11institutetext:  University of JyvÃskylÃ, Department of Physics, P.O. Box 35,
40014 University of JyvÃskylÃ, Finland 22institutetext: Helsinki Institute of
Physics, P.O. Box 64, 00014 University of Helsinki, Finland

# Collective Dynamics - theoretical overview

Yuuka Kanakubo 1122<EMAIL_ADDRESS>
###### Abstract

I overview the recent progress of phenomenological studies exploring
collective dynamics in relativistic nuclear collisions to understand various
QCD properties. Originally, collectivity was interpreted as a manifestation of
the hydrodynamic behaviour of the QGP as a response to the initial collision
geometry. Over the past decade, however, particularly following the
experimental observation of collectivity in small colliding systems,
pioneering studies have demonstrated the possibility of other interpretations.
In this talk, I highlight recent studies aimed at understanding various QCD
properties at different collision stages through the lens of collectivity and
emphasize the importance of establishing Monte Carlo event generators for
relativistic nuclear collisions.

##  1 Introduction

Collectivity, a momentum azimuthal anisotropy of final state hadrons, has been
interpreted as a hydrodynamic behaviour of the QGP in relativistic heavy-ion
(A–A) collisions Ollitrault:1992bk . Geometrical anisotropy in the initial
profile of quark-gluon plasma (QGP) gives an anisotropic pressure gradient,
which generates flow velocity as a consequence of hydrodynamic response
Molnar:2001ux ; Kolb:2003dz . The hydrodynamic response is controlled by the
viscous coefficients of QGP and the equation of state. Hence, the comparison
of momentum anisotropy between experimental results and hydrodynamic model
calculations is anticipated to provide insights into the properties of QGP.
This assumes that collectivity arises solely from the hydrodynamic expansion
against the initial geometrical anisotropy of QGP originating from the
collision geometry. However, the above interpretation of the emergence of
collectivity has undergone a significant change particularly following the
experimental observation of collectivity in proton–proton (p–p) collisions or
proton–ion (p–A) collisions. These collisions were previously regarded as
systems too small to produce QGP DobrigkeitChinellato:2023qm  and do not have
such a clear collision geometry compared to A–A collisions.

##  2 Origins of collectivity

A naive question to pose here is whether the mechanism giving rise to
collectivity in small systems is the same as in heavy-ion collisions or not.
This opened up an opportunity to excavate other possible origins of
collectivity in small systems Dusling:2015gta ; Nagle:2018nvi ;
Schenke:2021mxx . For instance, hot spots of a gluonic profile of a proton as
a seed of geometrical anisotropy in the initial state Schenke:2021mxx ,
collectivity arising from pre-equilibrium evolution Bozek:2022cjj , or
collectivity from string interactions in the initial state and hadronic
rescattering in the final state Bierlich:2021poz  etc. Also, it should be
noted that not only hydrodynamic models Schenke:2020mbo ; Zhao:2020wcd ;
Zhao:2022ugy  but also transport models Zhao:2021bef ; Oliva:2022rsv
reasonably describe the collectivity observed in experimental data. Given that
various physical mechanisms can potentially explain the observed collectivity
in experiments, qualitative and quantitative discrimination of contributions
from those physical mechanisms is essential for a comprehensive understanding
of dynamics in each collision stage.

An illustrative example of the discussion on the origin of collectivity can be
seen in the recent ATLAS experimental data in ultra-peripheral collisions
(UPC) ATLAS:2021jhn . The finite azimuthal anisotropy coefficient, $v_{n}$,
was reported as a function of transverse momentum $p_{T}$ in high-multiplicity
UPC events. There are two theoretical calculations which try to explain these
experimental data, a calculation with color glass condensate (CGC) Shi:2020djm
and hydrodynamics Zhao:2022ayk . Both show a reasonable agreement of $v_{2}$
with the data. In the former/latter calculation, the collectivity arises as a
consequence of the initial/final state effect Bozek:2016yoj ;
Giacalone:2020byk ; Lim:2021auv . Whether the collectivity in small systems
comes from the initial or final state effect is a long-standing problem and
still an open question.

The recent study with the (3+1)D IP-Glasma framework Schenke:2022mjv  tries to
tackle this question. In this framework, the longitudinal structure of
incoming nuclear gluon distributions is obtained by the JIMWLK equation. The
results demonstrate that the initial momentum anisotropy has a short-range
correlation in rapidity while the geometrical anisotropy has a long-range
correlation. This implies that the geometrical anisotropy can be a dominant
contribution to the collectivity considering that the collectivity in small-
system experimental data has a long-range correlation.

##  3 Nuclear structure

The nuclear structure holds information on the spatial correlation of nucleons
inside of a nucleus. Recent studies show that the nuclear structure can be
accessible also through the flow observables in relativistic heavy-ion
collisions Giacalone:2023cet . For instance, the difference of $v_{2}$ and
$v_{3}$ in Ru96+Ru96 and Zr96+Zr96 collisions can be explained by the
difference of quadrupole $\beta_{2}$ and octupole $\beta_{3}$ deformation in
Ru96 and Zr96 Jia:2021tzt ; Zhang:2021kxj . The deformation parameters such as
$\beta_{2}$ and $\beta_{3}$ are encoded in the generalized Woods-Saxon
distributions. The information of the deformation is reflected in the initial
geometry of QGP, which affects $v_{2}$ and $v_{3}$ in the final state.

A recent study emphasizes the potential necessity of using Woods-Saxon
parameters directly obtained from a rigorous calculation based on many-body
quantum mechanics for collision systems with deformed nuclei Ryssens:2023fkv .
One can extract the deformation parameters by fitting the generalized Woods-
Saxon distribution to the three-dimensional density profile from Skyrme
density functional calculations. The calculation with hydrodynamics using the
best-fit parameters shows a better agreement with experimental data for the
$v_{2}\\{2\\}$ ratio between Au+Au and U+U collisions compared to the result
with the previously obtained parameters.

There are still a lot of remaining tasks to be addressed in this study: the
(in)consistency of the nuclear structure between different energy scales
Mantysaari:2023prg ; Mantysaari:2023qsq ; Singh:2023qm , computational cost of
initial profiles with deformed nuclei Luzum:2023gwy , or non-equilibrium
effects in the flow analysis Kanakubo:2022ual  etc. However, it can be clearly
seen that the studies on nuclear structure have paved the way to investigate a
connection between high and low-energy nuclear physics.

##  4 Developments on statistical analysis

One of the primary goals of constructing dynamical models is to extract
physical parameters through comparisons between the model and data. Recently,
Bayesian parameter estimation has become the mainstream approach for this
purpose, propelled by pioneering works in the field Pratt:2015zsa ;
Bernhard:2016tnd . Although originally the transport coefficients of QGP were
the main target to study in the Bayesian analysis, the possibility of
investigating other physical phenomena has been explored too. For instance,
one can study the effect of neutron skins Giacalone:2023cet  or deuteron
production mechanisms JETSCAPE:2022cob . Also, the Bayesian parameter
estimation can be extended to a viscous anisotropic hydrodynamic framework
Liyanage:2023nds ; Heinz:2023qm , where $\eta/s(T)$ and $\zeta/s(T)$ can be
constrained up to higher temperatures compared to a framework with standard
viscous hydrodynamics.

However, despite the growing popularity of Bayesian parameter estimation, it
is crucial to acknowledge that there are always inherent uncertainties in a
theoretical model. Accurately estimating these uncertainties can be
challenging. In the previous results with Bayesian analysis, there has been a
common issue: the extracted nucleon width was $w\sim 0.8-1.0$ fm, which was
quite large compared to, for example, the proton radius that is generally
expected. This issue is reconciled by including total hadronic nucleus-nucleus
cross-section $\sigma_{AA}$ in the Bayesian analysis Nijs:2022rme  utilizing
the fact that $\sigma_{AA}$ is sensitive to $w$. The results show that the
estimated $w$ becomes below 0.7 fm and the $v_{2}\\{2\\}^{2}$-$\langle
p_{T}\rangle$ correlation with the best-fit parameters shows significant
improvement in the description of experimental data.

The significant computational cost of dynamical models is another problem to
be addressed to achieve precise model-to-data comparisons. A potential
solution to this issue is the application of deep convolutional neural
networks, which provides robust assistance for the model prediction
Hirvonen:2023lqy . The neural network is trained to predict the hydrodynamic
results event by event, $e.g.$ flow coefficients, average transverse momentum
and charged particle multiplicities, from an initial energy density profile.
It is shown that the trained neural network gives reliable predictions of
hydrodynamic results with just a fraction of computational time. Sufficient
statistics are easily attained compared to the calculation with full evolution
of hydrodynamics, which significantly reduces the statistical errors of the
computed flow observables, and of the rarest flow correlators in particular.

##  5 Monte Carlo event generators for relativistic nuclear collisions

As mentioned above, model-to-data comparisons in relativistic nuclear
collisions have become sophisticated because of the development of powerful
statistical methods. Nevertheless, a notable question remains: “Are the
results from dynamical frameworks comparable enough for direct comparisons
with experimental data?” There are two missing points in conventional hybrid
models. First, the description is limited to particle production in low
$p_{T}$ regions while it is hard to disentangle the soft and hard production
when there is an interplay between them Pablos:2022piv . Secondly, there is a
lack of serious consideration for energy-momentum conservation on an event-by-
event basis, particularly in obtaining the initial state and during the
conversion of fluids to particles at the end of the hydrodynamic simulation.
Both have been considered essential features for general-purpose Monte-Carlo
event generators in high-energy physics Buckley:2011ms .

Figure 1: (Left) Fraction of hadrons produced from the QGP fluids (core) at
midrapidity as a function of charged-particle multiplicity extracted from the
dynamical core–corona initialization framework Kanakubo:2021qcw . Results for
p–p collisions at $\sqrt{s}=7$, $13$ TeV, and Pb–Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV are shown. (Right) Four-particle cumulants
$c_{2}\\{4\\}$ for charged hadrons as a function of the number of charged
hadrons $N_{\mathrm{ch}}$ in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
Kanakubo:2022ual . Results calculated only with hadronic production from the
core component (triangles), the corona components (diamonds), and the
inclusive hadrons (squares) are shown.

The latest version of EPOS, EPOS4 Werner:2023zvo ; Werner:2023fne ;
Werner:2023jps , tries to tackle this problem. The key feature in the EPOS is
the core–corona picture – a two-component picture of equilibrated (core) and
non-equilibrated (corona) matter. By consistently including the particle
production from corona components in hydrodynamics, the applicable range of
the model is extended to high $p_{T}$ regions and small systems where local
equilibrium is not likely attained. Notably, the initial state model with
parallel scattering formalism has been significantly updated in the EPOS4.
Factorization/binary scaling is attained at high $p_{T}$ regions and
saturation regulates particle production at low $p_{T}$ regions, which makes
the model able to describe soft and hard production within a single framework
under the restriction of energy-momentum conservation. Also, it should be
noted that energy-momentum and charge conservation are achieved event-by-event
in the conversion of fluids to particles by implementing a microcanonical
sampling. Taking into account all the aforementioned features, the EPOS4 is an
MC-event generator modelled to cover the physics from lepton–lepton to heavy-
ion collisions with possible QGP formation.

The dynamical core–corona initialization framework (DCCI2) Kanakubo:2021qcw ;
Kanakubo:2022ual  is established by implementing the concept of the
core–corona picture in the dynamical initialization framework. In the DCCI2,
the source term of hydrodynamics is defined so that energy-momentum deposition
of non-equilibrated partons is proportional to their scattering rate evaluated
from the mean free path. Because of the core–corona picture, the framework
describes the particle yield ratios as a function of multiplicity from small
systems to heavy-ion collisions similarly to the EPOS4. Figure 1 (left) shows
the fraction of the number of hadrons produced from the core component,
$R_{\mathrm{core}}$, in the DCCI2. The clear scaling of $R_{\mathrm{core}}$
with multiplicity is seen for p–p collisions at $\sqrt{s}=7$, $13$ TeV, and
Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. The onset of QGP dominance
appears to be at $\langle dN_{\mathrm{ch}}/d\eta_{s}\rangle\sim 10$-$20$,
which is roughly consistent with the result from EPOS4 Werner:2023zvo . It is
also important to notice that the contribution from the corona components is
not negligible in heavy-ion collisions.

In Fig. 1 (right), the effect of the corona component on $c_{2}\\{4\\}$ is
investigated in Pb–Pb collisions. Comparisons of $c_{2}\\{4\\}$ calculated
only with hadronic production from the core component and with the inclusive
hadronic production (core+corona) reveal the non-equilibrium correction on
$c_{2}\\{4\\}$ that is purely obtained from hydrodynamics. The correction is
non-negligible especially in mid-central collisions even if $c_{2}\\{4\\}\sim
0$ in hadronic production from the corona component. This means that one
cannot simply compare the absolute value of $c_{2}\\{4\\}$ obtained from pure
hydrodynamic models with experimental data due to the existence of non-
equilibrium correction even if the contribution of “non-flow” correlation is
subtracted in $c_{2}\\{4\\}$. Therefore, the result urges that the
quantitative model-to-data comparisons of flow observable should be performed
considering both equilibrated and non-equilibrated components, ideally, within
a MC event generator.

There are pioneering works towards the establishment of MC-event generators
for QGP studies, $e.g.$ (3+1)D hydrodynamics with MC-EKRT minijets MCEKRT ,
sampling of thermal hadrons with energy-momentum and charge conservation
Oliinychenko:2019zfk ; Werner:2023jps , the frameworks with core–corona
pictures as discussed above Werner:2023zvo ; Kanakubo:2021qcw  etc.
Ultimately, a full 3D Bayesian parameter estimation with MC-event generators
Auvinen:2017fjw ; Shen:2023qm ; Mankolli:2023qm  would make it possible to
perform a quantitative extraction of physical parameters from rigorous model-
to-data comparisons. It also should be noted that the extension of the
applicability range of the models should be extended to lower collision
energies Du:2022yok ; Cimerman:2023hjw ; Pihan:2023qm ; Shen:2023qm ;
Almaalol:2022pjc  for the investigation of the structure of the full QCD phase
diagram.

##  6 Conclusions

In relativistic nuclear collisions, collectivity can arise from various
physical mechanisms. This implies a substantial opportunity to explore diverse
aspects of QCD physics through the study of collectivity. On the other hand,
this also indicates that one should disentangle the contribution from
different physics and needs to quantitatively identify each contribution in
collectivity. Developing Monte Carlo event generators for relativistic nuclear
collisions would be crucial for such quantitative model-to-data comparisons
with a full 3D Bayesian analysis.

Acknowledgement I would like to thank Giuliano Giacalone, Henry Hirvonen,
Mikko Kuha, Heikki MÃntysaari, Govert Njis, Wouter Ryssens, Wilke van der
Schee, Pragya Singh, and all members of the Centre of Excellence in Quark
Matter in JyvÃskylÃ, for discussions. Our research was funded as a part of the
Center of Excellence in Quark Matter of the Academy of Finland (Project No.
346325), the European Research Council Project No. ERC-2018-ADG835105
YoctoLHC, the Academy of Finland Project No. 330448, and the European Unionâs
Horizon 2020 research and innovation program under grant agreement No. 824093
(STRONG-2020).

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