# Photons from relativistic nuclear collisions

Hannah Vormanna Tom Reicherta,b Christian Spielesa Jan Steinheimere Marcus
Bleichera,b,c,d a Institut für Theoretische Physik, Johann Wolfgang Goethe-
Universität, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany  b
Helmholtz Research Academy Hesse for FAIR (HFHF), Campus Frankfurt, Max-von-
Laue-Str. 12, 60438 Frankfurt am Main, Germany  c GSI Helmholtz Center,
Planckstr. 1, 64291 Darmstadt, Germany  d John von Neumann-Institut für
Computing, Forschungszentrum Jülich, 52425 Jülich, Germany  e Frankfurt
Institute for Advanced Studies, Ruth-Moufang-Str. 1, D-60438 Frankfurt am
Main, Germany

###### Abstract

Collisions of atomic nuclei at relativistic velocities allow to recreate the
conditions encountered in neutron stars or in the early universe micro-seconds
after the Big Bang. These reactions are performed in today’s largest
accelerator facilities, e.g. at CERN in Geneva, at the Relativistic Heavy Ion
Collider at Brookhaven, NY or in the planned FAIR facility in Darmstadt
Germany. During such a collision the matter is heated up to hundreds of MeV
(billions of degrees) and compressed to densities of $3-10$ times the density
inside ordinary atomic nuclei (i.e. $10^{17}-10^{18}$ kg/m3). Usually these
collisions are studied via the measurement of a multitude of strongly
interacting particles, called hadrons, that are emitted at the end of the
collision. However, also some photons are created. These photons are of
special interest as they allow to look into the early stage of the collisions.
This is because they are only interacting electro-magnetically with the
hadrons of the created fireball which is a much weaker interaction compared to
the strong interaction. This paper elucidates the physics of the photons and
what can be learned from them.

photons,heavy-ion collisions

##  I Introduction

One of the most fascinating achievements in the last 50 years was the
development of accelerator technology to recreate the super hot matter present
a microsecond after the Big Bang or the extreme densities that are only found
in the interior of Neutron Stars. The largest of these particle accelerators
is CERN’s Large Hadron Collider (LHC), slightly smaller facilities exist in
the US, the Relativistic Heavy Ion Collider (RHIC) or are planned, e.g. in
Germany with the Facility for Anti-Proton and Ion Research (FAIR). These
facilities accelerate atomic nuclei (often Gold or Lead nuclei) to velocities
near the speed of light and smash them together. By this, one hopes to gain
new insights into the structure of matter below the scale of protons and
nucleons, namely into the dynamics of the Quarks and Gluons. The dynamics and
properties of these Quarks and Gluons, the building blocks of protons and
neutrons (and all other hadrons), are governed by the theory of strong
interaction, also called Quantum Chromo-Dynamics (QCD).

Unfortunately, Quarks and Gluons can not be directly observed as they are
bound to hadrons by a phenomenon called confinement. As the name “strong
interaction” already implies these hadrons interact intensely with each other
and may even be seen as a form of liquid Muller:2007rs . Indeed, the
experiments at RHIC have hinted that such matter has the lowest viscosity ever
observed. Unfortunately, this is a major disadvantage when one wants to obtain
information from the interior of the collision zone: Because the strongly
interacting hadrons are only emitted from the very late and very dilute stage
of the system’s evolution, when the interactions cease. Thus, they carry
mainly information from the end of the reaction when the system has already
cooled down.

How can one then obtain a deeper view into the most interesting hot and dense
region of the fireball created in such collisions? Here Photons, the quanta of
light, are perfect candidates. The main reason is that photons do not interact
strongly with the surrounding matter, but interact only via the
electromagnetic interactions, which is typically a factor $\alpha=1/137$
smaller than the strong interaction. Thus, once a photon is created it can
leave the fireball unscathed and can tell about the temperatures of the
fireball when it is hottest and most dense.

In this paper, we want to give some introduction into this exciting topic and
want to present the steps needed for such calculations in a pedagogical
manner. The paper is structured in the following way: First, we describe the
hydrodynamic set-up used for the calculations. Here, we used a standard (3+1)
dimensional relativistic hydrodynamic calculation, supplemented with a hadron
resonance gas equation of state to close the equations. In the next section,
we provide a short review on the motivation to use photons to explore the
properties of QCD-matter. Here, we focus on the production of real photons,
i.e. photons with vanishing mass (these are the photons one knows from daily
experience, albeit the energy of the photons is higher than usual) and omit
the discussion of virtual photons, i.e. massive photons which are also used to
explore hot and dense QCD-matter (we refer the interested reader to
vanHees:2007th ; Rapp:2009yu ). Then we give a brief overview on how to
calculate photon emission rates from hadronic collisions. For simplicity, we
will only discuss the most important channel ($\pi\rho\rightarrow\pi\gamma$)
and provide references for further detailed information. Finally, we show our
results for the photon rates for three different energies. The paper concludes
with a summary and a critical assessment of the omitted effects.

The whole study was carried out as part of a summer study program at the
Frankfurt University over a duration of 8 weeks.

##  II Technical set-up for the hydrodynamic evolution and method of solution

When one wants to explore the dynamics of QCD-matter one is confronted with a
multitude of complications. QCD, being a non-abelian gauge theory, practically
prohibits ab-initio calculation of dynamical processes due to the complexity
of the theory. However, some direct information can be obtained from large
scale lattice QCD simulations for static systems at very low baryon densities
Philipsen:2007rj . Here QCD predicts that the nuclear matter should undergo a
transition from hadron degrees-of-freedom (protons, neutrons, pions, etc.) to
quark and gluon degrees-of-freedom. The temperature of this transition depends
on the baryon density and is around $T_{c}\approx 150-160$ MeV for vanishing
baryon density. Thus, below the transition temperature the system consists of
hadrons which interact and form the dense matter observed at moderate
collision energies.

How does one then model the dynamics? Typically there are two choices:

  * •

A Boltzmann equation type simulation Bleicher:1999xi , where one uses
individual matrix elements to describe the interaction cross sections between
the various hadron species or

  * •

a hydrodynamic calculation, where one assumes that the QCD matter can be
treated as a liquid Kolb:2003dz .

Both approaches have advantages and disadvantages: The Boltzmann equation can
usually only be used for dilute systems and requires knowledge of all
individual matrix elements, in contrast hydrodynamics is only applicable if
the system is rather dense and can be assumed to be in local equilibrium.
Instead of individual matrix elements, the properties of the fluid enter via
an equation of state, i.e. the relation between pressure $p$ and energy
density $\varepsilon$. Here, we choose relativistic hydrodynamics, because the
calculational set-up and the numerical implementation is easier for the
present task.

Let us now review shortly the main ingredients into the hydrodynamic
simulation. For a more detailed description, we refer the reader to the very
pedagogical introduction to hydrodynamics of heavy ion collisions by
Ollitrault Ollitrault:2007du  and references therein.

In general, hydrodynamics represents local energy-, local momentum- and local
charge-conservation. The energy-momentum tensor $T^{\mu\nu}$ for an ideal
fluid111Using an ideal fluid instead of a viscous fluid is a good
approximation, because the viscosity of the matter produced in heavy ion
collisions is sufficiently small (see Muller:2007rs ; Reichert:2020oes  and
references therein). is

$T^{\mu\nu}=(\varepsilon+p)u^{\mu}u^{\nu}-g^{\mu\nu}p\,.$ (1)

Here and in the following, we use natural units, i.e. $\hbar=c=k_{B}=1$,
Einstein sum convention is implicit and the west coast metric, (i.e. the
spatial diagonal elements of $g^{\mu\nu}$ carry the minus sign), and $u^{\mu}$
is the velocity of the fluid cell while $p$ and $\varepsilon$ denote the
cell’s pressure and energy density, respectively.

The conservation of energy and momentum is described by

$\partial_{\mu}T^{\mu\nu}=0\,.$ (2)

In addition to the energy momentum conservation, we also include the
conservation of the baryon charge, which is described via the conservation of
the baryon current $j_{B}^{\mu}=\rho_{B}u^{\mu}$, with $\rho_{B}$ being the
baryon density in the local rest frame of a hydrodynamic cell
Ollitrault:2007du ; Rischke:1998fq .

The conservation equation for the baryon current is then

$\partial_{\mu}j_{B}^{\mu}=0\,.$ (3)

To close the system of equations, we also need to specify the relation between
energy density and pressure, $p(\varepsilon,\rho_{B})$ or $p(T,\mu_{B})$,
where $T$ is the local temperature in the cell and $\mu_{B}$ is the chemical
potential of the baryons in the cell needed to fix the baryon density. This
relation is known as the equation of state (EoS) and defines the properties of
the fluid. In Ollitrault:2007du  the simplifying assumption of a gas of
massless particles was made, leading to a simple EoS of $p=1/3\varepsilon$.
For the low energies discussed here, one can not make this assumption, but has
to employ a hadron resonance gas with massive particles.

It was shown in Dashen:1969ep  that an interacting gas of massive hadrons can
be approximated by treating unstable particles as explicit degrees of freedom
in the partition function of the gas. This is known as the hadron resonance
gas approximation. In this approximation, the logarithm of the partition
function ${\cal Z}$ is given as

$\displaystyle\ln{\cal Z}$ $\displaystyle=$ $\displaystyle\sum_{i}\ln{\cal
Z}_{i}(T,V,m_{i},\mu_{i})$ $\displaystyle=$ $\displaystyle\sum_{i}(\pm
1)\frac{g_{i}V}{2\pi^{2}}$
$\displaystyle\qquad\times\int_{0}^{\infty}\mathrm{d}k\,k^{2}\,\ln\left(1\pm
e^{\mu_{i}/T}e^{-\sqrt{m_{i}+k^{2}}/T}\right)\,,$

where the sum runs over all hadronic species in the particle data book
ParticleDataGroup:2018ovx . The upper (lower) signs are for fermionic
(bosonic) species, $g_{i}$ is the spin and iso-spin degeneracy factor of the
hadron, $m_{i}$ is the mass of the hadron, $V$ is the volume, and $\mu_{i}$ is
the chemical potential of species $i$.

The pressure is then given by

$p(T,\mu_{B})=\left.\frac{\partial\,T\ln{\cal Z}(T,V,\mu_{B})}{\partial
V}\right|_{T,\mu}\,.$ (5)

This can be done straightforward numerically and one obtains $p(T,\mu_{B})$
for the hadron resonance gas in tabulated form. More elaborate Equations of
State are available, see e.g. Motornenko:2019arp .

The hydrodynamic equations can be solved analytically for a system without
baryon current and for a (1+1) dimensional geometry (one longitudinal
direction plus time). This simple solution, known as the Bjorken solution
Bjorken:1982qr  (see Ollitrault:2007du  for a more pedagogical presentation),
is unfortunately only valid for very high collision energies. Towards lower
energies, an approximate analytical solution for a (2+1) dimensional system
(transverse and longitudinal direction plus time) was found by Belenkij and
Landau Belenkij:1955pgn . For the (3+1) dimensional case analytical solutions
do not exist and one needs to solve the equations numerically.

For the present calculation we employ a standard numerical scheme to solve the
equations on an Eulerian grid, known as the SHASTA (Sharp And Smooth Transport
Algorithm) Boris:1973tjt ; Rischke:1995ir . SHASTA is an Eulerian finite-
difference algorithm for solving the continuity equations employing a flux
correction technique. The choice of the algorithm is a matter of taste and
other solvers, e.g. of Gudanov type Karpenko:2013wva  or the Kurganov-Tadmor
method KURGANOV2000241 ; Schenke:2010nt , can also be applied. E.g. a
hydrodynamical code package can be downloaded from Code .

###  II.1 Evolution of the matter

Figure 1: [Color online] Evolution of the temperatures in the central region
of the reaction for three selected beam energies in the GSI, RHIC-BES region
of $E_{lab}=1.23A$ GeV (blue), $\sqrt{s_{NN}}=3$ GeV (orange) and $7.7$ GeV
(green).

After the description of the methodological set-up, let us come to the
physical set-up and physical assumptions used for the present investigation.
As initial condition for the (3+1) dimensional simulation, we set two gold
nuclei (target and projectile) on the hydrodynamic grid that approach each
other in longitudinal direction. Each gold nucleus is modeled as a sphere of
radius $r=1.2A^{1/3}$, where $A$ is the mass number, and volume $V=4/3\pi
r^{3}$ and constant $\rho_{B}=A/V=0.14$ fm3. When the gold fluids from target
and projectile come into contact they are immediately thermalized and their
kinetic energy is translated into thermal energy. Here a word of caution is
necessary: The assumption of instant local thermalization is rather crude and
is only justified as long as the thermal motion (average thermal velocity) of
the hadrons after the thermalization of the projectile and target fluids is
comparable to the longitudinal velocities of the incoming nuclei. While this
assumption is fulfilled at low energies, it becomes questionable at energies
above $\sqrt{s_{NN}}\approx 5-10$ GeV. Here, one would need to employ multi-
fluid simulations, which are however out of the scope of this investigation.

Let us now investigate the evolution of the temperatures encountered in
gold+gold (from now on denoted as Au+Au) reactions at various energies
currently studied experimentally at the Relativistic Heavy Ion collider (RHIC,
New York) and at the SIS100 at Darmstadt, Germany. In Fig. 1 we show the
evolution of the temperatures in the central region of the reaction for three
selected beam energies in the GSI, RHIC-BES region of $E_{lab}=1.23A$ GeV,
$\sqrt{s_{NN}}=3$ GeV and $7.7$ GeV. The central region here was defined as a
cube with a side length of 1 fm centered around the origin of the grid. We
observe that the system heats up tremendously leading to peak temperatures in
the order of $T_{max}=90$ MeV ($E_{lab}=1.23A$ GeV), $T_{max}=130$ MeV
($\sqrt{s_{NN}}=3$ GeV) and even $T_{max}=225$ MeV. The maximum of the curves
corresponds to the full overlap time of the nuclei, this is why it shifts in
time to lower values when the collision energy increases (the nuclei become
Lorentz contracted!).

In Fig. 2 we show the time dependence of the volume of “hot” cells, i.e. cells
with a temperature above $T=80$ MeV, $T=120$ MeV, and $T=150$ MeV for three
selected beam energies in the GSI, RHIC-BES region of $E_{lab}=1.23A$ GeV,
$\sqrt{s_{NN}}=3$ GeV and $7.7$ GeV. We observe that the hottest regions of
the system sustain only a short period of time during the evolution which
rapidly cease. At the largest investigated beam energy the region with $T>80$
MeV becomes more long-lived and the corresponding volume is increasing until
20 fm/c. In the next section we will turn to direct photon production to see
how the four-volume evolution of the temperature affects their production
rates.

Figure 2: [Color online] Time dependence of the volume of ’hot’ cells, i.e.
cells with a temperature above $T=80$ MeV, $T=120$ MeV, and $T=150$ MeV for
three selected beam energies in the GSI, RHIC-BES region of $E_{lab}=1.23A$
GeV (blue curves), $\sqrt{s_{NN}}=3$ GeV (orange curves) and $7.7$ GeV (green
curves).

##  III Photon production rates

The collision of two nuclei leads to the formation of a hot and dense fireball
due to the compression of nuclear matter and energy into a small volume in
space. The lifetime of this fireball is defined as the time until a final
state of freely streaming particles is reached as a result of the collective
expansion of matter. The trajectory of the fireball in the T-$\mu_{B}$-plane
of the QCD phase diagram depends on the energy of the collision. Above a
critical temperature $T_{c}$ a Quark-Gluon-Plasma (QGP), a deconfined state of
quarks and gluons, is expected to be formed. In order to probe possible phase
transitions one needs suitable observables which do not only carry information
about the final state after the freeze-out of the system but the whole
evolution of the fireball especially in the initial hot and dense phase.
Hadrons are unsuited due to typically short mean free paths. This is because
they interact strongly with the surrounding matter so that they only escape
from the fireball’s surface or after the expansion of the fireball when the
scattering rate is much smaller than the expansion rate. Contrary,
electromagnetic probes like photons and dileptons (virtual photons) are
promising candidates: Their mean free paths are typically larger than the
fireball itself meaning that they suffer little final state interaction (they
do not underlay the strong interaction) and their emission rates rapidly vary
with local intensive variables. In this paper we restrict ourselves to the
emission of real photons.

The production mechanisms of photons can be divided into three types
Arleo:2004gn : Firstly, photons can be produced primordially in so called hard
QCD interactions of two colliding nuclei (QCD Compton, annihilation, and
bremsstrahlung). They are called “prompt” photons. Rates of these processes
can be computed within the framework of perturbative QCD (pQCD) and dominate
at large transverse momenta $p_{T}$. Secondly, when a QGP forms, photons are
emitted from collisions of quarks and gluons. This radiation also dominates in
the large $p_{T}$ regime. Thirdly, the QGP expands and reenters the hadronic
phase. Here, hot hadronic resonances emit photons until the freeze-out of the
system whose rates dominate at small $p_{T}$. Together with the QGP photons
they are called “thermal” photons. Additionally, there are also “decay”
photons which are decay products of hadronic resonances. Unbiased experimental
spectra consist of the sum of all of these photons. In order to identify the
”thermal” photons, other competing components have to be subtracted by the
experiment. In the present paper we limit ourselves to real, thermal photons,
that is photons from secondary interactions of particles in the expanding
fireball, and focus on hadron gas emission. For the calculation of photon
emission rates we follow Turbide, Rapp and Gale (TRG) Turbide:2003si . They
present two different ways of defining thermal photon rates: In the context of
thermal field theory the differential photon emission rate from an
equilibrated system reads

$\displaystyle\frac{\mathrm{d}R_{\gamma}}{\mathrm{d}^{4}p}=P_{\mu\nu}W^{\mu\nu}$
(6)

with the polarization-summed photon tensor $P_{\mu\nu}$ and the hadronic
tensor $W^{\mu\nu}(p;\mu_{B},T)$ describing the effects of strong interaction.
This can also be written as follows

$\displaystyle
p_{0}\frac{\mathrm{d}R_{\gamma}}{\mathrm{d}^{3}p}=-\frac{\alpha}{\pi^{2}}\
f^{B}(p_{0};T)\
\operatorname{\mathbb{I}\mathrm{m}}\Pi^{\mu\nu}_{em}(p_{0}=p;T)$ (7)

where $f^{B}$ is the Jüttner distribution and $\Pi^{\mu\nu}_{em}$ denotes the
in-medium photon self-energy whose imaginary part is directly related to
$W^{\mu\nu}$.

Alternatively, within relativistic kinetic theory the photon rate of a given
reaction $1+2\rightarrow 3+\gamma$ can be written as

$\displaystyle p_{0}\frac{\mathrm{d}R_{\gamma}}{\mathrm{d}^{3}p}=\int$
$\displaystyle\frac{\mathrm{d}^{3}p_{1}}{2(2\pi)^{3}E_{1}}\frac{\mathrm{d}^{3}p_{2}}{2(2\pi)^{3}E_{2}}\frac{\mathrm{d}^{3}p_{3}}{2(2\pi)^{3}E_{3}}$
$\displaystyle\times\left((2\pi)^{4}\delta^{(4)}(p_{1}+p_{2}\rightarrow
p_{3}+p)\right.$
$\displaystyle\times\left.|\mathcal{M}|^{2}\frac{f(E_{1})f(E_{2})[1\pm
f(E_{3})]}{2(2\pi)^{3}}\right)$ (8)

with $\mathcal{M}$ being the invariant scattering matrix element for the
reaction under consideration.

In the equations above, the emission rate is the rate of production per unit
time and volume

$\displaystyle\mathrm{d}R_{\gamma}=\frac{\mathrm{d}N_{\gamma}}{\mathrm{d}^{4}x}\
.$ (9)

A meson gas consisting of pseudo-scalar, vector, and axial mesons ($\pi$, $K$,
$\rho$, $K^{*}$, $a_{1}$) can be characterized using an effective massive
Yang-Mills Lagrangian. For all interactions occurring in this Lagrangian that
produce photons, the emission rate can be calculated using Eq. III employing
the respective matrix elements $\mathcal{M}$.

Additionally, so-called hadronic vertex form factors have to be applied to
these rates to consider finite-size effects. This results in a suppression
compared to the bare rates. According to TRG they can be written in the form

$\displaystyle F(t)=\left(\frac{2\Lambda^{2}}{2\Lambda^{2}-t}\right)^{2}$ (10)

where $\Lambda=1$ GeV denotes a cutoff parameter and $t$ is the four-momentum
transfer in a given t-channel exchange. Its average $\overline{t}$ can be
approximated as

$\displaystyle\overline{t}\simeq-2Em_{\pi}$ (11)

for a t-channel pion exchange Arleo:2004gn . Here, $E$ is the energy of the
emitted photon and $m_{\pi}$ the pion mass.

With the help of this form factor TRG have parametrized the photon emission
rates for different production channels in the non-strange and strange sector
Turbide:2003si . These parametrizations are used here in order to obtain
thermal photon spectra of collisions of two gold atoms.

For the present calculation we use this parametrization for the
$\pi+\rho\rightarrow\pi+\gamma$ reaction rate Turbide:2003si :

$E\frac{\mathrm{d}R_{\pi+\rho\rightarrow\pi+\gamma}}{\mathrm{d}^{3}p}=F^{4}(E)\,T^{2.8}\exp\left(\frac{-(1.461T^{2.3094}+0.727)}{(2TE)^{0.86}}+(0.566T^{1.4094}-0.9957)\frac{E}{T}\right)\
[\mbox{fm}^{-4}\mbox{GeV}^{-2}]$ (12)

In the non-strange sector the reaction $\pi\rho\rightarrow\pi\gamma$ turned
out to be the dominating source of photons, in the strange sector it is
$K^{*}\pi\rightarrow K\gamma$ with a contribution of approximately 20% of the
former. We neglect the less important channels to keep the study simple.

When comparing their data for hadron gas emissions to QCD calculations of QGP
emissions, TRG could show a strong similarity of the two rates near the
expected phase transition. This is known as parton-hadron duality and means
that near the phase transition temperature both, quark and gluon degrees-of-
freedom and hadron degrees-of-freedom yield similar results. We hence focus on
the hadronic photon production in this paper and assume its validity also at
higher temperatures.

In Fig. 3 we show the rate of photon emission rate from the process
$\pi+\rho\rightarrow\pi+\gamma$ from a thermal hadron gas at four different
temperatures.

Figure 3: [Color online] The rate of photon emission rate from the process
$\pi+\rho\rightarrow\pi+\gamma$ from a thermal hadron gas at four different
temperatures from 80 MeV to 250 MeV (from bottom to top).

##  IV Results and discussion

Now all ingredients are ready and we are able to calculate the photon emission
from the simulation of a real collision of two heavy ions. The momentum
differential photon yield (rate $R_{i}\times V^{cell}_{i}\times\Delta t$) is
summed up for each individual cell $i$ on the hydrodynamic grid over the
course of the reaction. Thus it integrates the photon emission over the whole
volume and time of the fireball with different emission temperatures at each
space-time point.

In Fig. 4 we show the invariant photon spectra for three selected beam
energies in the GSI, RHIC-BES region of $E_{lab}=1.23A$ GeV, $\sqrt{s_{NN}}=3$
GeV and $7.7$ GeV. The first observation is that the number of photons
increases strongly with increasing beam energy. The main reason for this
behavior is the raise in temperature with the collision energy. This is in
line with the rates in Fig. 3 at fixed temperatures where an increase in
temperature of about $30-40$ MeV leads to emission rates that are up to two
orders of magnitude higher. Additionally, the space-time volume (volume
$\times$ lifetime of the system) of the source increases with increasing
energy. This is expected because at higher collision energies, more hadrons
are produced and the fireball becomes larger and lives longer until it decays
(cf. with Fig. 2). The second observation concerns the slope of the spectra.
Here one observes that the spectra become “hotter” with increasing beam
energy. That is also in line with our expectations that an increase in
collision energy leads to a hotter temperature.

Thus, the photon spectrum allows us to extract two important informations
about the collision: 1) the space time volume of the hot region and 2) the
average temperature during the course of the reaction. To show this, we have
fitted the curves in Fig. 4 (shown as dotted lines) with the photon yield, Eq.
12, and extracted the 4-volume of the emission source and the average
temperature. At $E_{lab}=1.23A$ GeV we obtain $V_{4}=$ volume $\times$
lifetime of the system $=2014$ fm4 and $T=89$ MeV, at $\sqrt{s_{NN}}=3$ GeV we
obtain $V_{4}=2197$ fm4 and $T=125$ MeV and at $\sqrt{s_{NN}}=7.7$ GeV we
obtain $V_{4}=2652$ fm4 and $T=200$ MeV. The obtained 4-volumes are consistent
with the volume evolution provided in Fig. 2 using these averaged
temperatures. It is also interesting to observe that the averaged temperatures
cross the QCD deconfinement temperature in the energy region around
$\sqrt{s_{NN}}=3-7.7$ GeV. This shows the power of the analysis of photon
spectra in contrast to hadron spectra as they allow to observe the hottest
regions of the fireball.

Figure 4: [Color online] Invariant photon spectra for three selected beam
energies in the GSI, RHIC-BES region of $E_{lab}=1.23A$ GeV (blue),
$\sqrt{s_{NN}}=3$ GeV (orange) and $7.7$ GeV (green). The calculations are
shown as full lines, the fits are shown as dotted lines.

In Fig. 5 we show the photon production rate from the process
$\pi\rho\rightarrow\pi\gamma$ from the simulation of a collision of two lead
nuclei at a beam energy of $E_{lab}=158A$ GeV in comparison to data from the
CERN WA98 experiment at the same beam energy. Our calculations show good
agreement with the experimental data within the statistical and systematical
errors of the measurement although we have only considered one of the numerous
possible hadronic production channels for photons here. However, as was shown
by TRG in Ref. Turbide:2003si  the reaction $\pi\rho\rightarrow\pi\gamma$ is
the dominant source of thermal photons at large transverse momenta which
explains why our rates are consistently shifted to slightly lower values
compared to the data. At low transverse momenta, where one expects higher
contributions of other production channels (see e.g. Turbide:2003si ), this
effect is even more pronounced.

Figure 5: [Color online] Invariant photon spectrum for a Pb+Pb collision with
$E_{lab}=158A$ GeV (red) and experimental data from the CERN WA98 experiment
(black) WA98:2000vxl . Error bars with downward arrows indicate unbounded
upper limits.

##  V Summary (and what can be improved)

We have studied the production of photons from nuclear collisions at
relativistic energies. We learned that thanks to their long mean free path
length, photons are perfect candidates to probe the hot and dense regions of a
fireball created in such a nuclear collision. Therefore, traces of a phase
transition of the QCD matter to a QGP can possibly be observed in photon
spectra. We shortly discussed that there are numerous production channels of
photons in such collisions and then focused on thermal photon production from
the most important reaction ($\pi\rho\rightarrow\pi\gamma$).

As a tool we used (3+1) dimensional hydrodynamics which is a common method of
modeling the evolution of QCD matter. Besides energy and momentum conservation
as well as conservation of the baryon current we saw that we need an EoS
(relation between energy density, pressure, and temperature) to close the
system of equations. Here, we used a hadron resonance gas EoS where particles
are modeled as point particles. For more details on aspects of the EoS that
should be considered, see the improvements list below. We found that the
photon yield is sensitive to the space-time volume of the matter created in
such collisions and that it allows to determine the average temperature of the
matter. After technical set-up, photon rates from Gale et al. were used to
calculate the spectra of photons at different collision energies.

We conclude that photons allow (as expected) to obtain a clear view on the
fireball that is superior to the analysis of hadron spectra.

Possible improvements: It is clear that the time scale of this exploration did
not allow to include all state-of-the-art physical phenomena and can only
serve as a teaser for the interested reader. Possible improvements are:

  * •

Information on further photon production rates are available, see e.g.
Turbide:2003si  and can be included in the calculation.

  * •

For a rigorous calculation, photon rates that depend also on the baryon
density should be employed. A parametrization of the rates is available and
can be taken from Heffernan:2014mla . This is of crucial importance because
low energy heavy ion collision can indeed create densities that are also found
in the interior of neutron stars or in neutron star mergers (where one expects
densities up to 3-6 times normal nuclear matter densities). This provides an
interesting link between laboratory experiments to explore the EoS of QCD
matter and the physics of neutron stars and gravitational waves (see e.g.
LIGOScientific:2018cki  where first constraints on the QCD EoS from the
observation of the gravitational wave GW170817 were presented and allowed to
constrain the radii of the neutron stars and the pressure as a function of
baryon density).

  * •

The EoS can be refined to include e.g. excluded volume effects (repulsive
interactions), a van-der-Waals gas or a phase transition to a deconfined (QGP)
state. This involves the construction of such an EoS or a parametrization of
the EoS from lattice QCD. Such a parametrization is available, see e.g.
Motornenko:2020vqm . Such modification will change the space time evolution of
the created matter and may lead to modified photon spectra. Such modifications
and how to learn about them from photon spectra are topics of current
research.

  * •

If a QGP EoS is used, one should also employ the exact photon rate for a quark
gluon gas. This can be obtained from Kapusta:1991qp .

  * •

Last but not least, one needs to include the experimental cuts if one aims to
compare to the data.

##  VI What can be learned from this project?

The topic is centered around the physics of today’s largest accelerator
facilities and fits very well into an advanced nuclear physics lecture. It
demonstrates how a well understood concept, namely thermal radiation (known
from Planck’s law), can be used to learn something about the interior of the
matter produced in such collisions. This opens the possibility to engage the
students in a general (or specific) discussion about the physics explored in
large accelerators (e.g. the quark-gluon-plasma Veleiev:2003 , constituents of
matter, creation of mass, relation to neutron stars, gravitational waves and
element synthesis) but also raises their interest for more theoretical
questions like the quark model of hadrons Hussar:1980  and gauge theories in
general Kenyon:2008  that are at the theoretical basis of the physics explored
with relativistic particle accelerators. A plethora of resources for Quantum-
Chromo-Dynamics (QCD) can be found in Kronfeld:2010 .

We have chosen a hydrodynamic approach Landau:1959  for this study, because
most students have already an intuitive idea about fluids. In addition, such
an approach allows to work with the known concepts of temperature and chemical
potentials which relates this study to the statistical mechanics lectures and
also to astrophysics questions like compact stars. Thus, the students can see
the interconnection between knowledge obtained from lectures and its
application in high energy physics and astrophysics Macher:2005 ; Sagert:2006
where concepts like the partition function are also used to obtain an
Equation-of-State.

In more detail the students learn about the connection to the physics of
Compton scattering Bartlett:1964 ; Wilkins:1992 , which they might have
studied before in the case of electrons. They learn that Compton scattering is
not only relevant in the field of electron-photon scattering, but is also
present in the scattering of other spin-1 bosons with other particles (e.g.
gluon+quark $\rightarrow$ photon+quark, where the gluon is converted to a
photon or $\rho$+pion $\rightarrow$ photon+pion, where the $\rho$ meson was
converted into a photon. Thus, the students learn that the Compton effect is
present in a wider range of physics. In addition, one touched the question of
massive (virtual) photons (e.g. in the case of the $\rho$ meson being
converted to a photon) which allows for a general discussion of virtual
particles.

Apart from the physics skills, the students do acquire technical knowledge in
solving a set of differential equations using a more complicated algorithm
than is usually employed.

###### Acknowledgements.

HV wants to thank the group at Frankfurt for their support where this work was
done as a summer study project.

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