $v_2$ depends on the
elliptic flow of parent particles.

\subsection{Thermal Photon $v_2$}

Elliptic flow co-efficient $v_2$ for photons is a much more powerful
tool than the $v_2$ of hadrons, to study the evolution history of
the system depending on the different emission time and production
mechanism of photons compared to hadrons. Photons are emitted from
all stages and throughout the evolution of the system, whereas,
hadrons are emitted only from the freeze-out surface at a relatively
much cooler temperature ($\sim 100$ MeV). The interplay of the photon
contributions from fluid elements at different temperatures with
varying radial flow pattern, gives the photon $v_2$ a richness,
which is not possible for hadrons. Also, photons emitted from QGP
phase as a result of $q \bar q$ annihilation and quark gluon Compton
scattering, carry the momenta of the parent quarks or anti-quarks,
which makes them an unique tool for QGP study.
\begin{figure}[t]
\centering
\includegraphics[height=4.2cm, width=5.50cm]{v2_7_cor.eps}
\includegraphics[height=4.2cm, width=5.50cm]{v2b_pt.eps}
\caption{Left panel: $v_2(p_T)$ for thermal photons along with the
$v_2$ for $\pi$ and $\rho$ mesons. Contributions the from quark matter
and  the hadronic matter are shown separately with the sum contribution.
Right panel: Thermal photon $v_2$ for different centrality bins [taken
from Ref.~\cite{CFHS}].}
\label{phot_v2_pt}
\end{figure}

Ideal hydrodynamic model has successfully predicted the elliptic flow
for different hadronic species upto a $p_T$ value of 1.5(2.3) GeV for
mesons (baryons)~\cite{QGP3} at RHIC. Experimental data show that,
$v_2(p_T)$ for hadrons saturates beyond that $p_T$ range. However,
ideal hydrodynamics predicted $v_2(p_T)$ rises monotonically with $p_T$.
The saturation at higher $p_T$ is is often explained as due to viscosity,
neglected in the discussion here.

Fig.~\ref{phot_v2_pt} shows the $p_T$ dependent $v_2$ for thermal
photons~\cite{CFHS} for a typical impact
parameter $b = 7$ fm. Contributions from quark matter (QM) and  hadronic
matter (HM) along with the sum of the two are shown separately (left panel
of Fig.~\ref{phot_v2_pt}). One can see that, $v_2(HM)$ rises monotonically
with $p_T$, similar to the hadronic $v_2$ predicted by
hydrodynamics. On the other hand, $v_2(QM)$ is very small at high $p_T$ or
early times, as very little flow is generated by that time. $v_2(QM)$
rises for smaller values of $p_T$ and after attaining a peak value around
$1.5 - 2.0$ GeV, it tends to 0 as $p_T$ $ \rightarrow 0$. The total flow or
$v_2$(QM+HM) tracks the $v_2(QM)$ at high $p_T$ in-spite of very large
$v_2(HM)$, as the yield of photons from hadronic phase is very small at
high $p_T$. It is well known and mentioned earlier that, for photon
energy  larger than the rest masses of the photon emitting  particles,
the photon production cross-section peaks when the photon momentum and
momentum of photon emitting particles become almost identical~\cite{wong}.
Thus, at high $p_T$ photon $v_2$ reflects the anisotropies of the quarks
and anti-quarks at early times. Also, the collision induced conversion
of vector mesons $(\rho)$ starts dominating the photon production for
$p_T \ge 0.4$ GeV, which gives rise to a structure at the transition
point in the photon $v_2$ curve~\cite{CFHS}. As HM contribution dominates
the $p_T$ spectrum and hydrodynamics is well applicable around that $p_T$
range, the structure is expected to survive in the experimental result also.

Recently Turbide {\it et al.}~\cite{TGFH} have shown that, total contribution
to photon $v_2$ from prompt fragmentation (small $+ \ ve \ v_2$) and
jets-conversion (small $- \ ve \ v_2$) photons is very small, almost
equal to zero. Also, we know that the prompt photons from Compton and
annihilation processes do not contribute to elliptic flow as their
emission is not subjected to collectivity and has azimuthal symmetry.
Thus, at low and intermediate
$p_T$ range, $v_2$ from thermal photons plays a dominant role in deciding
the nature of the direct photon $v_2$.
\begin{figure}[t]
\centering
\includegraphics[height=4.2cm, width=5.50cm]{v_2_b.eps}
\includegraphics[height=4.2cm, width=5.50cm]{v_2_t.eps}
\caption{ Left panel: $p_T$ integrated elliptic flow of thermal photons
from different phases at RHIC. Right panel: Time evolution of elliptic
flow from different phases, normalized by the final value at the time of
freeze-out [taken from Ref.~\cite{CHS}].}
\label{evolution}
\end{figure}

For central collision of two spherical  nuclei, the spatial eccentricity
($\epsilon$) of the overlapping zone is zero, thus the flow co-efficient
$v_2$ is also zero. The $v_2(p_T)$ for thermal photons for different
centrality bins are shown in the right panel of Fig.~\ref{phot_v2_pt},
where the value of $v_2$ rises from central towards peripheral collisions.
Now, the elliptic flow as well as spatial eccentricity both rise with
higher values of impact parameter. However, the ratio of the two remains
independent of impact parameter up to a very large value of b~\cite{CHS}.
The $p_T$ integrated $v_2$ for thermal photons from different phases as
a function of collision centrality shows an interesting behavior (left
panel of Fig.~\ref{evolution}). The $v_2(b)$ rises with b until
for very peripheral collisions. For these the system size itself becomes
very small to generate enough pressure gradient and elliptic flow, and
as a result $v_2(b)$ decreases. The $p_T$ dependent as well as $p_T$
integrated time evolution results~\cite{CHS} of thermal photon spectra
and elliptic flow show explicitly the gradual build up of spectra and
$v_2$ with time very well. The  photon  $v_2$ from QGP phase saturates
within about 4-5 fm/$c$ (right panel of Fig.\ref{evolution}),
whereas $v_2$ from hadronic phase is not very significant at early
times and it saturates much later.

\subsection{Thermal Dilepton $v_2$}
\label{sec:4}

Elliptic flow  of thermal dileptons is another very interesting
and illustrative observable which gives information of the different
stages of heavy ion collision depending on invariant mass $M$ and
transverse momentum $p_T$~\cite{dil}. At the $\rho$ and $\phi$
masses, the $p_T$ spectra and $v_2(p_T)$ show complete dominance
of hadronic phase. At these resonance masses radiation from quark
matter becomes significant only for very large $p_T ( \ge 4)$ GeV
(left panel of Fig.~\ref{rho_dn_v2}).  The $v_2(p_T)$ at $M = m_\rho$
also shows similar nature as spectra and the total $v_2$ is almost
similar to hadronic $v_2$  upto a large $p_T$. Right panel of
Fig.~\ref{rho_dn_v2} shows dilepton $v_2$ at rho mass along with
$v_2$ of rho meson where, $v_2(HM)$ tracks $v_2(\rho)$. Also, the
hadronic $v_2$ is little smaller than the $v_2(HM)$ for dileptons.
The effective temperature of dilepton emission is a little larger
than for hadrons, thus the spectra of the later is boosted by a somewhat
larger radial flow~\cite{dil}, which results in smaller elliptic flow.
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{rho_dn.eps}
\includegraphics[height=4.2cm, width=5.50cm]{rho_v2.eps}
\caption{$p_T$ spectra [let panel] and $v_2$ [right panel] for thermal
dileptons at $M=m_{\rho}$. $v_2(\rho)$ is also plotted in the $v_2$ curve
for comparison~\cite{dil}.}
\label{rho_dn_v2}
\end{figure}
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{dn_2.eps}
\includegraphics[height=4.2cm, width=5.50cm]{v2_2.eps}
\caption{$p_T$ spectra [let panel] and $v_2$ [right panel] for thermal
dileptons at $M=2$ GeV~\cite{dil}.}
\label{2_dn_v2}
\end{figure}

A totally different situation emerges for the $p_T$ spectrum
and $v_2$ for dilepton invariant mass $M > M_{\phi}$. For $M = 2$
GeV, the dilepton $v_2$ is  similar to $v_2(QM)$ inspite of the 20
times larger  $v_2(HM)$, as hadronic dileptons are very few compared
to QGP radiation at this M value (Fig.~\ref{2_dn_v2}).

The $p_T$ integrated spectra and $v_2$ as a function of invariant mass
$M$ show well defined peaks at the resonance masses ($\rho, \ \omega$
and $\phi$), as can be seen Fig.~\ref{dil_spec_v2}. For $M\le M_{\phi}$
the dilepton spectrum is totally dominated by hadronic phase and above
that it is dominated by contribution from QGP. $v_2(QM)$ is very small
at large $M$ and
rises for smaller $M$. It shows a  nature similar to the thermal photon $v_2$.
The $M$ dependent $v_2(HM)$ is significant only at the resonance masses
and beyond $\phi$ mass its contribution to total $v_2$ is negligible.
Thus, for large  values of $M$ or at early times flow comes from QGP
phase and its value is very small, whereas at later times or at low
$M$ values it is  from HM, which is very strong. The measurement of
flow parameter at high $M$ and/or high $p_T$ values can be very useful
to reveal a pure QGP signature.
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{dil_spec.eps}
\includegraphics[height=4.2cm, width=5.50cm]{dil_v2.eps}
\caption{Left panel: The mass spectra of thermal
dileptons from a hydrodynamical simulation of central $200A$ GeV $Au+Au$
collision. Quark and hadronic matter contributions are shown
separately. Right panel: $p_T$ integrated elliptic flow parameter
for dileptons and various hadrons~\cite{dil}.}
\label{dil_spec_v2}
\end{figure}

\subsection{Elliptic flow of decay photons}

We have already discussed that most of the produced photons in  heavy
ion collisions are from the 2-$\gamma$ decay of $\pi^0$ and $\eta$ mesons
and the direct photons contain a small fraction of the  inclusive
photon spectrum. Thus it is very interesting to know the nature of
$v_2$ of decay photons from pion and $\eta$ decays. The momentum
distribution of decay photons from $\pi^0$ decay in an invariant
form can be expressed as~\cite{cahn},
\begin{equation}
k_0\frac{dN}{d^3k}(p, k)=\frac{1}{\pi}\, \delta(p \cdot k-\frac{1}{2}
m^2)\, ,
\label{dec}
\end{equation}
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.5cm]{decay1.eps}
\includegraphics[height=4.2cm, width=5.5cm]{decay2.eps}
\caption{Left panel: Elliptic flow of photons from $\pi^0$ decay at
mid-rapidity. $f(p_T)$ stands for the momentum distribution of the
$\pi^0$. Right panel: Elliptic flow parameters for photons from
decay of $\pi^0$ obtained using the recombination model [taken from
Ref.~\cite{decay}].}
\label{layek}
\end{figure}
where p and k are the 4-momentum of the pion and photons and m is
the pion mass. Thus the Lorentz invariant cross-section of the decay
photons using the decay kinematics from the Eqn.~(\ref{dec}) is,
\begin{equation}
k_0\frac{d\sigma}{d^3k}=\int \frac{d^3p}{E}\left(E \frac{d\sigma}{d^3p}
\right)\frac{1}{\pi} \delta (p \cdot k -\frac{1}{2} m^2) \, .
\label{dsig}
\end{equation}
Layek {\it et al.}~\cite{decay} have calculated elliptic flow of
decay photons considering several azimuthally asymmetric pion
distributions (Fig.~\ref{layek}). They have shown that $k_T$
dependent $v_2$ of decay photons closely follows the $v_2(p_T)$
of $\pi^0$ evaluated at $p_T \sim k_T + \delta$ (where $\delta \sim
0.1 - 0.2$ GeV). Similar results were obtained for decay photons
from $\eta$ mesons also. This study could be useful in identifying
additional sources of photons as the $v_2$ of $\pi^0$ is similar
to that of $\pi^+$ and $\pi^-$. Also by using the property of
quark number scaling or the recombination model, the decay photon
$v_2$ can help to estimate the $v_2$ of constituent partons in the
$\pi^0$ or $\eta$ mesons (right panel of Fig.~\ref{layek}).
See Ref.~\cite{decay} for details.

\subsubsection{Experimental measurement of direct photon $v_2$}

PHENIX has measured direct photons and its $v_2$ by subtracting
$v_2$ of decay photons (2-$\gamma$ decay of $\pi^0$ and $\eta$
mesons) from inclusive photon $v_2$ using appropriate weight
factor~\cite{Sakaguchi}. The procedure followed by them to
estimate direct photon $v_2$ is as follows:
\begin{equation}
v_2^{\rm dir.} = \frac {R \times v_2^{ \rm incl.} - v_2^{\rm bkgd.}} {R - 1}
\end{equation}
where R is the direct photon excess over hadron decays defined as,
\begin{equation}
R= \frac {(\gamma/\pi^0)_{\rm incl.}} {(\gamma/\pi^0)_{\rm bkgd.}} \nonumber
\end{equation}
\begin{figure}
\centering
\includegraphics[height=7.5cm, width=10.50cm]{DirectPhotonv2.eps}
\caption{ Elliptic Flow of direct photons from $200 A$ GeV $Au+Au$
collision at RHIC~\cite{miki}.}
\label{phenix_v2}
\end{figure}
The factor $R$ is measured from spectral analysis. The inclusive and
hadron decay $v_2$ are measured using reaction plane method where the
background photons are measured using a Monte Carlo simulation. We have
already seen that the photon $v_2$ shows different signs and/or
magnitude depending on the production procedure of photons in heavy
ion collisions. Thus experimental measurement of photon $v_2$ can
be a very powerful tool to disentangle the intermix contributions
from different sources in different $p_T$ ranges. However, the
experimentally measured preliminary PHENIX data still contain a large
systematic error (Fig.~\ref{phenix_v2}), which need to be reduced
before making any specific conclusion about direct photon $v_2$.

\section{Photon tagged jets}

Jets are hard phenomenon characterized by a large momentum transfer
between partons and are characterized by several hadrons in a small
angle around a leading particle as a result of jet fragmentation.
The hadrons surrounding a jet are known as associated particles of the jet.

One of the key observables at RHIC energy is the strong suppression of
leading hadron yield at high values of transverse momentum compared to
expectations based on $p+p$ or $d+A$ collisions at same collision energy.
This is the celebrated phenomenon of `jet quenching'. It is assumed that the
hard partons lose a large fraction of their energy when passing
through the strongly interacting medium. A variety of qualitatively
different models are available  using collisional energy loss and
radiative energy loss mechanisms to describe the nuclear modification
factor $R_{AA}$, defined as,
\begin{equation}
R_{AA}(p_T,y)= \frac {d^2N_{AA}/dp_T dy} {T_{AA}(b) d^2 \sigma^{NN}/dp_T dy}
\end{equation}
Here, $T_{AA}$ is the nuclear overlap function and $\sigma_{NN}$ is
the nucleon-nucleon cross-section. The major problem for studying
jet properties arises from the fact that, conventional calorimetric
study can not measure the jet energy loss very accurately. Also it
is very difficult to directly measure the modification of jet
fragmentation function and jet production cross-section.

The `jet quenching' in heavy ion collisions can be studied by
measuring the $p_T$ distributions of charged hadrons in opposite
direction of a tagged direct photon. Quark-gluon Compton scattering
and quark anti-quark annihilation process are the main mechanisms
by which direct photons are produced at very high $p_T$, and jets
are produced in the opposite direction of these photons. From
momentum conservation law, the initial transverse energy of the
produced jets are equal to that of the produced photons, i.e,
$E_{\gamma} = E_{\rm jet}$. Thus uncertainties regarding the jet
cross-section can be avoided by tagging a direct photon in the
direction opposite to the jet (see. Ref.~\cite{WHS}).

Medium modified parton fragmentation function $D_{h/a}(z)$
( z is fractional momenta of the hadrons)
is used to study the jet energy loss. The differential $p_T$
distribution of hadrons from jet fragmentation in the kinematical
region $(\Delta\phi,\Delta y$) using $D_{h/a}(z)$ can be written as:
\begin{equation}
\frac{dN^{\rm jet}_{\rm ch}} {dyd^2p_T} = \sum_{r,h} r_a(E_T^{\gamma})
{\frac {D_{h/a}(p_T/E_T)} {p_T E_T}} {\frac {C(\Delta y \Delta \phi)}
{\Delta y \Delta \phi}},
\end{equation}
where, $ C(\Delta y \Delta \phi)=\int_{|y| \le \Delta y/2} dy
\int_{|\phi -\bar{\phi}_\gamma| \le \Delta \phi/2} d \phi
f ( y -|\phi -\bar{\phi}_\gamma|)$  is an overall factor and
$f(y,\phi)$ is the hadron profile around the jet axis and
$r_a(E_T^{\gamma})$ is the fractional production cross section
of a typical jet associated with the direct photons. Thus, comparison
between extracted fragmentation function in $AA$ and $pp$ collisions
can be used to determine the jet energy loss as well as the
interaction mean free path in the dense matter produced in high
energy heavy ion collisions. Thorston Renk~\cite{Renk_jet} has
shown that by gamma hadron correlation measurement, the averaged
probability distributions for quarks are accessible experimentally
and he has also explained an analysis procedure capable of
distinguishing between different energy loss scenarios leading
to the same nuclear suppression factor.
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{renk1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{renk2.eps}
\caption{ Left panel: Nuclear suppression factor $R_{AA}$ for
different toy models and hydrodynamical simulation compared with
PHENIX data. Right panel: Momentum spectrum of hard hadrons
correlated back to back with a photon trigger normalized to the
expectation of geometrical absorption (taken from
Ref~\cite{Renk_jet}).}
\label{renk_1_2}
\end{figure}
\subsubsection{Isolating the bremsstrahlung photons}

In order to measure direct photon cross section by suppressing the
hadron decay background or in particular to isolate bremsstrahlung
photons from accompanying hadrons, an interesting method known as `isolation
cut' can be used successfully. The basic assumption for performing the
cut is that the hadronic energy in a cone around the photon is less
than a certain fraction of the photon energy, i.e, a photon is
considered as isolated if the combined energy of the accompanying
hadrons is less than $\epsilon E_\gamma$ ( where, $E_\gamma $ is
the photon energy) inside a cone of half opening angle $\delta$
around the photon. The parameter $\epsilon$ is very small ( $\sim
0.1$) and is called the energy resolution parameter. The cone around
the tagged photon is known as isolation cone. The cone opening can
be related to the radius $R$ of a circle centered around the photon
in the centre of mass system, where R is defined in terms of rapidity
$\eta$ and azimuthal angle $\phi$ as,
\begin{equation}
R \ge \sqrt { {\Delta \eta}^2 + {\Delta \phi}^2}
\end{equation}
For small rapidities, the half opening angle $\delta$ equals the
radius $R$. This method was successfully implemented into the
theoretical study of isolated prompt photon production considering
fragmentation contribution also in next-to-leading order (NLO) by Gordon
and Vogelsang~\cite{Vogel}. A very good accuracy of this method over
a wide range of isolation parameters  for prompt photon production
was demonstrated in their calculation. Results from RHIC and LHC for this
would be very valuable.

\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{inter_1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{inter_2.eps}
\caption{ Left panel: The production rate (per event) of hard photons
in a central collision of gold nuclei at $\sqrt {s_{NN}} = 200$ GeV
as a function of time in the centre of mass system. Right panel:
Spectrum of photons from various sources. QM and HM denote quark
matter and hadronic matter contribution respectively  (figures
are from Ref.~\cite{prl_inter}.}
\label{interfero}
\end{figure}

\subsubsection{Dilepton tagged jets }

In the study of photon tagged jets, the main problem arises from the
jet pair production background, where a leading $\pi^0$ in the jet is
misidentified as a photon. For an event having a huge background
`isolation cut' is not a very efficient mechanism in the low $p_T$
range to study photon tagged jets in heavy ion collisions and it is
useful only for $pp$ collisions. In the high $p_T$ range although the
background contribution related problems are reduced, a substantial
problem in photon isolation is created by small opening angle. On
the contrary jets tagged by dileptons are not affected
by background and can be used to observe $p_T$ imbalance, a signal of
medium induced partonic energy loss. As we have mentioned earlier,
for dileptons not only $p_T$ but also invariant mass $M$ is another
equally important parameter which can be used accordingly to study
dilepton tagged jets in the medium. At high $p_T$ and high $M$, the
dilepton yield is much lower compared to low $p_T$ and low $M$ range,
however the relative background contribution is also lower in that
range.

At very high transverse momentum Drell-Yan process ($h_1 + h_2
\rightarrow l^+ l^- +X$) dominates the dilepton production from QGP
phase. Srivastava {\it et al.}~\cite{dil_tagged} have estimated the
results for dilepton tagged jets by studying
Drell-Yan process at NLO in relativistic heavy ion collisions at
RHIC and LHC energies. They have also shown that correlated charm and
bottom decay are unimportant as background for dileptons having large
transverse momentum or in the kinematical region of interest for jet
quenching. Lokhtin {\it et al.}~\cite{lok} have studied the dimuon +
jet production (including both $\gamma^*/Z \rightarrow \mu^+ \mu^- $
modes) at LHC energy and have shown the $p_T$ imbalance between
$\mu^+ \mu^-$ pair and a leading particle in a jet is clearly visible
even for moderate energy loss. It is directly related to absolute value
of partonic energy loss and almost insensitive to the angular spectrum
of emitted gluons and to experimental jet energy resolution.

\begin{figure}
\centering
\includegraphics[height=8.2cm, width=5.50cm]{1_osl.eps}
\includegraphics[height=8.2cm, width=5.50cm]{2_osl.eps}
\caption{ Left panel: The outward, side-ward and longitudinal
correlations of direct photon predicted by PCM at $K_T = 2$ GeV,
inclusion of thermal photons changes the results only marginally.
Right panel: Intensity correlation of photons at 1 GeV, considering
only PCM(BMS), only thermal (hyd), and all PCM+ thermal photons
(BMS+hyd) (see Ref.~\cite{prl_inter}).}
\label{HBT}
\end{figure}

\section{Intensity interferometry of thermal photons}

The quantum statistical interference between identical  particles
emitted in relativistic heavy ion collisions, provide valuable
insight about the shape and size of the particle emitting source.
We know that direct photons emitted from different stages of the
collision dominate the $p_T$ spectra depending on the range of
transverse momentum range. Thus, one can extract space time
dimension of the system at different stages of the collision by
measuring the correlation radii for photons at different transverse momentum.

Two particle correlation function $C(\vec{q},\vec{K})$ for photons
having momenta $\vec{k_1}$ and $\vec{k_2}$, emitted from a completely
chaotic source can be written as,
\begin{equation}
C(\vec{q},\vec{K}) =  1 + {\frac {1}{2}} {\frac {| \int d^4 x S(x,
\vec{K}) e^{iq.x}|^2} {\int d^4 x S(x,\vec{k_1})\int d^4 x S(x,\vec{k_2})}}
\end{equation}
In the above equation the factor 1/2 appears for averaging over spin
and $S(x,\vec{K})$ is the space time density function defined as,
\begin{equation}
E \frac{dN}{d^3K} = \int d^4 x  S(x,\vec{K})
\end{equation}
where,
\begin{equation}
\vec{q}= \vec{k_1} - \vec{k_2},  \vec{K}= (\vec{k_1} +\vec{k_2})/2
\end{equation}
The correlation function $C(\vec{q},\vec{K})$ can be expressed in
terms of outward, side-ward and longitudinal momentum difference
and radii ( $q_{\rm out}$, $q_{\rm side}$, $q_{\rm long}$ and
$R_{\rm out}$, $R_{\rm side}$, $R_{\rm long}$ respectively )as,

\begin{equation}
C(q_{\rm out},q_{\rm side},q_{\rm long}) \sim 1+ {\frac{1}{2}}
\exp[-(q_{\rm out}^2 R_{\rm out}^2 +q_{\rm side}^2 R_{\rm side}^2
+ q_{\rm long}^2 R_{\rm long}^2)/2 ]
\end{equation}
Photon four momentum in terms of transverse momentum $k_T$, rapidity
$y$ and azimuthal angle $\psi$ can be expressed as
\begin{equation}
k^{\mu}= (k_{T} \cosh y, k_{T} \cos \psi, k_{T} \sin \psi, k_{T} \sinh y)
\end{equation}
and
\begin{eqnarray}
q_{\rm long}& =&|k_{1z} - k_{2z}|= |k_{1T} \sinh {y_1} - k_{2T} \sinh {y_2}|
\nonumber\\
q_{\rm out}& =& {\bf{q_T}.k_T}/k_T, \nonumber\\
q_{\rm side}& =&| {\bf{q_T}} - q_{\rm out}{\bf{k_T}}/k_{T}|\nonumber
\end{eqnarray}
Photon interferometry in the QGP phase has been investigated by
several theoretical groups and the first experimental results on
direct photon was obtained by WA98~\cite{phot_wa98} collaboration.
Bass, M\"uller, Srivastava~\cite{prl_inter}, have calculated two body
quantum correlation of high energy photon using Parton Cascade Model
(PCM) and ideal hydrodynamic model for central $200A$ GeV $Au+Au$
collision at RHIC. They have shown that one can differentiate between
the direct photons from early per-equilibrium stage and the same from
later QGP and hadronic gas stages depending on features
of the correlation function. Left panel of Fig.~\ref{interfero} shows
that about 88\%  of the total photons having transverse momenta ($ 1
\pm 0.1$) GeV (produced by hard parton scattering in PCM~\cite{TGF})
are emitted
within a time period of 0.3 fm/$c$ at rapidity $y=0$. Right panel of the
same figure shows that at $k_T = 1$ GeV, the contribution of
pre-equilibrium phase and thermal photons are similar to the total
photon yield, whereas at $k_T =2$ GeV  hard photons from PCM outshine
the thermal ones by an order of magnitude. The emission time in PCM
is very small and hydrodynamic calculation with $\tau_0 = 0.3$ fm/$c$ allows
a smooth continuation of the emission rate. Intensity correlation for
$k_T \ge 2$ GeV (Fig.~\ref{interfero}) reveals a pre-thermal photon
dominated small size source of brief duration.  On the contrary, for
$k_T = 1$ GeV we can see much larger radii for an extended source and
suppression of pre-thermal contribution over thermal.
\begin{figure}
\centering
\includegraphics[height=4.2cm, width=5.50cm]{intensity_1.eps}
\includegraphics[height=4.2cm, width=5.50cm]{intensity_2.eps}
\includegraphics[height=4.2cm, width=5.50cm]{intensity_3.eps}
\includegraphics[height=4.2cm, width=5.50cm]{intensity_4.eps}
\caption{One dimensional correlation function for the kinematic
window used in WA98 experiment, assuming a fully source and emitting
only single photon. Outward side-ward and longitudinal correlation
function for thermal photons in central collision at RHIC energy.
(taken from Ref.~\cite{DKS_WA98}).}
\label{intensity}
\end{figure}

Intensity interferometry of thermal photons having transverse
momentum $k_T \sim 0.1 - 2.0$  GeV~\cite{DKS_WA98} provide an
accurate information about the temporal and spatial structure
of the interacting medium.

In reference~\cite{DKS_WA98} WA98 data is compared with theoretical
results and prediction are given for RHIC and LHC energies. One
dimensional correlation function in terms of $q_{inv}$ (invariant
momentum difference) for different $k_T$ and rapidity windows along
with WA98 data is shown in figure~\ref{intensity}. Theoretical
results are well fitted in the form $C = \ 1 \ + \ 0.5 \ a \exp[-q_{\rm
inv}^2 R_{\rm inv}^2/2]$ and are in reasonable agreement with the
experimental data. At RHIC energy, contribution from quark matter
increases and as a result the two source aspect in the outward
correlation radii become more clear. Similar results are obtained
for LHC energies. It is found that the transverse momentum dependence
of the different radii are  quite different from the corresponding
results for pions and do not decrease as $1/\sqrt{m_T}$. For SPS,
RHIC, and LHC energies the longitudinal correlations show similar
values, which can be explained as a result of boost invariance of
the flow pattern.

\section{Epilogue}

We have tried to give a reasonably complete introduction to the
exciting possibilities provided by radiation of direct photons
and dileptons for the study of the dynamics of relativistic heavy
ion collisions.

While low mass dileptons provide insights into the medium modification
of vector mesons, those having intermediate masses carry signatures
of thermal radiation from the quark gluon plasma. We have, due to
lack of space, left out the discussion of correlated decay of charm
and bottom mesons which give a large contribution to dileptons. These
are also important as they carry valuable information about the extent
of thermalization, elliptic flow, and energy-loss exhibited by heavy
quarks.

We have discussed sources of direct photons, consisting of prompt photons,
thermal
radiation from quark and hadronic matter and those due to passage of
jets through QGP. We have also seen that photons carry information
about the initial temperature, evolution of elliptic flow, and size
of the system. High statistics data at RHIC and LHC along with photon
or dilepton tagged jets will go a long way in seeing that we realize
the full potential of electromagnetic probes of quark gluon plasma.

\section{Acknowledgment}
This write up is based on the notes taken by RC and LB from the lectures
given by  DKS at the QGP Winter School 2008 at Jaipur. We thank the
organizers for their warm hospitality. The work discussed here has benefited
from discussions and collaborations with colleagues over many
years and from many countries, and we take this opportunity to thank all of
them and also to apologize to those whose work may not have found mention
in these introductory lectures as this write up is not intended to be a
review.

Finally we thank the organizers for their patience and for giving us
extra time to complete this write up.

\input{referenc}

\printindex
\end{document}