Analogous to the azimuthal correlation measurements of soft particles in an
event, the azimuthal anisotropy of jets $v_{n}^{\rm jet}$ can be measured with
respect to the second harmonic event plane, after separating jets from the
underlying event background. Displayed in Fig. 48 is a compilation [265] of
results on the jet $v_{2}^{\rm jet}$ and particle $v_{2}^{\rm part}$ for semi-
central collisions. The ATLAS calorimetric jet $v_{2}^{\rm calojet}$ and ALICE
charged jet $v_{2}^{\rm chjet}$ are consistent with each other and exhibit a
significant $v_{2}^{\rm jet}$ up to large $p_{T}$.999Note that for any initial
parton $p_{T}$, the particle $p_{T}$ will be less than that of a charged jet
and it is likewise less than that for a calorimetric jet due to the missing
initial parton energy in the particles and charged jets. Also shown are the
$v_{2}^{\rm part}$ of charged particles for comparison. The ALICE and ATLAS
$v_{2}^{\rm jet}$ measurements are indicative of pathlength-dependent parton
energy loss. A recent measurement by CMS [268] in pp collisions of the two-
particle $v_{2}$ (defined with respect to the jet axis) of constituent
particles within a jet as a function of $N_{\mathrm{part}}$ is consistent with
theoretical model predictions up to $N_{\mathrm{part}}$ $\sim 80$. There is an
intriguing increase in the two-particle $v_{2}$ within jets for
$N_{\mathrm{part}}$ from 80 to 110 suggesting possible collective or other
effects within jets in high-$N_{\mathrm{part}}$ pp collisions. If this is,
indeed, an effect of collective particle transport, it could alter our overall
view of the emergence of collectivity in AA collisions.

While the elliptic and higher flow harmonics are especially sensitive to the
shear viscosity of the QGP, the radially symmetric component of the transverse
flow, as expressed in the mean transverse momentum $\langle p_{T}\rangle$, is
primarily sensitive to the speed of sound of the QGP and the bulk viscosity.
The bulk viscosity is thought to be most important at lower collision
energies, where the initial temperature does not much exceed $T_{c}$, the
speed of sound is predicted to dominate at higher collision energies. One way
to avoid complications from changes in the overall geometry is to measure the
$\langle p_{T}\rangle$ in ultra-central collisions, where variations of the
particle multiplicity are thought to be indicative of different values of the
initial temperature $T_{\rm ini}$ caused by quantum fluctuations in the energy
deposition within the same maximal nuclear cross section. The proposal [269,
270] is to measure the speed of sound $c_{s}$ via the thermodynamic relation

$c_{s}^{2}=\frac{dP}{d\varepsilon}=\frac{d\ln T}{d\ln s}=\frac{d\ln\langle
p_{T}\rangle}{d\ln N_{\rm ch}},$ (14)

where the mean $p_{T}$ is used as a proxy for the average temperature and the
charged multiplicity $N_{\rm ch}$ as proxy for the average entropy density.
For comparison, the speed of sound $c_{s}(T)$ can be easily calculated by
lattice-QCD. Recent preliminary results from CMS [271] in ultra-central Pb+Pb
collisions at LHC yield a value $c_{s}^{2}/c^{2}=0.241\pm 0.002\pm 0.016$ at
the average temperature $T=219\pm 8$ MeV, which is in excellent agreement with
lattice-QCD results. It will be interesting to see whether the extracted value
for collisions at RHIC energies is lower as predicted by lattice-QCD.

An attribute of the QGP fluid that was not anticipated at the time of our 1996
review [1] is vorticity. Because of the very low specific shear viscosity of
the QGP any vorticity that is seeded into the fluid at early times can survive
for an extended period of time as Kelvin’s theorem states that circulation is
strictly conserved in an ideal fluid. The seeding of vorticity in non-central
heavy-ion collisions was first recognized in [272] where also global hyperon
polarization with respect to the collision plane was identified as an
experimental signature. Global $\Lambda$-hyperon polarization in the percent
range was subsequently observed in Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ $=7-200$ GeV [273]. The magnitude of the polarization
can be related to the average vorticity of the QGP at the moment of
hadronization and gives an average value $|\vec{\omega}|=(9\pm 1)\times
10^{-21}\,{\rm s}^{-1}$ for Au+Au collisions within the energy range studied
in [273]. The observed magnitude can be explained as the transfer of vorticity
into the QGP from the initial orbital angular momentum of the colliding nuclei
that results in a spin polarization of the QGP fluid [274, 275]. The detailed
vorticity pattern of the QGP fluid and the microscopic mechanisms of spin
transfer into the QGP and its equilibration are areas of active research. In
addition to spin polarization of hyperons, STAR and ALICE have also reported a
nonzero spin alignment of several vector mesons ($K^{*}$, $\phi$) [276, 277]
the origins of which are not yet well understood.

##  11 Equation of State

Interest in the equation of state of nuclear matter was the primary motivation
for our field of research and the inception of experiments utilizing
collisions of energetic heavy ions [278]. After initial studies of baryon-rich
nuclear matter in the GeV range [279], the interest became focused on
understanding the equation of state of excited QCD matter, which was
originally a centerpiece of the RHIC experimental program as exemplified by
the panel in Fig. 1 entitled “Temperature.” This interest has waned somewhat
in recent years as the definitive results of lattice gauge theory for zero to
moderate net baryon densities (see [280] for a recent comprehensive review)
have been shown to be in good agreement with hydrodynamic simulations for many
observables [281].101010This statement does not apply to the equation of state
at high net baryon density, which is relevant to astrophysical phenomena such
as neutron stars and neutron star mergers, where reliable lattice-QCD
simulations are still unavailable (see e. g. [282]). In this domain the
equation of state still remains a primary focus of research (see also below).
Instead of the equation of state, the analyses of relativistic heavy ion
collision data have recently focused mainly on dynamical properties, such as
the effects of viscosity on the collective flow.

Experimental interest in the equation of state of nuclear matter has now
shifted back to much lower collision energies, in the few-GeV range as
explored in the second RHIC beam energy scan, where the net baryon density of
the matter created is above that covered by reliable lattice calculations. A
main focus of this investigation is to determine whether the smooth crossover
between hadronic matter and QGP at low net baryon density gives way to a
first-order phase transition with a threshold critical point. The primary
experimental probes for a first- or second-order phase transition are large-
scale spinodal density fluctuations and critical net-baryon number
fluctuations, respectively. Hints of such critical behavior were observed in
net-proton number fluctuations in the first RHIC beam energy scan [110] but
require confirmation with much higher statistics data [111].

The recent detection of gravitational waves from binary neutron-star mergers
[283, 284] has sparked interest in connecting the equation of state governing
the collapse of binary-neutron star systems to the equation of state of matter
probed in heavy-ion collisions in the few-GeV energy range [285, 286]. The
shape of the gravitational wave signal is expected to be sensitive to the
degrees of freedom in the core of neutron stars. Calculations are currently
focused on exploring connections to the dynamical evolution of few-GeV heavy-
ion collisions in terms of the pressure, temperature, entropy, and isospin
[287, 288, 289, 290]. A first-order phase transition to quark matter is
expected to look very different than a smooth crossover, and the next
generation of gravitational wave observatories may be able to distinguish
between the two. Furthermore, the lowest energy probes in the second RHIC beam
energy scan and the future Compressed Baryonic Matter (CBM) experiment at FAIR
[291] are expected to provide the data necessary for a quantitative comparison
with neutron-star merger observations.

##  12 Small Systems

The motivation for colliding ultra-relativistic heavy ions at RHIC and the LHC
was that at such high energies large nuclei would be most likely to create hot
QCD matter in the thermodynamic limit. Notwithstanding this argument, there
was also an old idea that even high-energy proton-proton collisions could
produce a statistical system that might exhibit aspects of hydrodynamic
behavior [292, 293, 294]. After the advent of QCD, the question remained as to
whether a statistical system composed of locally deconfined quarks and gluons
could be produced in sufficiently energetic p+p collisions and behave as a hot
fluid, i.e. a QGP. However, attempts to find evidence for QGP formation in
high-multiplicity p+$\bar{\rm p}$ collisions at the TEVATRON remained
inconclusive [295].

The general consensus remains that minimum-bias p+p collisions do not involve
the formation of a QGP, and such events are commonly used as a baseline
against which nuclear modifications of hard probes are measured. This does not
rule out that a QGP fireball can be produced in rare high-multiplicity p+p
events. The first clear evidence for behavior that resembles a collective flow
pattern was observed by CMS in p+p events at $\sqrt{s_{\mathrm{NN}}}$ = 7 TeV
with more than 90 charged tracks [296]. Angular correlation measurements at
$\sqrt{s_{\mathrm{NN}}}$ = 2.76 and 13 TeV [297, 298] confirmed this
observation. Similar observations of collective flow patterns have been made
for p+Pb collisions at LHC [299, 300, 301, 302] and in p+Au, d+Au, and 3He+Au
collisions at RHIC [303] (see [304] for a review). The similarity of the
collective behavior seen in p+p, p+A, and A+A systems can be explained if a
strongly coupled QGP is formed in all these systems [305].

Surprisingly, on the other hand, no evidence has been found for the formation
of a QGP in p+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV in
modifications of hard probes, such as jets [306]. It is presently unclear how
the finding of apparent collectivity in soft particle emission can be
reconciled with the absence of evidence for jet quenching. One possibility is
that the soft collective behavior observed in p+p and p+A collisions is
generated without hydrodynamic flow (see e. g. [307]). It is well known in
other fields, e. g. plasma physics, that collective motion of particles can be
created by non-hydrodynamical mechanisms, such as the action of coherent
fields [308, 309]. If the origin of collective behavior in p+p and p+A
collisions were found to have an alternative explanation, our current
understanding of the origin of flow patterns in A+A collisions would have to
be revisited.

##  13 Summary and Outlook

Nearly three decades of experimental and theoretical research have affirmed
the scientific strategy aimed at the discovery and characterization of the
quark-gluon plasma that was described in [1]. Extensive measurements have
converted the qualitative expectations for the quark-gluon plasma signatures
summarized in Fig. 1 into quantitative knowledge. As with any preconceived
strategy, adjustments were made in reaction to new insights gathered along the
way. Some signatures have been found to be less useful or more difficult to
measure than originally thought. Others have proven to be immensely valuable
including several that were unanticipated or some that were known in principle
but underappreciated.

The average initial energy density reached in the most central heavy-ion
collisions in Fig. 2 exceeds the threshold for QGP formation above
$\sqrt{s_{\mathrm{NN}}}$ $\sim$ 10 GeV. In the high energy range,
$\sqrt{s_{\mathrm{NN}}}$ $>$ 50 GeV, this can be deduced from the measured
charged-particle multiplicity $dN_{\rm ch}/dy$ and the short hydrodynamization
time deduced from elliptic flow. At lower energies, it requires some
assumptions about the dynamics of energy deposition, which is no longer quasi-
instantaneous. The argument here is based in part on the continuity of the
valence quark number scaling of elliptic flow that is observed down to
$\sqrt{s_{\mathrm{NN}}}$ $=$ 11.5 GeV, although increasing deviations from the
scaling show up for $\sqrt{s_{\mathrm{NN}}}$ $<$ 39 GeV indicating a growing
contribution to flow from the hadronic phase [310, 311].

Identical particle (HBT) interferometry has revealed that a fireball of
nuclear size and a lifetime of $4-10$ fm/c, depending on the collision energy,
acts as the common source of the hadrons that are emitted. As already
mentioned above, the composition of the emitted hadrons and the fluctuations
of conserved quantities have been used to map the chemical properties of the
hadronizing fireball. Future experiments with extended pseudorapidity coverage
will allow balance functions of conserved quantities to reach farther back
into the history of the evolution of the fireball and track when chemical
equilibrium is first established.

The intense investigation of the collective flow patterns in experiments has
made it possible to quantitatively determine fluid properties of the QGP. The
specific shear viscosity of the QGP has been found to lie in the range
$0.05<\eta/s<0.2$ depending on $T/T_{c}$, establishing this novel QGP state of
matter as the most “perfect” fluid known. Furthermore, the valence quark
scaling of the flow pattern has provided strong evidence that the collective
flow is generated at the quark level in a fluid in which quarks are not
confined as hadrons. The spin polarization of hyperons adds a new dimension to
the exploration of the flow pattern by its sensitivity to the vorticity and
thermal shear of the fluid. In the future, more precise measurements of the
interaction of heavy quarks with this fluid will further probe the strongly-
coupled nature of the QGP by yielding quantitative determinations of its
diffusion constants.

Among soft signatures, the enhancement of strange hadron production with
respect to pp collisions and, more generally, the complete chemical
equilibration of all light hadron species at common thermodynamic conditions
have provided strong evidence for the transition from hadronic matter to a
deconfined state – the QGP – at a temperature $T_{c}\approx 155$ MeV, in
excellent agreement with lattice-QCD simulations. As shown in Fig.19, the
boundary between hadronic matter and the QGP has been mapped by two different
methods over a range of baryon chemical potentials $\mu_{B}$ up to at least
300 MeV and agrees well with expectations from lattice gauge theory.

The measured suppression pattern of heavy quarkonium states, especially the
$\Upsilon$ states, and their observed sequential melting provide further
confirmation for the deconfinement of quarks and gluons in the QGP, but the
mechanisms responsible for the suppression pattern are more complex than
originally thought. In particular, the reduced suppression, by regeneration at
the phase boundary, of the J/$\psi$ in A+A collisions at LHC compared to that
at RHIC energies provides clear evidence that charm quarks are deconfined in
the QGP.

Electromagnetically interacting and hard QCD signatures provide complementary
information about the properties of the QGP. Measurements of the spectrum of
direct photons and the invariant mass spectrum of dileptons have yielded lower
bounds for the temperature at which the QGP initially thermalizes. These
spectra exhibit thermal temperatures substantially above the transition
temperature $T_{c}$. The spectrum of dileptons in the mass region of the
$\rho$-meson confirm the hadronization (chemical freeze-out) temperature
deduced from the hadron yields.

An unambiguous detection of chiral symmetry restoration will require high-
precision measurements of the lepton pair spectrum in the mass region 1 GeV
$<M_{\ell^{+}\ell^{-}}<2$ GeV. Theoretical predictions indicate a difference
of approximately 15% between models that involve chiral symmetry restoration
in the QGP phase and models that do not. Measurements of this level of
precision require very precise knowledge of the background from semi-leptonic
charm decays and are out of reach for the existing detectors. The proposed
ALICE 3 [312] and NA60+ [313] experiments aim at reaching the required
precision to be able to detect the enhancement of the dielectron spectrum at
invariant masses above the $\phi$-meson peak characteristic of $\rho-a_{1}$
mixing that is the signature of chiral symmetry restoration.

The most versatile, but also the most complex probes of the QGP are energetic
quarks and gluons, created by hard scatterings during the first moments of the
nuclear collision. Such hard-scattered partons materialize as jets, in which
the initial momentum of the primary parton is shared among many hadrons. A
number of different observables have been found that encode the energy loss of
the primary parton on its path through the QGP, beginning with the suppression
of the inclusive yield of high pT hadrons in A+A collisions observed from the
mid-range of RHIC energies to those of the LHC and corroborated by the
observation of a strong suppression of the high-$p_{T}$ hadrons opposite in
azimuth to a high-$p_{T}$ trigger hadron.

These measurements involving individual hadrons were subsequently extended to
jets and di-jets, where a similar quenching of jets attributable to parton
energy loss was observed. More recently, differential measurements of jets and
their substructure have emerged as tools to investigate the mechanism that
causes parton energy loss and help determine the conditions under which energy
loss is primarily radiative or when elastic processes dominate. In parallel,
flavor tagging of jets has given evidence for a mass and color charge
dependence of the parton energy loss in the QGP.

According to our current understanding, the energy loss of the primary parton
and the redistribution of its momentum within the jet is controlled by just a
few parameters characterizing the medium. In a dilute or thin medium, they are
the density of scattering centers and the range of the color force in the
medium. In a dense, thick medium, the jet quenching parameter $\hat{q}$
encodes the transverse scattering power per unit length of the medium. The
suppression factor $R_{\rm AA}$ of inclusive hadrons provides a direct
measurement of $\hat{q}$ under the assumption that the energy loss of the
primary parton is predominantly caused by gluon radiation induced by
scattering in the medium. The dimensionless parameter $\hat{q}/T^{3}$ is found
to lie in the ($\pm 1\sigma$) range $3.4<\hat{q}/T^{3}<5.8$ at RHIC and
$2.4<\hat{q}/T^{3}<5.0$ at LHC [196], which is consistent with values for
$\hat{q}/T^{3}$ required to describe the inclusive jet suppression measured at
RHIC and LHC.

The values of $\eta/s$ and $\hat{q}/T^{3}$ deduced from the heavy ion data by
Bayesian model-data comparison are two examples where experimental data have
helped bracket fundamental transport coefficients of the QGP that cannot (yet)
be reliably calculated in QCD. A fundamental question that is still to be
resolved, is to what extent it is possible to probe the dynamical evolution of
the matter created in heavy-ion collisions from partons in the initial state
to the thermal quarks and gluons of the QGP and, finally, into hadrons. This
quest involves the investigation and understanding of the parton structure of
the initial state, of the energy sharing mechanisms that produce a thermal
plasma, and the response of the QGP to hard probes that are sensitive to a
range of different scales.

Future measurements with better resolution and higher statistics will probe
more deeply to reveal the various scales involved in the interactions of jets
with the QGP. Investigation of coherence effects, both theoretically and
through jet substructure measurements, will determine the extent to which the
medium is able to resolve the interactions of the parton as it propagates
through the QGP. By constraining the dependence on the color charge and mass
of the parton they can further confirm the scattering dynamics underpinning
parton energy loss. At the same time, these differential measurements become
effective probes of the shower evolution inside a jet and contribute to our
understanding of QCD.

Over the next few years, the new sPHENIX detector [314] at RHIC and the
existing RHIC and LHC experiments with upgraded detectors will make precision
measurements of jet modifications in heavy-ion collisions. In the future, a
newly proposed ALICE 3 [312] experiment is expected to join in that endeavor
at the LHC. Parallel advances in the theory of jet interactions with the QGP
medium will be required to turn the wealth of expected data into firm insights
into the structure and properties of the QGP and the internal dynamics of jet
formation. The remarkable success achieved for soft QGP probes, where
data–theory comparisons within well-defined frameworks have enabled
quantitative measurements of QGP bulk properties, can serve as a guide for the
scientific approach aimed at elucidating the microscopic structure of the QGP
over the wider range of scales that is accessible with hard QCD probes.

Another increasingly central direction of investigation is research into the
parton structure of cold nuclear matter. A better understanding of the
structure of the colliding nuclei is important as one attempts to understand
the initial conditions of a high-energy collision of nuclei. An example of
such investigations is the monitoring of sub-nucleonic proton shape
fluctuations by studying J/$\psi$ production in diffractive e+p collisions
[315]. Alternative experimental approaches utilize J/$\psi$ photo-production
in ultra-peripheral d+Au collisions [316] and coherent J/$\psi$ production in
ultra-peripheral Pb+Pb collisions [317].

Understanding the interaction of cold nuclear matter with hard probes is also
an essential aspect in the interpretation of the nuclear modification factor
$R_{\rm AA}$ as already discussed in conjunction with the physics of
quarkonium suppression and jet quenching. Phenomena that will benefit from
additional experimental investigations in p+A collisions include nuclear
suppression or enhancement effects at relatively low $p_{T}$ that are
alternatively attributed to shadowing of nuclear parton distributions,
momentum broadening of incident partons, or final-state absorption.

In the more distant future precision studies of the parton structure of
nucleons and complex nuclei will be the scientific focus of the electron-ion
collider (EIC) [318]. Generalized parton distributions and transverse momentum
dependent parton distributions will ve used to map the transverse parton
structure of the proton, while diffractive e+p and e+A collisions will provide
precise quantitative constraints on the saturation of gluon distributions at
small Bjorken-$x$. Besides being valuable in their own right, these results
will help reduce the model dependence of the initial state of relativistic
heavy-ion collisions.

In conclusion, the strategy for the investigation of hot QCD matter outlined
in [1] has been successful beyond expectations. As in any field of physics,
experimental and theoretical progress have gone hand-in-hand, leading to
changes in research emphasis and readjustments of the strategy. Many questions
at the core of the initial RHIC research program have been answered and given
way to new ones [13]. Among those most important are the following. How does
the partonic microscopic structure of the QGP evolve into a “perfect” fluid at
longer distance scales? How small can a QGP that behaves fluid-like be? What
is the structure of the QCD phase diagram at high net baryon density? We can
be optimistic that improved experimental techniques, supported by theoretical
advances, and combined with creative and novel approaches will provide
information over the next decade that will help answer these questions.

Acknowledgments

We thank Roberta Arnaldi, Steffen Bass, Hannah Bossi, Helen Caines, Charles
Gale, Marek Gazdzicki, Laura Havener, Joseph Kapusta, Raghav Kunnawalkam
Elayavalli, Andras Laszlo, Yen-Jie Lee, Michael Lisa, Rongrong Ma, Ian Moult,
Jean-François Paquet, Ralf Rapp, Lijuan Ruan, Mike Sas, Jürgen Schukraft,
Enrico Scomparin, Alba Soto-Ontoso, and Willam Zajc for valuable input during
the writing of this article. We especially thank Hannah Bossi for assistance
in various aspects of the preparation of figures for this manuscript.

We appreciate helpful comments on a draft version of the manuscript made by
Yasuyuki Akiba, Frank Geurts, Peter Jacobs, Georgios Konstantinos Krintiras,
Krishna Rajagopal, Lijuan Ruan, Björn Schenke, Jürgen Schukraft, Andre Ståhl,
Marco Van Leeuwen, and Urs Wiedemann.

We are indebted to the ALICE, ATLAS, CMS, PHENIX and STAR collaborations for
their extensive experimental results.

We acknowledge support from the Office of Science of the U.S. Department of
Energy, JH from grant DE-SC004168 and BM from grant DE-FG02-05ER41367. BM also
acknowledges support by Yale University during Spring 2022 and Spring 2023.

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