# Quark-Hadron Transition and Entanglement

Berndt Müller  Department of Physics, Duke University, Durham, NC 27708-0305,
USA  Andreas Schäfer  Institut für Theoretische Physik, Universität
Regensburg, D-Regensburg, Germany

###### Abstract

The dual holographic description has enjoyed many successes in explaining
fundamental properties of the early stages of relativistic heavy ion
collisions up to the formation of a minimal-viscosity quark-gluon fluid.
However, there have been few attempts to extend its application beyond this
stage. Here we explore the prospects for such an extension beyond the time of
hadronization. Our discussion makes use of recent insights into the duality of
entanglement properties of field theory states in the edge of Anti-de Sitter
space and non-trivial topologies of horizons in the bulk, often referred to as
ER = EPR duality. We discuss this topic from the point of view of heavy-ion
phenomenology, review several relevant concepts, and map out a path toward
combining them into a comprehensive, at least semiquantitative description of
relativistic heavy ion collisions. We outline possible next steps in this
direction.

##  I Introduction

One of the fundamental questions of high energy physics is how apparently
thermal behavior emerges in relativistic heavy ion collisions. A large number
of measurements show such thermal behavior and are best described when one
assumes that the produced transient Quark-Gluon Plasma (QGP) is in a fully
thermalized state. Yet the nuclear reaction, which occurs in isolation, cannot
produce a state of high von Neumann entropy because the reaction is governed
by the laws of quantum chromodynamics (QCD) which assures unitarity of the
S-matrix. The transient QGP and all the hadrons measured in the final state
must exist in a highly entangled quantum state that mimics the properties of a
thermal ensemble that is in interaction with an external heat bath.

One is immediately led to the question whether there are any observable
differences between a thermal ensemble and such a highly entangled excited
state. The answer is not as simple as one might think as local observables
and, more generally, observables that depend on only a fraction of the
complete many-particle final state have difficulty differentiating between
these two alternatives.

Whether entropy can be generated in an isolated system in a process that is
governed by time reversal invariant laws or not, depends on the definition of
generalized, non-equilibrium “entropy”. Several proposals for such a notion
exist and, unsurprisingly, the answer depends on the entropy definition used.
In addition, as the deviation from equilibrium is time dependent, the most
suitable notion of entropy may yield an answer that depends on the time
elapsed after the start of the collision.

The complexity of this issue is illustrated by the fact that the Anti-de
Sitter space-conformal field theory (AdS/CFT) duality allows to map the
production of a QGP to the formation of a black hole in five-dimensional Anti-
de Sitter (AdS5) space. Any description of entropy production will share the
subtleties of the black hole information paradox expressed most succinctly by
the Page curve Page (1993a, 2013). For any pure quantum state of an isolated
many-particle system the von Neumann entropy is zero and remains zero under
unitary time evolution. If, however, only part of the system is observed, i.e.
if the information residing in the rest of the system is discarded, the
reduced system appears thermal with the corresponding thermal entropy.
Although the Page curve was specifically introduced to resolve the information
problem of black holes, it is understood to be the characteristic property of
any generic many-body quantum system Page (1993b). Its existence has been
confirmed experimentally in various experiments, see, e.g., Fig. 4 in Kaufman
et al. Kaufman  _et al._ (2016).

Our goal here is to outline a program which aims at describing essential
information theoretical properties of high energy nuclear collisions,
including those properties resulting in a Page curve, using concepts from
AdS/CFT duality. Obviously this is an ambitious goal, and in the present
article we can only outline the first steps. We will provide a list of open
issues at the end of this work that may help define a systematic pathway
towards a more precise understanding of how the apparent thermal properties of
the final state of a relativistic heavy ion collision can be reconciled with
the unitarity of the S-matrix in QCD.

There are many quantum field theories for which holographic duals are known to
exist. In the realm of gauge theories the original prototype, the ${\cal N}=4$
supersymmetric conformal Yang-Mills theory, is by far the most widely explored
example, because for large $N_{c}$ and large ’t Hooft coupling $\lambda$ it
has a tractable holographic dual, i. e. classical supergravity in AdS space.
The exact holographic dual for QCD is not known, although models that share
important features with QCD have been constructed Witten (1998); Sakai and
Sugimoto (2005). Therefore, any attempt to apply AdS/CFT phenomenology to QCD
is problematic from the start and can only be successful when combined with
other field theory and string theory techniques which allow to correct for
some of the differences. For example, the violation of conformal symmetry can
be treated perturbatively in QCD, see, e.g. Braun  _et al._ (2003, 2019);
Kumericki  _et al._ (2007), the finiteness of the ’t Hooft coupling can be
corrected by string perturbation theory, see e.g. Waeber  _et al._ (2015), and
the calculation of $1/N_{c}$ expansions on the field theory side has evolved
into a broad research field of its own ’t Hooft (1974).

However, QCD has also certain properties, like the confinement/deconfinement
transition, which cannot be treated perturbatively and thus require a more
incisive approach. As we are interested in the hadronization of the quark-
gluon plasma we cannot avoid studying holographic models that incorporate a
confinement/deconfinement transition into their basic framework through some
versions of the Hawking-Page transition Hawking and Page (1983). Examples of
such models include the ${\cal N}=4$ super-Yang-Mills theory on a compactified
spatial volume, such as $S^{3}$, models including a compactified additional
dimension of AdS space Aharony  _et al._ (2006), and models that break
conformal invariance through the introduction of a dilaton field Gürsoy  _et
al._ (2009a); Mandal and Morita (2011). Although there are reasons to believe
that such holographic models can describe many relevant properties of QCD
qualitatively, it is far from clear whether they can be refined to decribe
these relevant features with sufficient precision to allow for a quantitative
comparison with experimental data.

In view of these caveats the success of holographic methods in modeling
certain aspects of relativistic heavy ion collisions came as a pleasant
surprise. They have been used extensively to understand rapid thermalization
Lin and Shuryak (2008); Balasubramanian  _et al._ (2011a, b); Shuryak (2012);
Chesler and Yaffe (2011, 2014); Hubeny  _et al._ (2013), the fast transition
to hydrodynamic flow Janik and Peschanski (2006); Heller  _et al._ (2012a),
the small value of the specific shear viscosity Kovtun  _et al._ (2005), and
much more. In fact, the last two decades have witnessed such continuous
progress that today a much larger assortment of powerful techniques and
insights exists than ever before. For example, semiclassical quantum gravity
and applications to black hole physics have recently conceptually resolved the
information paradox Maldacena and Susskind (2013); Maldacena and Qi (2018);
Akers  _et al._ (2020); Almheiri  _et al._ (2020a, 2019, 2021); Penington  _et
al._ (2022); Penington (2020); Anderson  _et al._ (2020). All these
encouraging successes motivate us to try to understand aspects of the breakup
of the QGP into hadrons, which have only rarely been explored using
holographic methods.

Our effort to expand the use of holographic duality to a description of the
complete heavy ion collision is also motivated by the recognition that a field
theoretical description of hadronization on the quantum level is far too
complex to be tractable. Without tractable holographic descriptions that
capture salient features of this transition, even if they cannot reproduce all
aspects quantitatively, we simply lack a perspective to fully understand the
many-body quantum mechanics of a heavy ion collision.

Let us start by reminding the reader of a few salient phenomenological
properties of heavy ion collisions that any comprehensive description must
address:

  * •

There exist two distinct ways in which the particles that are ultimately
detected, hadrons, are produced from the evaporating QGP. Beginning
immediately after formation of the QGP ”fireball”, hadrons are emitted from
its surface. As the expanding QGP cools down to the pseudocritical temperature
$T_{c}$, hydrodynamics ceases to be valid, and the fireball converts into
hadrons throughout the remaining volume. The thermal model Andronic  _et al._
(2018) describes the production rates of all hadrons and nuclei remarkably
well with a single universal temperature parameter (the chemical freeze-out
temperature $T_{\rm CF}=156.6\pm 1.7$ MeV Andronic  _et al._ (2021)) that is
determined to percent-level accuracy. Because of the closeness of this value
to the central temperature of the QCD crossover transition determined by
lattice QCD ($T_{c}=158\pm 0.6$ MeV Borsanyi  _et al._ (2020) or
$T_{c}=156.5\pm 1.5$ MeV Bazavov  _et al._ (2019)) $T_{\rm CF}$ is interpreted
as QCD deconfinement temperature. The precision of the coincidence between
$T_{\rm CF}$ and $T_{c}$ itself is remarkable because the QCD crossover
transition, as measured by the chiral susceptibility, has an intrinsic width
of $15\pm 1$ MeV Borsanyi  _et al._ (2020).

  * •

The fact that the thermal model works so well is quite astonishing in view of
the strong interactions among the constituents of the system before and after
the hadronization transition. A focus of the discussion on this point is
provided by the hypertriton He${}^{3}_{\Lambda}$ whose yield in Pb-Pb
collisions at LHC agrees with the prediction of the thermal model for the
universal temperature $T_{\rm CF}$ despite the fact that its binding energy is
only 0.4 MeV, i. e. much smaller than $T_{\rm CF}$, and its size is comparable
to the size of the collision system. Recently, however, it was found Acharya
_et al._ (2022a) that this is no longer true for a smaller collision system
(p-Pb) with earlier hadronization time, adding to the conundrum of
hadronization in heavy ion collisions.

  * •

Heavy ion collisions create thermal conditions that resemble those prevailing
in the early universe, but the time scales are vastly different. In the
cosmos, different particle species fell out of thermal equilibrium at
different times due to cosmic expansion and cooling. For example, photons and
neutrinos have different cosmic background radiation temperatures. The same
could be expected for heavy ion collisions but is not observed for Pb-Pb
collisions. As we argued in Müller and Schäfer (2017) one framework that can
potentially provide an explanation for the lack of thermal differentiation is
the Eigenstate Thermalization Hypothesis (ETH) Deutsch (1991); Srednicki
(1994); D’Alessio  _et al._ (2016). The ETH may explain not only the success
of the thermal model but also the difference between Pb-Pb and p-Pb collisions
as the lifetime of the QGP in p-Pb collisions may be too short to apply the
full ETH formalism, which is based on energy eigenstates (see Section III).

Our manuscript is organized as follows. Section II contains some general
remarks about entanglement entropy in the context of relativistic heavy ion
collisions, and we introduce the concept of the Page curve. Because the ETH
plays an important role in our arguments we present a brief review in Section
III of those aspects that are relevant for us. In Sections IV and VI we will
further discuss some relevant aspects of heavy ion collisions and provide more
details on the properties mentioned above. In particular, we will argue in
Section IV that the usual numerical AdS calculations in the Poincaré patch are
insufficient to obtain the Page curve. In Section V we will review these
holographic model calculations of the early stages of heavy ion collisions up
until apparent thermalization. In this time interval they contain the relevant
physics and give phenomenologically satisfactory results.

To extend the holographic description to later times we propose two distinct
hadronization mechanisms: surface radiation and bulk hadronization. In Section
VII we adapt recent ideas Aharony  _et al._ (2006); Mateos  _et al._ (2006);
Bigazzi  _et al._ (2020) for the Hawking-Page phase transition to bulk
hadronization. To describe the quantum entanglement of hadrons emitted from
the fireball surface requires a very different approach. As hadrons cannot
exist within the QGP fireball, its surface acts as a boundary for hadron
states. In Section VIII we discuss how hadron emission can be treated as the
production of particle-hole states at the QGP surface in analogy to Hawking
radiation from the black hole horizon. This allows us to adapt the arguments
for the formation of ”islands” within black holes which are entangled with the
outgoing Hawking radiation to the QGP fireball. The increasing entanglement of
the remaining QGP with the emitted hadrons is holographically mapped onto the
black hole evaporation process. This will make it possible to transcribe the
arguments for the occurrence of a Page curve to the QGP hadronization process.
We synthesize these considerations into a schematic holographic description of
the hadronization process in Section IX.

Our arguments are admittedly speculative, but they have the advantage that
their underlying assumptions can be confirmed or refuted by detailed
calculations. In Section X we present a limited list of possible future
studies.

##  II Entanglement Entropy

In high-energy heavy ion collisions one never measures the complete final
state but detects only a subset of all emitted particles. Detector constraints
typically limit the detection to hadrons in a certain rapidity interval
$\Delta y\sim 1$ that is much smaller than the full beam rapidity range
$-y_{\rm beam}\leq y\leq y_{\rm beam}$.111The acceptance of detectors usually
covers a certain pseudorapidity window rather than a rapidity window. At high
energy, however, by far the largest number of emitted particles are pions
whose mass is smaller than their average transverse momentum, the effect of
the difference between pseudorapidity and rapidity is small for most global
observables. Because the subsystem $\Phi_{\Delta y}$ within the kinematic
range $\Delta y$ is entangled with the rest of the final state
$\Phi^{\prime}_{\Delta y}=\Phi\setminus\Phi_{\Delta y}$ outside this rapidity
window, it carries a certain entanglement entropy:

$S(\Phi_{\Delta y})={\rm Tr}[\rho(\Phi_{\Delta y})\ln\rho(\Phi_{\Delta y})].$
(1)

Here

$\rho(\Phi_{\Delta y})={\rm Tr}_{\Phi^{\prime}}[\rho(\Phi)],$ (2)

denotes the reduced density matrix of the final state constrained to the
rapidity window $\Delta y$, $\Phi$ denotes the complete final state, and
$\rho(\Phi)$ is the complete final-state density matrix. Unitarity of the
S-matrix in QCD ensures that $S(\Phi)=0$, if the initial state is a pure
quantum state, which is satisfied to a high degree of precision in the
collision of two heavy ions.

For $\Delta y\ll 2y_{\rm beam}$ the entanglement entropy $S(\Phi_{\Delta y})$
grows with the size of the rapidity window. Boost invariance at midrapidity,
realized at high collision energy, dictates that $S(\Phi_{\Delta y})/\Delta y$
is approximately independent of $\Delta y$ for small rapidity windows. The
complementarity law of quantum information also dictates that $S(\Phi_{\Delta
y})=S(\Phi^{\prime}_{\Delta y})$. Accordingly, when $\Delta y$ exceeds half
the total rapidity range of the final state, $\Delta y>y_{\rm beam}$, the
entanglement entropy has to decrease again and approach zero when $\Delta y\to
2y_{\rm beam}$. The entanglement entropy associated with a rapidity window
thus follows a Page curve Page (1993b, a), as sketched by the red solid line
in Fig. 1.

Figure 1: Page curve (red solid line) for the entanglement entropy associated
with a rapidity window $\Delta y$.

Particle detectors are not capable of measuring all information contained
within $\Delta y$, e.g., they generally do not record the relative phases
among all emitted particles. Although many modern detectors are capable of
recording the full set of kinematic correlations among hundreds or thousands
of detected particles, a large part of this information is usually discarded
in the data analysis and only selected few-body correlations are retained
222There are exceptions to this statement, e.g., in the measurement of the
collective flow vector or in fully resolved jet measurements, but even these
analyses ignore the full correlation information that is, in principle,
contained in the recorded data.. The reduction to a subset of data within the
rapidity window $\Delta y$ further changes the concept of entropy into a
notion that has been aptly called “entropy of ignorance” about the full
details of the final state Duan  _et al._ (2020). What is commonly called
“entropy” of the final state of a relativistic heavy ion collision is derived
from the single-particle distribution in momentum space; here we denote this
single-particle entropy as $S_{\rm sp}(\Delta y)$. The single-particle entropy
grows at a different rate than the entanglement entropy (see Duan et al. Duan
_et al._ (2020) for a detailed analysis in a slightly different context) and
continues to grow as $\Delta y$ exceeds $y_{\rm beam}$, as illustrated by the
blue dashed line in Fig. 2.

Figure 2: Schematic diagram of the final-state entropy $S_{\rm sp}(\Delta y)$
in a rapidity window $\Delta y$ derived from the single-particle momentum
distribution, shown as blue dashed line, in comparison with the entanglement
entropy in the same window, shown as red solid line.

A number of estimates of the single-particle entropy have been presented in
the literature. The estimates are based on the measured spectra of produced
particles and calculate the entropy of an equilibrated Boltzmann (or
Bose/Fermi) gas with the same momentum and mass distribution. This is usually
done per unit of rapidity (at midrapidity), resulting in a measured value of
$dS_{\rm sp}/dy=S_{\rm sp}(\Delta y)/\Delta y$, which corresponds to the slope
of the dashed line in Fig. 2. At top RHIC energy, $dS_{\rm sp}/dy\approx
5,000$ Pal and Pratt (2004); Müller and Rajagopal (2005); at top LHC energy
$dS_{\rm sp}/dy\approx 11,500$ Hanus  _et al._ (2019).

##  III The Eigenstate Thermalization Hypothesis

ETH Deutsch (1991); Srednicki (1994); D’Alessio  _et al._ (2016) posits that
even a single energy eigenstate can appear like a thermal system for most, if
not all, observables of practical interest. It is thought to apply generally
to systems that exhibit chaos at the quantum level. The ETH represents a
generalization of Random Matrix Theory (RMT) Wigner (1967) and, like RMT, is
formulated in terms of the matrix elements of an observable ${\cal A}$ in the
energy basis:

$A_{\alpha\beta}=\langle E_{\alpha}|{\cal A}|E_{\beta}\rangle\,.$ (3)

The ETH goes beyond RMT by assuming that the off-diagonal elements of the
matrix $A_{\alpha\beta}$ can be written as normalized elements
$R_{\alpha\beta}$ a random matrix modified by factors involving the
microcanonical entropy $S(E)$ and the spectral function $f(E,\omega)$ of the
system:

$A_{\alpha\beta}=A(E)\delta_{\alpha\beta}+e^{-S(E)/2}f(E,\omega)R_{\alpha\beta}\,,$
(4)

where $E=(E_{\alpha}+E_{\beta})/2$ and $\omega=E_{\alpha}-E_{\beta}$. The
exponential factor implies that the individual off-diagonal elements are
suppressed relative to the diagonal elements for a system with a high energy
level density, i. e., with $S(E)\gg 1$. Then the diagonal matrix elements
alone determine the thermal average of the observable:

$\displaystyle\langle{\cal A}\rangle_{T}$ $\displaystyle=$ $\displaystyle
Z(T)^{-1}\int\frac{dE}{E}\,e^{S(E)-E/T}\,A(E),$ $\displaystyle Z(T)$
$\displaystyle=$ $\displaystyle\int\frac{dE}{E}\,e^{S(E)-E/T},$ (5)

with corrections that are exponentially suppressed.

On the basis of the assumption (4) it is possible to show that the system
prepared in an energy eigenstate behaves like a thermal system. A few
particularly noteworthy properties are D’Alessio  _et al._ (2016):

  * •

The long-time average of an observable equals the thermal average:
$\overline{A}=\langle{\cal A}\rangle_{T}$.

  * •

The quantum fluctuations of ${\cal A}$ are equal to the thermal fluctuations
with corrections of $O(1/N)$, where $N$ is the number of degrees of freedom of
the system.

  * •

The time correlation function of finite-time expectation values $\langle{\cal
A}\rangle_{t}$ obeys a Kubo relation with the function $f(E,\omega)$ as the
spectral density.

These properties assure that the system, when monitored through the observable
${\cal A}$, is indistinguishable from a thermal system.

The Thouless energy $E_{\rm Th}$ is defined as the energy difference $\omega$
below which the factor $f(E,\omega)$ in (4) can be replaced with a constant
for most observables, and ETH reaches the RMT limit. In a dynamically evolving
system, the relevant range of energy differences $\omega$ is inversely related
to the evolution time by the uncertainty relation. It is often assumed that
the behavior of a chaotic quantum system for times longer than the Thouless
time $t_{\rm Th}=\hbar/E_{\rm Th}$ is described by RMT. However, it was found
in numerical studies of discrete quantum systems that the time until full ETH
behavior is established, $t_{\rm ETH}$, can be much longer, depending on the
operator under investigation Dymarsky (2019); Richter  _et al._ (2020). In
fact, can be parametrically longer than $t_{\rm Th}$ by a factor proportional
to the system size.

The intuitive interpretation of this latter time scale is that the
establishment of quantum entanglement requires causal connection across the
whole system, and full entanglement is only reached asymptotically. The most
important prediction of ETH for a fully entangled system is that if only a
small part of the density operator enters an observable, i.e. if the trace is
taken over more than half of the states, the observable behaves as for a
system in contact with a heat bath. It is tempting to assume that this
property explains the experimental success of the thermal model in
relativistic heavy ion collisions mentioned in the Introduction.

The operator dependence of the onset of ETH behavior might actually allow us
to understand why the measured yield of hypertritons in p+Pb collisions
differs from the prediction of the thermal model. The question is whether the
time of hadronization, $t_{\rm H}$, is larger or smaller than $t_{\rm ETH}$.
As the hypertriton wave function is much more extended than that of the proton
one should expect a strong form-factor suppression of the corresponding matrix
element, resulting in an especially large $t_{\rm ETH}$. At the same time
$t_{\rm H}$ is especially small for the proton size fireball produced in p+Pb.
Both effects suggest that ETH-behavior has not yet established itself when
hypertritons are formed in p+Pb collisions, which would explain the observed
suppression compared to the thermal model.

The current consensus is that ETH applies to most, if not all, systems that
exhibit quantum chaos. This conjecture has been confirmed in tractable model
systems where precise numerical calculations are possible. It leads to the
question whether QCD exhibits chaos at the level of the full quantum field
theory. We know that nonabelian gauge theories are chaotic at the classical
level and exhibit ergodic properties Bolte  _et al._ (2000). Unfortunately,
however, we are still unable to construct highly excited energy eigenstates of
a system governed by the laws of QCD and confirm the validity of ETH for such
systems. The validity of ETH for QCD thus remains a conjecture.

##  IV Time scales in the collision

The phenomenology of thermalization in heavy ion collisions is characterized
by multiple different time scales. The initial state is very far from thermal
equilibrium, but the time available to reach it is limited by the rapid
longitudinal expansion of the created quark-gluon system. Hadronization, a
confinement-deconfinement transition, takes place during this short time span
and drastically changes the properties of the system. This transition is
usually modeled as an early ”chemical freezout”, to be distinguished from
complete kinematic freeze-out, which is assumed to occur later.

The very large momenta of the colliding nuclei imply large Lorentz factors
$\gamma$ in the center-of-mass (CM) system, up to $\gamma\sim O(1000)$.
Accordingly, the transverse extent of the colliding nuclei at the initial
collision instant in the CM system is much larger (by a factor $\gamma$) than
their longitudinal thickness. In addition, transverse fluctuations are
substantial, of order 50% for the energy density Müller and Schäfer (2012), on
length scales between the inverse saturation scale $1/Q_{s}\sim 0.2$ fm and
the nucleon radius (0.7 fm). This generates, in combination with the velocity
of light $c$ for ballistic processes or the velocity of sound
$v_{s}=c/\sqrt{3}$ for diffusive processes, additional time scales. The
various time scales pose a challenge to any holographic description spanning
the entire collision.

However, upon closer inspection one realizes that all these time and length
scales can be traced back to three primary scales related to the underlying
field theory (QCD), and the initial and spatial boundary conditions:

  * •

The energy density $\varepsilon_{0}$ initially deposited in the collision; in
the glasma model $\varepsilon_{0}\sim Q_{s}^{4}$, where $Q_{s}$ is the gluon
saturation scale Lappi (2006).

  * •

The initial transverse extent $R_{T}$ of the collision region, which depends
on the nuclear radii $R$ and the impact parameter $b$.

  * •

The QCD confinement scale $\Lambda_{\rm QCD}$ reflected in the nucleon radius
$r_{N}$ and the pseudocritical temperature $T_{c}$.

There are three analogous independent scales that can be present in the
AdS/CFT dual description:

  * •

The energy density $\varepsilon_{0}$ initially deposited in the shock wave
collision.

  * •

The initial transverse extension $R_{T}$ of the collision region, which
depends on the transverse width of the shock waves and the impact parameter
$b$.

  * •

In holographic models, which contain an extraneous length scale, this can set
the scale for a transition of the Hawking-Page type 333We refer to any
transition between AdS space with a black brane and thermal AdS space as
“Hawking-Page type” transition, independent of the mechanism that sets the
temperature scale at which the transition occurs. that marks the boundary
between a confined and a deconfined phase. For example, in the strong coupling
limit of the simplest holographic model dual to ${\cal N}=4$ super-Yang-Mills
theory Maldacena (1998a), such a transition occurs when the dual theory is
considered in global AdS space Hawking and Page (1983). Similar transitions
exist in more realistic holographic models of QCD even when considered on the
Poincaré patch.

When the dual description is constrained to the Poincaré patch of AdS space
Bayona and Braga (2007), as it is necessary to describe a heavy ion collision,
the third scale must be introduced by some modification of the holographic
dual that breaks the conformal symmetry, if hadronization is to be described.
Examples include models with imbedded D-branes Kruczenski  _et al._ (2004);
Sakai and Sugimoto (2005) and the Scherk-Schwarz compactification of a dual
theory with an additional dimension considered by Witten Witten (1998) and
used in the study of Aharony et al. Aharony  _et al._ (2006).

In recognition of these primary scales we can divide the progression of a
relativistic heavy ion collision into four stages, which are summarized in
Table 1.

time | HIC phenomenology | Holographic dual model
---|---|---
Stage I | Hydrodynamization | Numerical AdS simulations
$t\leq 1$ fm/c | hydrodynamic attractors lead from | entropy production mapped by apparent horizon
| transient large amplitude fluctuations |  $t\leq 0.2$ fm/c equilibration without fluctuations
| to viscous hydrodynamic expansion |  $t\leq 1-2$ fm/c equilibration with fluctuations
Stage II | Expansion of the QGP fireball | Collision of localized shocks Waeber and Yaffe (2022)
$t\leq O(15)$ fm/c | hadronization at QGP surface | (Sections V and VII)
Stage III | Bulk hadronization | Smoothed HP transition
$t\approx O(15)$ fm/c | chemical freeze-out | (Sections VII and IX)
Stage IV | Expansion of HRG | Entangled hadrons
$t\geq O(15)$ fm/c | kinetic freeze-out | network of ER bridges (Section VIII)
Table 1: The different stages of a heavy ion collision (HIC) and their
proposed holographic modelling.

We now explain the reasoning behind Table 1 and describe the roles the primary
scales play in the different stages.

Stage I: At very short times neither the confinement scale nor the transverse
extent of the reaction region are important. During this stage the local
energy density far exceeds the critical density $\varepsilon_{c}$ at the
deconfinement transition, and information about the finite transverse size has
not spread widely because of causality. The initial energy density
$\varepsilon_{0}$ is the only relevant scale both on the QFT and holographic
side of the duality. Both statements may not apply to very small collision
systems, such as those produced in proton-proton or proton-nucleus collisions.

Stage II: The resulting hydrodynamic QGP expands during the second stage.
During this expansion a small fraction of the QGP hadronizes on the surface of
the expanding fireball. By construction, this surface is at the confinement-
deconfinement transition temperature $T_{c}$, which is of the order of
$\Lambda_{\rm QCD}$. Thus, the expansion dynamics of the QGP fireball depends
on the two scales $\varepsilon_{0}$ and $R_{T}$, while the hadronization
processes at its surface depend, in addition, on $\Lambda_{QCD}$. Therefore,
the latter aspect cannot be described in the standard framework of AdS
modeling on the Poincaré patch, while the hydrodynamical expansion can.

Stage III: At the end of the second stage the remaining fireball becomes
thermodynamically and hydrodynamically unstable and completely hadronizes into
a quantum entangled hadron gas. The duration of this process depends on the
nature of the phase transition. In models with a very large number of colors
and a first-order transition the conversion from a gauge plasma to a hadron
(glueball) gas takes an extended period of time because the formation rate of
color singlet hadrons is small Aharony  _et al._ (2006). In QCD, where the
transition is a rapid, but smooth crossover, the process is deemed to be so
fast that it is usually described as instantaneous. Because time evolution in
QCD is unitary no thermal entropy is produced, but ETH applies to this stage,
explaining the validity of the “thermal” model. (For this to apply the time
from the start of the collision until this third stage must be long enough for
global ETH behavior to be established Dymarsky (2019), which may not apply to
very small collision systems. Numerical simulations in the AdS Poincaré patch
are also unable to describe this stage adequately.)

The transition from (slow) surface hadronization to (rapid) bulk hadronization
means that the Page curve for a heavy ion collision has an asymmetric shape.
Before the final hadronization transition only a small fraction of the QGP (in
Section VI we will argue for roughly 20%) has decayed, while the rest decays
on a much shorter time scale during Stage III. Figure 3 shows a schematic
sketch of the Page curve for this scenario.

Figure 3: Schematic representation of the Page curve for the hadronizing
quark-gluon plasma in a relativistic heavy ion collision. The growth rate of
the entanglement entropy of the emitted hadrons is low during Stage II because
hadrons are only emitted from the surface. The growth rate then increases
quickly during Stage III due to bulk hadronization before dropping equally
rapidly after half of the QGP has hadronized. At the end of the hadronization
transition, the entropy again equals the initial entropy because all hadrons
are entangled with each other.

Stage IV: The final stage of a heavy ion collision, which features an
expanding cloud of entangled hadrons, resembles the final stage of a decaying
black hole which comprises an expanding cloud of entangled photons without a
black hole remnant at the center. Again, numerical simulations limited to the
Poincaré patch of AdS space are insufficient to capture this entanglement.

While each stage of a heavy ion collision has a well motivated holographic
dual, is is not clear how well holography can model a heavy ion collision in
quantitative terms. This is not a concern for the very early stage, when the
quark-gluon system is far away from the confinement scale, and a holographic
description can be justified by its approximate conformal symmetry and the
limited sensitivity of hydrodynamics to details of the initial state. It is
unclear whether tractable holographic models can provide for a quantitative
description at later times where more time scales are relevant.

Even if the holographic model cannot mirror a real heavy ion collision
quantitatively, in can potentially provide important conceptual insights with
parametric validity. As an example, we consider the final state when the whole
system consists of hadrons that are freely streaming toward the detectors.
There are two possibilities: Either the hadrons are in thermal equilibrium and
described by a density matrix with high entropy, which is the standard
assumption made in heavy ion physics, or the hadrons form a highly entangled,
nearly pure quantum state with very little entropy, as we argue here. In the
first case there exist well established techniques that permit detailed
theoretical predictions, whereas any such calculation requires additional
assumptions in the latter case. We will argue in the next Section, building on
work by van Raamsdonk et al. Van Raamsdonk (2020), that the holographic dual
of an entangled state of many hadrons differs only in subtle ways from the
holographic dual of a QGP fireball and that, therefore, a holographic
description can extend smoothly beyond the hadronization transition.

Another important question is whether the fireball reaches ETH-type behavior
before hadronization. This cannot be true for all observables, as the
following argument shows: As already noted, transverse variations in locally
conserved quantities equilibrate by diffusion. The associated time scale
$R_{T}^{2}/D$, where $D$ is the relevant diffusion constant in the QGP, is
identical to the one introduced in the previous Section as Thouless time
$t_{\rm Th}=1/E_{\rm th}$. It is useful to estimate this time constant for a
typical volume of QGP created in a heavy ion collision. The typical value of a
diffusion coefficient in QCD is $D\sim(\pi T)^{-1}$ Ding  _et al._ (2012).
With $R_{T}\approx 5$ fm and $T\approx 300$ MeV one finds $t_{\rm Th}\sim 120$
fm/c, which is much longer than the lifetime of the QGP.

The crucial question in this context is, how long it takes until few-particle
observables that are amenable to experimental measurement are well
approximated by their thermal value. The time it takes for entanglement to
propagate throughout the fireball provides for a lower bound. Quantum
information propagates with velocity $v_{E}<c$ yielding this information time
scale to be of order $R_{T}/v_{E}$. At strong coupling holographic models in
$d=4$ dimensions give $v_{E}\approx 0.62c$ Liu and Suh (2014), which means
that the speed of information transport is similar to that for energy
transport, which is given by the speed of sound $c_{s}=c/\sqrt{3}$. This means
that entanglement can spread through the entire fireball before the onset of
hadronization.

In this work we discuss certain ideas that may ultimately make it possible to
extend the successful holographic description of heavy ion collisions beyond
the Page time. The crucial feature of any such attempt is the preservation of
full quantum coherence which, in turn, requires the addition of novel concepts
to the holographic description that become relevant at later times in the
collision. As we discussed above, this implies that numerical simulations of
AdS gravity must be expanded to cover the global AdS geometry, not just the
Poincaré patch.

The underlying idea is that a heavy ion collision proceeds from the highly
entangled many-body wave functions of the colliding nuclei to a highly
entangled QGP which evolves continuously into a highly entangled multi-hadron
state but never to a truly thermal state with large von Neumann entropy. In
other words, the collision can be viewed as a unitary S-matrix mapping between
the initial and final hadron states. In this language, it is completely
irrelevant that in some intermediate state certain local observables look
thermal. It is ultimately the interaction of the emitted hadrons with the
detectors, which act as incoherent environment, that creates the thermal
entropy. This view implies that the establishment of full ETH behavior and
thus the success of the thermal model is far less of a mystery, because
$t_{\rm Th}$ is much shorter than the time that elapses until final-state
hadrons hit the first detectors.

Elucidating the microscopic processes which govern thermalization of many-
particle quantum systems is relevant for many subfields of physics. For
example, in Popescu  _et al._ (2006) it was argued that thermodynamics can be
based on the assumption of a pure state wave function of the universe with
which all systems investigated in the laboratory are entangled. Clearly, such
an assumption requires experimental verification. HICs involving two nuclei
colliding in the ultra-high vacuum of the beam-pipe, may be the “cleanest”
system to obtain such verification or refutation as the evolution of the
system can be followed without environmental influence over times many orders
of magnitude longer than the intrinsic time scales and with a dynamics which
is completely defined by QCD. A huge amount of precise data has already been
accumulated and is readily available for theoretical analysis.

It is not clear whether measurements that can detect the presence of complex
entanglement features among the emitted hadrons are feasible in heavy ion
collisions, just as it is unlikely that a practical test of the entanglement
pattern of Hawking radiation could be conducted even if we had access to an
isolated evaporating black hole. Examples of observables that might be both,
sensitive to the entanglement properties of the emitting system and amenable
to measurement, are quantum correlations among hadron spins, especially in
small collision systems Gong  _et al._ (2022), or among isospins, so-called
disoriented chiral condensates Anselm and Ryskin (1997); Mohanty and Serreau
(2005), for which possible indications have been recently observed Acharya
_et al._ (2022b).

There is hope that such experiments are possible with atomic physics analogues
of black holes Steinhauer (2016); Kolobov  _et al._ (2021). As QED is also
time reversal invariant, QED and QCD processes should have similar information
theoretical properties and, therefore, the experimental verification of ETH
behavior and Hawking radiation for QED systems suggest that QCD systems behave
in a similar manner. However, while these insights are intellectually
satisfying, their practical utility depends on whether or not a holographic
description of entanglement and ETH behavior is feasible in practice. This
will be discussed in Section VII.

##  V Holographic Simulations of Early Collision Stages

Starting with the pioneering papers Chesler and Yaffe (2009, 2011, 2014)
numerical AdS calculations have played an ever more important role to inform
our understanding of the early phase of high-energy heavy ion collisions. Over
time the numerical techniques have been steadily improved, such that today
realistic three-dimensional collisions can be studied Waeber and Yaffe (2022)
instead of somewhat schematic collisions of infinite plane shock waves. Also,
more and more details and special cases were studied Heller  _et al._ (2012b);
Casalderrey-Solana  _et al._ (2014); Chesler  _et al._ (2015); Ecker  _et al._
(2016); Casalderrey-Solana  _et al._ (2016); Endrodi  _et al._ (2018); Waeber
_et al._ (2019); Müller  _et al._ (2020). In parallel, calculations of the
entropy density Gubser  _et al._ (1998), shear viscosity Buchel (2008),
conductivity Waeber and Schäfer (2018) and inverse equilibration times Waeber
and Schäfer (2018) at NLO in the string coupling were found to be in good
agreement with QGP phenomenology. As these corrections can be encoded in
higher derivative terms of classical gravity, there is a possible path to more
quantitative holographic simulations of the early collision stages.

Numerical solutions of the classical Einstein equations with AdS boundary
conditions share the difficult problem of having to deal with the
diffeomorphism invariance of general relativity. Using a suitable metric as an
ansatz one can significantly reduce the diffeomorphism freedom. Chesler and
Yaffe used Eddington-Finkelstein coordinates and managed to rewrite the
Einstein equation as a system of nested ordinary differential equations, which
they solved using functional methods (Chebychev functions). A nice review of
early developments can be found in Chesler and van der Schee (2015). In recent
years activity has somewhat abated although there is still continuous
progress, see e.g. Waeber and Yaffe (2022). Already early on these simulations
showed that AdS dynamics leads to rapid hydrodynamization van der Schee  _et
al._ (2013); Chesler  _et al._ (2015) with sizeable transverse flow Chesler
and Yaffe (2015).

The prevailing strategy has been to merge these holographic simulations of the
early stages of heavy ion collisions with statistical descriptions of the
later stages of the collisions van der Schee  _et al._ (2013). Early
calculations focused on highly symmetric settings in order to reduce the
computational demands, but more recently asymmetric settings have been
studied, and by now semi-realistic heavy ion collisions can be simulated
Heller  _et al._ (2012b); Chesler  _et al._ (2015); Ecker  _et al._ (2016);
Casalderrey-Solana  _et al._ (2016); Endrodi  _et al._ (2018); Waeber  _et
al._ (2019); Müller  _et al._ (2020). In particular, these studies have shown
that holographic simulations reproduce relativistic viscous hydrodynamics,
validating the hybrid holographic-transport approach, see Schenke (2021) for a
recent review. In the present context it is relevant that hydrodynamization
occurs at a fixed proper time Waeber  _et al._ (2019) and that AdS dynamics
and viscous hydrodynamics are indistinguishable for hydrodynamic observables
close to hydrodynamization such that the whole QGP fireball shows hydrodynamic
behavior at nearly the same proper time.

Holographic calculations are attractive because they allow to treat problems
that are too difficult to solve using QFT techniques. The hope is that the
successful application of holographic techniques, which has been realized for
the early stages of heavy ion collisions, can be extended to the later
collision stages. This hope is nurtured by several observations documented in
the literature, which we will discuss now.

The insight that quantum entanglement in the field theory is holographically
encoded in features of the higher dimensional geometry was most clearly
expressed by Maldacena and Susskind Maldacena and Susskind (2013) who argued,
building on ideas formulated in Israel (1976); Maldacena (2003), that the
maximally extended (eternal) AdS-BH geometry is dual to a thermofield double
state in the quantum field theory, which means that the quantum states of the
field theory in the two asymptotic regions are maximally entangled. More
generally, in this picture, the entanglement between the quantum states in two
different space-time regions is geometrically represented by an Einstein-Rosen
(ER) bridge between two regions in AdS space. We refer the reader to Almheiri
_et al._ (2020b) for a review of the many publications building on this
paradigm and as well as its relation to recent progress in understanding the
role of entanglement for BH decay by Hawking radiation.

In Maldacena and Susskind (2013) the label “ER = EPR” was coined for this
connection, where EPR stands for Einstein-Podolsky-Rosen and, more generally,
for any form of entanglement in the quantum field theory. For our discussion
Fig. 13 in Maldacena and Susskind (2013) is most stimulating as it suggests
that the emission of entangled Hawking radiation can be described as creation
of higher dimensional Einstein-Rosen bridges between the decaying BH and the
emitted particles. This picture, however, also displays an obvious problem of
this idea: In a high-energy heavy ion collison thousands of hadrons are
produced allowing for a plethora of entanglement patterns, far more than the
number of simple geometries available for the dual picture.

In this context the recent ideas advanced by van Raamsdonk et al. Van
Raamsdonk (2017, 2020, 2021); Raamsdonk and Waddell (2021); May and Van
Raamsdonk (2021) are relevant, which posit that one large connected region of
conformal field theory (CFT) and several smaller entangled domains of the same
CFT have a nearly indistinguishable holographic dual. If it were legitimate to
interpret the large region as QGP fireball and the many small domains as
individual hadrons this would imply that the crossover transition from QGP to
hadron gas could be continuous in a holographic dual description. This notion
instills hope that a holographic description of the hadronization process is
feasible and practical.444Our concept inverts van Raamdonk’s reasoning. His
aim was to show that continuous space can emerge from the entanglement of
isolated domains; we want to describe entanglement among isolated components
of the final state geometrically. The dual description would be the
disintegration of one large AdS black hole into several small ones as
discussed, e.g., in Maldacena and Susskind (2013) in the contact of black hole
evaporation.

The analogy is now clear. Quarks and gluons are confined to the interior of
hadrons. Therefore, a diluted gas of thousands of hadrons can be interpreted
in analogy to many isolated conformal field theories (CFT) on the edge. As
each of these covers only a small faction of the fireball or expanding hadron
cloud, in the bulk AdS direction this leads only to noticeable effects very
close to the asymptotic region of AdS space. (In order to distinguish this
asymptotic region from spatial boundaries of localized states, we will use the
term “AdS edge”. Such localized systems are usually modeled in the framework
of boundary conformal field theory (BCFT) and its holographic dual description
Takayanagi (2011).) Deeper inside the bulk, all hadronic regions will remain
connected via a network of ER bridges. As a result, the dual geometry will be
very similar to that of the QGP fireball which is just one large AdS-BH. This
means that from the information theoretical point of view there is no marked
difference between entangled quarks and gluons with large relative momentum in
the early fireball phase, a mixed QGP plus hadron gas state at intermediary
times, and an entangled purely hadronic state at late times and that,
therefore, many properties of this system can be calculated from the dual
description valid at early times. In Sections VII and VIII we will expand on
this from the AdS perspective.

To summarize, we argued that a successful extension of AdS calculations to the
description of hadronization requires their extension from the Poincaré patch
to the global AdS geometry and the inclusion of the scale $L$. We will suggest
in Sections VII and VIII simplified models as first steps towards this long-
term goal.

##  VI Hadronization in Relativistic Heavy Ion Collisions

At very high energies, the valence quarks of two colliding nuclei effectively
pass through each other and deposit some of their energy on a time scale much
shorter than 1 fm/c, a process that can be modeled as a quantum quench.
Following this quench. the deposited energy thermalizes and forms a quark-
gluon plasma on a time scale of the order $\tau_{\rm th}\sim O(1/T)$ where $T$
denotes the temperature after thermalization. The thermalization process can
be studied in holographic models, either by energy shell collapse in
5-dimensional Anti-de Sitter space (AdS5) Balasubramanian  _et al._ (2011a, b)
or by numerically solving shock front collisions in AdS5 Chesler and Yaffe
(2011, 2014).

The quark-gluon plasma then expands hydrodynamically while hadronization
occurs in regions where the temperature reaches $T_{c}\approx 150$ MeV. At
high collision energies the evolution of the matter near the center of
momentum of the two nuclei is approximately boost invariant and can be
described in terms of Milne coordinates $\tau,\eta,x,y$, where
$\tau=\sqrt{t^{2}-z^{2}}$ is the proper time and
$\eta=\frac{1}{2}\ln[(t+z)/(t-z)]$ is called space-time rapidity. Massless
particles moving along lines of constant $\eta$ also have rapidity $\eta$ in
momentum space. Boost invariance means that to a good approximation the
evolution is a function of $\tau$ and the transverse coordinates $x,y$ only
for moderate values $|\eta|\ll\eta_{\rm beam}$, where $\eta_{\rm beam}$ is the
beam rapidity in the center-of-momentum frame.

Hydrodynamics simulations show that the hypersurface where the transition from
quark-gluon plasma (deconfined matter) to hadrons (confined matter) occurs is
composed of two main domains (see Fig. 4). One domain is time-like and located
approximately at constant $\sqrt{x^{2}+y^{2}}\equiv r\approx R$, where $R$ is
the nuclear radius, and stretches from $\tau_{\rm th}$ to a time $\tau_{c}$
when the quark-gluon plasma converts to hadrons in bulk. The bulk
hadronization proceeds along a space-like hypersurface given by $r<R$ and
$\tau=\tau_{c}$, which defines the second domain.

Figure 4: Contour plot of the evolution of the energy density in a midcentral
Au+Au collision at the highest RHIC energy $\sqrt{s_{\rm NN}}=200$ GeV. The
horizontal axis shows the proper time $\tau$, the vertical axis shows one of
the transverse coordinates. The black dashed line delineates the hadronization
hypersurface $T=T_{c}=150$ MeV Gale  _et al._ (2021).

The hadronization processes in these two domains are different: On the time-
like boundary of the quark-gluon plasma, hadron emission can be visualized as
surface radiation. The emission rate is given by

$dE_{\rm had}^{\rm surface}/d\tau=\int dx\,dy\,\tau
d\eta\,\delta(r-R)\,T^{0i}n_{i},$ (6)

where $n_{i}$ is the outward directed (space-like) normal vector on the
hadronization hypersurface. Introducing the notation $S_{c}=T^{0i}n_{i}$ for
the energy flow density at the hadronization temperature $T_{c}$, one finds

$dE_{\rm had}^{\rm surface}/d\eta=2\pi R\int_{\tau_{\rm
th}}^{\tau_{c}}\tau\,d\tau\,S_{c}\approx\pi R\,\tau_{c}^{2}\,S_{c},$ (7)

since $\tau_{c}\gg\tau_{\rm th}$. The energy converted into hadrons on the
space-like hadronization surface is similarly given by

$dE_{\rm had}^{\rm
bulk}/d\eta=2\pi\int_{0}^{R}r\,dr\,\tau_{c}\,\varepsilon_{c}\approx\pi
R^{2}\,\tau_{c}\,\varepsilon_{c},$ (8)

where $\varepsilon_{c}$ is the energy density of matter at temperature
$T_{c}$.

For noninteracting hadrons with mass $m$ at temperature $T=1/\beta$,
neglecting quantum statistics, one has

$\displaystyle S(T,m)$ $\displaystyle=$
$\displaystyle\frac{T^{4}}{4\pi^{2}}\left(3(1+\beta m)+(\beta
m)^{2}\right)e^{-\beta m}$ (9) $\displaystyle\varepsilon(T,m)$
$\displaystyle=$ $\displaystyle\frac{T^{4}}{2\pi^{2}}\left(3(\beta
m)^{2}K_{2}(\beta m)+(\beta m)^{3}K_{1}(\beta m)\right).$

When summed over all well established hadron species in the Particle Data Book
weighted by their statistical degeneracies $d_{i}$, one finds

$\frac{S_{c}}{\varepsilon_{c}}\equiv\frac{\sum_{i}d_{i}S(T_{c},m_{i})}{\sum_{i}d_{i}\varepsilon(T_{c},m_{i})}\approx
0.17.$ (10)

Putting everything together and considering that $\tau_{c}\approx R$, one
obtains

$\frac{dE_{\rm had}^{\rm surface}/d\eta}{dE_{\rm had}^{\rm
bulk}/d\eta}\approx\frac{\tau_{c}\,S_{c}}{R\,\varepsilon_{c}}\approx
0.15-0.20.$ (11)

This means that approximately $80-85$% of the hadrons produced by
hadronization of the quark-gluon plasma are created during the bulk
transition.555As some regions at the fringe of the nuclear fireball may never
become hot enough for deconfinement to occur, a small fraction of the final
state hadrons may be produced directly without going through an intervening
plasma phase. This phenomenon is described in so-called core-corona models
Werner (2007). The relative magnitude of the corona contribution to hadron
production shrinks with increasing size of the collision region and increasing
collision energy Petrovici  _et al._ (2017).

Figure 5: Intensity plots of the emission hypersurface for hadron pairs in
Au+Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV (from Plumberg and Heinz
(2015)). Upper panel: Hadron pairs with $K_{T}=0$, selectively weighting
hadrons from bulk hadronization. Lower panel: Hadron pairs with $K_{T}=2$
GeV/c, selectively weighting hadrons from surface radiation. The emission
intensity is highest in the dark red regions and lowest in the dark blue
regions.

The two hadronization mechanisms can be experimentally separated by measuring
the emission of hadron pairs instead of single hadrons. Hadron pairs radiated
off the surface of the outward flowing fireball carry a large outward directed
total momentum $K_{T}=p_{T,1}+p_{T,2}$. On the other hand, hadron pairs formed
when the QGP hadronizes in bulk have on average $K_{T}=0$. This is nicely seen
in Fig. 5. where the emission hypersurface of hadron pairs at midrapidity for
the same Au+Au collisions at RHIC is shown in the $(x,y,t)$ space, where $x,y$
are the transverse coordinates Plumberg and Heinz (2015). Just as in Fig. 4
the hypersurface resembles a capped cylinder. The upper panel shows the
emission points of hadron pairs with $K_{T}=0$; the lower panel shows the
emission points with $K_{T}=2$ GeV/c. Red color indicates the highest rate of
emission; blue color indicates the lowest emission rate.

##  VII Hadronization as an AdS phase transition

The Einstein equations with a negative cosmological constant have two
solutions with asymptotic AdS geometry. One is plain AdS space; the other is a
black hole (AdS-BH) imbedded in AdS space. The black hole geometry is
parametrized by the Schwarzschild horizon (we will use the notation $r_{h}$).
As we shall discuss, $r_{h}$ is related to the color screening distance in the
dual gauge theory. At thermal equilibrium, $r_{h}$ is uniquely determined by
the temperature $T=1/\beta$ encoded in the period of the Euclidean version of
the geometry Gibbons and Hawking (1977).

The Euclidean action plays the role of free energy in the space of geometries;
its minimum thus corresponds to the stable equilibrium state at a given
temperature. At low values of $T$, plain AdS space ($r_{h}=0$) has the lowest
Euclidean action; above a certain temperature $T_{c}$, which depends on the
details of the dual gravity theory, the AdS-BH geometry has the lowest free
energy. As mentioned in a footnote in Section IV, we here refer to such
transitions as “Hawking-Page type” transitions Hawking and Page (1983). In the
simplest holographic model, the pure $(d+1)$-dimensional Einstein action with
negative cosmological constant $\Lambda=-d/L^{2}$ on global AdS space, the
transition is discontinuous with $r_{h}(T_{c})=L$, which means that the
Hawking-Page transition is a first-order phase transition. For a detailed
derivation and the exact relation between $r_{h}$ and $T$, see e. g. Ref.
Witten (1998).

In the dual gauge theory, the Hawking-Page transition corresponds to the
confinement-deconfinement transition Witten (1998); Aharony  _et al._ (2004).
This can be seen in multiple ways. The most commonly used argument is that the
free energy of the thermal plain AdS geometry is described by a Hagedorn
spectrum of gauge-singlet excitations, whereas the free energy of the AdS-BH
geometry is proportional to $N_{c}^{2}$. Maybe the most intuitive argument is
obtained by considering the potential of a heavy quark-antiquark pair, i. e.,
static objects carrying color charge in the fundamental representation of the
gauge group Maldacena (1998b). In the gauge theory, this potential is
determined by a Wilson loop connecting the world lines of the quark-antiquark
pair. In the holographic dual representation the potential is determined by
the Nambu-Goto action of a string connecting the quark-antiquark pair through
the bulk (see left panel of Fig. 6).

Figure 6: String configurations in the AdS bulk connecting a static quark-
antiquark pair separated by distance $\ell$. The left panel shows the string
in the plain AdS geometry; the right panel shows the two disconnected strings
reaching from the quark (antiquark) to the black hole horizon $r_{h}$ in the
AdS-BH geometry.

For the conformally invariant ${\cal N}=4$, large-$N_{c}$ super-Yang-Mills
theory and for the plain AdS geometry in the bulk, the potential is found to
be Maldacena (1998b)

$U(\ell)=-\frac{4\pi^{2}\sqrt{2\lambda}}{\Gamma(1/4)^{4}\,\ell},$ (12)

where $\lambda=g^{2}N_{c}$ is the ’t Hooft coupling and $\ell$ is the quark-
antiquark separation. (There is no confining potential because the theory is
conformally invariant.) For the AdS-BH geometry one finds that the potential
$U(\ell)$ vanishes for separations

$\ell>\ell_{s}\approx 0.869L^{2}/r_{h},$ (13)

which defines the screening distance of the color force in the gauge theory
Rey  _et al._ (1998); Brandhuber  _et al._ (1998); Liu  _et al._ (2007). The
screening occurs, because for $\ell>\ell_{s}$ the lowest energy string
configuration corresponds to a disconnected pair of strings stretching
straight from the quark (antiquark) to the black hole horizon $r_{h}$ (shown
in the right panel of Fig. 6). We can thus consider $r_{h}$ as a geometric
parameter related to the color screening length in the gauge theory. Since for
$T\gg T_{c}$ the black hole radius is related to the temperature $T$ at
thermal equilibrium by $r_{h}=(4\pi/n)L^{2}T$, this translates into a thermal
color screening length $\ell_{s}\approx 0.869/(\pi T)$in $d=4$ space-time
dimensions.

It is possible to relax the firm connection between AdS-BH radius $r_{h}$ and
the temperature $T$ by allowing for geometries with a conical singularity Eune
_et al._ (2013). Such geometries do not correspond to a minimum of the free
energy but they allow for a smooth interpolation between the equilibrium AdS-
BH geometry and the thermal AdS geometry. This is illustrated in Fig. 7 for
AdS5. The solid lines show the free energy $F(r_{h},T)$ for several different
values of $T$, and the dashed line traces the location of the minima. The
uppermost solid (red) curve is for the lowest temperature $T_{\rm min}$ for
which an AdS-BH solution of Einstein’s equations exists, $T_{\rm
min}=\sqrt{2}/(\pi L)$, the middle solid (blue) curve represents the free
energy at the temperature of the Hawking-Page phase transition, $T_{c}=3/(2\pi
L)$, and the lowest solid (black) curve shows the free energy for
$T=1.2T_{c}$.

Figure 7: The solid curves show the free energy $F(r_{h},T)$ for AdS5 as
function of the black hole radius $r_{h}$ for three different temperatures
$T$: $T=T_{\rm min}$ (red upper curve), $T=T_{c}$ (blue middle curve), and
$T=1.2T_{c}$ (black lower curve). See text for the definition of $T_{\rm min}$
and $T_{c}$. The dashed line shows the equilibrium (on-shell) free energy
$F(r_{h},T_{0}(r_{h}))$ which traces out the extrema in the family of free
energy curves. The free energy is shown in units of $L/G$ where $G$ is the
gravitational constant.

As is evident from Fig. 7, the Hawking-Page transition for the conformal,
large-$N_{c}$ super-Yang-Mills theory is a first-order phase transition. This
is mirrored in the behavior of the pure non-supersymmetric SU($N_{c}$) gauge
theory, which also exhibits a first-order deconfinement transition for
$N_{c}\geq 3$ Panero (2009). Gravity dual models that resemble pure
SU($N_{c}$) gauge theory more quantitatively can be constructed by adding a
dilaton field to 5-dimensional Einstein gravity, which describes the running
of the gauge coupling constant. In general, dilaton gravity duals of confining
gauge theories exhibit a first-order deconfinement phase transition similar to
that of the conformal super-Yang-Mills theory Gürsoy  _et al._ (2009a).

On the other hand, the deconfinement transition in QCD is known to be a smooth
crossover Bazavov  _et al._ (2019); Borsanyi  _et al._ (2020). The thermal
properties of dual dilaton gravity models including fundamental matter
(quarks) that mimic the running coupling and chiral properties of QCD have
been studied Mandal and Morita (2011), and models that can change from a
first-order phase transition to a crossover transition have been constructed
Attems  _et al._ (2018).

The dynamics of the bulk transition depends on the rate at which the
temperature drops. If the cooling rate is slow compared with the microscopic
times scales, the transition will proceed at or near $T_{c}$ through a mixed
phase via bubble formation Aharony  _et al._ (2006); Bigazzi  _et al._ (2020);
Janik  _et al._ (2021). In the case of rapid cooling, the transition occurs
via a Gregory-Laflamme instability Gregory and Laflamme (1993) from a
supercooled phase at $T_{\rm min}$ Hubeny and Rangamani (2002); Mandal and
Morita (2011); Buchel and Lehner (2015); Dias  _et al._ (2016); Yaffe (2018).
The second scenario prevails in the limit $N_{c}\to\infty$, but for
holographic models of QCD with $N_{c}=3$ it is likely that the first scenario
is realized under the conditions of a heavy ion collision. For a smooth
crossover certainly the first scenario is realized.

##  VIII Hadron emission as analogue of Hawking radiation

Decades of work by many theorists were needed to reach the present level of
understanding of the black hole information puzzle. Understanding the
mechanisms ruling the generation of entanglement in hadronization is most
probably a problem of comparable difficulty. Therefore, it is tempting to
profit from the insights of the black hole community by constructing a model
which treats hadronization in analogy to Hawking radiation. In this analogy
the hadrons correspond to the Hawking radiation outside of the black hole
horizon as hadrons cannot exist within the QGP, so its surface presents a
horizon for them. Photon pair creation at the black hole horizon corresponds
to hadronic particle-hole production at the QGP surface where the ingoing hole
state gets absorbed but transfers its entanglement with the outgoing hadron to
the interior of the (shrinking) QGP fireball in analogy to island formation in
black hole decay Almheiri  _et al._ (2021), see the sketch in Fig.8.

Figure 8: Sketch of the transition from a highly entangled QGP fireball to a
highly entangled hadron gas based on the fact that the time-reversal
invariance of QCD forbids creation of von Neumann entropy.

The main idea motivating this analogy is that black hole decay leads to a Page
curve for the entropy of Hawking radiation outside of the horizon which is
what we also expect for the hadronic state outside of the QGP. Complete
entanglement among the hadrons is only reached at the end of hadronization.

There exist various phenomenological observations on the QCD side which could
fit into our model. As explained in the Introduction the uniform temperature
of all observed hadron yields, which is well described by the thermal hadron
resonance gas model, is difficult to understand. A possible line of argument
was suggested by us in Ref. Müller and Schäfer (2017) based on the
phenomenologically very successful quark recombination model Fries  _et al._
(2008). In this model quarks and gluons from the fireball coalesce at its
surface to form hadrons. In doing so their energies $E_{i}$ add up and their
probability densities multiply, generating for any Fock-state of the hadron
$h$ with energy $E_{h}$ the common factor Müller  _et al._ (2005)

$\prod_{i}e^{-E_{i}/T_{\rm ch}}=e^{-(\sum_{i}E_{i})/T_{\rm
ch}}=e^{-E_{h}/T_{\rm ch}},$ (14)

where the subscript “ch” indicates that the temperature parameter is derived
from the chemical composition of the hadron gas.

An ad hoc feature of this model, dictated by phenomenology, is that the
temperature of the QGP fireball and that of all produced hadrons has the same
value $T_{\rm ch}$. This feature is not easily understood without
entanglement, because different hadrons scatter differently and thus are not
expected to decouple at the same time from the expanding fireball. This could
be naturally explained if a highly entangled QGP state and a highly entangled
hadronic state are basically indistinguishable as we argued above based on the
ideas of van Raamsdonk Van Raamsdonk (2020).

##  IX A holographic picture of hadronization

As long as information about the state of a time reversal invariant quantum
system is not lost by any kind of measurement and the associated (partial)
collapse of the complete many-body wave function, it must be possible “to run
the movie backwards”. The highly complex initial state $\Phi_{\mathrm{i}}$ of
a relativistic heavy ion collision has nearly zero entropy.666The ground state
of a colliding nucleus is unique, and any interaction of the nucleus with the
accelerator structure is completely negligible on the scale of nuclear
excitations. Finally, although the two nuclei can be Coulomb excited on their
approach to each other, the excitation is coherent and does not change the
fact that the nuclear quantum state is pure. In the parton (quark-gluon)
basis, this state is characterized by a density matrix with two blocks
describing the two nuclei approaching each other. This entangled initial state
evolves by a unitary transformation into another highly entangled final state
$\Phi_{\mathrm{f}}$ characterized by a full density matrix in the parton
basis. Eventually, this final many-parton quantum state is projected onto
hadron states by experimental measurements, which identify the asymptotic
eigenstates of the many-parton system. This happens at a time of order
$O(10^{-9}~{}{\rm s})$, much longer than the duration of the nuclear reaction
which is of order $O(10^{-23}~{}{\rm s})$. Since the detector acts as a heat
bath, this leads to decoherence and thus to entropy production.

For a realistic holographic description of a relativistic heavy ion collision
that includes hadronization it is crucial to describe entanglement at each
time. This is not possible in numerical solutions of the classical Einstein
equations, which depend only on the classical, local energy momentum tensor on
the AdS edge.777We remind the reader that use the term “AdS edge” to avoid
confusion of the asymptotic region of AdS space with the boundary of the QCD
fireball. It is thus unclear whether an approximately valid AdS model that
keeps track of entanglement during a transition of the Hawking-Page type
exists. We argue in this Section that it does.

Our argument rests crucially on the fact that for a many-particle quantum
state, such as the final state of a heavy ion collision, the effects of
entanglement are only relevant if the complete wave function is considered,
but are negligible for observables involving only a few hadrons. This is a
consequence of the ”monogamy of entanglement” which is well established in
quantum information theory.

Lots of text

Figure 9: Illustration of the different stages of a high energy heavy ion
collison. A) After the QCD fireball is formed hadrons are emitted from its
surface in analogy to Hawking radiation. B) This leads to volume growth of the
whole QCD system on the edge. In parallel the fireball cools, e.g. the dual
black hole sinks deeper into the AdS throat. However, the 3-dimensional volume
of the QGP fireball and thus the 3-dimensional volume of the AdS BH remains
roughly constant, see Fig. 4. C) When the temperature of the fireball, which
is identical to that of the AdS black hole, reaches $T_{c}$ the Hawking-Page-
like transition occurs corresponding to complete hadronization of the
remaining fireball on the AdS edge. The picture illustrates the moment of the
transition. D) Due to the monogamy of entanglement any pair of hadrons in the
state after the hadronization transition can only share a very small fraction
of entanglement, on average proportional to $O(1/N_{h})$ with $N_{h}$ being
the number of hadrons. Figure 10: E) As the Hawking-Page-like transition for
QCD at small baryon number density is a cross-over the difference between the
situation just before (at $T=T_{c}+\epsilon$, see Fig. 9C) and after (at
$T=T_{c}-\epsilon$, see Fig. 9D) is small. Therefore, E), which is strictly
valid only above $T_{c}$, is expected to provide a good approximation. Note
that because hadrons on the AdS edge are maximally entangled with states on
the horizon, they are uncorrelated with all other edge hadrons such that all
of them form a thermal ensemble with temperature $T_{c}$. This is just one of
the crucial elements of ETH. Note also, that the difference in this respect to
Fig. 9D is only of order $O(1/N_{h})$ and thus negligible. For more
discussions, please see the main text.

There exists a fundamental constraint, closely related to the no-cloning
theorem Wootters and Zurek (1982), which states that quantum entanglement
cannot be freely shared among many objects Wootters (1998); Coffman  _et al._
(2000); Osborne and Verstraete (2006). This has been analyzed in detail for
systems of qubits for which quantitatively precise statements can be made and
formal proofs are possible. One can define a quantity $\tau(\rho_{AB})$,
called ”tangle”, which quantifies entanglement between the elements of bi-
partitions of multi-particle quantum states $A$, $B$, described by a density
matrix $\rho_{AB}$. $\tau$ can have values between 0 (no entanglement) and 1
(complete entanglement). For a quantum state of $n$ subsystems $A_{1}$,
$A_{2}$, …$A_{n}$ the following constraint holds Osborne and Verstraete
(2006):

$\sum_{k=2}^{n}\tau(\rho_{A_{1},A_{k}})\leq\tau(\rho_{A_{1},(A_{2}A_{3}...A_{n})})\leq
1.$ (15)

We identify the $A_{i}$ in our case with the $N_{h}$ individual hadron states
and assume that a bound of the type (15) exists that limits the average
entanglement between any two hadrons to a value of order $O(1/N_{h})$. We will
use this argument only in a very generic manner, arguing that for, e.g. a
heavy ion collision at LHC, in which thousands of hadrons are produced, the
effects of entanglement are negligible as long as only few-hadron observables
are measured.

As we have argued in Section VI heavy ion phenomenology provides compelling
evidence for the existence of two hadronization mechanisms: the emission of
individual hadrons from the QGP fireball surface ($\sim 20$%) and the
instantaneous hadronization of the remaining fireball at $T_{c}$ ($\sim 80$%).
Both mechanisms are characterized by the same temperature $T_{c}$. We
described specific models in the AdS/CFT context in Section VII and VIII. In
this Section we combine these models to an overall scenario.

On one hand, the presence of two mechanisms with quite different time
dependence complicates our endeavour compared with the usual black hole
information puzzle. On the other hand, we can make use of detailed
phenomenological knowledge based on a huge amount of high-precision data
amassed by heavy ion experiments. We hope that this advantage outweighs the
disadvantages.

There exist many concepts in the literature which could be adapted to our
situation. For example, Maldacena and Susskind suggested in Maldacena and
Susskind (2013) (see Fig. 13) a geometric, holographic interpretation of the
entanglement of Hawking radiation in terms of ER bridges for a Minkowski space
black hole, which is somewhat similar to ours. The difference between their
picture and ours is that we do not study a horizon in the edge but in the bulk
of AdS space and that we treat hadrons as entangled with the AdS horizon dual
to the QGP fireball rather than photons as entangled with a black hole
horizon.

Figures 9 and 10 present conceptual illustrations of our ideas. We will
outline in Section X some steps that could be taken to make these more
quantitative.

As shown in Fig. 4, the transverse size of the QGP fireball stays nearly the
same as function of time, as the internal cooling caused by the hydrodynamic
expansion is nearly compensated by evaporation from the surface. The surface
temperature of the QGP fireball stays fixed at $T_{c}$ during this period
while the volume of the hadron resonance gas (HRG) outside of the QGP fireball
grows continuously. When the QGP fireball temperature reaches $T_{c}$
throughout, the remaining volume hadronizes quickly. In the AdS dual
description, the spatially bounded region on the AdS edge corresponding to the
QGP fireball extends into the AdS bulk until it reaches an also spatially
bounded BH horizon. It appears plausible that this region in AdS space, which
is schematically depicted as a cylinder in Fig. 9 A), should be modeled as the
holographic dual of a BCFT Takayanagi (2011). The emitted hadrons are
entangled with states on the BH horizon via ER bridges, as illustrated in Fig.
13 of Ref. Maldacena and Susskind (2013), such that no entropy is produced.

As discussed in Section I, hadronization in heavy ion collisions is usually
modeled as “chemical freeze out”, which is hypothesized to explain the
observation that all hadron yields agree with the predictions of the thermal
model although hadronic interactions continue in the expanding hadron gas. As
we argue that the success of the thermal model is a consequence of quantum
coherence, e. g. in the form of ETH, we do not need any such additional
mechanism. However, even our mechanism requires a rapid transition from the
hydrodynamic QGP phase to the hadronic phase. In fact, microscopic simulations
indicate a rapid growth of the specific viscosity at temperatures below 150
MeV signalling a rapid breakdown of hydrodynamics when the expanding fireball
cools below $T_{c}$ Yang and Fries (2022).

Holographically, this transition corresponds to the transition between Figs. 9
B and D, where Fig.9 C is a schematic representation of an intermediate stage.
We do not yet have a reliable quantitative description of these intermediate
stages, neither in QCD nor in holography, but one future goal stated in
Section X is to develop such a description in the holographic model of the
transition.

With respect to phenomenology, however, we can argue that the mapping to a
purely hadronic late stage should be a smooth one, i. e. a cross-over rather
than a first order phase transition, in line with the fact that the
deconfinement/confinement phase transition of QCD is a cross-over.

We conclude this Section with the following remark: If a holographic dual of
QCD exists, it must be possible to develop a quantitative description of the
hadronization of a quark-gluon plasma. At present we do not know whether this
is the case. However, even if if it does not or turns out to be unachievable
in practice, much could be learned about the quantum physics of the
confinement transition by investigation of a holographic model that
incorporates salient features of the quark-hadron transition in QCD. In the
next Section we will discuss some steps that can be taken in this direction.

##  X (Some) Open Questions

As already emphasized our reasoning is speculative and must be consolidated or
refuted by detailed investigations. Because the problem of decoherence and
thermalization of many-particle quantum states is a very generic one, there
exists a large and rapidly expanding literature on it, where the ETH plays a
prominent role. This leads to our first set of open questions:

  * •

Does QCD exhibit ETH behavior?

We interpret the success of the thermal model as heuristic indication that it
does show ETH behavior, but it would be desirable to affirm this with rigorous
methods, such that the success of the thermal model becomes a prediction
rather than merely an observation. To make contact with the experimental data,
it is not only necessary to confirm ETH properties in principle, but to also
establish the time scale at which ETH behavior becomes manifest (see Dymarsky
(2019)).

In fact, this question is much discussed in quantum information theory and
answering it could become an early success of quantum simulators (see, e. g.,
Ref. Schuckert and Knap (2020) and references therein). There are two crucial
advantages of quantum computing in this context. First, for the arguments
given in Section III the extensive nature of the microcanonical entropy leads
to a strong exponential suppression of all off-diagonal matrix elements except
for very small systems, which are easiest to simulate. Second, the simulation
of realistic hadrons is not required to test the validity of ETH as long as
the Hamiltonian is well enough modeled. Also, showing ETH behavior for the
gauge group SU(2), which is easier to simulate, should be sufficient.

Another promising approach is lattice gauge theory. Just as RMT behavior was
established for the QCD Dirac operator by simulations on small lattices with
classical computers Berbenni-Bitsch  _et al._ (1998); Verbaarschot and Wettig
(2000) the same may be possible for ETH.

  * •

Is ETH necessary or only sufficient to explain success of the thermal model?
If it is necessary the relevant question becomes: How long does it take until
few particle observables look in good approximation thermal?

Recently a number of observations suggest that ETH is less universal than
often assumed and may not even be necessary for a system to thermalize Harrow
and Huang (2022). In Ref. Majidy  _et al._ (2022) it was shown that non-
Abelian charges reduce entropy-production rates and may enhance finite-size
deviations from eigenstate thermalization. The occurrence of many-body scars
may also negate ergodic behavior in certain systems Turner  _et al._ (2018)
and slow down thermalization Michailidis  _et al._ (2020).

In general, one should be aware that ETH postulates very restrictive
constraints for matrix elements of generic operators between energy
eigenstates. It could well be that less restrictive requirements already imply
the emergence of thermal behavior to such an extent that the differences are
unobservable in realistic experiments. For example, it would be interesting to
understand whether monogamy of entanglement in many-particle systems Wootters
and Zurek (1982); Wootters (1998); Coffman  _et al._ (2000); Osborne and
Verstraete (2006) is already sufficient to mimic thermalization as long as
only small subsystems of the complete, entangled quantum states are observed.
Obviously, the search for alternatives to, and generalizations of, the ETH
paradigm is a vast and rapidly developing area of research.

A second set of questions revolves around the search for useful holographic
models of the quark-hadron transition.

  * •

Do recent advances in numerical solutions of gravity in AdS space make it
possible to better understand the dynamics of a Hawking-Page type transition?

Over the years much effort has been invested to better understand the
phenomenological implications of the original Hawking-Page transition without
leading to definitive conclusions Hubeny and Rangamani (2002); Mandal and
Morita (2011); Buchel and Lehner (2015); Dias  _et al._ (2016); Yaffe (2018);
Janik  _et al._ (2021). In the large-$N_{c}$ limit, the HP transition will
proceed by spinodal decomposition of a deeply supercooled phase unless the
cooling process is very slow. One interesting question is whether the system
re-equilibrates after the transition is complete Yaffe (2018); another is
whether quantum gravity corrections at finite ’t Hooft coupling Gubser  _et
al._ (1998) help to smoothen out the HP transition. In a very slowly cooling
scenario, the transition will proceed via bubble nucleation. It is an
interesting question which of these scenarios is applicable to pure
SUN($N_{c}$) gauge theories at moderate $N_{c}$ where the equilibrium
deconfinement transition is known to be of first order.

Progress in the numerical treatment of matrix models Hanada and Watanabe
(2022); Pateloudis  _et al._ (2022); Buividovich (2022); Buividovich  _et al._
(2019) as well as AdS/CFT initial value problems Waeber and Yaffe (2022) may
make it possible to explore the real-time dynamics of Hawking-Page
transitions. However, none of these efforts addresses the question how the
entanglement structure of the quasi-thermal QGP is reflected in the
entanglement structure of the produced hadrons. In the holographic dual this
raises the challenge of computing the geometric form of an ER bridge between
the asymptotic (“edge”) regions of a holographic thermofield double state for
general initial conditions.

  * •

Would holographic models in which the transition is a smooth crossover as in
QCD allow for an approximate identification of the HP transition with the
hadronization transition of a quark-gluon plasma?

The possible application of the HP transition in nonconformal dilaton-gravity
models Gürsoy  _et al._ (2009a, b); Megias  _et al._ (2011) as a holographic
model of the dynamical deconfinement transition in QCD was explored in Ref.
Gürsoy  _et al._ (2013). More recently, the dynamical nature of the transition
including domain formation Attems  _et al._ (2020) and domain wall propagation
Ecker  _et al._ (2021) have been investigated. Even shock wave collisions were
studied in a holographic model that parametrizes the evolution of the quark-
hadron transition from a smooth crossover to a first-order phase transition
with increasing baryon chemical potential Attems (2021).

  * •

Can hadron emission from the surface of a quark-gluon plasma be modeled as
splitting quenches?

There exist low-dimensional toy models for the splitting of one domain of
conformal field theory into two Shimaji  _et al._ (2019); Caputa  _et al._
(2019) or the formation of an Einstein-Rosen (ER) bridge between two horizons
Anderson  _et al._ (2020). Can these models be generalized to multiple black
hole splittings, correponding to the production of multiple hadrons and to
higher dimensions? Is AdS/BCFT the adequate framework to describe
hadronization? If it is, does it imply, in the spirit of Ref. Van Raamsdonk
(2020), that the holographic description of the quark-gluon plasma and an
interacting hadron gas are quite similar? In fact this is exactly what is
observed on the field theoretical side of the duality. The properties of a
hadron resonance gas and the QGP are indistinguishable close to hadronization
(see, e. g., Bellwied  _et al._ (2021).

Finally, one would like to know whether there are experimental signatures of
the entanglement structure among hadrons emitted in the decay of quark-gluon
plasma.

  * •

How can the validity of the analogy between hadronization and black hole decay
proposed in Section VII be tested?

As discussed in Section IX the monogamy property of entanglement Osborne and
Verstraete (2006) implies that the states of any pair of hadrons among the $N$
emitted hadrons is only entangled at the $O(1/N)$ level. If one would know
what the most accessible entanglement signature is, one could test this
conclusion as a function of the QGP fireball size. One possibility is to look
for quantum correlations among the spins of hadrons Gong  _et al._ (2022).

##  XI Conclusions

The application of concepts from the AdS/CFT duality to describe the early
stage, spanning about $1-2$ fm/$c$, of high energy heavy ion collisions as
they are studied experimentally at LHC and RHIC has been remarkably successful
and brought many important insights. This suggests that one should try to
extend the dual holographic description to later times including the
hadronization stage. As we explained, any such effort has to face the full
complexity of the black hole information problem in the dual desription.

The time evolution of both the AdS black hole and QCD is unitary. No
information gets lost and no entropy is produced. Consequently, thermal
ensemble in the strict sense cannot be produced. However, what can be produced
is a system for which all realistic measurements produce results that are
indistinguishable from strictly thermal ones. Recent progress on black hole
evaporation has substantiated this conclusion. For QCD the corresponding field
theoretical calculation, which would have to treat quantum entanglement during
each step of the collision appears impractical. If, however, the AdS- and QCD-
based descriptions of a heavy ion collision were holographically dual up to
calculable corrections, a QCD-based calculation would be unnecessary, and one
could simply adapt the insights from black holes to heavy ion collisions.

Here we focused on the question whether and how the AdS dictionary could be
extended to times beyond the hadronization of the quark-gluon plasma and how
the phenomenological properties of heavy ion collisions up to and beyond the
Page time could be represented holographically. In doing so, we have not
reached a conclusive answer, but we also did not encounter obvious roadblocks,
and we formulated topics for future work that could help answer some of the
many remaining questions.

Acknowledgments: We thank M. Kaminski, L. Yaffe, and S. Waeber for many
valuable discussions and comments on a draft of this manuscript. We also thank
T. Hartman for insightful discussions on Euclidean wormholes. B. M.
acknowledges support from the U.S. Department of Energy Office of Science
(Grant DE-FG02-05ER41367) and from Yale University during a sabbatical stay in
Spring 2022.

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