# Dilepton Radiation from Strongly Interacting Systems

P. Salabura  J. Stroth  Jagielonian University, Cracow, Poland  Goethe
University, Frankfurt am Main, Germany  GSI Helmholtzzentrum für
Schwerionenforschung GmbH, Darmstadt, Germany  Helmholtz Research Academy for
FAIR, Germany

###### Abstract

We review the current understanding of time-like virtual photon emission from
QCD matter. The phenomenology of dilepton emission is discussed and basic
theoretical concepts are introduced. The experimental findings are presented,
grouped into production of lepton pairs in elementary processes, production
off cold nuclear matter and emission from heavy-ion collisions. The review
emphasizes the role of dilepton emission as tool for studying exotic phases of
QCD matter. Open questions and a route to probe the QCD phase diagram with
dileptons are outlined.

###### keywords:

Heavy-ion phenomenology, thermal radiation, dileptons, in-medium
modifications, chiral symmetry restoration.

###### Contents

  1. 1 Introduction
  2. 2 QCD matter under extreme conditions
    1. 2.1 The Initial Stage
    2. 2.2 The Hot and Dense Phase
    3. 2.3 Freeze-out
    4. 2.4 The phase diagram of QCD matter
  3. 3 Experimental aspects of dilepton spectroscopy
    1. 3.1 The dilepton signal
    2. 3.2 Extraction of the dilepton signal
      1. 3.2.1 Muons
      2. 3.2.2 Electrons
    3. 3.3 Determination of the combinatorial background.
  4. 4 Basic Theoretical Concepts
    1. 4.1 Electromagnetic decays of hadrons
      1. 4.1.1 Vector Meson Dominance
      2. 4.1.2 Two body decays of vector mesons
      3. 4.1.3 Dalitz decays
    2. 4.2 Nucleon-Nucleon bremsstrahlung
    3. 4.3 Dileptons from hard processes
    4. 4.4 Thermal radiation
    5. 4.5 Vector mesons and chiral symmetry restoration
    6. 4.6 In-medium spectral function in microscopic models
    7. 4.7 Mass modifications in transport codes
  5. 5 Survey on Experimental Results
    1. 5.1 Nucleon-nucleon collisions
      1. 5.1.1 RHIC and LHC
      2. 5.1.2 SPS
      3. 5.1.3 BEVALAC/SIS18
    2. 5.2 Proton and Photon Induced Reactions on Nuclei
      1. 5.2.1 Vector Meson Line Shapes.
      2. 5.2.2 Production Experiments
    3. 5.3 Heavy Ion Collisions
      1. 5.3.1 BEVALAC/SIS18
      2. 5.3.2 SPS
      3. 5.3.3 RHIC and LHC
  6. 6 Status and Open Questions
    1. 6.1 In-medium properties of the $\rho$ meson
    2. 6.2 Multi-differential dilepton observables
  7. 7 Outlook

##  1 Introduction

States of QCD matter at extreme temperatures and densities can be created in
the laboratory by colliding heavy ions at (ultra-)relativistic energies. The
strong interaction provides sufficient stopping to convert all or a large
fraction of the kinetic beam energy into excitation and deposit it around the
center-of-mass of the collision system. Once the energy density in the
collision system exceeds a critical value $\epsilon_{c}\approx
1\,\mathrm{G}e\mathrm{V}/\mathrm{fm}^{3}$ deconfined matter occurs which can
be characterized as a liquid of strongly coupled quarks and gluons. Due to the
high pressure such “nuclear fireballs” expand rapidly and the system crosses
over from partonic degrees of freedom to a system of hadrons. This evolution
resembles features of the universe about 10 $\mu$s after its birth. While the
universe was cooling down to temperatures of $kT~{}160$ M$e$V little droplets
of the matter were formed of the smallest size possible allowing color-neutral
states. Since, the interior of the nucleons contains the “hot soup” of the
early universe as the energy density in the interior of a nucleon is of the
order of $\epsilon_{c}$.

Neutron stars (NS) can be considered giant nuclei111Despite the fact that
neutron star matter is electrically neutral and that the neutron density
exceeds the proton density substantially. with radii of the order of $10^{4}$
m and core densities factors higher than nuclear ground state density. The
high energy density in the NS core is represented by mass and governed by the
immense gravitational forces pulling the matter inward. It is not known if
under such high pressures the hadrons (neutrons) still survive as quasi-
particles or if they are squeezed into each other forming a cold partonic
phase. Similar states of QCD matter are expected to be realized in heavy-ion
collisions at a few G$e$V per nucleon pair. The reaction is that of a bunch of
nucleons interacting with a bunch of nucleons which, during the course of a
central collision, are almost stopped. In contrast, at ultra-relativistic
energies, the interaction is rather dominated by the violent interaction of
two giant clouds of very low-x gluons, while the“stripped” valence quarks
mostly escape the interaction zone along the collision axis. That explains why
the net-baryon density, i.e. $\rho_{B}-\rho_{\bar{B}}$, is high in case of low
collision energies and essentially close to zero at mid-rapidity for collision
energies in excess of $\sqrt{s_{\mathrm{NN}}}\,=100\,\mathrm{G}e\mathrm{V}$.
Common to both situations is that the matter formed in the collisions are
transient states with lifetimes of a few fm/$c$ only.

The experimental challenge is to measure with sufficient precision probes
which are sensitive to the properties of the high-density and high-temperature
phase. Virtual photons are such a case. They occur as intermediary objects
which couple lepton pairs (i.e. $\mathrm{e}^{+}\mathrm{e}^{-}$,
$\mu^{+}\mu^{-}$) to the electromagnetic current of strongly interacting
systems. An important advantage of dileptons over hadronic probes is their
immediate decoupling from the strongly interacting system allowing them to
escape nearly undisturbed during all stages of the collision. Moreover, they
carry rich information since the four-momentum of the intermediary virtual
photon and its spin orientation can be reconstructed. This is an important
difference to real photons. But at the same time dileptons also pose
challenges: First, the signal is rare and has to be reconstructed with
sufficient purity. Second, experiments integrate the radiation emitted
throughout the evolution of the four-volume of the transient states. To
disentangle the cocktail of all sources contributing to the signal finally
involves also the inspection of the hadronic final state of the collision.
Last, pair spectroscopy gives rise to combinatorial background which has to be
determined with high precision and accuracy.

Dilepton spectroscopy in relativistic heavy-ion collisions has been pioneered
at SPS (CERN) in the mid eighties by NA34-HELIOS and NA38 and, soon later,
also investigated at much lower beam energies provided by the BEVALAC (LBNL)
using the two-arm spectrometer DLS. With the availability of lead beams at the
CERN SPS, dedicated second generation $4\pi$-experiments started operation.
CERES was designed to measure electrons with two cylindrical RICH detectors
and NA50, the successor experiment of NA38, used a 4 m hadron absorber in
front of a cylindrical spectrometer for muon detection. The dilepton program
at SPS has been temporarily concluded with a high statistics run with the NA60
dimuon spectrometer, an upgraded version of NA50. Meanwhile, dilepton
spectroscopy is pursued at almost all operational heavy-ion facilities, i.e.
from relativistic energies with beams of one G$e$V per nucleon on stationary
targets with HADES at the SIS18 of GSI, to ultra-relativistic energies as
provided by the colliders RHIC using the detectors PHENIX and STAR and at LHC
by CMS, ATLAS and ALICE. At LHC all the three experiments with central
detection are in principle able to investigate thermal dilepton radiation,
although continuum radiation in the invariant mass range below one
G$e$V/$c^{2}$ is mostly in the focus of the ALICE experiment.

In this paper, we review the current status of dilepton spectroscopy in
relativistic heavy-ion collisions. We in particular address continuum
radiation as an observable for QCD matter under conditions of high temperature
and energy density as formed in (ultra-)relativistic collisions of heavy ions.
An important aspect for the understanding of such continuum radiation is the
coupling of virtual photons to hadrons via intermediary vector mesons which is
also discussed in some detail. To cover the current experimental situation we
report recent measurements concluded at GSI/SIS18, BNL/RHIC and CERN/SPS/LHC.
Dileptons are also the proper tool to study medium modifications of hadrons in
cold (i.e. nuclear ground state) matter. The extraction of medium-
modifications requires a detailed knowledge of the coupling of virtual photons
to hadrons in vacuum. This aspect is related to electromagnetic transition
form factors of hadrons in the time-like region, which governs the
electromagnetic decay of excited hadronic states. We will also touch this
topic and will discuss the relevance/connection to the heavy-ion program. In
the following section we will start with a short survey about the
phenomenology of relativistic heavy-ion collisions. We will introduce the so-
called “standard model” of heavy-ion collisions and characterize various
stages of the reaction. In Sec. 3 we will address experimental aspects of
dilepton spectroscopy and discuss the difference between dimuon and dielectron
detection. An introduction to the theoretical foundations of dilepton
spectroscopy is presented in Sec. 4. Sec. 5, finally, is devoted to
experimental results and comparisons to model calculations. We finish with an
status and outlook.

##  2 QCD matter under extreme conditions

High-energy heavy-ion collisions are characterized by the collision energy
defined by the total center of mass energy per nucleon-nucleon pair
$\sqrt{s_{\mathrm{NN}}}$, the “size” of the collision system, determined by
the combination of ions brought to collision and the centrality of the
collision estimated i.e. by the event activity. The latter is typically
evaluated by the multiplicity of charged particles reconstructed for a given
event. The heaviest systems routinely chosen are the symmetric systems Au+Au
or Pb+Pb with around 400 nucleons participating in a central collision.

In an experiment at the BEVALAC utilizing a medium-heavy collision system
(Ta+Ta) at 400 $\,A$ M$e$V it was observed for the first time that the
particles expelled from the reaction zone exhibit collective behavior [1]. In
semi-central collisions protons and fragments showed a distinct emission
pattern, relative to a plane spanned by the beam axis and the impact parameter
then called side-splash or bounce-off and squeeze [2]. Observed was a
preferred in-plane flow in forward (near projectile) rapidity while at mid-
rapidity the flow was directed perpendicular to the reaction plane and
symmetrically back-to-back. Such a pattern is characteristic for a system in
which pressure is built up and the constituents move along the gradient of the
local density profile. Flow patterns are nowadays characterized by the Fourier
components $v_{n}$ of the modulating function

$F(\phi)=F_{0}\left(1+\sum_{n=1}^{N}2v_{n}cos(n\phi)\right)$ (1)

which can be investigated in multi-differential manner, like for different
particle species, and as a function of the centrality, the rapidity and the
transverse momentum. Here $\phi$ measures the azimuth (w.r.t. to the beam
axis) relative to the event plane. The second component $v_{2}$ (“squeeze”) is
particularly interesting. As a function of beam energy the sign of $v_{2}$
changes two times. At low beam energies, in the fusion-fission or multi-
fragmentation regime, fragment and nucleon emission is preferred in plane and
caused by attraction and resulting angular momentum of the combined nuclear
system. For BEVALAC, SIS18, AGS energies, $v_{2}$ turns negative (out of
plane) because the projectile and target like residues do not separate fast
enough from the fireball and block the in-plane emission of particles from the
hot and dense zone. At ultra-relativistic energies, $v_{2}$ is positive again
and a direct measure for the pressure of the initial, almond-shaped QGP
droplet [3]. The change of $v_{2}$ as a function of the collision energy
reveals collective behavior of the matter formed in heavy-ion collisions at
any energy between $\sqrt{s_{\mathrm{NN}}}\,=2\,\mathrm{G}e\mathrm{V}$ and
$\sqrt{s_{\mathrm{NN}}}\,=5\,\mathrm{T}e\mathrm{V}$. Moreover, in [4] it has
been conjectured that $v_{1}$ would signal a “softest point” if QCD feature a
first order phase transition to a deconfined medium in the region of high net-
baryon density. At the same time $v_{4}$ would exhibit a maximum [5]. Yet, the
quantification of the pressure is model dependent [6] and the quest for the
precise knowledge of the initial condition a key program of the heavy-ion
community. We will come back to the consequences of collective effects for the
emission of dileptons in Sec. 5.3.2 and 7.

For the discussion of dilepton emission the course of a heavy-ion reaction is
conveniently subdivided into three stages: An initial stage, which is
characterized by the violent first encounter of the constituents of the
nuclei, an intermediate stage when the system has build up transverse motion
and which by large can be understood as an extreme state of matter, and a late
stage characterized as dilute gas of excited hadrons which changes its
chemical composition solely by decays. In the first stage the system’s energy
density is determined by the degree of stopping, then the system converts
pressure into collective motion while it is evolving as a locally thermalized
state, and finally the system decouples and the chemical composition and phase
space-distribution of the particles traversing the detectors is determined.
This approach is indeed a qualitative picture and in reality the different
stages cannot be strictly separated. Moreover, the assumption that the
fireball has homogeneous properties at a given stage is not necessarily
justified. Rather, from the Glauber picture for colliding nuclei it follows
that less stopping occurs in the periphery. In phenomenological descriptions
of such reactions it has been tried to accommodate this fact by separating the
fireball into a core and a corona [7]). In the following, we will discuss in
more detail the three stages of the collision with emphasis on the relevance
for dilepton production and how an increasing collision energy draws on the
microscopic structure and dynamics of the matter.

###  2.1 The Initial Stage

Evidently, this stage is most affected by the collision energy. The nucleon
wavelength in the N-N rest frame shrinks from a size of the order of the
nucleon itself at SIS18 to dimensions where the softest partons of the
nucleons are resolved at LHC. At the same time also the penetration time, i.e.
the instant it takes until both nuclei overlap completely, changes
substantially. Compared to the typical hadron formation time (or a baryon
resonance lifetime $\tau_{\mathrm{R}})$ the penetration takes almost ten times
longer at SIS18 energies down to a fraction of only a few per mill at LHC
energies. At near relativistic collisions (SIS18), which we would like to
consider first, the initial collisions are of “nucleon nucleon type” and
proceed elastically and inelastically through resonant pion production via
intermediate baryonic resonances (mostly $\Delta$’s). Since the life times of
hadronic resonances are substantially shorter than the time it takes until the
fireball freezes out, they can be absorbed and recreated in subsequent
collision processes. Indeed, several generations of the resonances can be
created until complete overlap is achieved in central collision of heavy
nuclei. The built-up of the pressure is going in-line with the excitation and
decay of resonances and hence the separation between the initial and
intermediate stage is not clearly defined. While the first N-N collisions have
already taken place, more and more nucleons stream into the interaction
region. Hence, the in-streaming nucleons not only impinge on the counter-
streaming nucleons of the collision partner but also collide with (excited)
baryons which already “piled up” at mid-rapidity. This situations is referred
to as the formation of resonance matter featuring secondary collisions
including baryonic resonances [8]. With increasing beam energy, higher-lying
baryonic resonances are excited and the “frequency” of the collision processes
increases due to the growing Lorentz contraction. Once full overlap is
reached, a substantial fraction of the beam energy is converted to excitation
and pressure. The latter is composed of thermal as well as static pressure due
to the nuclear mean field. At beam energies of around 10 $\,A$ G$e$V(top AGS),
the time for full penetration is already as short as the lifetime of a
baryonic resonance. In the central region of such collisions baryons might
collide subsequently so frequently that the resonance states are not even
fully established between subsequent scatterings.

Dilepton radiation from this stage can in general be described as multiple N-N
bremsstrahlung (cf. Sec. 4.2). In most general terms, the emission process of
a bremsstrahlung photon includes amplitudes for quasi-elastic as well as
inelastic collisions involving intermediary baryonic resonances. One way to
theoretically describe this stage is to study the time evolution of the
nucleon collisions by means of microscopic transport models (see e.g. [9]).
However, the baryon densities can be high and the baryon meson-clouds start to
overlap substantially. This situation is expected to give rise to strong
modifications of the baryonic resonance properties or even might constitute
limiting conditions for a resonance gas. A consistent numerical treatment of
in-medium effects governing the colliding baryons and mesons is based on
kinetic theory and has been formulated by Kadanoff and Baym in the 60’s [10].
Several microscopic transport models have been developed which use
approximations of the Kadanoff-Baym equations of various sophistication (cf.
4.7). In cascade mode, i.e. with many-body effects essentially turned off, the
nucleons flow on straight trajectories between collisions and the individual
scattering processes are described on the basis of known, extrapolated or
calculated cross sections. This is a good description as long as the average
time between collisions $\tau\sim\left(\sigma\rho\beta\right)^{-1}\;$ is
longer than the time it takes until the scattered hadron has reached its
asymptotic state [11].

Towards higher collision energies the situation is fundamentally different.
Even a p-p collision can turn into a complex process since parton-parton
interactions are at action. The high energy and strong interaction gives rise
to a vast range of possible final states. Moreover, due to the Lorentz
contraction essentially all parton-parton interactions happen at the same
time. Parton-parton collisions range from the very soft (non-perturbative) to
the hard (perturbative) regime. The discussion of the many facets of such
calculations carried out to describe proton proton collisions at collider
energies goes way beyond the scope of this paper. We will focus on
observations which are relevant for dilepton production.

Hard initial state parton-parton collisions are responsible for the production
of charm and heavier flavor and to direct virtual photon production through
quark anti-quark annihilation via the Drell-Yan process. Both processes give
rise to hard virtual photon production dominating , depending on beam energy,
different regions of the dilepton invariant mass spectrum. It is commonly
understood that these contributions can be calculated on the basis of
perturbative QCD taking into account next to leading order graphs 4.3.
Perturbative methods, however, cannot straightforwardly be applied to beam
energies below $\sqrt{s_{NN}}\lesssim 10$ G$e$V. The calculation of charm
production down to threshold energies might become possible, though, since
long distance dynamics in the hard process described by so-called Sudakov
corrections seem to dominate over the higher order corrections [14]. In
contrast, the large energy density generated in the initial stage of the
heavy-ion collision is by large due to multiple collision of soft partons,
i.e. mostly gluons. In this context it is interesting to note that pomeron
exchange has recently found interest again to describe the non-perturbative
(soft) interactions [15]. Pomerons can be understood as coherent multi-gluon
states carrying the quantum numbers of the vacuum. In particular they do not
carry charge and consequently do not radiate (virtual) photons. In the NEXUS
model, the multi-parton collisions are separated into hard collisions (those
appearing at large momentum transfer) and de-localized, soft collisions
treated in an effective manner via string formation [16]. The stochastic
treatment of the multi-parton collisions allows to study also the energy
density variations over the initial fireball volume. These distributions,
which fluctuate event by event, are in particular responsible for the higher
order angular modulations observed in the particle emission [20].

While many observables in heavy-ion collision scale with the number of
nucleons participating in the collision, hard processes do not trivially scale
with the the number of hard binary collisions due to shadowing effects [18].
nevertheless, the number of participating nucleons is a key quantity to
classify heavy-ion reactions at high energies. Centrality allows a good
classification of the transverse energy production, which in turn is a good
measure of the entropy of the system. As the centrality increases, more and
more nucleons take part in the formation of the nuclear fireball. In
experiment, the number of charged particles in the acceptance of the detector
system is usually used to approximate the centrality. Glauber Monte Carlo
simulations can be used to relate a certain multiplicity class to the
respective centrality class defined by a mean number of participating
(primordial) nucleons $\langle A_{part}\rangle$ [19]. Its has been shown, that
this strategy can be applied for experiments with beams of a few $\,A$ G$e$V
on stationary targets [21] up to a few $\,A$ T$e$V energies at the LHC [22].
At ultra-relativistic energies the initial stage of the collision does not
contribute substantially to the yield of soft (virtual) photons. However, the
high pressure in the initial state and the flow generated out of it has some
effect on the momentum distribution of the (virtual) photons (cf. sec. 6.2).

###  2.2 The Hot and Dense Phase

The intermediate stage evolves from a state of maximal energy density. The
values achieved depend on the collision system and energy as well as on the
impact parameter. In the most general case, the energy density is not constant
across the interaction volume but will drop towards its periphery and will
exhibit fluctuations. However, there is good evidence that throughout the
evolution during this stage the system is locally equilibrated. The true
nature of the “thermalization” process leading to this situation is still a
matter of research. In particular, at lower beam energies, rapidity
distributions of protons suggest that the fireball is not fully equilibrated.
Yet, it is conceivable that different degrees of freedom, e.g. the baryon
excitation spectrum versus the baryon momentum distribution, will have
different equilibration times. Fast thermalization in general can be achieved
if three or many-body collisions are at work, or, arguing in the language of
quantum mechanics, if a certain degree of entanglement is realized such that
the combined wave function of a number of constituents probes the phase space
according to the level density available (Fermi’s golden rule). At the high-
energy frontier, recently a pre-equilibrium stage has been introduced, which
connects the initial stage characterized by multi-gluon interactions, to a
later stage, when the system can be safely treated as hydrodynamic evolution.
This stage describes the evolution of the system to equilibration using
kinetic theory for gluons [17].

Throughout the “hot and dense” stage, the microscopic structure of the matter
can primarily be of hadronic or partonic character. It is said to have
hadronic character if the constituents are good quasi particles identifiable
by their pole masses and partonic, if quasi-free quarks and gluons are the
relevant degree of freedom. This are somewhat idealized pictures resembling
features of a dilute hadron resonance gas and perturbative QGP, respectively,
both not precisely realized in nuclear collisions. We would like to give an
intuitive picture how a dense hadronic phase could be understood based on the
cloudy bag model. In a sufficiently dense baryonic system, meson clouds of
neighboring baryons will substantially overlap. If such a system is heated,
the average kinetic energy of a baryon increases as does the probability to
find baryons in an excited state of growing excitation energy. In an excited
baryon the energy can be carried as well by the mesonic excitations of the
cloud or by the dynamics of the valence quarks r pure quark model
configurations. In other words, the wave function of an excited baryon can
contain contributions of baryon-meson bound states. Such a separation into
core and cloud effects has recently been put forward in describing the
structure of baryonic resonances probed by electron scattering
experiments[23]. Here it has been observed that in particular the soft, i.e.
long wave length, perturbations are carried by the cloud. Kinetic energies
governing the hadronic interactions in a thermalized system formed in heavy-
ion collision are generally of soft character. In a baryon-dense medium,
however, it will be difficult to attribute the mesonic excitations to a single
baryon core only. Rather, mesonic states might develop a different spectral
structure due to the simultaneous coupling to several baryonic cores. In such
a case, one can speak of a de-localization of the mesonic excitation which,
however, would be strongly modified due to the coupling to neighboring
hadrons. As will be discussed in Sec. 4.4, vector mesons states propagating to
a dense hadronic medium are believed to be the prime source of dileptons with
invariant masses up to one G$e$V/$c^{2}$.

Figure 1:  The solid blue lines depict conjectured first order phase
transition lines ending in critical points. The phase regions are the quark
gluon plasma (QGP), hadron gas (HG) and nuclear liquid (NL). The (black)
squares represent freeze-out parameters derived from hadronic final states
while circles depict mean fireball temperatures extracted from dilepton excess
radiation. Results from lattice QCD calculation for the location of the HR-QGP
cross over is shown as yellow band The dashed yellow line marks the limit of
validity of the Taylor expansion used for deriving thermodynamic parameters
from lQCD.

This picture resembles ideas developed by Hagedorn to interpret particle
spectra observed in violent hadronic collisions, known as the bootstrap model
(see [24] for a comprehensive discussion and historical aspects). The basic
idea is that a nuclear fireball contains in itself little fireballs, each
populated with densities governed by Boltzmann weights (cf. Eq. 4). As a
consequence, already in the 60’s Hagedorn suggested that a system composed of
hadrons cannot be heated beyond a critical temperature since at this
temperature mass production would be preferred over the increase of the
average kinetic energy. These constituent fireballs are nowadays interpreted
as high-lying hadron resonances, which in turn can have internal structures
comprising multi-quark components. Examples for such types of baryonic
resonances at lower excitation energies are e.g. the $\Lambda(1405)$, the
Roper resonance $N^{*}(1440)$, which have strong meson-core components in
their wave functions.

At the highest collider energies, the initial energy density reached in the
collision zone is well beyond $1\,\mathrm{G}e\mathrm{V}/\mathrm{fm}^{3}$ and
net-baryon density practically zero. It is known from lattice QCD calculations
that at such energy densities QCD matter resides in a partonic phase.
Moreover, from the evaluation of the chiral susceptibility [32], which is
obtained from the second derivative of (the logarithm of) the partition
function w.r.t. the quark mass a pseudo-critical temperature of
$T_{c}=156.5\pm 5$ M$e$V is found [33],[34]. If quarks were precisely
massless, QCD matter would undergo a second order phase transition to a
chirally restored phase in the vicinity of this temperature. There is evidence
[26] that a fireball created at collider energies, will freeze-out into a
hadronic medium at a temperature just around the pseudo critical temperature
derived from QCD. There is still debate, to what extend residual inelastic
collisions will still occur in the hadronic system. This question is of some
importance for the understanding of the dilepton emission and will be
addressed later.

While virtual photon radiation from a QGP can be understood as being due to
abundant radiative $q\bar{q}$ annihilation (cf. Sec. 4.4), the cloudy bag
model is in particular helpful to understand virtual photon radiation expected
from a dense, baryon-rich hadronic medium. Indeed, the Vector Meson Dominance
Model (VDM) suggests that all photons couple to hadrons through intermediate
vector mesons, which in turn can be regarded as an excitation of the cloud
carrying the quantum numbers of the photon. As it will be discussed in detail
in Sec. 4.1.1, VDM is very successful in the description of electromagnetic
transitions of hadrons in vacuum (Dalitz and two-body decays). Applying this
model to means to study in order Investigating the spectral distribution of
virtual photons emitted from a dense hadronic system in the context of VDM
means studying in-medium vector meson propagators, most importantly the one of
the $\rho$ meson. As we will discuss in Sec 4.4, various model calculations
demonstrate strong modifications of the in-medium $\rho$ spectral function due
to the coupling of the $\rho$ to baryon-resonance hole states. The salient
feature of these calculations is the ansatz of VDM for baryon-photon coupling
stressing the importance of a sound microscopic understanding of the
underlying transitions (cf. Sec. 4.1).

To extract model predictions for the baryon densities and temperatures reached
in the various stages of heavy-ion collisions one can use “coarse graining”
methods. For this, multiple simulated heavy-ion collision events, each
computed with the same collision energy, system size and impact parameter, are
sampled on a 3+1 dimensional space-time grid [12]. The densities of a given
space-time cell can then be derived by boosting each cell in its local rest-
frame. The degree of thermalization can be assessed by inspecting the stress-
energy tensor in each cell. For example, using UrQMD it has been found that
the density in central Au-Au collisions at
$\sqrt{s_{\mathrm{NN}}}\,=2.4\,\mathrm{G}e\mathrm{V}$, averaged over a central
volume of $7^{3}\,\textrm{fm}^{3}$, reaches already
$\rho=0.5\;\mathrm{fm}^{-3}\simeq 3\,\rho_{0}$ about $12\;\textrm{fm}/c$ after
first contact of the nuclei. With a similar approach, densities of
approximately $1$ and $2\;\mathrm{fm}^{-3}$ where found for
$\sqrt{s_{\mathrm{NN}}}\,=3.5\,\mathrm{G}e\mathrm{V}$ and
$\sqrt{s_{\mathrm{NN}}}\,=8.8\,\mathrm{G}e\mathrm{V}$ [13], respectively.
Using a 3-fluid hydro-simulations densities reach even about 20% higher
values. It can therefore be conjectured that maximum net-baryon densities 20
times higher than the ground state density are produced in the central region
of heavy-ion collisions. However, the densities derived in this way have to be
taken as upper values since the short-range part of the nucleon-nucleon
potential is not explicitly treated in most microscopic transport models.

###  2.3 Freeze-out

In this conceptional approach to dilepton emission from QCD matter created in
heavy-ion collisions it is logical to define a third phase, which spans the
time after the system’s chemical composition of particles is frozen and only
changed by electroweak, or strong decays with ($c\tau\gg t_{\mathrm{f.o.}}$).
The idea behind this classification is to identify contributions to the
integrated dilepton yield which can be understood as being due to dilepton or
Dalitz decays of hadrons in vacuum and proportional to the hadron yields at
freeze-out. With this definition, the yield connected to this stage of the
collision can in principle be derived independently if the respective hadron
multiplicities of a certain reaction class are measured and the decay
processes is well understood. Based on observed multiplicities and phase space
distributions of hadrons, the expected dilepton yield from this stage can be
calculated using the respective branching ratios for decay channels with a
dilepton in the final state. The resulting invariant mass distribution
containing dileptons from different hadronic sources is called hadronic
cocktail, or in brief cocktail. This approach can be verified experimentally
by comparing a derived cocktail to the measured dilepton invariant mass
distribution in elementary reactions (cf. Sec. 5.1), or by inspecting a very
peripheral heavy-ion reaction class.

Figure 2:  Hadron densities at chemical freeze-out assuming a thermalized
hadron resonance gas according to Eqs. 2, 3 and 4 as function of the collision
energy. For the $\rho$ a hypothetical “off-shell” $\rho$ with mass 500
M$e$V/$c^{2}$ is chosen representing the LMR medium radiation. Width is
neglected throughout. The dashed-doted line shows the temperature at chemical
freeze-out.

Moreover, at high collision energies it is well established that the (charged)
particle multiplicities can be described by grand canonical partition
functions assuming that the system suddenly hadronizes out of a thermalized,
deconfined phase (QGP) [25, 26]. While details are still under discussion (see
e.g. [27]), this observation gives straight access to particle abundances as
they are solely determined by a few thermodynamical parameters like volume,
temperature and chemical potentials. Indeed, from the inspection of hadron
multiplicities observed for various collision energies and spanning from
$\sqrt{s}\,=5.13\,A\,\mathrm{T}e\mathrm{V}$ (top LHC energies) down to
$\sqrt{s}$$\;=7$ $\,A$ G$e$V (lowest RHIC energies in collider mode) it was
found that the derived freeze-out temperatures ($T$) and baryo-chemical
potentials ($\mu_{b}$) all line up on a narrow band in the QCD phase diagram
(see Fig. 1). As an example, in [28] a phenomenological functional dependence
has been reported for central collision of heavy collision systems (i.e. Au-Au
or Pb-Pb) as

$T(\sqrt{s_{\mathrm{NN}}})=\frac{T_{\mathrm{lim}}}{\left(1+\exp\left(2.6-\ln\left(\sqrt{s_{\mathrm{NN}}}/(0.45\cdot\mathrm{G}e\mathrm{V}\right)\right)\right)}$
(2)

and

$\mu_{B}(\sqrt{s_{\mathrm{NN}}})=1303\cdot\mathrm{M}e\mathrm{V}\cdot\left(1+0.286\cdot\mathrm{G}e\mathrm{V}\cdot\sqrt{s_{\mathrm{NN}}}\right)^{-1}.$
(3)

The parameter $T_{\mathrm{lim}}$ defines the plateau in temperature for high
beam energies observed in experiment (see cf. Fig. 1). Other definitions of a
universal freeze-out line assume fixed energy per particle or constant entropy
density etc. [30]. Using the above parameterization (Eqs. 2 and 3), the
particle multiplicity densities $n_{i}$ in central collisions can be
calculated for a given collision energy using the approximate formula [29]

$\displaystyle n_{i}\left(T,\\{\mu\\}\right)$ $\displaystyle=$
$\displaystyle\frac{g_{i}T^{3}}{2\pi^{2}}\sum_{n=1}^{\infty}\frac{\left(\pm\lambda_{i}\right)^{n}}{n^{3}}\left(\frac{nm_{i}}{T}\right)^{2}K_{2}\left(\frac{nm_{i}}{T}\right)$
(4) $\displaystyle=$
$\displaystyle\frac{g_{i}\lambda_{i}}{2\pi^{2}}m_{i}^{2}TK_{2}\left(\frac{m_{i}}{T}\right)\pm\mathrm{h.o.}\>.$

Here $n_{i}$ denotes the number density for the particle of type $i$,
$\\{\mu\\}=\\{\mu_{B},\mu_{Q},\mu_{S}\\}$ the chemical potentials of the
emitting system connected to the conserved quantum numbers: baryon number
($B$), charge ($Q$) and strangeness ($S$) and $m_{i}$ the mass of the
particle. $K_{2}$ is the Bessel function of second kind. The minus sign has to
be used for fermions (baryons) while the plus sign is for bosons (mesons).
Moreover,

$\lambda_{i}\left(T,\\{\mu_{i}\\}\right)=e^{\mu_{B}B_{i}+\mu_{Q}Q_{i}+\mu_{S}S_{i}}$
(5)

is the fugacity factor of particle $i$ with the chemical potentials
$\\{\mu_{i}\\}$ and

$g_{i}=2J_{i}+1$ (6)

the spin degeneracy of the particle. The formula breaks the series expansion
(4) after the first term and is precise to within 10% for not too small
$m_{i}/T$. The pion to nucleon ratio along the freeze-out line can now readily
be calculated (see Fig. 2). As can be seen, matter created at colliders
freezes out clearly as pion dominated fireballs while at SIS18/BEVALAC
energies the system has only about 10% pions in the final state. Generally, a
substantial additional amount of pions is produced by rapid decays of (heavy)
mesonic or baryonic resonances. In general, such a freeze-out scenario can
describe the yields and the soft part of the transverse momentum distributions
of measured particles, once in addition collective expansion of the fireball
is taken into account.

As it has already been mentioned, in semi-central collisions a strong $v_{2}$
signal is observed. At beam energies of a few $\,A$ G$e$V particle emission in
the reaction plane is partially suppressed due to absorption of particles
emitted from the collision zone (participant matter) in the bypassing
projectile and target (spectator) matter. The latter effect results in a
negative $v_{2}$. With increasing beam energy the $v_{2}$ signal turns
positive (to in-plane) with increasing strength. At the highest RHIC energies
it has been found that $v_{2}$ [56] for identified baryons and mesons can be
collapsed to a unique curve if $v_{2}$ is scaled to the number of constituents
quarks $n_{q}$ of the identified particle (i.e. $v_{q}=2$ for mesons and
$v_{q}=3$ for baryons) and plotted in bins of transverse kinetic energy per
constituent quark $E_{T}/n_{q}=\left(m_{T}-m_{0}\right)/n_{q}$, where $m_{T}$
is the transverse mass and $m_{0}$ the rest mass of the identified hadron.
This observation is commonly called quark number scaling. The success of the
hydrodynamical description of the fireball evolution at ultra-relativistic
collision energies supports the assumption of a very rapid thermalization in
the initial stage. In particular the strong flow effects and quark number
scaling give very clear evidence for a deconfined phase (partonic) phase of
the matter which is strongly coupled. Moreover, the flow develops throughout
the whole expansion until the system freezes out. Hence, it is to be expected
that any penetrating probe, as dileptons, which on average decouple from the
system early on, should not follow this flow systematics.

The validity of this expansion scenario has been studied at RHIC by STAR in
the the so-called beam-energy scan [57]. No major non-monotonic behavior of
observables characterizing the fireball evolution has been found down to
collision energies of $\sqrt{s}\,=7.7\,A\,\mathrm{G}e\mathrm{V}$ so far.
Exceptions maybe the excitation function of anti-hyperons [48], which show
indeed a change it the centrality dependence, and the excitation function of
Gaussian source radii extracted from two-pion interferometry [49]. The main
purpose of this energy scan, which is currently repeated with more statistics
and an upgraded detector system, is to scrutinize this search for non-
monotonic excitation functions of observables and/or for the disappearance of
signatures evidently attributed to the existence of a partonic phase in this
intermediate stage.

As it was already said, a strict division of a heavy-ion reactions into well
separated stages is an idealization. Yet, it eases the way to the
interpretation of dilepton emission in (ultra-) relativistic heavy-ion
collisions. At the high energy frontier dileptons emitted in the initial stage
(cf. Sec. 2.1) are characterized by partonic (hard) processes. Although at a
few $\,A$ G$e$V first chance collisions are not hard, in the sense that
perturbative QCD would apply, yet typical values of $\sqrt{s_{\mathrm{NN}}}$
(taking the Fermi momentum into account) soon significantly depart from those
characteristic for binary N-N collisions in the thermalized fireball. The
contribution from this first chance collisions or pre-equillibrium stage can
be experimentally accessed by studying elementary (reference) systems (cf.
Sec. 5.1).

Finally, one might like to question that a thermalization of the fireball is
achieved at low beam energies. In that case, microscopic (transport) theory
would be the only way to theoretically address the dilepton continuum.
However, the recent development of ”coarse graining” methods (cf. Sec. 2.1)
for the description of dilepton production offers yet another possibility
merging both approaches. We will come back to this point in the discussion of
experimental results cf. Sec. 6.

###  2.4 The phase diagram of QCD matter

The present understanding of the phase structure of QCD matter is illustrated
in Fig. 1. The properties of QCD matter at vanishing baryo chemical-potential,
i.e. for baryon anti-baryon symmetric matter, is well understood from lattice
QCD calculations. A cross over between a hadron gas at low temperature to a
strongly coupled liquid of quarks and gluon has been located around a pseudo-
critical temperature of $T_{pc}=156$ M$e$V [33, 35]. Below the pseudo-critical
temperature thermodynamic quantities derived from QCD lattice calculations are
in line with respective quantities computed in the hadron resonance gas model
[36]. Above this temperature they deviate and lattice QCD only gradually
approaches the properties of an ideal gas of quarks and gluons at very high
temperatures. For temperatures likely reached in heavy-ion collisions, i.e.
below 300 MeV, the quantities reflect the properties of a strongly couple
fluid. The transition from the chirally broken situation to its full
restoration is a smooth cross over and falls approximately in place with the
transition from a confined to deconfined state.

In contrast, for matter with large net-baryon density (high $\mu_{B}$) the
deconfinement transition is expected to be of first order. The region in the
phase diagram where the transition turns to be first-order, characterized by
the occurrence of a critical point, has not been located yet. Thermodynamic
properties of strongly interacting matter in the region of finite baryo-
chemical potential cannot easily be assessed with lattice QCD. The reason is a
sign problem which causes a complex Fermi determinant once $\mu_{B}$ takes
finite values. Using Taylor expansion techniques it became possible to extent
lattice results to $\mu_{B}\simeq 0.5T$, showing no indication for critical
point indicating that the hadron-QGP transition would turn into first order.
The search for the critical point, or likewise for indications of a first
order deconfinement transition, is a key program of relativistic heavy-ion
experiments. In parallel, theoretical approaches to solve QCD based on
effective field theories, like Functional Renormalization Group methods or
Schwinger-Dyson calculations, are developed to narrow done the uncertainties.

McLerran and Pisarski have proposed to “organize” the phase diagram of QCD
matter by inspecting QCD properties in the limit of a large number of colors
($N_{c}\rightarrow\infty$ limit) [31]. Obviously, in the hadronic world the
number of degrees of freedom do not change with the number of colors since
hadrons are color neutral objects and need to contain all colors to neutralize
color charge. The number of degrees of freedom in the partonic phase would
scale like $N_{c}^{2}$, because the number of gluons species carry
combinations of color and anti-color. They argued that there should exist a
third phase, termed quarkyonic, which would be characterized by a linear
scaling of the degrees of freedom with the number of colors. It would occur at
high net-baryon densities and moderate temperatures and could be considered as
a gas of confined quarks surrounded by hadronic excitations and glueballs.
However, a rigorous derivation of the phase structure in the real world with
three colors only is not available.

The quest is to locate “trajectories” in the phase diagram along which the
collision systems evolve during the intermediate stage of heavy-ion collisions
and before the system freezes out. Since the abundancies and phase space
distributions of hadrons are fixed only after the system has frozen out, the
determination of such trajectories based on hadron spectra is close to
impossible. Guidance where to locate fireballs formed in heavy-ion collisions
probe the QCD phase diagram is given by the so-called freeze-out parameters
which are depicted in Fig. 1 as black “data points” (cf. Sec. 2.3). They are
obtained from fits of the hadron resonance gas (Eq. 4) to freeze-out
abundancies of a large number hadrons measured in the experiments. The dashed
line connecting the data points is defined by Eqs. 2 and 3.

Electromagnetic probes are currently the only way to obtain direct information
from the dense system before freeze-out. Two such data points are included in
Fig. 1 as circles. They refer to temperatures obtained from the slope of
dilepton invariant mass spectra. The approximate baryo chemical-potentials are
taken from the respective freeze-out point. Both lie above the freeze-out
curve as is expected due to the penetrating nature of dileptons. The procedure
to extract such information from dileptons emitted from heavy-ion collision is
a central subject of this review.

##  3 Experimental aspects of dilepton spectroscopy

The central goal of continuum dilepton spectroscopy is to reconstruct the so-
called excess radiation, which is emitted out of the dense and hot stage of a
heavy-ion reaction. This is a formidable task since on the one hand we have to
deal with pair reconstruction, which requires to form all possible opposite
sign pairs from all leptons (electrons or muons) reconstructed in a given
event and hence generates combinatorial pairs not originating from the same
virtual photon. On the other hand, the expected distribution of the excess
radiation is rather structure-less and has to be extracted as one of all
signal contributions which all are adding to a eventually huge background of
combinatorial pairs. Moreover, lepton-pair decays of hadrons or pair emission
from annihilation processes are suppressed of order $\alpha^{2}$ and hence are
truly rare probes. The reconstruction procedure for the dilepton signal can be
grouped to the following steps:

  1. 1.

Reconstruction of sufficiently pure lepton track candidates.

  2. 2.

Rejection of background leptons on the single track level.

  3. 3.

Background rejection on the pair level.

  4. 4.

Estimation and subtraction of the combinatorial and correlated background.

  5. 5.

Identification of the excess radiation.

  6. 6.

Efficiency and acceptance correction of spectra.

In the following we first introduce the dilepton observables. Thereafter, we
address experimental aspects of dimuon and dielectron signal extraction and
then discuss how the unknown combinatorial background can be approximated
based on random combinations of leptons. At the end, we introduce selected
experiments which have been used, or are still in use, for dilepton continuum
spectroscopy.

###  3.1 The dilepton signal

The 6-fold differential production probability for dileptons in covariant form
reads

$E_{+}E_{-}\frac{\mathrm{d}^{6}P}{\mathrm{d}^{3}p_{+}\mathrm{d}^{3}p_{-}}=\frac{\mathrm{d}^{6}P}{q_{\perp}\,\mathrm{d}M_{ll}\,\mathrm{d}y\,\,\mathrm{d}q_{\perp}\,\mathrm{d}\varphi\,\mathrm{d}\theta\,\mathrm{d}\phi}\,.$
(7)

Figure 3:  Angles characterizing the decay of virtual photons in its rest
frame with the z axis pointing along the photon direction. The production
plane is spanned by the beam axis and the momentum vector of the virtual
photon in the laboratory frame. The definition of the two angels $\theta$ and
$\phi$ depends on the polarization axis used.

Here $E_{+-}$ and $p_{+-}$ denote the lepton energies and momenta,
respectively. The common choice for the six independent coordinates, which
characterize the emission of the virtual photon in the beam system, are
typically taken to be the invariant mass $M_{\ell\ell}$, the transverse
momentum $q_{\perp}$, the rapidity $y$, the polar $\theta$ and the azimuthal
emission $\varphi$ angles of the virtual photon relative to the beam axis, and
the decay axis of the lepton pair relative to the emission axis of the virtual
photon in the photon rest frame $\alpha_{h}$ (helicity angle). In case
electrons are observed ($m_{\ell}=m_{e}\ll M_{\ell\ell}$), the invariant mass
can be well approximated using the geometric mean of the lepton momenta as

$M_{ee}\simeq
2\sqrt{|p_{\ell}|\,|p_{\ell}|}\;\sin{\left(\frac{\alpha_{\ell\ell}}{2}\right)}$
(8)

with $\alpha_{\ell\ell}$ denoting the opening angle of the lepton pair in the
laboratory. This equation demonstrates that dileptons with small invariant
mass but sufficient momentum to traverse the detector systems appear with
small opening angle. This topology can be used to reduce combinatorial
background (cf. Sec. 3.2.2). It may be convenient to use the transverse mass
$m_{\perp}$ instead of the transverse momentum. For the lepton pair it is
defined through

$m_{\perp}^{2}=M_{\ell\ell}^{2}+q_{\perp}^{2}=q_{\mu}q^{\mu}+q_{\perp}^{2}\;$
(9)

where $q$ is the virtual photon four-momentum.

It is common to distinguish dilepton invariant mass regions according to the
sources dominating them: the Low Mass Region (LMR) below $1.1$ G$e$V/$c^{2}$,
i.e. including the $\phi$ meson pole mass, the Intermediate Mass Region (IMR)
$1.1<M_{l^{+}l^{-}}<2.9$ G$e$V/$c^{2}$, spanning between the low-mass vector
mesons and charmonium, and a High Mass Region (HMR) above. The relative
abundancies of pairs in these regions depends primarily on the collision
energy per nucleon pair. While in experiments at LHC, RHIC and SPS, dileptons
are populating all regions, experiments at SIS18/BEVALAC up to today could
only explore the LMR.

The polar and azimuthal emission angles of individual leptons are connected to
the photon polarization, which in turn depends on the production mechanism.
The respective angular distributions are given in the rest frame of the
virtual photon and can be decomposed in the following way (see [37] and
references therein):

$\displaystyle\frac{\mathrm{d}P}{\mathrm{d}\Omega_{\ell}}=1$
$\displaystyle+\lambda_{\theta}\cos{\theta_{\ell}}^{2}+\lambda_{\phi}\sin^{2}\theta_{\ell}\cos{2\phi_{\ell}}+\lambda_{\theta\varphi}\sin{2\theta_{\ell}}\cos{\phi_{\ell}}$
(10)
$\displaystyle+\lambda_{\varphi}^{\perp}\sin^{2}\theta_{\ell}\sin{2\phi_{\ell}}+\lambda_{\theta\varphi}^{\perp}\sin{2\theta_{\ell}}\sin{\phi_{\ell}}\,.$

The factors
$\lambda_{\theta},\lambda_{\phi},\lambda_{\theta\varphi},\lambda_{\varphi}^{\perp}$
and $\lambda_{\theta\varphi}^{\perp}$ are the anisotropy coefficients where
$\lambda_{\varphi}^{\perp}$ and $\lambda_{\theta\varphi}^{\perp}$ are non-zero
only if the emission is asymmetric with respect to the scattering plane and
$\mathrm{d}\Omega_{\ell}=\mathrm{d}\cos{\theta_{\ell}}\>\mathrm{d}\phi_{\ell}$.
Depending on the choice of the reference frame, the coefficients can take
different values. A good choice for thermal radiation is the helicity frame
(cf. Fig. 3). Here, the quantization axis (z) is taken to be along the
momentum direction of the virtual photon in the heavy-ion collision (or
hadron-hadron) center of mass. The x-axis is taken in line with the reaction
plane. For the discussion of Drell-Yan radiation or quarkonium production also
the Collins-Soper frame is used for which the bisector between the two hadron
(heavy-ion) momentum directions in the virtual photon rest frame is taken as
the quantization axis. The photon polarization and consequently lepton angular
distributions can provide essential information about the dilepton production
mechanism as well in vacuum as in hot and dense medium. This topic is
discussed in Sec. 4.

###  3.2 Extraction of the dilepton signal

Signal dileptons are pairs which originate from the same virtual photon
(mother particle). From the physics point of view it does a priori not matter
if electrons or muons are used to reconstruct the virtual photon spectrum,
aside from a different phase space cut-off at low invariant masses due to the
by a factor $\simeq 200$ higher rest mass of muons. Experimentally, however,
there are substantial differences which are related to both, the detection of
the signal with sufficient purity as well as the rejection of background.

####  3.2.1 Muons

The detection of muons has advantages over electron detection if the
laboratory momenta of the leptons are sufficiently high. Using hadron
absorbers in front of a muon detection system enables that essentially only
muons reach these detectors. Such absorber technique can, in certain cases,
allow to easily trigger on the production of muons or even on muon pairs which
is in most cases impossible for electrons. A disadvantage of the absorber
technique is that hadron identification in the acceptance of the absorber
requires placing of respective detectors in front of the absorber which
enlarges the whole detection system accordingly. Hadron absorbers, moreover,
introduce collisional energy loss and multiple scattering to muons.
Consequently, absorber techniques are advised if the laboratory momenta (or
energy) of the muons are not too small. This can be assessed by computing the
the mean muon energy loss $<dE/dx>$ of an absorber material over one nuclear
interaction length $\lambda_{I}$ in this material:

$<E_{\mathrm{loss}}>=\int_{0}^{\lambda_{I}}\frac{\mathrm{d}E}{\mathrm{d}x}\mathrm{d}l\approx\lambda_{I}\;\frac{\mathrm{d}E}{\mathrm{d}x}\biggr{\rvert}_{\mathrm{MIP}}\,.$
(11)

For Iron, using for simplicity the mean energy loss of a minimum ionizing
(MIP) muon of $1.45$ M$e$V /(g/cm${}^{2})$, a muon has lost $\approx 200$
M$e$V over a distance after which the hadron’s survival probability has
dropped to $P_{\mathrm{surv.}}=1/e$. This is a rough approximation in case of
low hadron momenta where hadronic cross sections strongly vary because of
resonant scattering.

The success of such a detection strategy depends on the configuration of the
spectrometer. NA50 used a stand-alone muon spectrometer behind a thick
absorber and no magnet in front of it. As a consequence, the muon tracks had
to be extrapolated backward to the target (primary vertex) region through the
absorber in order to gain information about the origin of the muons. Multiple
scattering of muons in the absorber, however, blur the vertex resolution of
such tracks. This situation was overcome after the upgrade to the NA60
experiment ([38] and references herein) when a dipole magnet was added in
front of the absorber. The region in the dipole field was filled with high-
resolution silicon strip and pixel detectors. This enabled matching of track
segments reconstructed in the muon spectrometer to track candidates obtained
from the silicon tracker by matching the candidates in configuration and
momentum space. The “double match” provided sufficient selectivity to reach
high purity in the muon reconstruction by avoiding fake matches of tracks
produced by different particles in front of and behind the absorber.

Moreover, the precision of the track reconstruction right behind the target
empowered a partial rejection of off-vertex tracks, which are characteristic
for daughter muons escaping after weak decays of charmed hadrons near to the
primary vertex. A similar approach will also be realized in the upgrade of the
ALICE muon forward detector for Run 3 and 4 [39]. A third option is to segment
the absorber and to interlace it with tracking detectors. This allows to
substantially reduce the negative effect of multiple scattering by
continuously matching the proper track segments to a global track. Such a
solution has been chosen by the CMS experiment at CERN [40] which in addition
uses the return yoke of the solenoid as hadron absorbers and the reversed
field herein as second spectrometer. This technique is also foreseen for the
muon detection system of the upcoming CBM experiment at FAIR [41], albeit not
having a strong magnetic field in the absorber region. The challenge here is
the high track density in the detector behind the first absorber which “sees”
substantial low-energy secondary particles at a stage where not all primary
hadrons are stopped yet.

A background source of muons, when using absorber techniques, are weak decays
of charged pions or kaons in front of the absorber or punch-through hadrons,
i.e. hadrons which escape the absorber including hadrons produced through
interactions in the absorber. In the case of a weak decay e.g. a pion will
decay into a muon by radiating off a neutrino and carrying on most of the
pion’s momentum with gradual changes in the direction. It then will penetrate
the hadron absorbers in the same way as signal muons do. The fraction of pions
(or kaons) decaying before an absorber placed at a distance $d$ away from the
interaction point is

$P_{\downarrow}=1-e^{-\gamma\frac{d}{c\gamma\tau}}$ (12)

for a pion (kaon) with laboratory energy $E$, lifetime $\tau$ and
$\gamma=\frac{E}{m_{\pi,K}}$. Evidently, muon detection is advantageous for
fixed target experiments at ultra-relativistic energies, in collider
experiments at large forward or backward rapidities or for high-$p_{\perp}$
probes in the central rapidity region due to the large $\gamma$ factors in the
laboratory frame. In all cases signal impurities due to muons from weak decays
can additionally be suppressed by active detection of the momentum change to
the trajectory of the decaying hadron (both a directional change and a sudden
kinetic energy loss) introduced by the escaping neutrino. For the latter, the
“mass” of the tracking system plays an important role since the ability to
detect such a kink in the trajectory is again strongly effected by multiple
scattering. Even if the track reconstruction is not able to spot the kink in
the trajectory, it is still possible to draw on the Distance of Closest
Approach (DCA) of the extrapolated track to the primary interaction vertex.

####  3.2.2 Electrons

The detection of electrons is not affected by above considerations but faces
other challenges. Below momenta of around $300$ M$e$V, the precise limit
depends on the detector geometry and the resolution of the time-of-flight
measurement, electrons can effectively be separated from hadrons (and muons)
by their high velocity ($v_{e}\simeq c$). The most crucial “background” hadron
is the pion which, due to its small mass, is the first to reach near $c$ and,
moreover, is typically much more abundant than the electron. The pion
rejection power, the fraction of falsely identified pions (adding to the
electron sample) over the total number of detected charged pions, should
ideally reach at least $10^{-3}-10^{-4}$. Additional detectors are needed to
provide sufficient electron hadron separation for higher momenta. Commonly
used are Ring Imaging Cerenkov (RICH) Detectors based on gaseous radiators, as
for example in the CERES [42] and the HADES spectrometers [43]. However, RICH
detectors need to be placed in regions without magnetic field to guarantee
straight trajectories of the electrons inside the radiator which avoids
blurring of the imaged ring. Hence, the introduction of RICH detectors limits
the flexibility in the detector arrangement. Depending on the index of
refraction of the radiator gas, pions can be effectively discriminated from
electrons up to momenta of around of 4-5 G$e$V/$c$ [44, 45]. The other option
is to combine time-of-flight with a precise dE/dx (energy loss) measurement.
This strategy is successful if Time-Projection-Chambers (TPC) are used for
tracking, as demonstrated by STAR experiment at RHIC [46] and a ALICE at LHC
[39]. Continuous tracking in the large active volumes of these detectors
provide a large number of quasi-independent energy loss measurements. As
observable a truncated mean is typically used to minimize the effect of
fluctuations governed by a highly skewed probability distribution with a long
tail [68].

Additional electron-to-pion separation at higher momenta is provided by
Transition Radiation Detectors (TRD). These detectors make use of the
transition radiation produced by charged particles crossing boundaries between
materials with different refractive indices. Since the emission probability is
small, the particles have to cross many boundaries and consequently low-mass
foams or compounds are the material of choice. The difficulty is to detect the
transition radiation, which is emitted sparsely in the soft x-ray domain and
typically together with the signal induced by the charged electron traversing
the same region of the detector where the x-rays are converted. Yet another
option to identify electrons is to combine momentum (tracking) information
with the signal induced in an electromagnetic calorimeter. At reasonably high
energy the resolution is sufficient to identify the leptons by simultaneous
measurement of momentum and energy222The relative energy resolution of a
calorimeter improves like $E_{e}^{-\frac{1}{2}}$. Again, precise values depend
on the type and quality of the calorimeter.

To maximize the reconstruction efficiency for leptons in situations where
several detectors can contribute, it became common to exploit neural networks
in the decision process. The input layer to the networks is given by the
signals of all detector systems which can contribute to the identification.
This includes also the momenta of the track candidates selected for the
decision process. The output layer is a single quantity ranging between zero
and unity and providing a probabilistic information on the track candidate’s
nature. The advantage is that there are no strict cuts required for each
signal response. Otherwise, each condition would impose a probability
$\epsilon_{i}$ that the candidate survives this condition. For a particle
candidate to survive all conditions, a “survival” probability of
$P=\prod_{i=1}^{N}\epsilon_{i}$ would result where $n$ is the number of
conditions applied. Hence, any fluctuation of an individual signal response
out of the the selection range would reject the respective track.

While weak decays of hadrons are not of relevance as possible background
sources in dielectron measurements, external photon conversion is. In heavy-
ion collisions abundant photons come from $\pi^{0}$ decay. In fixed target
experiments, a high conversion probability can already occur in the target.
This is in particular a problem for measurements using high-$Z$ targets as the
conversion probability increases quadratic with the atomic number of the
target material. This effect can be mitigated with segmented targets, for
which the total material budget is distributed over a number of target slices
arranged at distances significantly larger than the diameter of the target. In
such an arrangement, the average distance a prompt photon propagates in the
target material is reduced.

There are two important strategies to control the combinatorial background in
dielectron spectroscopy. One is to reconstruct as many as possible conversion
and $\pi^{0}$ Dalitz-pairs by tracking down to low momentum and applying
conditions on invariant mass of the reconstructed pair. While for the
conversion both, the opening angle of the two electron tracks and the
invariant mass are close to zero by all practical means, also in case of
Dalitz decays a large fraction of the electron pair has opening angles below a
few degree. The condition can also be placed as 2-dim selection window
evaluating independently the mean momentum and the opening angle, i.e. the two
factors defining the invariant according to Eq. 8. This provides a measure to
optimize the signal survival probability for a given background rejection
power. With some detector configurations a “close pair rejection” can already
be achieved without tracking by detecting such pairs by respective double-
signals in the Cerenkov detector. This is possible if the radiator volume of
the RICH is placed close to the target (primary vertex) and in a field-free
region. However, the separation power between detector signals due to a single
electron and electron-positron close pair is critically depending on the total
amount of Cerenkov photons reconstructed.

Indeed, conversion pairs and a good fraction of the Dalitz pairs have opening
angles below a few degree owing to the small invariant mass of the
intermediate virtual photon. Such opening angles are less likely for random
pairs. If a combination is found with $\theta^{+-}<\theta^{+-}_{cut}$ a pair
is assumed to come from a $\pi^{0}$ decay (photo conversion or Dalitz),
eventually is filled to the histogram but the individual electron and
positrons are furthermore not used for other combinations. It has been found
that this strategy can reduce background without substantially reducing signal
efficiency. The rejection can even be improved if also incompletely
reconstructed tracks are included in the search of a close by partner lepton.
A fully identified electron or positron would then be rejected if a “track
fragment” is found in close vicinity of it. The virtue of this method will,
however, critically depend on the two-track separation power of the tracking
system. Dielectrons with small opening angle can also be identified if a RICH
detector is placed right behind the interaction region. The Cherenkov photons
produced by the the close by track can either be directly detected by counting
the fired cells in the ring region or by identifying ring distortions or clear
signatures of overlapping double rings. The success of such rejection cuts
depends on details of the spectrometer configuration and the performance of
its detectors, in particular of the RICH.

A second handle to reduce background contributions is to reject electron
tracks which have transverse momenta below a certain threshold. Electrons from
$\pi^{0}$ Dalitz decays or from conversion of the $\pi^{0}$’s decay photons
peak at low transverse momentum. This cuts, however, has to be handled with
caution as it also reduces the phase space of the detected signal pairs in the
region of small invariant mass.

###  3.3 Determination of the combinatorial background.

Once pure samples of lepton tracks have been obtained, the signal
reconstruction is rather similar for muon and electron pair spectroscopy.
Since in heavy-ion experiments we are in particular interested in the dilepton
continuum sources, an important task is to find the best approximation for the
a priori unknown combinatorial background. In fact, the true signal
distribution is obtained from subtracting two smooth distributions in
situations where the signal eventually amounts to less than a percent of the
reconstructed unlike-sign spectrum. Such situations are typically reached in
environments with highest charged particle multiplicity. Hence, background
rejection strategies are key to dilepton continuum spectroscopy in heavy-ion
collision experiments.

Generally, the bulk of the background pairs can be grouped into three
categories: (1) wrong combinations of leptons from two fully reconstructed
true signal pairs, (2) pairs which contain at least one lepton from an
incompletely detected pair or (3) pairs which contain at least one falsely
identified hadron. To the latter, we count also muons originating from weak
decays. The term combinatorial is chosen here for pairs which result from
“uncorrelated” processes, i.e. do not originate from the same virtual photon.
The probability for observing such combinatorial pairs

$P(M^{+},M^{-})=P(M^{+})\times P(M^{-})$ (13)

factorizes, where $P^{+}(M^{+})$ and $P^{-}(M^{-})$ are the probability
density functions to observe $M$ leptons in an event. In the most general
case, $P^{+/-}$ depends on the detector configuration and the event
characteristic like, most importantly, collision energy, system size and
centrality. Moreover, it can be defined as multi-differential probability
w.r.t. the leptons phase space. Indeed, signal spectra have been derived in
situations with signal-to-background as low as 1% [47], which then requires a
precision in the background determination of a few per mill. For step (4) of
the reconstruction procedure the lepton samples for both polarities with
multiplicities $M^{+}_{i}$ and $M^{-}_{i}$ in a given event $i$ are then
paired taking all possible combinations which results in

$M^{+-}=\sum_{i}{M^{+-}_{i}}=\sum_{i}{\left(M^{+}_{i}\times M^{-}_{i}\right)}$
(14)

unlike-sign pairs. Most pairs are of combinatorial type and their multiplicity
increases about quadratically with the charged particle multiplicity of the
event class. This is easily understood from the fact that pions, which are the
lightest and hence most abundant hadrons coupling to leptons, contribute to
background either through weak decays (muons) or through their Dalitz or two
photon decays. In the latter case, electrons emerge from the external pair
conversion of these decay photons. The true signal pair distribution $S^{+-}$
is hence obtained from the total pair yield333The total (unlike-sign same-
event) yield is sometimes also called foreground although, in the true sense
of the word, foreground does not include the background. $M^{+-}$ by
subtracting the best estimate for the a priori unknown background $B^{+-}$:

$S^{+-}=M^{+-}-B^{+-}\;.$ (15)

To derive the signal with an anticipated statistical precision $\delta S/S$,
the background has to be known with an superior precision of

$\frac{\delta B^{+-}}{B^{+-}}\approx\frac{\delta
S^{+-}}{S^{+-}}\cdot\frac{S^{+-}}{B^{+-}}\;.$ (16)

To minimize the statistical uncertainty, the true background is typically
estimated using both, event mixing and like-sign same event methods. Either
one of these methods have their drawbacks and caution has to be taken. The
strategy for event mixing is to subdivide a given number of events into event
classes such that events with similar characteristics like e.g. the number of
participant nucleons (centrality) or the location of the event plane relative
to the detector are combined to form the event classes (pools). Then pairs are
generated by combining each a lepton from a different event of the same pool.
In this way it is guaranteed that there are no correlations between the two
leptons while they still were reconstructed from events with the same event
characteristic. The latter is needed to mitigate effects in event mixing as
the true combinatorial background is originating from exactly the same event.
Since a given lepton from one event can be group with oppositely charged
leptons of any other event from the same event pool the statistical error can
in principle be brought to the required level (cf. 16). This is in particular
helpful to generate the background in sparsely populated region of the phase
space. However, it was found that statistical limitations in the total event
sample can still cause remaining structures in the event mixed pair sample
[51]. Also, the topology of pairs has to be chosen such, that the
reconstruction efficiency of the pair can be assumed to be given by the
product of the single track reconstruction efficiencies. This is to exclude
“correlations” in the lepton reconstruction induced by detector effects like
e.g. a dropping reconstruction efficiency if two tracks are propagating too
close to each other, in respect to the granularity of the detection system.
This is not always guaranteed if a pair has a small opening angle. Depending
on the type of tracking detector used the reconstruction efficiency might be
effected by the local track density.

A draw back of this multi-event sampling is that the generated background is a
priori not normalized. Several methods have been developed to provide proper
normalization. Most common is to exploit like-sign same-event pairs which
cannot originate from a single virtual photon due to charge conservation. In
this method, for each event like-sign (charged) pairs $M^{++}$ and $M^{--}$
are formed. Since in this procedure the same events are used for which the
unlike-sign pairs are generated, a proper normalization is in reach. To
increase the statistics and to cope with asymmetries in the detection of
leptons with opposite charge the best approximation is obtained from an
average of the two charged pair samples. However, the exact procedure to
normalize this charged pair spectrum depends on the nature of dominant pairs
producing the background. In case of dimuon spectroscopy, dominant background
pairs contain candidates originating from weak decays of pions and kaons. If
the emitting source is truly thermal, the production of these mesons is
uncorrelated and as a consequence also their decay muons. In dielectron
spectroscopy the dominant contributions stem from photo conversion and
$\pi^{0}$ Dalitz decay. Here, the background is mostly produced in pairs.
These are two extreme cases and in practice one can assume that the true
background pairs are always a mix of the two. For example, in dielectron
spectroscopy background can also originate from misidentified hadrons which
then also come to large extend as uncorrelated singles. Vice versa, in the
dimuon case background can occur due to simultaneous weak decays of two pions
or kaons originating from the same mother particle.

For the case of fully uncorrelated background tracks it has been shown that
the combinatorial background is given by the geometric mean of the two charged
pair samples:

$B^{+-}_{\mathrm{LS}}=2\sqrt{N^{++}\cdot N^{--}}.$ (17)

This equation holds strictly if the multiplicities show Poissonian probability
distributions. It has been demonstrated in [50], however, that different
multiplicities for positively and negatively charged background muons can
indeed cause deviations of the estimated from the true background. Hence, an
additional factor has to be multiplied on the right side of Eq. 17 which takes
care of this asymmetry. It can be derived by comparing the mixed-event like-
sign with the mixed-event opposite-sign background as

$R=\frac{N^{+-}_{\mathrm{mixed}}}{2\sqrt{N^{--}_{\mathrm{mixed}}\cdot
N^{++}_{\mathrm{mixed}}}}\;.$ (18)

Moreover, as mentioned above, such like-sign electron pairs include quasi
correlated pairs if both leptons originate from each one of the two
intermediary photons after internal or external conversion. Common sources of
this kind are $\pi^{0}$ or $\eta$ (via two photon or Dalitz decays) and, to a
smaller amount, $\omega\rightarrow\pi^{0}e^{+}e^{-}$ decays. For the latter,
there occur three photons as intermediary states due to the dominant $\pi^{0}$
decay into two photons. To avoid that the normalization is hampered by quasi
correlated pairs one has to remove tracks from external conversion most
efficiently and restrict the normalization to a region where these abundant
sources do not contribute.

At high collision energy, quasi correlated like-sign pairs can also arise from
correlated open heavy-flavor mesons [52]. For example a pair of charged $D$
mesons produced in a collision can both decay semi-leptonically thus producing
a quasi-correlated lepton pair. Due to the finite life-time of the $D$ mesons
such possible background sources can be mitigated by rejecting lepton tracks
not originating from the primary vertex. Similar background is produced by
correlated $B$ meson production.

Last, it has to be ensured that the background determination is not hampered
by selective trigger conditions used to record the events in the experiment.
For example, the reconstruction of proper like-sign pairs is impossible if the
trigger was set on detection of at least two leptons with opposite charge as
it is often done in two-arm spectrometers [53]. But care has also to be taken
in case the trigger on two leptons is not charge sensitive as pointed out in
detail for the NA60 experiment in [54].

##  4 Basic Theoretical Concepts

Time-like virtual photons are an ideal probe to study the microscopic
properties of strong-interaction matter. They couple directly to the
electromagnetic current of hadronic processes and probe the spectroscopic
properties of the involved hadrons. Sufficient understanding of the various
radiative decay or scattering processes is an important prerequisite to enable
the separation of the total dilepton yield obtained in experiments into
various contributing sources. In the following, we will outline the
theoretical approaches to model dilepton radiation. We will focus on soft
radiation as this is relevant for dilepton emission of hadronic systems
emphasized in this review.

###  4.1 Electromagnetic decays of hadrons

Leptonic and semi-leptonic two-body and three-body decays of hadrons are the
important processes contributing to the hadronic cocktail in the LMR (cf. Sec.
2.3). They can be grouped into three cases, each featuring an intermediary
virtual photon decaying into a pair of leptons: The first one is the
transition of a hadron into other hadrons with smaller mass and a virtual
photon. A second group is a special case of the first type but with a neutral
meson decaying into a real and a virtual photon. The last case is the direct
transition (annihilation) of vector mesons into virtual photons.

In the most general case the partial width for a decay of particle ($P$) into
a n-body final state can be expressed by [68]

$d\Gamma=\frac{\left(2\pi\right)^{4}}{2M}\,\mid\mathcal{T}\mid^{2}\,d\Phi_{n}\left(P;p_{1}\dots
p_{n}\right)\,,$ (19)

where $M$ is the mass of the decaying particle and $d\Phi_{n}$ the $n$-body
final state phase space. In case of a Dalitz decay of a hadron into a daughter
hadron and a lepton pair, the invariant transition amplitude $\mathcal{T}$ is
defined by [60]

$\sum_{pol}\mid\mathcal{T}\left(P\rightarrow
P^{\prime}\ell^{+}\ell^{-}\right)\mid^{2}\,=e^{2}\mathcal{W^{\mu\nu}}\frac{1}{q^{4}}L_{\mu\nu}\,,$
(20)

where the sum is running over the spin states of the initial and final state
particles. The squared transition matrix element has three parts: (1) the
hadronic tensor

$\mathcal{W^{\mu\nu}}=\sum_{spins}\mathcal{M}^{\mu}\mathcal{M}^{\nu}\,,$ (21)

which describes the $P\rightarrow P^{\prime}\gamma^{*}$ transition, (2) the
square of the photon propagator $1/q^{2}$ and (3) the lepton tensor

$L_{\mu\nu}=\sum_{spins}j_{\mu}j_{\nu}$ (22)

accounting for the photon-dilepton pair conversion. The leptonic current

$j^{\mu}=\bar{u}(k_{1})\gamma^{\mu}\upsilon(k_{2})$ (23)

is a vector current coupling to photons and contains the respective Dirac wave
functions $u(k_{1})$ and $\upsilon(k_{2})$ of leptons with four momenta
$k_{1}$, $k_{2}$ and mass $m_{l}$. The leptonic tensor can explicitly be
calculated as

$L^{\mu\nu}=4(k_{1}^{\mu}k_{2}^{\nu}+k_{2}^{\mu}k_{1}^{\nu}-(k_{1}\cdot
k_{2}+m_{l}^{2})g^{\mu\nu})\,,$ (24)

and also controls the proper infrared cut-off due to the lepton phase space.

The hadronic tensor includes all information about the interaction between the
involved hadrons and the coupling to the virtual photon produced in the
radiative process. The structure of the hadron-photon vertex is encoded in
electromagnetic Transition Form Factors (eTFF) which depend on the four-
momentum transfer $q$. In case of dilepton decays with $q^{2}>0$ the eTFF’s
are defined in the time-like region. In the space-like region ($q^{2}<0$ )
eTFF’s can be studied via electron scattering experiments. In view of dilepton
spectroscopy most interesting are time-like eTFF for light neutral mesons,
which have been extracted with good precision from dilepton invariant mass
distributions. Similar results on baryon Dalitz decays are very scarce. The
results will be reviewed in the remainder of this section.

As it has already been stressed in Sec. 3, photon polarization affects the
lepton angular distributions. Indeed, the general form of the transition
matrix element $\mathcal{T}$ depends on the polarization of the virtual photon
and includes non-trivial angular dependencies of the emitted leptons. To
better visualize this dependence one can rewrite the matrix element (Eq. 20)
by introducing spin density matrix elements [61, 62] as

$\sum_{pol}\mid\mathcal{T}\mid^{2}=\frac{e^{2}}{q^{4}}\sum_{\lambda,\lambda^{\prime}}\rho^{had}_{\lambda,\lambda^{\prime}}\rho^{lep}_{\lambda,\lambda^{\prime}}\,,$
(25)

where the hadron $\rho^{had}_{\lambda,\lambda^{\prime}}$ and the lepton
$\rho^{lep}_{\lambda,\lambda^{\prime}}$ spin density matrix elements are
defined in similar fashion as the hadronic and leptonic tensors. They can be
written using the four-vector of the virtual photon
$\epsilon^{\mu}(q,\lambda)$ with the helicities $\lambda=\pm 1$, $\lambda=0$
and the four-momentum $q$ as

$\displaystyle\rho^{lep}_{\lambda,\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle\epsilon^{\mu}(q,\lambda)\,L_{\mu,\nu}\,\epsilon^{\nu}(q,\lambda^{\prime})^{*}\,,$
(26) $\displaystyle\rho^{had}_{\lambda,\lambda^{\prime}}$ $\displaystyle=$
$\displaystyle\epsilon^{\mu}(q,\lambda)\,\mathcal{M}_{\mu,\nu}\,\epsilon^{\nu}(q,\lambda^{\prime})^{*}\,.$
(27)

The asymmetry coefficients given in Eq. 10 can be expressed as combinations of
spin density parameters (see e.g. [62]). While the hadronic part
($\rho^{had}_{\lambda,\lambda^{\prime}}$) depends in general on the reaction
dynamics and on the type of transition, the leptonic part can be calculated
explicitly in QED. In the virtual photon rest frame (leptons are emitted back-
to-back) it is given by [62]

$\rho^{lep}_{\lambda,\lambda^{\prime}}=4\mid\mathbf{k}\mid^{2}\times\left(\begin{array}[]{ccc}1+\cos^{2}\theta_{e}+a&-\sqrt{2}\cos\theta_{e}\sin\theta\,e^{-i\phi_{e}}&\sin^{2}\theta_{e}\,e^{-2i\phi_{e}}\\\
-\sqrt{2}\cos\theta_{e}\sin\theta\,e^{i\phi_{e}}&2(1-\cos^{2}\theta)+a&\sqrt{2}\cos\theta_{e}\sin\theta\,e^{-i\phi_{e}}\\\
\sin^{2}\theta_{e}\,e^{2i\phi_{e}}&\sqrt{2}\cos\theta_{e}\sin\theta
e^{i\phi_{e}}&1+\cos^{2}\theta_{e}+a\,,\end{array}\right)$ (28)

where $a=2m_{\ell}^{2}/\mid\mathbf{k}\mid^{2}$ and $\mathbf{k}$ is the
momentum vector of one of the two leptons. The polar ($\theta_{e}$) and the
azimuthal ($\phi_{e}$) angles are given in the rest frame of the virtual
photon with respect to a quantization axis (cf. Fig. 3). In this review we
will use the helicity reference frame defined such that the $z$ axis points
along the direction of the virtual photon in the CM frame . The diagonal
elements $\rho_{11}=\rho_{-1-1}$ correspond to transversely polarized photons
($\sim cos^{\theta}_{e}$), while the $\rho_{00}$ ($\sim sin^{2}\theta_{e}$) to
the longitudinally polarized ones. As we shall see below, these components
appear in the expressions for the angular distributions of electrons emitted
in Dalitz decays.

####  4.1.1 Vector Meson Dominance

Successful theoretical approaches describe the radiative hadron decays by
applying the Vector Meson Dominance Model (VDM), which was already mentioned
as an important concept for the description of thermal radiation of QCD matter
(cf. Sec. 2.2). Vector Meson Dominance has been suggested by Sakurai already
in the sixties [63] and provides a very intuitive picture of the hadron-photon
interaction. The conjecture is that the coupling is actually mediated by the
light vector mesons $\rho$, $\omega$ and $\phi$, which carry the same spin and
parity as the photon and play the role of an interpolating field between
strong and electromagnetic interactions. With their intrinsic quark structure
they act as the microscopic electric dipole to which the photon couples, very
much like nucleon-hole excitations in the description of the giant iso-vector
dipole resonance of nuclei. According to VDM the hadronic electromagnetic
current can straight-forwardly be written as

$\mathcal{M}^{\mu}=J^{\mu}=e\sum_{V}\frac{m_{V}^{2}}{g_{V}}V^{\mu}\,,$ (29)

where $V^{\mu}$ are the vector meson fields ($\rho,\omega,\phi$) and $g_{V}$
and $e^{2}=4\pi\alpha$ ($\hbar,c=1$) are the vector meson and the
electromagnetic coupling constants, respectively. This current operator is
acting on the QCD vacuum and creates antiquark-quark pairs overlapping with
the vector meson states. An important conjecture of VDM is that the strong
interaction between vector mesons and other hadrons is proportional to exactly
the same coupling constant $g_{V}$ as defined in Eq. 29.

Another important aspect of VDM are interferences between the amplitudes for
different intermediary vector meson states. Besides the well known effect of
the $\rho/\omega$ interferences in the $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$
channel entering the pion electromagnetic form factor [64], much stronger
effects due intermediary baryonic resonances have been predicted for the
dilepton invariant masses below the vector meson poles [65, 66, 67]. This
mechanism is closely connected to Dalitz decays of low-mass $N^{*}$, $I=1/2$,
$\Delta,I=3/2$ baryon resonances as will be discussed below (Sec. 4.1.3). In
heavy-ion collisions at beam energies of a few $\,A$ G$e$V temperatures
reached in the fireballs are about a factor two lower than at SPS or collider
energies. Consequently such effects may play an important role since baryon
resonances may dominate over mesonic excitations (cf. Fig. 2).

####  4.1.2 Two body decays of vector mesons

The partial decay width $d\Gamma$ for a two body decay of a particle with mass
$M$ into particles $a$ and $b$ is given as [68]

$d\Gamma=\frac{1}{32\pi^{2}}\mid\mathcal{T}(V\rightarrow\ell^{+}\ell^{-})\mid^{2}\frac{p}{M^{2}}d\Omega$
(30)

where

$p=\frac{\left((M^{2}-(m_{a}+m_{b})^{2})(M^{2}-(m_{a}-m_{b})^{2})\right)^{1/2}}{2M}\,$
(31)

is the photon momentum in the CM frame and $(p/M^{2})d\Omega$ is the phase
space factor including the solid angle $d\Omega=d\phi\,d(\cos\theta)$ of one
of the decay particles. The partial decay width of the vector mesons
$\rho,\omega$ and $\phi$ into a pair of leptons with masses
$m_{\ell^{+}},m_{\ell^{-}}$ is calculated as two-step process of the meson-
virtual photon transition and its subsequent conversion to a dilepton pair.
Thus the matrix element (Eq. 20), averaged over the vector polarization and
electron spins, factorizes as shown in [69, 70, 71] to

$\displaystyle\mid\mathcal{T}(V\rightarrow\ell^{+}\ell^{-})\mid^{2}$
$\displaystyle=$
$\displaystyle\mid\mathcal{W}(V\rightarrow\gamma^{*})\mid^{2}\cdot\frac{4}{M^{4}}\cdot\mid
L(\gamma^{*}\rightarrow\ell^{+}\ell^{-})\mid^{2}$ (32) $\displaystyle=$
$\displaystyle\frac{16\pi^{2}\alpha^{2}}{3}\frac{M_{V}^{4}}{g_{V}^{2}}(M^{2}+2m_{\ell}^{2})\frac{1}{M^{4}}\,.$
(33)

Note that $q^{2}=M^{2}$ with $M$ the invariant mass of the dilepton. The
matrix element

$\mid\mathcal{W}(V\rightarrow\gamma^{*})\mid^{2}=4\pi\frac{\alpha}{g^{2}_{V}}M_{V}^{4}$

is given by (Eq. 29) at the meson pole $M_{V}$ . The conversion of the virtual
photon to a lepton pair is written as

$\mid
L(\gamma^{*}\rightarrow\ell^{+}\ell^{-})\mid^{2}=\frac{4\pi\alpha}{3}(M^{2}+2m^{2}_{\ell})\,.$
(34)

Finally, using Eq. (30), the dilepton decay width $\Gamma(M)$ can be obtained
as a function of the dilepton mass as

$\Gamma(M)_{\ell^{+}\ell^{-}}=\frac{4\pi\alpha^{2}}{3g^{2}_{V}}\frac{M_{V}^{4}}{M^{3}}\left(1-\frac{4m_{\ell}^{2}}{M^{2}}\right)^{1/2}\left(1+\frac{2m_{\ell}^{2}}{M^{2}}\right)\,.$
(35)

Table 1 summarizes the total decay widths, the branching ratios to dileptons
and the coupling constants $g_{V}$ of the light vector mesons defined at the
pole positions ($M=M_{V}$).

| Decay | BR |  $\Gamma$ [ M$e$V] | $g_{\mathrm{V}}$
---|---|---|---|---
$\rho$ | $e^{+}e^{-}$ | $(4.72\pm 0.05)\cdot 10^{-5}$ | $149.1\pm 0.8$ | $5.03$
$\rho$ | $\mu^{+}\mu^{-}$ | $(4.55\pm 0.28)\cdot 10^{-5}$ |  |
$\omega$ | $e^{+}e^{-}$ | $(7.28\pm 0.14)\cdot 10^{-5}$ | $8.49\pm 0.08$ | $17.1$
$\omega$ | $\mu^{+}\mu^{-}$ | $(9.0\pm 3.1)\cdot 10^{-5}$ |  |
$\phi$ | $e^{+}e^{-}$ | $(2.954\pm 0.03)\cdot 10^{-4}$ | $4.27\pm 0.03$ | $-12.9$
$\phi$ | $\mu^{+}\mu^{-}$ | $(2.87\pm 0.19)\cdot 10^{-4}$ |  |
Table 1: Branching ratios for dilepton decays, total widths $\Gamma$ and
coupling constants $g_{\mathrm{V}}$ of light vector mesons.

One should note that the ratios of the coupling constants $g_{V}$, extracted
from the measured branching ratios [72], are in good agreement with the SU(3)
flavour symmetry predictions:
$g_{\rho}:g_{\omega}:g_{\phi}=1:3:-3/\sqrt{(}2)$. Hence, in the context of
VDM, they can be expressed in terms of a single universal vector meson-photon
coupling constant $g_{\gamma}\simeq 5.6$.

The shape of the dilepton invariant mass distribution
$dN/dM_{\ell^{+}\ell^{-}}$ of the two-body decay of the broad $\rho$ meson
depends on the production process. This is an important aspect as will become
evident later in the discussion of the dilepton spectra obtained in hadronic
collisions. In general, the mass distribution $dN/dM$ of a resonance can be
approximated by the relativistic Breit-Wigner formula, multiplied with
$dN_{p}(M)/dM$ quantifying the number of initial states populating the
resonance at mass $M$ as

$\frac{dN}{dM}=\frac{dN_{p}(M)}{dM}\frac{\Gamma_{in}\Gamma_{\ell^{+}\ell^{-}}M^{2}}{(M^{2}-M_{\mathrm{V}}^{2})^{2}+M^{2}_{\mathrm{V}}\Gamma_{\mathrm{tot}}^{2}}\,.$
(36)

Here, $\Gamma_{in}$ and $\Gamma_{\ell^{+}\ell^{-}}$ account for the partial
widths related to the production and the decay process, respectively. The
number of states $dN_{p}/dM$ depends on the production process and available
phase space. In a baryon dominated hadron resonance gas $\rho$ mesons are
likely produced via excitation and decay of baryonic resonances. For such a
two-step production process $NN\rightarrow N^{*}(\Delta)N\rightarrow
NN\rho\rightarrow NN\ell^{+}\ell^{-}$ the mass distribution of the baryon
resonance will strongly affect the dilepton mass distribution. Indeed, low
mass resonances like $N^{*}(1520)$ or $\Delta(1720)$ have a significant
branching to the $N\rho$ channel. The production of $\rho$ mesons in such two-
step processes appears as a special type of (baryon) Dalitz decay as will be
discussed in Sec. 4.1.3. In an equilibrated hadron resonance gas such baryon
resonances will be excited with preference of the lower-side of their mass
distribution. This limits also the phase space for the decay and in turn
favours decays populating the low-mass tail of the $\rho$ meson, an effect,
which is further enhanced by the $1/M^{3}$ dependence of decay width
$\Gamma_{\ell^{+}\ell^{-}}$ (cf. Eq. 35).

One should note, however, that the latter only holds for strict VDM, which
assumes that the coupling $g_{V}$ is constant. For narrow resonances like the
$\omega$ and the $\phi$ meson such an assumption is justified but the
situation is more complex in case of the broad $\rho$ meson state. For
example, the extended VDM model of [73] uses two coupling schemes for the
hadron-photon transition: a direct coupling to photons and a coupling via
intermediate vector mesons which vanishes at $q^{2}=0$. Consequently, a linear
dependence on the dilepton mass is predicted [74].

An important channel for dilepton production in heavy-ion reactions at ultra-
relativistic energies is the annihilation of two pions in the fireball. The
respective elementary cross section has been calculated using a Breit-Wigner
resonance description in [69] and [71]:

$\displaystyle\sigma_{\pi\pi\rightarrow\rho\rightarrow\ell^{+}\ell^{-}}(M)$
$\displaystyle=$
$\displaystyle\frac{4\pi}{p_{\pi}^{2}}\frac{M\,\Gamma_{\pi\pi}\,M\,\Gamma_{\ell^{+}\ell^{-}}}{(M^{2}-M^{2}_{\rho})-M^{2}_{\rho}\Gamma_{\mathrm{tot}}^{2}}$
(37) $\displaystyle=$
$\displaystyle\frac{8\pi\alpha^{2}p_{\pi}}{3M^{3}}\frac{M_{\rho}^{4}}{(M^{2}-M_{\rho}^{2})^{2}+M_{\rho}^{2}\Gamma^{2}_{\mathrm{\mathrm{tot}}}(M)}\left(1-\frac{4m_{\ell}^{2}}{M^{2}}\right)^{1/2}\left(1+\frac{2m_{\ell}^{2}}{M^{2}}\right)$
(38) $\displaystyle=$
$\displaystyle\frac{4\pi}{3}\frac{\alpha^{2}}{M^{2}}\left(1-\frac{4m_{\pi}^{2}}{M^{2}}\right)^{1/2}\left(1-\frac{4m_{\ell}^{2}}{M^{2}}\right)^{1/2}\left(1+\frac{2m_{\ell}^{2}}{M^{2}}\right)\mid
F_{\pi}\mid^{2}\,,$ (39)

It demonstrates the dependence of the cross section on the $\pi\pi$ center of
mass collision energy $\sqrt{s}=M$ with $F_{\pi}$ denoting the eTFF of charged
pions and

$\Gamma_{\pi\pi}=\frac{g^{2}_{\rho}}{2\pi}\frac{p^{3}_{\pi}}{M^{2}}$

defining the two pion decay width of the $\rho$ meson
($\Gamma_{\ell+\ell^{-}}$ is given by Eq. 35). The proper threshold behaviour
is guaranteed by $\Gamma_{\pi\pi}$ as it cuts off the dilepton mass
distribution from pion annihilation at $2m_{\pi}$.

In contrary, for dileptons from the previously discussed baryon resonance
Dalitz decays such a cut-off does not exist since the mass of the $\rho$ meson
is generated from the excitation energy of the baryonic resonance. These
production mechanisms have been applied in the transport codes GiBUU [75],
UrQMD [76], (P)HSD [77, 78], RQMD [79, 81] and SMASH [80]. The threshold
behaviour is particularly visible in calculations carried out for
SIS18/BEVALAC energies. We will return to this aspect in the discussion of the
data (cf. Sec. 6).

For dilepton decays of vector mesons lepton angular distributions were
calculated in [82, 83, 84] and expressed, in agreement with Eq. 20, by

$\frac{d\sigma}{d\Omega}\propto
1+\lambda_{\theta}\,\cos^{2}\theta_{l}+\lambda_{\theta\phi}\,\sin
2\theta_{l}\,\cos\phi_{l}+\lambda_{\phi}\,\sin^{2}\theta_{l}\,cos2\phi_{e}\,.$
(40)

The anisotropy coefficients $\lambda_{\theta},\lambda_{\theta,\phi}$ and
$\lambda_{\phi}$ are related to the spin density matrix elements (cf. Eq. 28).
Compared to Eq. 10, the coefficients related to the angle relative to the
scattering plane vanish. For example, in the case of pion annihilation and the
quantization axis oriented in the direction along the pion momentum, the
angular distribution integrated over azimuthal angle further simplifies to
[62]:

$\frac{d\sigma}{d\Omega}\propto(1+cos^{2}\theta_{e})(\rho_{-1,-1}+\rho_{1,1})+2(1-cos^{2}\theta_{e})\rho_{0,0}\propto\rho_{0,0}sin^{2}\theta_{e}\,.$
(41)

Note that the only non-vanishing element of the spin density matrix is
$\rho_{00}$. Hence, the only relevant asymmetry coefficient is
$\lambda_{\theta}=-1$ (longitudinal polarization). For the Drell-Yan process
similar considerations lead to $\lambda_{\theta}=+1$, i.e. the virtual photon
is fully transversely polarized.

####  4.1.3 Dalitz decays

The decay of a particle with mass $M$ into a three particle final state with
momenta $p_{i}$ and $m_{i}^{2}=p_{i}^{2}$ can be expressed by its partial
decay width $d\Gamma$, which is determined by the Lorentz invariant matrix
element $\mathcal{T}$ [68]

$d\Gamma=\frac{1}{(2\pi)^{5}}\frac{1}{16M^{2}}\mid\mathcal{T}\mid^{2}\,\mid\vec{p_{1}}^{*}\mid\;\mid\vec{p_{3}}\mid
dm_{12}\,d\Omega_{1}^{*}d\Omega_{3}\,.$ (42)

In this expression $\vec{p_{1}}^{*}$ and $d\Omega_{1}^{*}$ define the momentum
and solid angle element of particle 1 in the rest frame of the two-body
subsystem (1-2), respectively. The invariant mass of this subsystem is
$m_{12}$ and the momentum $p_{1}^{*}$ can be calculated with with Eq. 31 by
setting $M=m_{12}$. The solid angle element $d\Omega_{3}$ is defined in the
rest frame of the decaying mother particle with mass $M$ into particle 3 and
the two-particle subsystem (1-2), while the momentum $p_{3}^{*}$ can also be
obtained from Eq. 31. All three momenta of the daughter particles lie in a
plane. For dilepton decays ($P\rightarrow P^{\prime}\ell^{+}\ell^{-}$) it is
convenient to relate the system $1-2$ to the dilepton (equivalently to the
virtual photon $\gamma^{*}$) with the invariant mass $M_{\ell^{+}\ell^{-}}$.
Next, assuming that the decay is independent of the azimuthal emission angle
of $\gamma^{*}$ w.r.t. the decay plane, one can introduce three angles to
characterize the decay process: the polar angle related to the emission of the
virtual photon and the polar and azimuthal angles describing the lepton decay
axis in the $\gamma^{*}$ rest frame (cf. Eq. 40). The latter two are related
to the polarization of the virtual photon and are in particular important for
the characterization of the decay source. Meson decays
Dileptons from Dalitz decays of scalar mesons make up a significant fraction
of the LMR. Electron pairs with masses below 140 M$e$V/$c^{2}$ come dominantly
from $\pi^{0}\rightarrow\gamma e^{+}e^{-}$. Strong contributions to dileptons
with masses up to the 500 M$e$V/$c^{2}$ originate from
$\eta\rightarrow\gamma\ell^{+}\ell^{-}$. At even higher masses pairs from
$\eta^{\prime}\rightarrow\gamma\ell^{+}\ell^{-}$ as well as from the vector
mesons $\omega\rightarrow\pi^{0}\ell^{+}\ell^{-}$ and
$\phi\rightarrow\pi^{0}\ell^{+}\ell^{-}$ contribute, yet with less strength.
In the calculations of the respective transition amplitude (Eq. 20) the
$P\rightarrow P^{\prime}\gamma^{*}$ vertex for the decays of interest is
identical for pseudoscalar-vector to vector and vector-pseudoscalar to vector
transitions. In both cases the respective hadronic current (cf. Eq. 21) can be
written as [60]

$\mathcal{M^{\mu}(P\rightarrow
P^{\prime}\gamma^{*})}=f_{PP^{\prime}}(q^{2})\varepsilon^{\alpha\beta\gamma\mu}p_{\alpha}q_{\beta}\epsilon_{\gamma}$
(43)

where $p_{\alpha}$ and $q_{\beta}$ are the four-momenta of the daughter
particle $P^{\prime}$ and the virtual gamma $\gamma^{*}$, respectively. The
polarization of the vector particle is denoted with $\epsilon_{\gamma}$ and
$\varepsilon^{\alpha\beta\gamma\nu}$ is the totally antisymmetric tensor. The
function $f_{PP^{\prime}}(q^{2})$ is the eTFF, which describes the virtual
photon-hadron vertex and encodes the electromagnetic structures of the
involved hadrons. For a point-like particle it is just a constant while for
extended objects it generally depends on the four momentum transfer which, in
the time-like region, is equivalent to the dilepton invariant mass
$M_{\ell^{+}\ell^{-}}=\sqrt{q^{\beta}q_{\beta}}$.

Figure 4:  Mass dependence of eTFF’s derived from measurements of the dilepton
Dalitz decays of $\eta$ (left), $\omega$ (middle) [87, 88] and $\pi^{0}$
(right) [90] mesons. Lines are results of fits using Eq. 48. performed at MAMI
(A2 collaboration) [85, 87], BESIII [86], SPS/NA60 [88] and NA62 [90] for
$\eta$, $\omega$ and $\pi^{0}$ mesons together with older data from Lepton-G.

Therefore, measurements of the invariant mass of dileptons from Dalitz decays
enables extracting this important quantity from data. The differential decay
widths (Eq. 42) can be calculated using a similar factorization scheme as
given by Eq. 33 (see [60, 69, 72] for details):

$d\Gamma(P\rightarrow
P^{\prime}\ell^{+}\ell^{-})=\frac{\alpha}{3\pi}d\Gamma\left(P\rightarrow
P^{\prime}\gamma^{*}\right)\left[1-\frac{4m^{2}_{l}}{M^{2}_{\ell^{+}\ell^{-}}}\right]^{1/2}\left[1+\frac{2m^{2}_{l}}{M^{2}_{\ell^{+}\ell^{-}}}\right]\frac{dM^{2}_{\ell^{+}\ell^{-}}}{M^{2}_{\ell^{+}\ell^{-}}}\,.$
(44)

Using the invariant transition matrix element $\mathcal{T}$ given by Eq. 20,
and after normalization to the partial width for a decay with a real photon in
the final state, one arrives to

$\displaystyle\frac{d\Gamma(P\rightarrow
P^{\prime}\ell^{+}\ell^{-})}{dM^{2}_{\ell^{+}\ell^{-}}\Gamma(P\rightarrow
P^{\prime}\gamma)}$
$\displaystyle=\frac{\alpha}{3\pi}\left[1-\frac{4m^{2}_{l}}{M^{2}_{\ell^{+}\ell^{-}}}\right]^{1/2}\left[1+2\frac{m^{2}_{l}}{M^{2}_{\ell^{+}\ell^{-}}}\right]\,\frac{1}{M^{2}_{\ell^{+}\ell^{-}}}\;$
(45)
$\displaystyle\times\left[\left(1+\frac{M^{2}_{\ell^{+}\ell^{-}}}{M^{2}-M^{2}_{p^{\prime}}}\right)^{2}-\frac{4M^{2}M^{2}_{\ell^{+}\ell^{-}}}{(M^{2}-M^{2}_{p^{\prime}})^{2}}\right]^{3/2}\left|\frac{f_{PP^{\prime}}(M^{2}_{\ell^{+}\ell^{-}})}{f_{PP^{\prime}}(0)}\right|^{2}$

which can be written in a compact way as

$\frac{d\Gamma(P\rightarrow
P^{\prime}\ell^{+}\ell^{-})}{dM^{2}_{\ell^{+}\ell^{-}}\Gamma(P\rightarrow
P^{\prime}\gamma)}=[QED]\times
F^{2}_{pp^{\prime}}(M^{2}_{\ell^{+}\ell^{-}})\,.$ (46)

The function $F_{PP^{\prime}}(M^{2}_{\ell^{+}\ell^{-}})$ is the eTFF
normalized to its value at $M^{2}_{\ell^{+}\ell^{-}}=0$ corresponding to the
emission of a real photon (i.e. at the photon point). With the help of the VDM
the eTFF ca n be expressed as weighted sum of the vector meson’s Breit-Wigner
distributions over all low-mass vector mesons with mass $M_{V}$

$F_{PP^{\prime}}(M^{2}_{\ell^{+}\ell^{-}})=\left[\sum_{V}\frac{g_{PP^{\prime}V}}{g_{V}}\frac{M^{2}_{V}}{M^{2}_{V}-M^{2}_{\ell^{+}\ell^{-}}-i\Gamma_{V}M_{V}}\right]\,\left[\sum_{V}\frac{g_{PP^{\prime}V}}{g_{V}}\right]^{-1}\,,$
(47)

where the weights are the ratio of the coupling strength’ $g_{PP^{\prime}V}$
and $g_{V}$ characterizing the $P\rightarrow P^{\prime}V$ and
$V\rightarrow\gamma^{*}$ vertices, respectively. They can be fixed from quark
models, as described in [60] (note that Eq. 3.10 in [60] uses constants with
extra factor 2) or experimental data on meson decays. In the limit of small
width’ and using the pole approximation, the expression Eq. 47 can be expanded
in terms of $q^{2}$ to arrive at the known dipole form

$F_{PP^{\prime}}(M^{2}_{\ell^{+}\ell^{-}})\simeq\frac{1}{1-q^{2}/\Lambda^{2}}\simeq
1+\frac{q^{2}}{\Lambda^{2}}=1+\frac{1}{6q^{2}<r^{2}_{PP^{\prime}}>}\,.$ (48)

| Decay | BR |  $\Lambda^{-2}$ (GeV${}^{-}2$) | Reference
---|---|---|---|---
$\pi^{0}$ | $e^{+}e^{-}\gamma$ | $(1.174\pm 0.035)\cdot 10^{-2}$ | $2.020\pm 0.031$ | [90]
$\eta$ | $e^{+}e^{-}\gamma$ | $(6.9\pm 0.4)\cdot 10^{-3}$ | $1.97\pm 0.13$ | [87]
$\eta$ | $\mu^{+}\mu^{-}\gamma$ | $(3.1\pm 0.04)\cdot 10^{-4}$ | $1.934\pm 0.084$ | [88]
$\eta^{{}^{\prime}}$ | $\mu^{+}\mu^{-}\gamma$ | $(1.08\pm 0.27)\cdot 10^{-4}$ | $1.7\pm 0.4$ | [60]
$\eta^{{}^{\prime}}$ | $e^{+}e^{-}\gamma$ | $(4.73\pm 0.3)\cdot 10^{-4}$ | $1.6\pm 0.19$ | [86]
$\omega$ | $e^{+}e^{-}\pi^{0}$ | $(7.7\pm 0.6)\cdot 10^{-4}$ | $1.99\pm 0.22$ | [87]
$\omega$ | $\mu^{+}\mu^{-}\pi^{0}$ | $(1.3\pm 0.4)\cdot 10^{-4}$ | $2.223\pm 0.045$ | [88]
$\phi$ | $e^{+}e^{-}\pi^{0}$ | $(1.35\pm 0.11)\cdot 10^{-5}$ | $2.02\pm 0.11$ | [89]
Table 2: Summary decay parameters extracted from Light Meson Dalitz decays.
For a definition see text.

The root mean square radius $<r^{2}_{PP^{\prime}}>^{1/2}$ defines a
characteristic size of the transition which reflects the hadron size and
$\Lambda^{-2}$ is the derivative of the eTFF w.r.t. $q^{2}$ at the photon
point. The above approximation can be applied in the time-like and space-like
regime. Fig. 4 shows results of recent high precision measurements of eTFF of
mesons and fits to the data using Eq. 48. The extracted values for
$\Lambda^{-2}$ are summarized in Tab. 2. They generally confirm a growth of
the eTFF as a function of the invariant mass and agree well with the
prediction of VDM (given in Tab. 4 of [60]), except for the $\omega$ and
$\phi$ mesons. A particularly significant disagreement can be seen for the
$\omega$ meson.

State of the art calculations of eTFF’s are performed using various
approaches: effective Lagrangians [91, 92], Dyson-Schwinger formalism [93,
94], quark models [95, 96] and dispersion relations [97, 98, 99] (for a recent
review see also [100]). Recently, substantial progress has been made in
extracting eTFF’s from lattice-QCD calculations, both on the mesonic (pionic)
[101, 102] and baryonic sector [103, 104]. In particular the $\omega$ eTFF has
triggered a lot of interest due to apparent violation of strict VDM. The
description has been significantly improved by leading order chiral Lagrangian
calculations [92] and dispersive theories [98], all of which still failing to
account for its strong rise at very large $q^{2}$.

A different motivation to study eTFF’s is related to physics beyond the
standard model. The results from measurements of the muon $(g-2)$ differ
w.r.t. the Standard Model predictions [105, 106]. It appears that the eTFF’s
of light pseudo-scalar mesons enter in the hadronic light-by-light scattering
calculation and, together with the vacuum polarization, are currently the most
prominent part of systematic uncertainties of Standard Model predictions
([100]). Furthermore, Dalitz decays constitute an irreducible background in
searches for a possibly existing dark photon, an extra U(1) gauge boson which
mediates the interaction between Standard Model photons and dark matter.
Several measurement were performed searching for signals from a two-body decay
of the dark photon in meson and baryon Dalitz-decays and in heavy-ion
collisions with negative outcome but providing important limits on its
coupling strength to the SM photon (see [107] for further information).

The angular distributions of the leptons in light neutral meson Dalitz-days
can be calculated using Eq. 41. For such transitions the spin density matrix
$\rho^{had}_{\lambda,\lambda^{\prime}}$ contains only the elements
$\rho_{-1-1}$ and $\rho_{11}$ and consequently only contributions from
transversely polarized photons are selected. This leads to the general
expression for the differential cross section defined in the helicity frame

$\frac{d\sigma}{d\Omega}\propto
p_{\pi}^{2}\left[2m_{\ell}^{2}+\mid\mathbf{k}\mid^{2}(1+cos^{2}\theta_{l})\right]\,.$
(49)

Figure 5: Electron angular distributions in the helicity frame for the
$\pi^{0}\rightarrow e+e-\gamma$ (left) [108] and the $\eta\rightarrow
e^{+}e^{-}\gamma$ (right) [87] Dalitz decays. Red lines denote the
$(1+B*cos^{2}\theta_{e})$ anisotropy.

Experimental results for the decays $\pi^{0}\rightarrow e^{+}e^{-}\gamma$ [85,
108] and $\eta\rightarrow e^{+}e^{-}\gamma$ [87, 109], shown in Fig. 5,
corroborate the expected $1+cos^{2}\theta_{l}$ distribution (please note that
for electrons one can neglect the electron mass term in Eq. 49).

Baryon decays
The matrix element for a radiative transition of an excited baryon to its
ground state $\mathcal{M}(R\rightarrow N\gamma^{*}$) can be expressed as
coherent sum of helicity amplitudes $A_{3/2}(q^{2})$, $A_{1/2}(q^{2})$ and
$S_{1/2}(q^{2})$, defined in the resonance decay frame [110, 111]. The
amplitudes describe transitions for a resonance of given spin $J$ and helicity
$\lambda^{*}$ to a nucleon with the helicity $\lambda=\pm 1/2$ and a photon
with $\lambda_{\gamma}=1,0$. The projection axis is defined to be along the
photon momentum and hence $\lambda^{*}=-\lambda+\lambda_{\gamma}>$ (please
note that one can consider only $\lambda_{\gamma}\geq 0$ due to conserved
parity). The first two amplitudes are related to the transverse photon
polarization ($\lambda_{\gamma}=1$) while the last one is related to the
longitudinal photon polarization ($\lambda_{\gamma}=0$). Transverse helicity
amplitudes have been determined in the space-like region for several
resonances in electro- and photo-production experiments. However, they are
unknown in the time-like region which is only accessible by their Dalitz
decays $R\rightarrow N\gamma^{*}$ (for a review see [111]). The respective
differential decay width $d\Gamma(R\rightarrow
Ne^{+}e^{-})/dM_{\ell^{+}\ell^{-}}$ can be expressed similar to Eq. 44 but
here with the $d\Gamma(R\rightarrow N\gamma^{*})$ given as a function of three
covariant or Magnetic ($G_{M}(q^{2})$), Electric ($G_{E}(q^{2})$) and Coulomb
($G_{C}(q^{2})$) eTFF’s. The form factors are related to the helicity
amplitudes discussed earlier (respective relations can be found in [110]). The
relations for the $\Delta(1232)$ resonance were introduced by Jones and
Scadron [112] and for higher mass resonances by Devenish, Eisenschitz and
Körner [113]. It should be noted that the Coulomb form factor couples only to
the $S_{1/2}$ while the electric and the magnetic form factors couple to all
three helicity amplitudes.

The partial decay widths $d\Gamma(R\rightarrow N\gamma^{*})$ for a resonance
with mass $M_{R}$ and a spin $J\geq 3/2$ or $J=1/2$, as well as with normal
($J^{P}=1/2^{-},3/2^{+},..$) and abnormal parities
($J^{P}=1/2^{+},3/2^{-},..$), are given by Krivoruchenko and Fässler in [110].
Equivalent expressions based on covariant form factors were obtained by
Zetenyi and Wolf [114]). The expressions read:

$\displaystyle d\Gamma($ $\displaystyle R_{J\geq 3/2}\rightarrow
N\gamma^{*})=F(M_{R},M_{\ell^{+}\ell^{-}})^{\pm}$ (50) $\displaystyle\times$
$\displaystyle\left(\frac{l+1}{l}\,|G_{M/E}^{\pm}(M_{\ell^{+}\ell^{-}})|^{2}+(l+1)(l+2)\,|G_{E/M}^{\pm}(M_{\ell^{+}\ell^{-}})|^{2}+\frac{M^{2}_{\ell^{+}\ell^{-}}}{M_{R}^{2}}|G_{C}(M_{\ell^{+}\ell^{-}})|^{2}\right)\,,$
$\displaystyle F(M_{R},M_{\ell^{+},\ell^{-}})^{\pm}$
$\displaystyle=\frac{9\alpha}{16}\,\frac{(l!)^{2}}{2^{l}\,(2l+1)!}\;\frac{(M_{R}\pm
M_{N})^{2}}{M_{R}^{2l+1}M^{2}_{N}}\;\Big{[}(M_{R}\pm
M_{N})^{2}-M^{2}_{\ell^{+}\ell^{-}}\Big{]}^{l-1/2}$ (51)
$\displaystyle\times\Big{[}(M_{R}\mp
M_{N})^{2}-M^{2}_{\ell^{+}\ell^{-}}\Big{]}^{l+1/2}$

and

$\displaystyle d\Gamma(R_{J=1/2}\rightarrow N\gamma^{*})$
$\displaystyle=H(M_{R},M_{\ell^{+}\ell^{-}})^{\pm}\,(2\,|G_{E/M}^{\pm}(M_{\ell^{+}\ell^{-}})|^{2}+\frac{M_{\ell^{+}\ell^{-}}^{2}}{M_{R}^{2}}|G_{C}(M_{\ell^{+}\ell^{-}})|^{2})\,,$
(52) $\displaystyle H(M_{R},M_{\ell^{+}\ell^{-}})^{\pm}$
$\displaystyle=\frac{\alpha}{8\,M_{R}}\;\Big{[}(M_{R}\mp
M_{N})^{2}-M^{2}_{\ell^{+}\ell^{-}}\Big{]}^{1/2}\times\Big{[}(M_{R}\pm
M_{N})^{2}-M^{2}_{\ell^{+}\ell^{-}}\Big{]}^{1/2}\,.$ (53)

Note that for $J=1/2$ only two eTFF are defined and that there is no
contribution from the Coulomb form factor at $M_{\ell^{+}\ell^{-}}^{2}=0$. The
$+$ and $-$ signs stand for resonances with normal and abnormal parity,
respectively ($P=P_{intr}(-1)^{l}$, $l=0$ for $J=1/2$ and $l=J-1/2$ for
$J\geq/3/2$). The functions $F$ and $H$ depend on the resonance mass ($M_{R}$)
and the virtual photon mass $M_{\ell^{+}\ell^{-}}$ [110]). For a point like
particle the form factors $G_{M/E}$ are constant and can be fixed at the
$M_{\ell^{+}\ell^{-}}^{2}=0$ (photon point) from the existing data.

Figure 6:  (Left panel) Dependency of $G_{M}$ on $M^{2}_{\ell^{+}\ell^{-}}$
for the $\Delta(1232)\rightarrow R\gamma^{*}$ transition for various $\Delta$
masses obtained from the covariant quark model of [118]. (Right panel) Ratio
of the measured dielectron yield to model predictions as a function of the
invariant mass for $\Delta(1232)\rightarrow pe^{+}e^{-}$ (see text and [108]
for details).

In such cases, the mass dependence of the decay width is given by the
functions $F$ and $H$ and can be regarded as ”QED” reference, as in case of
the meson decays. One should however note, as pointed out in [115], that there
were also other expressions used in model calculations of dilepton production
which provided inconsistent results. The influence of various expressions on
$d\Gamma(R\rightarrow Ne^{+}e^{-})$ and the parameterization of resonance mass
distributions for dielectron spectra has been investigated in detail for the
$\Delta(1232)$ in [78].

Actually, the $q^{2}$ dependence of the eTFFs is still an open issue as well
from the theory as from the experiment viewpoint. For example, calculations
presented in [116] assume strict VDM for the $\Delta(1232)\rightarrow
N\gamma^{*}$ transition and a dominance of the $\rho$ meson in the baryon-
photon coupling, which strongly affects the mass dependence of the eTFF.
However, this approach overestimates the radiative decay widths of baryonic
resonances. The extended VDM (eVDM) developed in [110, 117] includes the
coherent contribution of several excited vector meson states to calculate the
decay transition rates like $\rho^{{}^{\prime}}$, $\rho^{{}^{\prime\prime}}$
and $\omega^{{}^{\prime}}$, . The model has several parameters which have been
fixed by the quark counting rules, by available data on helicity amplitudes in
the space-like region and based on known branching rations for $R\rightarrow
N\rho$ decays. This approach allows to reduce the otherwise too large
branching ratios of the $R\rightarrow N\gamma$ transitions computed with
strict VDM models.

The covariant constituent quark model by Ramalho and Peña [95, 118] provides a
description with a dominant $G_{M}$ form factor for the
$\Delta(1232)\rightarrow N\gamma^{*}$ transition while the other two form
factors come out to be small, in accordance with data obtained in the space
like region. The eTFF is build from two contributions originating form the
quark core and the pion cloud. The quark core component describes the
resonance as a quark-diquark system in $S$-wave with a small admixture ($1\%$)
of a $D$ state. This contribution is determined from lattice QCD and also in
agreement with the data in the space-like region at high $q^{2}$. The
comparison with time-like data visualizes the role of the meson cloud
component which is dominantly of pionic type. The contribution of the pion
cloud in the time-like region is parameterized using two terms: a photon,
either directly coupling to a pion or to an intermediate baryon states. The
eTFF of the pion is fixed from available data. In their most recent work
[118], in contrast to previous implementations [119, 95] based on the
Iachello-Wann model developed for the nucleon eTFF [120, 121], the correct
position of the meson pole and the width has been included. A similar model
has been applied for the $N^{*}(1520)$ resonance [96]. The striking feature of
both calculations is an important and increasing role of the pion cloud
contribution in the time-like region.

For the $\Delta(1232)$ a rapid enhancement of the $|G_{M}|^{2}$ as a function
of the dielectron invariant mass is predicted, particularly significant for
higher resonance masses (see Fig. 6 left panel). The pion cloud plays a
dominant role, as can be seen from the comparison to the separately plotted
(thin lines) quark-core contribution. The latter provides an almost constant
strength below the $\rho$ meson pole which rapidly increases when the
invariant mass approaches the pole region. In the same figure (right panel)
the first experimental data from HADES on the $\Delta(1232)\rightarrow
pe^{+}e^{-}$ transition are shown [108]. Plotted is the ratio of the measured
dielectron yield to the one expected for a decay of a point-like
$\Delta(1232)$ (denoted as ”QED”) as a function of the invariant mass. An
increase of the ratio with increasing mass is clearly visible, similarly to
the observation for Dalitz decays of mesons discussed in the previous section.
The data are also compared to predictions of the covariant quark model of
Ramahlo-Peña [118] (blue solid line). The increase in the measured ratio can
be interpreted, within this model, as predominately due to pion a cloud effect
(red dotted line) since the core contribution (black dashed line) shows an
almost flat behavior.

###  4.2 Nucleon-Nucleon bremsstrahlung

In the most general case N-N (nucleon-nucleon) bremsstrahlung can be defined
as a process associated with production of a photon (real or virtual) in a N-N
interaction, regardless of the number of particles in the final state. Ultra-
relativistic N-N collisions are governed by partonic interactions and can
result in very hard photons through $q\bar{q}$ annihilation (Drell-Yan) or to
very complex states with charged particle multiplicities reaching rapidity
densities in excess of ten and even giving rise to thermal radiation. Such
contributions from hard processes are usually not subsumed under the term
bremsstrahlung.

At much lower collision energies, in the range of a few G$e$V beam energy per
nucleon, the dominant contribution to dielectron production from N-N
collisions originate from the $\Delta(1232)$ isobar Dalitz decay as discussed
in the previous section. However, also other amplitudes contribute to the
scattering process $NN\rightarrow NN\gamma^{*}$. In the non-resonant (”quasi-
elastic”) process one or both nucleons go off-shell for some time after or
before the photon is emitted. Their virtuality is lifted by strong interaction
between the nucleons, conveniently calculated in the one-boson exchange
approximation. Such bremsstrahlung is suppressed in p-p over n-p collisions
due to the absence of a dipole moment in the time-dependent electromagnetic
field generated by the colliding protons. In microscopic transport model
calculations “quasi-elastic” bremsstrahlung is often calculated using the soft
photon approximation [122, 123]. It assumes photon emission following elastic
nucleon-nucleon interactions with an appropriate phase space modification
induced by the produced virtual photon; any interference processes are
neglected. Contributions from the $\Delta$ isobar and higher resonances are
treated as separate processes and added incoherently as an extra source of
lepton pairs.

Various calculations of N-N bremsstrahlung were performed in the framework of
the One Boson Exchange model in the nineties [116, 124, 125], all treating the
resonant and non-resonant channel coherently. The results show a dominance of
$\Delta$ contributions and the importance of interference terms originating
from various meson exchange graphs. Moreover, the calculations demonstrate a
strong sensitivity of the dielectron yield on off-shell electromagnetic form
factors of nucleons, and on the eTFF of the $\Delta(1232)$. Also the modelling
of the nucleon-nucleon interaction requires form factors to dress the
respective vertices and also various nucleon-meson coupling constants, which
all need to be fixed from experimental data. While the coupling constants and
hadronic form factors can be extracted from meson production data reasonably
well, the electromagnetic form factors can only be inferred from dilepton data
which are very scarce.

The comparison of model calculations to dielectron data from p-p collisions
generally confirms the dominance of $\Delta$ Dalitz decays for dilepton
invariant masses above the pion mass. In p-p collisions only a tiny
contribution originates from non-resonant radiation [108]. This situation
changes dramatically in case of n-p collisions where the non-resonant n-n
bremsstrahlung contribution plays a significant role [126]. There are two
reasons for that. First, the non-vanishing dipole moment and, second, the
additional channel of charged pion exchange not possible in p-p collisions.
The internal charged current gives rise to substantial extra radiation.
Moreover, the respective eTFF at the photon (internal) pion vertex can
introduce significant changes in the dielectron invariant mass distribution
towards the $\rho$ meson pole mass. The respective one-boson-exchange (OBE)
calculations differ in how gauge invariance is implemented, in particular in
graphs related to the production of lepton pairs from the internal (charged)
pion exchange line. This particular mechanism of dielectron production has
been realized already in the $90^{\prime}es$ [125]. The role of charged pion
exchange was emphasized by more recent calculations of Kaptari and Kämpfer
[127] who predicted cross sections by a factor 2-4 larger for the ”quasi-
elastic” contribution as compared to former results. After implementing these
newly obtained cross sections for bremsstrahlung in transport calculation
performed by the HSD group [77] it became possible to explain the long
standing ”DLS puzzle” [128], related to unexplained dielectron yield measured
in low energy light ion collisions. However, the subsequent calculations of
Shyam and Mosel, using a similar effective Lagrangian model but a different
recipe to conserve gauge invariance, did not confirm such a large increase of
the ”quasi-elastic” contribution but stressed the importance of off-shell
$\rho$ contributions in eTFF of the internal charged pion [129, 130].

In order to differentiate between various calculations and mechanisms of
dilepton production in N-N collisions information on the lepton angular
distributions are very valuable. As already discussed for the Dalitz decays of
neutral meson, the distribution of the lepton decay angle w.r.t. the
$\gamma^{*}$ axis (in the helicity frame) is expected to show a
$1+B\,cos^{2}(\theta_{l})$ distribution with $B=1$ (see Eq. 49). OBE
calculations of [131] also predict significant anisotropies of the electron
distributions from N-N collisions. For the $\Delta$ Dalitz decay a similar
anisotropy as for the $\pi^{0}$ Dalitz with $B\sim 1$, with some slight
dependence of $B$ on the dielectron mass, was obtained. This can be understood
as being due to a dominance of transverse photon polarization. On the other
hand, for the non-resonant neutron-proton bremsstrahlung a smaller anisotropy
of $B\sim 0.4$, decreasing with increasing invariant mass, was predicted. The
only experimental results on electron angular distributions from N-N
bremsstrahlung was recently obtained by the HADES experiment and will be
discussed in (cf. Sec. 5.1.3).

###  4.3 Dileptons from hard processes

With increasing collision energy partonic processes start to dominate the
initial N-N reactions. As a result, the collision partners can fragment into
quark diquark states and the final states are characterized by an increasing
number of produced particles. A particular process of relevance for dilepton
production is the annihilation of a quark with an antiquark, each from one of
the two collision partners. The cross section for this Drell-Yan process is
readily written down in case of very high collision energies, where the
parton-parton annihilation can be calculated perturbatively. The cross section
for a purely electromagnetic transition of two quarks (partons) can be
inferred from the respective Feynman diagram of an annihilation process into a
dilepton (one-photon exchange):

$\frac{d\hat{\sigma}}{dQ^{2}}=\frac{4\pi\alpha^{2}}{3Q^{2}}e^{2}\delta\left(Q^{2}-\hat{s}\right)\,.$
(54)

Here $\hat{\sigma}$ is the “off-shell” cross section to produce a dilepton
with squared invariant mass $Q^{2}=\hat{s}^{2}$ and $\alpha$ is the fine
structure constant. To arrive at a cross section for production in a p-p
collision one uses the factorization concept of QCD. Along this concept, the
annihilation is treated as hard process while the probability to find a parton
with given momentum fraction $x$, determined by non-perturbative QCD, is given
by the experimentally accessible parton distribution (structure) functions
[133]. In that way the parton-parton (off-shell) cross section has to be
weighted with the probabilities to find partons with the proper momenta
$p_{q}$ and $p_{\hat{q}}$ given by the respective structure functions $f_{q}$
and $f_{\hat{q}}$

$\frac{d\sigma}{dQ^{2}}=\beta_{c}\sum_{q}\int dx\int
dy\>f_{q}\left(x\right)f_{\hat{q}}\left(y\right)\frac{d\hat{\sigma}}{dQ^{2}}\,.$
(55)

Here $x$ and $y$ are denoting the momentum fractions carried by the quark and
antiquark, respectively. Then, the $q\bar{q}$ collision energy can be
expressed in terms of the p-p center of mass energy $\sqrt{s}$ as
$\hat{s}=(xp_{1}+yp_{2})^{2}\simeq xys$. The fact that the Drell-Yan cross
section at high energies is determined solely by a QED cross section times the
structure functions makes this process a formidable tool to investigate the
internal structure of nucleons. The comparison of such leading order results
with experiment have revealed that higher order effects do play a role. This
has lead to the introduction of so-called K-factors, included in the
calculation to force agreement with data. More care has to be taken in the
evaluation of Drell-Yan cross sections in $p+A$ and $A+A$. Not only the
structure functions are modified when nucleons are embedded in a medium, but
also additional effects occur like gluon shadowing and a $p_{\perp}$
enhancement caused by parton multiple-scattering (Cronin effect). The precise
knowledge of in-medium partition functions, in particular those of the gluons,
is of great importance to determine the initial conditions of an ultra-
relativistic heavy-ion collision (cf. Sec. 2.1 (see e.g. [134] for a recent
review on this topic).

Both hard processes important for dilepton continuum physics in the IMR, open-
charm production and Drell-Yan, can be calculated using PYTHIA [156] tuned in
p-p collisions and scaled by the number of binary collisions. The correlated
charm production can also be determined experimentally by measurements of
$D$-meson production. Unfortunately, such measurement of correlated $D$ meson
production are not easy because of their short decay path and have so far been
obtained at SPS only by the $NA60$ experiment. The production of open heavy-
flavour hadrons has been measured via their hadronic and semi-leptonic decays
at mid- and forward-rapidity in pp collisions at the RHIC and LHC. The results
on $D$ meson production are in agreement (within factor 2) with pQCD
calculations performed FNOLL, PYTHIA of POWHEG. Nuclear modification factors
for charm mesons, identified by their hadronic decay channels, have been
recently also measured at LHC in $Pb+Pb$ collisions by ALICE [136] (for recent
overview of LHC results on heavy flavour see [137], ). An analysis of dilepton
emission in Au-Au collisions at $\sqrt{s_{\mathrm{NN}}}$200 based on STAR data
obtained with the Heavy Flavor Tracker (HFT) has been ongoing at the time this
review was written. THE HFT should help to reduce the contribution of
correlated open charm decay while at the same time it produced additional
conversion background due to its material budget. Excellent tracking
performance of the TPC can help to identify electrons and positrons emerging
from conversion in the HFT.

At beam energies below around $\sqrt{s_{\mathrm{NN}}}$$=10$ G$e$V perturbative
approaches are difficult as ever higher-order terms have to be taken into
account. As of today, the calculation of prompt dilepton radiation from
initial semi-hard444We distinguish semi-hard process from hard processes where
perturbative calculation are applicable and soft processes, which are driven
by baryon resonance production and meson exchange. scattering processes have
not been investigated from first principles. An attempt to provide a
consistent picture of particle production going from soft to hard processes is
the NEXUS project introduced in Sec. 2.1 and described in [15, 16]. However,
radiative (electromagnetic) processes have not been treated yet.

The TAMU group has investigated possible contributions of semi-hard processes
to the dilepton yield by studying $\rho$ meson production in a schematic jet-
quenching model [153]. The initial transverse momentum spectrum of $\rho$
mesons is assumed to follow a power law with

$\frac{1}{p_{\perp}}\frac{dN_{\mathrm{prim.}}}{dp_{\perp}}=A(1+Bp_{\perp})^{-a}\,.$
(56)

Here $A$, $B$ and $a$ are parameters adjusted to N-N scattering data. Second,
the transverse momentum distribution is modified to implement the Cronin
effect for which the parametrization is adjusted such as to reproduce
measurements of direct photon production in $p+A$ collisions. Next, the
fraction of $\rho$ mesons which are destroyed by re-scattering before they can
leave the surrounding medium is evaluated. For this, hadronic as well as pre-
hadronic cross sections and the local particle density are used. Finally, the
remaining $\rho$ mesons are decayed into dileptons according to vacuum
branching ratios. Such a model can account for the contribution of hard
dileptons around the $\rho$ pole mass as observed e.g. by the NA60 experiment
cf. Sec. 5.3.2.

Non-equilibrium models with partonic degrees of freedom can trace the dilepton
emission from initial state dynamics consistently by following the collision