# The unexpected uses of a bowling pin: exploiting 20Ne isotopes for precision
characterizations of collectivity in small systems

Giuliano Giacalone<EMAIL_ADDRESS>Institut für
Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120
Heidelberg, Germany  Benjamin Bally  ESNT, IRFU, CEA, Université Paris-Saclay,
91191 Gif-sur-Yvette, France  Govert Nijs  Theoretical Physics Department,
CERN, CH-1211 Genève 23, Switzerland  Shihang Shen  Institute for Advanced
Simulation and Institut für Kernphysik, Forschungszentrum Jülich, D-52425
Jülich, Germany
Thomas Duguet  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette,
France  KU Leuven, Instituut voor Kern- en Stralingsfysica, 3001 Leuven,
Belgium  Jean-Paul Ebran  CEA, DAM, DIF, 91297 Arpajon, France  Université
Paris-Saclay, CEA, Laboratoire Matière en Conditions Extrêmes, 91680 Bruyères-
le-Châtel, France  Serdar Elhatisari  Faculty of Natural Sciences and
Engineering, Gaziantep Islam Science and Technology University, Gaziantep
27010, Turkey  Helmholtz-Institut für Strahlen- und Kernphysik and Bethe
Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany
Mikael Frosini  CEA, DES, IRESNE, DER, SPRC, LEPh, 13115 Saint-Paul-lez-
Durance, France  Timo A. Lähde  Institut für Kernphysik, Institute for
Advanced Simulation and Jülich Center for Hadron Physics, Forschungszentrum
Julich, D-52425 Jülich, Germany  Center for Advanced Simulation and Analytics
(CASA), Forschungszentrum Julich, D-52425 Jülich, Germany
Dean Lee  Facility for Rare Isotope Beams and Department of Physics and
Astronomy, Michigan State University, MI 48824, USA  Bing-Nan Lu  Graduate
School of China Academy of Engineering Physics, Beijing 100193, China  Yuan-
Zhuo Ma  Facility for Rare Isotope Beams and Department of Physics and
Astronomy, Michigan State University, MI 48824, USA  Ulf-G. Meißner
Helmholtz-Institut für Strahlen- und Kernphysik and Bethe Center for
Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany  Institute for
Advanced Simulation, Institut für Kernphysik, and Julich Center for Hadron
Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany  Tbilisi State
University, 0186 Tbilisi, Georgia  Jacquelyn Noronha-Hostler  Illinois Center
for Advanced Studies of the Universe, Department of Physics, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Christopher Plumberg  Natural Science Division, Pepperdine University, Malibu,
CA 90263, USA  Tomás R. Rodríguez  Departamento de Estructura de la Materia,
Física Térmica y Electrónica and IPARCOS, Universidad Complutense de Madrid,
E-28040 Madrid, Spain  Robert Roth  Institut für Kernphysik, Technische
Universität Darmstadt, 64289 Darmstadt, Germany  Helmholtz Forschungsakademie
Hessen für FAIR, GSI Helmholtzzentrum, 64289 Darmstadt, Germany  Wilke van der
Schee  Theoretical Physics Department, CERN, CH-1211 Genève 23, Switzerland
Institute for Theoretical Physics, Utrecht University, 3584 CC Utrecht, The
Netherlands  NIKHEF, Amsterdam, The Netherlands  Vittorio Somà  IRFU, CEA,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France

###### Abstract

Whether or not femto-scale droplets of quark-gluon plasma (QGP) are formed in
so-called small systems at high-energy colliders is a pressing question in the
phenomenology of the strong interaction. For proton-proton or proton-nucleus
collisions the answer is inconclusive due to the large theoretical
uncertainties plaguing the description of these processes. While upcoming data
on collisions of 16O nuclei may mitigate these uncertainties in the near
future, here we demonstrate the unique possibilities offered by complementing
16O16O data with collisions of 20Ne ions. We couple both NLEFT and PGCM ab
initio descriptions of the structure of 20Ne and 16O to hydrodynamic
simulations of 16O16O and 20Ne20Ne collisions at high energy. We isolate the
imprints of the bowling-pin shape of 20Ne on the collective flow of hadrons,
which can be used to perform quantitative tests of the hydrodynamic QGP
paradigm. In particular, we predict that the elliptic flow of 20Ne20Ne
collisions is enhanced by as much as 1.170(8)stat.(30)syst. for NLEFT and
1.139(6)stat.(39)syst. for PGCM relative to 16O16O collisions for the 1% most
central events. At the same time, theoretical uncertainties largely cancel
when studying relative variations of observables between two systems. This
demonstrates a method based on experiments with two light-ion species for
precision characterizations of the collective dynamics and its emergence in a
small system.

††preprint: CERN-TH-2024-021

#### Introduction.

A central motivation for the program of ultra-relativistic nuclear collisions
is to access bulk properties of QCD matter that emerge in conditions similar
to those found in the early Universe or in extreme astrophysical objects [1].
A prime example is the quark-gluon plasma (QGP), the hot phase of QCD matter
that behaves like a near-perfect fluid [2]. Hydrodynamic behavior is inferred
from the harmonic spectrum of the azimuthal distributions of final-state
hadrons [3],

$\frac{dN_{\rm ch}}{d\eta\,d^{2}{\bf p}}=\frac{dN_{\rm
ch}}{d\eta\,dp_{T}}\frac{1}{2\pi}\left(1+2\sum_{n=1}^{\infty}v_{n}\cos
n(\phi-\phi_{n})\right),$

where $dN_{\rm ch}/d\eta\,d^{2}{\bf p}$ is the charged hadron distribution
differential in pseudorapidity, $\eta$, and transverse momentum, ${\bf p}$,
with $p_{T}=|{\bf p}|$ and $\phi$ the azimuthal angle. The coefficients
$v_{n}$ quantify the anisotropic flow. In hydrodynamics, $v_{n}$ arise as a
response of the system created in the interaction region to the anisotropy of
its geometry, as dictated by an emergent pressure-gradient force [4], the
hallmark of hydrodynamic behavior. An elliptical deformation of the
interaction region leads to elliptic flow, $v_{2}$, a triangular deformation
to $v_{3}$, and so on [5].

Observations of anisotropic flow in small systems [6, 7], such as proton-
nucleus and proton-proton collisions, have triggered tremendous efforts
investigating whether a QGP description is appropriate even in regimes where
applying hydrodynamics becomes hard to justify [8, 9]. Theoretical studies
have either pushed hydrodynamic simulations to extreme situations [10, 11, 12,
13, 14, 15], analyzed in detail the transition from kinetic theory to
hydrodynamics [16, 17, 18, 19, 20, 21, 22, 23], or studied the emergence of
collectivity via other mechanisms [24]. Small systems pose, thus, a
fundamental challenge rooted in the issue of the thermalization and
hydrodynamization of QCD matter [25, 26, 27, 28, 29].

To advance our knowledge of small systems, one has to isolate in the
experimental data information able to discriminate theoretical approaches. A
breakthrough in this direction would be the identification of a correlation
between the final-state anisotropy in momentum space ($v_{n}$) and the
deformation of the initial-state geometry, supporting an underlying
hydrodynamic-type scenario. This strategy has been pursued at the Relativistic
Heavy Ion Collider (RHIC) in a system-geometry scan comparing $p{}^{197}$Au
and $d{}^{197}$Au collisions at the same beam energy. Defining
$v_{2}\\{2\\}\equiv\sqrt{\langle v_{2}^{2}\rangle}$ as the elliptic flow at a
given multiplicity, both the PHENIX collaboration [30] and the STAR
collaboration [31, 32] observe

$v_{2}\\{2\\}_{d{\rm{}^{197}Au}}>v_{2}\\{2\\}_{p{\rm{}^{197}Au}}.$

This constitutes a plausible signature of the elliptical geometry of the
system formed when a deuterium impinges onto a large gold target. Similarly,
at the Large Hadron Collider (LHC) one observes [33, 34, 35]

$\displaystyle v_{2}\\{2\\}_{{\rm{}^{208}Pb}{\rm{}^{208}Pb}}$
$\displaystyle>v_{2}\\{2\\}_{p{\rm{}^{208}Pb}},$ $\displaystyle\hskip
15.0ptv_{3}\\{2\\}_{{\rm{}^{208}Pb}{\rm{}^{208}Pb}}$ $\displaystyle\approx
v_{3}\\{2\\}_{p{\rm{}^{208}Pb}}.$

The enhancement of elliptic flow in 208Pb208Pb collisions is interpreted as
coming from the intrinsic ellipticity of the overlap area for off-central
collisions, i.e. collisions that are not head-on. These observations hint at
the role played by the collision geometry, but employ proton-nucleus
collisions as a baseline of a system that does not present any intrinsic
shape. This presents two drawbacks. First, proton-nucleus collisions have a
different longitudinal structure than nucleus-nucleus collisions (including
$d{}^{197}$Au collisions [36, 37]), whose geometry is better correlated across
rapidities [38, 39]. Second, the geometry of proton-nucleus collisions largely
depends on the proton structure at low values of the Bjorken $x$ variable,
which is poorly understood [7]. Thus, it would be desirable to isolate
signatures of the geometry of the initial states in the scattering of actual
ions, presenting a well-defined notion of an interaction region.

Upcoming data on collisions of 16O isotopes is expected to mitigate these
issues [40]. Preliminary data from 16O16O collisions were recently presented
at the Quark Matter 2023 conference by the STAR collaboration [41]. At the
CERN LHC, a run of 16O16O collisions is expected to take place in 2025.
Comparing peripheral 208Pb208Pb (or 129Xe129Xe) collisions and central 16O16O
collisions should reveal [42, 43]:

$v_{2}\\{2\\}_{\rm{}^{208}Pb^{208}Pb}>v_{2}\\{2\\}_{\rm{}^{16}O^{16}O}.$

However, comparing highly peripheral 208Pb208Pb collisions with central 16O16O
collisions is suboptimal, as it does not resolve issues related to the
definition of an overlap area and the longitudinal structure.

In this Letter, we demonstrate an alternative more robust approach to isolate
the impact of the initial-state geometry. We study central collisions of light
ions presenting different shapes. Differences in the collective flow between
two collision systems would demonstrate the influence of the nuclear geometry,
a technique akin to that used to infer nuclear deformation effects in isobar
collisions at RHIC [44, 45, 46, 47]. The advantage with light species is that
we benefit from an advanced knowledge of their geometries coming from ab
initio calculations of nuclear structure [48, 49]. The drawback with light-ion
collisions is instead that the anisotropy induced by nuclear shapes is only a
small correction to the anisotropy induced by large density fluctuations
caused by the small numbers of participant nucleons. In other words, with
light ions extreme nuclear shapes are required for their fingerprints to be
detectable in the final state.

Here, we overcome this issue. We exploit the fact that the stable isotope
presenting the most extreme ground-state geometry in the Segrè chart, namely
20Ne, is close in mass to 16O. We argue that having 20Ne20Ne data in
conjunction with 16O16O data leads to the observation of unambiguous imprints
of the initial-state geometry on the collective flow. This in turn enables one
to perform quantitative tests of hydrodynamics in a small system.

#### Nuclear structure inputs.

Modern _ab initio_ approaches to the nuclear many-body problem aim at solving
as exactly as possible Schrödinger’s equation for nuclear Hamiltonians
constructed through chiral effective field theories of low-energy QCD. Such
approaches are routinely used to describe the structure of light- and medium-
mass nuclei [50, 51, 52, 53, 54, 55] and first applications to 208Pb were even
recently reported [56, 55]. In this work, we employ results for the structure
of 16O and 20Ne derived within the the framework of Nuclear Lattice Effective
Field Theory (NLEFT) simulations and the ab initio Projected Generator
Coordinate Method (PGCM).

The NLEFT framework [57, 58, 59] combines the principles of effective field
theory with lattice Monte Carlo methods, and is well suited to probe
clustering and other collective phenomena in the ground states of nuclei [60].
NLEFT simulations implement a Euclidean time evolution coupled with auxiliary-
field Monte Carlo simulations to produce ground-state configurations of
nucleons for each realization of the nuclear wave function. The pin-hole
algorithm [60] enables one to keep track of the positions of the nucleons
during the Euclidean time evolution while preserving the information about
their center-of-mass. The produced nuclear configurations carry, thus, many-
body correlations to all orders as dictated by the ground state of the
Hamiltonian. We employ a minimal pion-less EFT Hamiltonian with a periodic
lattice of eight sites with spacing $a=1.315$ fm [61], which successfully
reproduces measured binding energies and charge radii for the isotopes under
study. For 16O, the pinhole configurations are taken from Ref. [62], while a
new set is calculated for 20Ne. Due to the larger mass number, these
configurations contain a larger fraction of nuclei with a non-unique center-
of-mass due to the periodicity, as well as a higher number of negative-weight
states [60, 58] than the 16O ones. These issues are addressed in the
evaluation of our uncertainties for the subsequent hydrodynamic study (see the
Supplemental Material (SM)). Lastly, we distribute nucleons at each lattice
site uniformly between $-a/2$ and $a/2$ while maintaining a minimum inter-
nucleon distance, $d_{\rm min}$, to mimic the effect of short-range repulsion.

Figure 1: Point-nucleon densities of 16O and 20Ne obtained from particle-
number-projected Hartree-Fock-Bogoliubov states with deformations constrained
to the predictions of the ab initio PGCM framework. The background plots show
slices of the densities through the origin. The black dots and lines show the
centers and boundaries of the regions used in the clustered sampling method
(see text and SM for details).

Figure 2: The deformed shape of 20Ne impacts the hydrodynamic flow of its
collisions as compared to 16O16O collisions. Here we show results for charged
particle multiplicity $dN_{\text{ch}}/d\eta$ (top left), mean transverse
momentum $\langle p_{T}\rangle$ (top middle), relative fluctuations of
transverse momentum $\delta p_{T}/\langle p_{T}\rangle$ (top right), elliptic
flow $v_{2}\\{2,|\Delta\eta|>1\\}$ (bottom left), triangular flow
$v_{3}\\{2,|\Delta\eta|>1\\}$ (bottom middle) and the Pearson correlation
coefficient $\rho(v_{2}\\{2\\}^{2},\langle p_{T}\rangle)$ (bottom right). In
each panel, we show the 16O16O and 20Ne20Ne results, as well as their ratio,
using both PGCM and NLEFT as nuclear structure inputs. For
$\rho(v_{2}\\{2\\}^{2},\langle p_{T}\rangle)$ a difference is taken instead of
a ratio in the lower panel. We show statistical uncertainties (error bars),
the total systematic uncertainty (solid bands) as well as its components being
_Trajectum_ (hatched) and nuclear structure (dotted).

The _ab initio_ PGCM formalism [63, 64, 65, 66, 67, 68] is also adapted to
describe collective correlations, e.g. quadrupolar and octupolar deformations
that appear in doubly-open-shell systems such as 20Ne. In particular, it was
shown in Ref. [67] that this method captures experimental data on the ground-
state rotational band and the charge density of this nucleus, employing a
recent N3LO chiral EFT Hamiltonian [69] which we also use here. We first
perform PGCM calculations exploring simultaneously the triaxial quadrupole
($\beta^{v}_{20},\beta^{v}_{22}$) and octupole
($\beta^{v}_{30},\beta^{v}_{32}$) degrees of freedom to determine average
intrinsic deformations for the correlated ground states of 16O and 20Ne. The
resulting shape parameters align with the results of empirical frameworks such
as the energy density functional approach [70, 71, 72] or the antisymmetrized
molecular dynamics approach [73]. Then, we compute an intrinsic Hartree-Fock-
Bogoliubov state constrained at those average deformations, and we evaluate
the particle-number projected one-body density of the resulting system. To
quantify the systematic uncertainty on the procedure, the average deformations
of the ground states are computed from pure mean-field states as well as from
particle-number-projected states (more details in the SM). The results in the
latter case are shown in Fig. 1. We note deformed geometries with well-
separated clusters. In 16O they form an irregular tetrahedron with two short
and two long edges of 2.30 and 2.55 fm respectively (see [74] for recent work
employing a regular tetrahedron). For 20Ne we observe a characteristic
bowling-pin-like 16O+$\alpha$.

For the hydrodynamic simulations, the densities in Fig. 1 are randomly
oriented and used to sample either 16 or 20 coordinates of nucleons for each
realization of the nucleus. Unlike the NLEFT simulations, PGCM does not
provide us with correlated samplings of nucleon positions. Sampling nucleons
capturing the ground-state correlations of the N3LO Hamiltonian is therefore
ambiguous. We use two methods as a quantification of this systematic
uncertainty. One samples nucleons independently (as in [75, 76]), whereas the
second divides up space into four or five regions (see Fig. 1) and samples
exactly two protons and two neutrons from each (see also SM). Lastly,
configurations are rejected if nucleons are closer than $d_{\rm min}$.

#### Hydrodynamic simulations.

We perform event-by-event hydrodynamic simulations of 20Ne20Ne and 16O16O
collisions by means of the Trajectum framework [77, 78, 43, 79]. The
calculations start with configurations of nucleons in the colliding nuclei,
taken from either the PGCM or the NLEFT results.111For all profiles we provide
20k configurations as part of the submission. Each collision is then assigned
to an impact parameter, participant nucleons are selected, and energy density
is deposited in the transverse plane. Following a brief pre-equilibrium phase,
the system is evolved as a relativistic viscous fluid. Hydrodynamic cooling
lasts until the local temperature reaches a critical value ($T\sim 154$ MeV),
below which hadronization occurs. Subsequent strong decays and rescattering of
hadrons are computed by the SMASH code [80, 81, 82], leading to the particle
distributions in the final state. These are analyzed to construct multi-
particle correlations following the experimental protocols. We define the
collision centrality from the multiplicity of charged particles with
$p_{T}\geq 0.4\,\text{GeV}$ and $|\eta|\leq 2.4$, with 0% centrality
corresponding to the limit of small impact parameters.

The parameters of the model are chosen probabilistically by sampling from the
posterior distribution inferred in a Bayesian analysis of 208Pb208Pb
collisions, within the same model [83]. We use twenty different samples from
the parameter space to quantify the uncertainty on the results coming from
wide parameter variations. This represents the largest part of the _Trajectum_
systematic uncertainty, which in addition also takes into account effects of
finite grid spacing (as discussed in the SM).

Our results for $p_{T}$-integrated observables that characterize the
collective flow of hadrons are displayed in Fig. 2. Our first remark concerns
the cancellation of uncertainties we observe when a relative variation of
observables, e.g. a ratio, is taken between 16O16O and 20Ne20Ne collisions.
The dominant uncertainty on the absolute magnitude of the results (upper two
plots in each panel) is the systematic one. However, in the relative
variations (lowest plots) the contribution from the systematic error becomes
nearly equal to that from the statistical error. This enables us to make
robust predictions for percent-level variations of observables across the two
systems. The larger uncertainty affecting the PGCM results is due to the
ambiguities of the empirical method used to extract the correlated
distributions of nucleons.

We discuss now those observables that are more strongly impacted by the
bowling-pin shape of 20Ne. The first is the rms elliptic flow, $v_{2}\\{2\\}$,
in the lower-left panel of Fig. 2. We find:

$\frac{v_{2}\\{2\\}_{\text{NeNe}}}{v_{2}\\{2\\}_{\text{OO}}}=\begin{cases}1.170(8)_{\text{stat.}}(30)_{\text{syst.}}^{\textit{Traj.}}(0)_{\text{syst.}}^{\text{str.}}&\text{(NLEFT)},\\\
1.139(6)_{\text{stat.}}(27)_{\text{syst.}}^{\textit{Traj.}}(28)_{\text{syst.}}^{\text{str.}}&\text{(PGCM)},\end{cases}$

in the 0–1% most central events. This is nearly identical for both nuclear
structure inputs, implying that the enhancement of fluctuations in the second
harmonic predicted by the NLEFT simulations for 20Ne20Ne collisions is largely
captured by the (randomly-oriented) bowling pin predicted by the PGCM
calculation. The $v_{2}\\{2\\}$ ratio between 20Ne20Ne and 16O16O collisions
is as large as that expected between peripheral ($\sim$60% off-central)
208Pb208Pb collisions and central 16O16O collisions [42, 43]. However, the
cancellation of uncertainties that we achieve here is only possible because we
consider experiments with two ions close in mass.

Another probe of the bowling-pin shape of 20Ne is the correlation between the
mean squared elliptic flow, $v_{2}\\{2\\}^{2}$, and the mean transverse
momentum, $\langle p_{T}\rangle$. It is quantified via a Pearson coefficient
denoted by $\rho_{2}\equiv\rho(v_{2}\\{2\\}^{2},\langle p_{T}\rangle)$ [84],
which reflects the correlation between the shape and the size of the produced
QGP droplets [85, 42, 86]. Results for $\rho_{2}$ are reported in the lower-
right panel of Fig. 2. The suppression of the observable in central 20Ne20Ne
collisions relative to 16O16O, observed for both nuclear structure inputs, is
a generic signature of the elongated nuclear shape [87, 88, 89, 75, 90]. The
same effect has been reported in 238U238U [91] and 129Xe129Xe [92, 93]
experiments.

The $\rho_{2}$ correlator is strongly sensitive to several hydrodynamic model
parameters, and thus plagued by a large systematic uncertainty which makes
16O16O and 20Ne20Ne results overlap. Neglecting the triaxiality of these
nuclei, and dubbing $\beta_{2}$ the nuclear quadrupole deformation (where
$\beta_{2,\rm^{20}Ne}>\beta_{2,\rm^{16}O}$ from spectroscopic data [94], as
well as from the densities shown in Fig. 1), the $\rho_{2}$ observable roughly
follows at a given centrality: $\rho_{2}=a-b\beta_{2}^{3}$, where $a$ and $b$
are positive coefficients [95, 96, 91]. Model studies suggest that both $a$
and $b$ are nearly independent of the collision system at the same centrality
[95, 75]. As a consequence, we expect the difference $\rho_{2,\rm
Ne+Ne}-\rho_{2,\rm
O+O}\propto\left(\beta_{2,\rm^{16}O}^{3}-\beta_{2,\rm^{20}Ne}^{3}\right)$ to
isolate the imprint of the nuclear deformation. This is confirmed in Fig. 2
(lower-right panel), where the evaluated difference cancels most of the
systematic uncertainties. A comment is in order. In hydrodynamics, the
$\rho_{2}$ of ultra-central 16O16O collisions is about the same as that of
peripheral 208Pb208Pb collisions at the same multiplicities [97, 79].
Therefore, contrary to the enhancement of $v_{2}\\{2\\}$ relative to 16O16O
systems, which occurs in both central 20Ne20Ne and peripheral 208Pb208Pb
collisions, the suppression of $\rho_{2}$ represents a geometry-driven effect
only accessible by colliding 20Ne isotopes.

Four more observables are in Fig. 2, namely the charged multiplicity, $dN_{\rm
ch}/d\eta$, the mean transverse momentum, $\langle p_{T}\rangle$, the
fluctuations thereof, and the triangular flow, $v_{3}\\{2\\}$. Significant
differences appear between PGCM and NLEFT for $dN_{\rm ch}/d\eta$ and $\langle
p_{T}\rangle$ in the ratio plots. These can be understood from the respective
nuclear radii.222Here the charge radii equal the matter radii for NLEFT since
the computation is isospin symmetric. For PGCM the matter radii are about
0.012 fm smaller than the charge radii. The NLEFT charge RMS radii are 2.76
and $3.17\,$fm for 16O and 20Ne respectively (ratio 1.14), whereas clustered
PGCM has 2.87 and $3.09\,$fm with ratio 1.08. For both NLEFT and PGCM we use a
Gaussian nucleon charge distribution of width 0.84 fm [98, 99]. This compares
well with the experimental values 2.6955 and 3.0055 fm (ratio 1.11) [100]. We
note that for PGCM the independent sampling method gives 0.05 and 0.03 fm
smaller radii for 16O and 20Ne respectively. The $d_{\text{min}}$ parameter
has negligible effect when smaller than 0.5 fm, but increases especially the
PGCM radii for larger values. Due to the relatively larger difference in size
comparing 20Ne and 16O, the NLEFT results lead to a smaller $\langle
p_{T}\rangle$ for 20Ne20Ne as compared to the PGCM results due to a reduced
radial expansion. Similarly, the larger size of the PGCM oxygen leads to an
increased 16O16O cross section and consequently per collision a lower
multiplicity, affecting the $dN_{\rm ch}/d\eta$ ratio (see also [83]). For the
fluctuations of $\langle p_{T}\rangle$ the observed mild enhancement in
central 20Ne20Ne collisions is a generic consequence of the more deformed 20Ne
shape, which enhances fluctuations in the overall size of the overlap region
[95, 101].

#### Conclusion & Outlook.

We have showcased the possibility of reducing theoretical systematic
uncertainties in hydrodynamic model calculations of small systems. One needs
experiments with two light-ion species presenting sufficiently different
geometries to perform quantitative tests of the QGP paradigm for shape-induced
modifications of the collective flow. As 16O16O collisions are essentially
already available at colliders, the extreme shape of 20Ne makes this proposal
realizable in practice.

Our predictions are based on the same hydrodynamic picture used in the
description of collisions of heavy nuclei. They do not include additional
ingredients, e.g., out-of-equilibrium corrections due to the expected partial
thermalization of the interaction region [23, 102, 103], the breakdown of the
equations used in our simulations due to causality constraints [104, 105,
106], features such as the escape mechanism [107] that would affect the
elliptic flow in a transport-based approach, or even the impact of momentum
anisotropies originating in the initial states [24]. Therefore, testing our
predictions in experiments will provide unprecedented quantitative insights
into the applicability of a QGP paradigm and the emergence of collective
dynamics in QCD matter.

As an outlook, additional research avenues are opened by the study of high-
energy 20Ne20Ne collisions. They will be the subject of future works.

The elongated 20Ne shape may help reveal hard-probe modifications in a small
system via the study of path-length-dependent effects in the comparison
between 20Ne20Ne and 16O16O collisions. These can be studied experimentally by
triggering on ultra-central events presenting large final-state ellipticities
[108], and estimated theoretically from the analysis of the path lengths
traversed by the hard probes [109].

Second, both 16O and 20Ne can be injected in the SMOG2 system of the LHCb
detector to perform fixed-target experiments at $\sqrt{s_{\rm NN}}\approx
0.07$ TeV in the center-of-mass frame [110]. The LHC could thus deliver
20Ne20Ne and 16O16O collisions at both $\sqrt{s_{\rm NN}}\approx 7$ and 0.07
TeV, as well as fixed-target 20Ne+208Pb and 16O+208Pb collisions. This wealth
of experimental information combined with the possibility of canceling
uncertainties via the study of relative observables would provide a unique
handle on the manifestations of small-$x$ dynamics and nonlinear QCD evolution
and how they impact the collective structure of nuclei [111].

Finally, several $\gamma$-mediated processes in ultra-peripheral nucleus-
nucleus collisions (UPCs) are aimed at imaging the gluonic content of nuclei
at high energy. As showcased in Ref. [112] for the diffractive production of
J/$\psi$, the shape of 20Ne can leave distinct signatures on the cross
sections on top of a $\gamma$+16O background. Ratios of observables in UPCs
would allow one to cancel uncertainties and obtain a more transparent view of
the gluon geometries and their modification at high energy.

#### Acknowledgements.

We acknowledge the participants of the EMMI Rapid Reaction Task Force “Nuclear
physics confronts relativistic collisions of isobars” and of the ESNT workshop
“Deciphering nuclear phenomenology across energy scales” for fruitful
discussions about the applications of 20Ne, which have triggered this project.
G.G. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) (German Research Foundation) – Project-ID 273811115 – SFB 1225
ISOQUANT, and under Germany’s Excellence Strategy EXC2181/1-390900948 (the
Heidelberg STRUCTURES Excellence Cluster). We also acknowledge support from
the U.S. Department of Energy (DE-SC0024586, DE-SC0023658, DE-SC0013365, DE-
SC0023175) and the U.S. National Science Foundation (PHY-2310620). This work
is supported in part by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (ERC AdG EXOTIC, grant
agreement No. 101018170), by DFG and NSFC through funds provided to the Sino-
German CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant
No. 11621131001, DFG Grant No. TRR110). The work of UGM was supported in part
by the CAS President’s International Fellowship Initiative (PIFI) (Grant No.
2018DM0034). R.R. is supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) – Projektnummer 279384907 – SFB 1245. T.R.R. is
supported by the Spanish MCIU (PID2021- 127890NB-I00). The PGCM calculations
were performed using HPC resources from GENCI-TGCC (Contracts No. A0130513012
and A0150513012) and CCRT (TOPAZE supercomputer). The NLEFT code development
and production calculations utilized the following computational resources:
the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for computing
time on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC) and
special GPU time allocated on JURECA-DC; the Oak Ridge Leadership Computing
Facility through the INCITE award “Ab-initio nuclear structure and nuclear
reactions”; and the TUBITAK ULAKBIM High Performance and Grid Computing Center
(TRUBA resources).

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##  I Nuclear structure calculations

###  I.1 NLEFT

Nuclear Lattice Effective Field Theory (NLEFT) is a method to solve the
nuclear $A$-body problem in a finite volume, more precisely on a hypercubic
lattice of volume $L\times L\times L\times L_{t}$, with $L$ the spatial extent
and $L_{t}$ the temporal one. It also employs corresponding lattice spacings
$a$ and $a_{t}$. Here, the lattice spacing $a$ is typical chosen in the range
from 1 to 2 fm, corresponding to a UV cutoff of $p_{\rm max}=\pi/a=314$ to
$628$ MeV. In this work, we use $a=1.3155$ fm.

There are various formulations of the underlying action. Here, we employ the
so-called minimal nuclear model [61], which has been successfully used to
describe the gross properties of light and medium-mass nuclei and the equation
of state of neutron matter to a few percent accuracy. It has also been
successfully applied to studies of nuclear thermodynamics [113], studies of
clustering in hot dilute matter [114] and the puzzling 4He transition form
factor [115]. This minimal nuclear interaction is given by the SU(4)-invariant
leading-order effective field theory based on smeared contact terms,

$H_{{\rm SU(4)}}=H_{\rm
free}+\frac{1}{2!}C_{2}\sum_{\mathbf{n}}\tilde{\rho}(\mathbf{n})^{2}+\frac{1}{3!}C_{3}\sum_{\mathbf{n}}\tilde{\rho}(\mathbf{n})^{3},$

where $\mathbf{n}=(n_{x,}n_{y},n_{z})$ are the lattice coordinates, $H_{\rm
free}$ is the free nucleon Hamiltonian with nucleon mass $m=938.9$ MeV. The
density operator $\tilde{\rho}(\mathbf{n})$ is defined as [60],

$\tilde{\rho}(\mathbf{n})=\sum_{i}\tilde{a}_{i}^{\dagger}(\mathbf{n})\tilde{a}_{i}(\mathbf{n})+s_{L}\sum_{|\mathbf{n}^{\prime}-\mathbf{n}|=1}\sum_{i}\tilde{a}_{i}^{\dagger}(\mathbf{n}^{\prime})\tilde{a}_{i}(\mathbf{n}^{\prime}),$

where $i$ is the joint spin-isospin index, $s_{L}$ is the local smearing
parameter, and the non-locally smeared annihilation and creation operators
with parameter $s_{NL}$ are defined as

$\tilde{a}_{i}(\mathbf{n})=a_{i}(\mathbf{n})+s_{NL}\sum_{|\mathbf{n}^{\prime}-\mathbf{n}|=1}a_{i}(\mathbf{n}^{\prime}).$

The summation over the spin and isospin implies that the interaction is SU(4)
invariant. The parameter $s_{L}$ ($s_{NL}$) controls the strength of the local
(non-local) part of the interaction. The low-energy constants $C_{2}$ and
$C_{3}$ give the overall strength of the two-body and three-body interactions,
respectively. We take values from [61], $C_{2}=-3.41\cdot 10^{-7}\,$MeV-2,
$C_{3}=-1.4\cdot 10^{-14}\,$MeV-5, $s_{NL}=0.5$, and $s_{L}=0.061$.

Figure 3: Point-nucleon densities of 16O (top) and 20Ne (bottom) obtained from
NLEFT. The panels on the left show densities obtained from configurations
aligned as described in the text, whereas the panels on the right show
baseline densities obtained from configurations of which the angular
information for each nucleon was randomized before the alignment of the
nucleus. The background plots show slices of the densities through the origin.

Central to the studies done here is the pinhole algorithm [60], which performs
a Monte Carlo sampling of the $A$-body density of the nucleus in position
space. It allows to pin down the center-of-mass of a given nucleus, which is a
basic ingredient to calculate charge or matter distributions as done here. It
was e.g. successfully used to investigate the emergent geometry and intrinsic
cluster structure of the low-lying states of 12C [116]. We obtain the pinhole
algorithm by including the $A$-nucleon density operator in the projection
amplitudes according to

$\displaystyle Z_{f,i}^{\rm pinhole}$ $\displaystyle\equiv
Z_{f,i}(i_{1},j_{1},\ldots,i_{A},j_{A};\vec{n}_{1},\ldots,\vec{n}_{A};N_{t})$
$\displaystyle=\langle\Psi_{f}|M^{N_{t}/2}\rho_{i_{1},j_{1},\ldots,i_{A},j_{A}}(\vec{n}_{1},\ldots,\vec{n}_{A})\>$
$\displaystyle\qquad\qquad\times M^{N_{t}/2}|\Psi_{i}\rangle,$

which amounts to the insertion of a “screen”, with pinholes located at the
positions $\vec{n}_{1},\ldots,\vec{n}_{A}$ and spin-isospin indices
$i_{1},j_{1},\ldots,i_{A},j_{A}$, at the midpoint of the Euclidean time
evolution. Here, the NLEFT transfer matrix $M$ is applied $N_{t}$ times to the
initial and final states. With this, one is able to pin down the center-of-
mass of a given nuclear state with spin $J$ and parity $P$ and consequently
derive charge and matter distributions. It should also be noted that the
resolution of the pinhole algorithm in coordinate space is $a/A$.

The pinhole positions contain many-body correlations up to all possible
orders. To reduce them to a one-body intrinsic density distribution, a certain
alignment for each configuration is needed, and one should keep in mind that
such a procedure may not be unique. For example, one can align configurations
according to the long- or short-axis of the moment of inertia matrix [117,
116]. Here we try to align the pinhole configurations of 16O and 20Ne
according to the symmetry found by PGCM calculations, i.e., a tetrahedron and
bowling pin shape, respectively. We first identify $\alpha$ clusters by
requiring that the distance among the four nucleons (proton spin up/down, and
neutron spin up/down) is (approximately) the smallest. For 16O, we randomly
choose one cluster $\mathbf{c}_{1}$ and align it to one of the tetrahedron
symmetry axis $\mathbf{a}_{1}$; then we randomly choose another cluster
$\mathbf{c}_{2}$ and symmetry axis $\mathbf{a}_{2}$, and rotate along
$\mathbf{a}_{1}$ so that the inner product of $\mathbf{c}_{2}$ and
$\mathbf{a}_{2}$ is the largest. For 20Ne, we first align the long principal
axis to $z$ direction and require $\sum_{i=1}^{A}(z_{i}-z_{\rm c.m.})^{3}$ to
be positive. Then we choose the cluster that has the smallest $|z|$ value to
put it in the $\pm x$ directions.

The corresponding point-nucleon densities for the NLEFT calculation are
displayed in Fig. 3 (left). The overall shape as well as the inner cluster
structure of both nuclei look quite similar to the PGCM results. The bottom
part of 20Ne looks thinner and the top part looks fatter than the one obtained
by PGCM, this could be due to the effect of aligning the long principal axis
to $z-$direction, that the density is strengthened too much along this
direction [117]. To study the effect of the alignment procedure, we also show
in Fig. 3 (right) the same densities, but using configurations where the
angular information has been randomized. Without any bias coming from the
alignment procedure, these distributions should be exactly spherical. One can
therefore see that at least some of the non-trivial shape seen in Fig. 3
(left) is due to bias from the alignment procedure. However, the
$\alpha$-clustering seen on the left is much more pronounced, as well as the
bowling-pin shape of 20Ne.

Figure 4:  We show radial density profiles of 16O (left) and 20Ne (right), for
both NLEFT and PGCM. The PGCM results are computed using the density
$\rho_{m,2}$ as given in the text, while the NLEFT results come from smeared
lattice configurations with $d_{\text{min}}=0.5$ fm.

Figure 5: We show $v_{2}\\{2,|\Delta\eta|>1\\}$ (left),
$v_{3}\\{2,|\Delta\eta|>1\\}$ (middle) and $\rho(v_{2}\\{2\\}^{2},\langle
p_{T}\rangle)$ (right), with kinematic cuts as used by CMS (left and middle)
and ATLAS (right).

###  I.2 PGCM

The calculations described here were performed using the numerical suite
TAURUS [118, 119] in a model space with $e_{\text{max}}=6$,
$e_{\text{3max}}=18$ and $\hbar\omega=12$ MeV. This model space proved to be
sufficient to converge systems with similar masses in previous calculations
[67, 119].

Concerning the nuclear Hamiltonian, $H$, we employ the recent $\chi$EFT-based
interaction at N3LO published in Ref. [69] and apply the rank-reduction scheme
proposed in Ref. [63] to obtain an effective two-body operator.

The first step of our PGCM calculations is the generation of a set of
reference states. Here, we consider real general Bogoliubov quasi-particle
states optimized at the variation-after-particle-number projection level. In
addition, the Bogoliubov states $|\Phi(q)\rangle$ are constrained to have, on
average, a given deformation
$q\equiv(\beta_{20},\beta_{22},\beta_{30},\beta_{32})$, where

$\beta_{lm}=\frac{4\pi}{3R_{0}^{l}A}\langle
r^{l}\left(Y_{lm}+(-1)^{m}Y_{l-m}\right)\rangle,$

with $A$ being the number of nucleons, $R_{0}=1.2A^{1/3}$, $r$ being the
position and $Y_{lm}$ being a spherical harmonic. For a more efficient
exploration of quadrupole deformations, the constraints on $\beta_{20}$ and
$\beta_{22}$ are reformulated as constraints on the usual triaxial parameters

$\displaystyle\beta$ $\displaystyle=\sqrt{\beta_{20}^{2}+2\beta_{22}^{2}},$
$\displaystyle\gamma$
$\displaystyle=\arctan\left(\frac{\sqrt{2}\beta_{22}}{\beta_{20}}\right),$

and we use a parallelogram mesh as described in Ref. [120]. For 16O, we
considered the intervals: $\beta\in[0,1.2]$ with a spacing $\Delta\beta=0.2$,
$\gamma\in[0,60^{\circ}]$, $\beta_{30}\in[0,1.2]$ with a spacing
$\Delta\beta_{30}=0.3$, $\beta_{32}\in[0,1.2]$ with a spacing
$\Delta\beta_{32}=0.3$. For 20Ne, we considered the intervals:
$\beta\in[0,1.5]$ with a spacing $\Delta\beta=0.3$, $\gamma\in[0,60^{\circ}]$,
$\beta_{30}\in[0,1.5]$ with a spacing $\Delta\beta_{30}=0.3$,
$\beta_{32}\in[0,0.9]$ with a spacing $\Delta\beta_{32}=0.3$.

To pick the states to be included in the configuration mixing, we project all
(converged) quasi-particle states $|\Phi(q)\rangle$ on $J^{\pi}=0^{+}$ and the
appropriate number of protons and neutrons, and select the states with a
projected energy

$E(0^{+},q)=\frac{\langle\Phi(q)|HP^{\pi=+1}P^{J=0}_{00}P^{Z}P^{N}|\Phi(q)\rangle}{\langle\Phi(q)|P^{\pi=+1}P^{J=0}_{00}P^{Z}P^{N}|\Phi(q)\rangle},$

lower than a threshold of $E_{\text{th}}=4$ and 8 MeV for 16O and 20Ne,
respectively, above the absolute minimum. Here, $P^{\pi}$, $P^{J}_{MK}$,
$P^{Z}$ and $P^{N}$ are the projection operators onto a good parity, total
angular momentum, number of protons and number of neutrons, respectively [121,
122]. To say it differently, we consider the mixing set

$\Sigma\equiv\left\\{|\Phi(q)\rangle,E(0^{+},q)-\min\left[E(0^{+},q)\right]<E_{\text{th}}\right\\}.$

The PGCM ansatz considered for the ground state takes the form

$|\Psi^{0^{+}ZN}_{1}\rangle=\sum_{q}^{\Sigma}f_{1q}^{0^{+}ZN}P^{\pi=+1}P^{J=0}_{00}P^{Z}P^{N}|\Phi(q)\rangle,$

with the weights, $f_{1q}^{0^{+}ZN}$, being determined variationally by
requiring the energy to be minimal. After solving the variational equation, we
use the weights to determine an “average” deformation for the PGCM correlated
state, $|\Psi^{0^{+}ZN}_{1}\rangle$, following the procedure described in Ref.
[75]. The average deformation is calculated using either the deformation of
the Bogoliubov states ($\bar{q}_{1}$) or the deformation of the associated
particle-number projected states ($\bar{q}_{2}$). Then, we build a new
Bogoliubov state constrained to have the deformation $\bar{q}_{1}$,
$|\Phi(\bar{q}_{1}\rangle)$, as well as a particle-number projected state
constrained to have the deformation $\bar{q}_{2}$,
$|\Phi^{ZN}(\bar{q}_{2})\rangle$. Finally, we compute the particle-number
projected spatial densities

$\displaystyle\rho_{m,1}(x,y,z)$
$\displaystyle=\sum_{st}\frac{\langle\Phi(\bar{q}_{1})|a^{+}_{xyzst}a_{xyzst}P^{Z}P^{N}|\Phi(\bar{q}_{1})\rangle}{\langle\Phi(\bar{q}_{1})|P^{Z}P^{N}|\Phi(\bar{q}_{1})\rangle},$
$\displaystyle\rho_{m,2}(x,y,z)$
$\displaystyle=\sum_{st}\frac{\langle\Phi^{ZN}(\bar{q}_{2})|a^{+}_{xyzst}a_{xyzst}|\Phi^{ZN}(\bar{q}_{2})\rangle}{\langle\Phi^{ZN}(\bar{q}_{2})|\Phi^{ZN}(\bar{q}_{2})\rangle},$

where $a^{+}_{xyzst}$ ($a_{xyzst}$) creates (annihilates) a particle with spin
$s=\pm 1/2$ and isospin $t=\pm 1/2$ at position $(x,y,z)$. The two densities
are used in the subsequent hydrodynamic simulations to partially estimate the
uncertainty related to our empirical procedure.

###  I.3 Comparison of configurations

In Fig. 4, we compare the orientation-averaged radial distributions of nuclei
generated using both NLEFT and PGCM. For PGCM, we show the density
$\rho_{m,2}$ as described above, and for NLEFT we show densities reconstructed
from the lattice configurations after they have been smeared and after a
minimal distance $d_{\text{min}}=0.5$ fm has been imposed. We take the origin
to lie at the center of mass of either the density (PGCM) or at the center of
mass of the sampled nucleons (NLEFT). Overall agreement between the two
calculations is good, especially for 20Ne.

##  II Kinematic cuts

In Fig. 5, we show $v_{2}\\{2,|\Delta\eta|>1\\}$,
$v_{3}\\{2,|\Delta\eta|>1\\}$ and $\rho(v_{2}\\{2\\}^{2},\langle
p_{T}\rangle)$ with different kinematic cuts from those used in the main text.
In the main text, cuts as used by ALICE were used, whereas here we use CMS
cuts for $v_{n}\\{2,|\Delta\eta|>1\\}$, and ATLAS cuts for
$\rho(v_{2}\\{2\\}^{2},\langle p_{T}\rangle)$. One can see that while the
individual curves for 20Ne and 16O have a dependence on the cuts, the
ratios/differences shown in the bottom panel for each observable are
unchanged.

##  III Statistical and systematic uncertainties

Figure 6: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O collisions, using NLEFT
nuclear structure input. The gray curves represent 20 different parameter
choices taken from a Bayesian analysis [83]. The blue curve is the mean of the
gray curves, and the error bars shown on it represent the statistical
uncertainty. The blue band represents the systematic uncertainty, and is
constructed from the spread of the gray curves.

The observables in the main text are shown with statistical and systematic
uncertainties, where the latter are broken up into _Trajectum_ and nuclear
structure parts. In this section, we discuss in detail how these uncertainties
are computed.

To quantify the systematic uncertainty due to the remaining uncertainty in the
parameters of the _Trajectum_ model after the Bayesian fit, we take 20 likely
choices of parameters, which are randomly drawn from our posterior. We
subsequently evaluate our observables on all of these choices, and use the
spread of the parameters to generate a systematic uncertainty. Since this
essentially requires doing the computation 20 times, we choose to do this
computation using settings which evaluate relatively quickly. This requires
some approximations, which we then subsequently identify and correct for. In
this way, the computation is naturally split up into a ‘main’ computation
where we use the 20 choices of parameters to generate an ensemble of
predictions, and a number of corrections. This section is organized in the
same way.

###  III.1 _Trajectum_

First, let us discuss how the ensemble of 20 predictions is used. As an
example, in Fig. 6, we show in gray the 20 predictions in 16O16O (NLEFT) for
$v_{2}\\{2,\text{OC}\\}$, an observable related to
$v_{2}\\{2,|\Delta\eta|>1\\}$ (more on this below). Each prediction comes with
its own statistical uncertainty as estimated by the _Trajectum_ code. We then
average the values to produce the mean value (shown as a blue curve), which we
take as our central value. The statistical uncertainty is estimated by
averaging the statistical variances of the 20 predictions.333We average the
variances instead of the standard deviations because the variance is an
additive quantity. We then compute the standard deviation of the ensemble of
predictions, which gives us the total uncertainty. The systematic uncertainty
is then computed as the quadratic difference between the total and the
statistical uncertainties. The systematic uncertainty coming from this source
is counted as part of the _Trajectum_ systematic uncertainty.

Figure 7: We show both $v_{2}\\{2,|\Delta\eta|>1\\}$ and
$v_{2}\\{2,\text{OC}\\}$ for 16O16O (NLEFT) (top). Also shown is the ratio
$v_{2}\\{2,|\Delta\eta|>1\\}/v_{2}\\{2,\text{OC}\\}$ (bottom), which serves as
the correction factor discussed in the text. It can be seen that the
correction factor is small but significant.

We now turn to a discussion of the corrections, where we start with
corrections common to the PGCM and NLEFT simulations. Firstly, we point out an
important distinction between the $\rho(v_{2}\\{2\\}^{2},\langle
p_{T}\rangle)$ observable and the others. Since $\rho(v_{2}\\{2\\}^{2},\langle
p_{T}\rangle)$ can and does cross zero, we treat its corrections additively,
whereas the other corrections are treated multiplicatively. In the discussion
that follows, we will be using multiplicative corrections as an example, which
will always involve a division of some sorts. The additive corrections work
the same way, but replacing division by subtraction.

The first correction we encounter occurs only for
$v_{2}\\{2,|\Delta\eta|>1\\}$ and $v_{3}\\{2,|\Delta\eta|>1|\\}$. These
observables are statistically hard due to their large $\eta$ gap, but there is
a cheaper alternative, which we have called $v_{2}\\{2,\text{OC}\\}$, where
the OC stands for oversampled cumulants. The observable $v_{n}\\{2\\}$ is
defined by

$v_{n}\\{2\\}^{2}=\langle\langle\exp(in(\varphi_{i}-\varphi_{j}))\rangle\rangle,$

where the double $\langle\cdot\rangle$ means first an average over particle
pairs $(i,j)$ inside each event, and then an average over events. Which
particle pairs are used then determines the subtle differences between various
related observables. For $v_{n}\\{2\\}$, one simply uses all pairs within the
kinematic cuts, whereas for $v_{n}\\{2,|\Delta\eta|>1\\}$ one uses only pairs
where one particle has $\eta<-0.5$ and the other has $\eta>0.5$. The latter
choice suppresses non-flow by using only particle pairs which are far apart in
rapidity. Since _Trajectum_ is a theory model, we can do one thing which in
experiment is impossible; we can generate multiple particle sets from a single
hydrodynamic evolution, which we call oversamples. If one analyzes these
oversamples separately in the same way as described above, then using
oversamples simply improves statistics without introducing a bias. What we can
also do, however, is to use the oversamples as a cheap way to mimic the effect
of having a (large) $\eta$ gap, while still using all available pairs from an
event. The way the observable $v_{n}\\{2,\text{OC}\\}$ does this is by letting
go of the requirement that the partners in a particle pair need to be from the
same oversample. This dilutes the effect of non-flow (in particular resonance
decays), thereby mimicking the $v_{n}\\{2,|\Delta\eta|>1\\}$ observable, but
with much better statistical uncertainty.

Figure 8: We show the same correction factor as in the bottom panel of Fig. 7,
but for the ratio 20Ne20Ne/16O16O (NLEFT). We show both the correction for
$v_{2}\\{2,|\Delta\eta|>1\\}$ and for $v_{3}\\{2,|\Delta\eta|>1\\}$. It can be
seen that whereas the former is a relatively small correction, the latter can
be as much as 20%.

Returning to the same example used in Fig. 6, in Fig. 7 we show both
$v_{2}\\{2,|\Delta\eta|>1\\}$ and $v_{2}\\{2,\text{OC}\\}$ for 16O16O using
NLEFT, as well as the ratio between both observables. This ratio represents
the correction factor with which one needs to multiply the
$v_{2}\\{2,\text{OC}\\}$ result to obtain $v_{2}\\{2,|\Delta\eta|>1\\}$. We
treat the uncertainties in this correction factor as statistical. Note that
whereas for the 20Ne20Ne/16O16O ratio of $v_{2}\\{2,|\Delta\eta|>1\\}$ this
correction factor is well under 1% for most centralities, the same ratio for
$v_{3}\\{2,|\Delta\eta|>1\\}$ shows a correction of up to 20% (see Fig. 8).

Figure 9: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O (NLEFT), using three
different grid spacings for the hydrodynamic calculations. In addition, we
show the extrapolation to zero grid spacing, where the error bars show
statistical uncertainty, and the bands show systematic uncertainty.

There is one other correction applied to both PGCM and NLEFT results, and that
is a finite grid spacing correction. The hydrodynamic simulations are
performed at a finite grid spacing, which can bias the results. To correct for
this, we perform calculations using 0.125, 0.083 and 0.063 fm grid spacings.
Continuing to use our example of $v_{2}\\{2,|\Delta\eta|>1\\}$, we show these
different computations for $v_{2}\\{2,\text{OC}\\}$ of 16O16O for NLEFT in
Fig. 9. We then take a continuum limit by performing a least squares fit of
these results to

$v_{2}\\{2,\text{OC}\\}=c_{1}+c_{2}a^{2},$

where $c_{i}$ are constants, and $a$ is the grid spacing. The constant $c_{1}$
then contains our extrapolated result, which is also shown in Fig. 9. Using
$c_{1}$ and $c_{2}$ together, we can then obtain a factor which translates
from the grid spacing that the ensemble of 20 predictions was made with to the
extrapolated value. We use the prior uncertainties from the fit as statistical
uncertainties on the correction factor, and if the posterior uncertainties are
larger (which happens if $(\chi^{2})_{\text{red}}>1$), then we use the squared
difference of the posterior and prior uncertainties as a systematic, which we
count as part of the _Trajectum_ systematic uncertainty.

###  III.2 NLEFT

We now describe the corrections which are applied only to the NLEFT results.
The first of these pertains to the weights of the NLEFT configurations. For
16O (20Ne), around 26% (33%) of configurations have a weight of $-1$, whereas
the rest has weight $+1$. To take this into account properly, we must assign
each collision weight $+1$ if the weights of the two nuclei are both positive
or both negative, and $-1$ otherwise. The number of collisions with weight
$+1$ is therefore only 62% (56%), and this makes the simulations much more
expensive statistically. For this reason, we compute the ensemble of 20
predictions using only the configurations with positive weight, and then
perform a correction using a dedicated high statistics run using all
configurations. In Fig. 10, we show the results for $v_{2}\\{2,\text{OC}\\}$
for 16O16O using either only the positive weight configurations or all
configurations, together with the ratio between the two results, which is used
as the correction factor. We then count the statistical uncertainty of this
ratio as part of the total statistical uncertainty. We note that the weighting
effect is in fact rather important for e.g. the charge radius. Indeed for 20Ne
the positive and negative weight radii are 3.205 and 3.244 fm respectively,
which then leads to an overall radius of 3.17 fm quoted in the main text.

Figure 10: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O (NLEFT), where we
either take only configurations with positive weight, or all configurations.
In the latter case, we propagate the weights through the entire computation.
The resulting correction factor is shown in the bottom panel.

The other correction applied to NLEFT only deals with resolving the
periodicity induced by the lattice. We obtain the configurations on non-
periodic boundaries by examining all possible places to put the boundaries
between adjacent unit cells, and selecting the choice which leads to the
smallest nuclear rms size. In a small number of cases (1.1% for 16O and 6.3%
for 20Ne), this is ambiguous, as there is more than one solution which
minimizes the rms size. In this case, we have two choices for resolving this
further. The first choice is to just pick randomly, whereas for the second
choice we pick such that after clustering into $\alpha$-clusters, the clusters
have a smaller size. In other words, the second choice favors
$\alpha$-clustering. Given that the unambiguous configurations appear to be
$\alpha$-clustered, we believe that the second choice should be better.
However, since this is a choice we impose and not something that comes from
the Hamiltonian, we view this as a source of systematic uncertainty. We show
both methods for $v_{2}\\{2,\text{OC}\\}$ in 16O16O in Fig. 11, as well as the
ratio of choice 2 over choice 1. This ratio is used as a correction factor, as
the ensemble of 20 predictions was made with choice 1, whereas we believe that
choice 2 is better. The standard deviation between the two choices divided by
choice 1 is used as a systematic uncertainty after quadratically subtracting
the statistical uncertainty of the ratio. This systematic uncertainty is
counted as the sole part of the nuclear structure part of the total systematic
uncertainty for NLEFT. From Fig. 2 one can see that this uncertainty is almost
everywhere negligible compared to the _Trajectum_ systematic uncertainty.

Figure 11: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O (NLEFT), where we
resolve the ambiguity with both choices described in the text. The resulting
correction factor is also shown in the bottom panel, where the error bars show
statistical uncertainty, and the bands show systematic uncertainty.

Figure 12: We show the radial profiles of 16O (left) and 20Ne (right) for
PGCM, where we show both sampling methods discussed in the text, alongside the
original densities these configurations were sampled from.

###  III.3 PGCM

Figure 13: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O (PGCM), where we either
sample independently from the densities provided by PGCM, or we sample such
that we enforce that each region (corresponding with an alpha cluster)
contains exactly 4 nucleons. The resulting correction factor is shown in the
bottom panel.

We then move on to describing the two sources of nuclear structure systematic
uncertainty for the PGCM results. The first of these sources comes from the
fact that PGCM produces a density function, whereas what _Trajectum_ needs as
its input are explicit configurations. Taking a finite number of samples from
a density invariably alters the shape of the sampled distributions compared to
the density they were sampled from. As an extreme example of this, when
sampling a single nucleon from any density, the resulting shape will always be
spherical. Given that the entire effect we study in this work relies on
precisely knowing the shapes of the colliding nuclei, it is important to
quantify how the shape distortion from sampling affects the final results. To
this end, we use two different sampling methods. The first (independent
sampling) simply samples 16 (20) independent nucleons from the density. The
second (clustered sampling) divides the space up into 4 (5) regions centered
on the local maxima defined by the $\alpha$-clusters, where the region
boundaries are determined by equidistance between region centers.444We move
the centers slightly compared to these initial ansätze to ensure each region
has approximately equal integrated density inside. These regions can be seen
in Fig. 1, where the centers are shown as black points on the 3D view, and the
boundaries between the regions are shown in the 2D cross sections. We then
sample 4 nucleons from each region according to the density. The clustered
sampling method enforces a more even distribution of sampled nucleons
throughout the density, suppressing fluctuations where one of the regions
contains more nucleons than another. For this reason, we believe that
clustered sampling should yield configurations closer in shape to the original
density than independent sampling and importantly is more realistic in
approximating the physical sampling where all nucleon-nucleon correlations
would have been taken into account. Fig. 12 shows the radial profile of both
16O (left) and 20Ne (right) using both of these methods, and compares against
the original density they were sampled from (shown as well in Fig. 4). For
16O, the clustered method is near the original density from around 1 fm
outwards, whereas for the independent method this is about 1.5 fm. Below those
values the clustered method underestimates the original density, whereas the
independent method overestimates it. For 20Ne, the difference between both
methods is more pronounced, with the clustered method being closer to the
original density than the independent method almost everywhere.

Given that both methods are approximations, we take an agnostic approach, and
use their mean as our central value, whereas we use the standard deviation
between them as part of the nuclear structure systematic uncertainty (after
quadratically subtracting the statistical uncertainty). The ensemble of 20
predictions is made using clustered sampling, by which we normalize both the
central value and the systematic uncertainty to obtain the correction factor.
Fig. 13 shows the $v_{2}\\{2,\text{OC}\\}$ of 16O16O for both of the sampling
methods, as well as the correction factor obtained from them.

Figure 14: We show $v_{2}\\{2,\text{OC}\\}$ for 16O16O (PGCM), where we use
either the density $\rho_{m,1}$ or $\rho_{m,2}$ as inputs to _Trajectum_. The
resulting correction factor is shown in the bottom panel.

The last source of nuclear structure systematic uncertainty for PGCM is
constructed from the two previously discussed methods to constrain the
configurations to the ground state, resulting in the two densities
$\rho_{m,1}(x,y,z)$ and $\rho_{m,2}(x,y,z)$. The results obtained from these
two densities give us a handle on the effect of these variations in the PGCM
computation on the final state observables. Therefore, similarly to the
previous source of systematic uncertainty, we use the mean of the results
obtained from both densities as our central value, and use the standard
deviation between them as part of the nuclear structure systematic error. As
the computation using $\rho_{m,2}(x,y,z)$ is used for the ensemble of 20
predictions, we use it to normalize both the central value and the systematic
error. Fig. 14 shows the $v_{2}\\{2,\text{OC}\\}$ of 16O16O for both of these
versions, as well as the correction factor obtained from them.