Evidence of the triaxial structure of 129Xe at the Large Hadron Collider

Benjamin Bally,1 Michael Bender,2 Giuliano Giacalone,3 and Vittorio Soma4

1Departamento de Fısica Teorica, Universidad Autonoma de Madrid, 28049 Madrid, Spain2Universite de Lyon, Institut de Physique des 2 Infinis de Lyon,

IN2P3-CNRS-UCBL, 4 rue Enrico Fermi, 69622 Villeurbanne, France3Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

4IRFU, CEA, Universite Paris-Saclay, 91191 Gif-sur-Yvette, France

The interpretation of the emergent collective behaviour of atomic nuclei in terms of deformedintrinsic shapes [1] is at the heart of our understanding of the rich phenomenology of their structure,ranging from nuclear energy to astrophysical applications across a vast spectrum of energy scales.A new window onto the deformation of nuclei has been recently opened with the realization thatnuclear collision experiments performed at high-energy colliders, such as the CERN Large HadronCollider (LHC), enable experimenters to identify the relative orientation of the colliding ions in a waythat magnifies the manifestations of their intrinsic deformation [2]. Here we apply this technique toLHC data on collisions of 129Xe nuclei [3–5] to exhibit the first evidence of non-axiality in the groundstate of ions collided at high energy. We predict that the low-energy structure of 129Xe is triaxial(a spheroid with three unequal axes), and show that such deformation can be determined fromhigh-energy data. This result demonstrates the unique capabilities of precision collider machinessuch as the LHC as new means to perform imaging of the collective structure of atomic nuclei.

A key signature of the formation of quark-gluon plasma(QGP [6, 7]) in nuclear collision experiments performedat high-energy colliders is the observation of sizable an-gular anisotropy in the emission of hadrons in the planeorthogonal to the collision axis (dubbed transverse plane,or (x, y) in Fig. 1). If N hadrons are detected in agiven collision event, their transverse angular distribu-tion, dN/dφ, where φ is the azimuthal angle, presents aquadrupole (elliptical) component [8]:

dN/dφ ∝ 1 + 2v2 cos(2(φ− φ2)

)(1)

where v2, dubbed elliptic flow, is the magnitude of thequadrupole asymmetry. v2 is engendered in nuclear col-lisions by the pressure gradient force, ~F = −~∇P , drivingthe QGP expansion that converts the spatial anisotropyof the system geometry, which has in general some ellip-ticity [9, 10], into an anisotropic flow of matter, carriedover to the detected hadrons. For nearly head-on (cen-tral) collisions (Fig. 1), elliptic flow is naturally sensitiveto the intrinsic quadrupole deformation that character-izes the ground state of colliding ions, i.e., the ellipsoidaldeformation of their surface,

R(θ, ϕ) = R0 {1 + β [cos γY20(θ, ϕ) + sin γY22(θ, ϕ)]} ,(2)

where R0 parametrizes the nuclear radius, the Ylm arespherical harmonics, and the positive coefficients β andγ encode the ellipsoidal shape. The former gives the mag-nitude of deformation, with well-deformed nuclei havingβ ≈ 0.3, while the latter indicates the length imbalanceof the axes of the spheroid (e.g., if the nucleus is prolate,like a rugby ball, or oblate, flattened at the poles), andvaries between γ = 0 and γ = 60◦, as shown by Fig. 2(a).

Spectroscopy of nuclei at low energy provides accessto the nuclear charge quadrupole moment that can alsobe parameterized with coefficients βv and γv of similar

FIG. 1. Illustration of a head-on collision between two atomicnuclei performed in a collider experiment. The nuclei, de-formed in their ground state, are randomly oriented as theyrun in the beam pipe, and the shape of their area of overlapcan range from circular to elliptical. A quark-gluon plasma(QGP) is formed in the this area. The hydrodynamic ex-pansion of this medium in the plane transverse to the beam,(x, y), is driven by a force field, ~F , which carries the samequadrupole anisotropy as the QGP geometry, i.e., as the over-lap area. Note that in the frame of the laboratory both nucleiwould look like thin pancakes, squeezed in beam direction, z,due to a strong effect of Lorentz contraction.

size as (albeit not equivalent to [12]) β and γ in Eq. (2).For well-deformed nuclei, βv can be determined throughthe measurement of a single transition probability [13],whereas access to several transitions is required for γv[14]. Identifying these parameters with a nuclear shaperequires assuming that the nuclear wave function can

arX

iv:2

108.

0957

8v1

[nu

cl-t

h] 2

1 A

ug 2

021

2

FIG. 2. Meaning of the triaxial parameter, γ, and its influence on the geometry of head-on collisions of nuclei at low meantransverse momentum, 〈pt〉. (a) Spheroids with a quadrupole deformation, β > 0. Depending on the value of γ, they caneither present two axes of the same length, and be either prolate (γ = 0) or oblate (γ = 60◦), or present three axes of differentlengths and be triaxial (0 < γ < 60◦), with the maximum triaxiality being reached for γ = 30◦. Values of γ between 60◦ and360◦ correspond simply to rotations of the same types of shapes. (b) Head-on collisions at small values of the mean transversemomentum of detected hadrons, 〈pt〉, permit to isolate configurations that maximize the overlap area. Depending on the valueof γ, collision configurations at low 〈pt〉 can thus vary between geometries that maximally break azimuthal symmetry (γ = 0)and geometries that are azimuthally symmetric (γ = 60◦).

be factorized into an intrinsic state describing the geo-metrical arrangement of the nucleons, and a state giv-ing the orientation of this structure in the laboratoryframe [1, 15]. Such factorization is not always mean-ingful, for example because of large shape fluctuations.Additionally, for odd-mass and odd-odd nuclei, quantummechanics entangles the contributions from collective in-trinsic deformation and individual single-particle statesto the quadrupole moment, such that the former cannotbe uniquely determined from a spectroscopic experiment.

Measurements of v2 at colliders provide an alterna-tive access route to the intrinsic deformation of all nu-clei, even and odd. The impact of βv at high energy hasbeen established in 238U+238U collisions at the Relativis-tic Heavy Ion Collider (RHIC) [11], and in 129Xe+129Xecollisions at the LHC [3–5]. Our goal is to show that suchexperiments open as well a new window onto γv. The keyfeature is the possibility of selecting events for which therelative orientation of the colliding ions maximizes thebreaking of azimuthal symmetry induced by their shapes[2]. One needs the mean hadron momentum,

〈pt〉 =1

N

N∑

i=1

pt,i, (3)

where pt,i ≡ |pt,i| for particle i with transverse momen-tum pt,i = (px,i, py,i). For collisions at fixed N , 〈pt〉provides a measure of the (inverse) size of the transversearea where the QGP is formed [16], such that events car-rying abnormally small values of 〈pt〉 correspond to largeoverlap areas. Following Fig. 2(b), when the ions haveβ > 0, low-〈pt〉 configurations correspond to overlap ge-ometries ranging from maximally elliptic, for γ = 0, toazimuthally isotropic, for γ = 60◦. At low 〈pt〉, then,the magnitude of v2 depends on γ, so that the depen-dence of v2 on 〈pt〉 probes γ. For strongly prolate 238Unuclei with β ≈ 0.3 and γ ≈ 0, the effectiveness of thismethod in probing β has been recently demonstrated in238U+238U collisions [17]. Here we employ this techniqueto reveal for the first time signatures of γ, the triaxialityof nuclei. We use LHC measurements in 129Xe+129Xecollisions, which are ideal candidates for such a study, asthe ground state of all even-mass xenon isotopes around129Xe are triaxial in low-energy nuclear models [18, 19].

We determine now in the framework of energy-densityfunctional methods [20] applied to the nuclear many-body problem the triaxiality of the lowest 1/2+ state of129Xe, corresponding to the experimental ground state.We first perform a set of symmetry-breaking constrained

3

FIG. 3. Structure of the ground states of 129Xe and 208Pb.Left: 208Pb. Right: 129Xe. The upper panels representbeyond-mean-field potential energy surfaces in the (βv, γv)plane, where we plot the energy shift, ∆E, with respect tothe energy minimum. The lower panels show the functionsg2(βv, γv) normalized to unity at their maximum. Star mark-ers label the coordinates of the average intrinsic quadrupolemoments for both nuclei, namely, βv = 0.06, γv = 25.3◦ for208Pb, and βv = 0.19, γv = 23.6◦ for 129Xe. We note thatthe white regions in the displayed color maps correspond tothe minimum of ∆E(βv, γv) in the upper panels, and to themaximum of g2(βv, γv) in the lower panels.

Hartree-Fock-Bogoliubov (HFB) calculations that pro-vide intrinsic states covering the (βv, γv) plane. The low-est state with good quantum numbers in the laboratoryframe is then constructed using the projected generatorcoordinate method (PGCM), i.e., we consider a many-body wave function that is a linear superposition of theintrinsic states across the (βv, γv) plane, projected ontoquantum numbers reflecting the symmetries of the nu-clear Hamiltonian. From there, we compute the so-calledcollective wave function g(βv, γv) [20], which squaredgives roughly the contribution of each (βv, γv) point tothe final PGCM state. The same effective Skyrme-typenucleon-nucleon interaction, SLyMR1 [21, 22], is used atall stages of the calculations, done for 129Xe and 208Pb, aswe shall look at high-energy results for both these species.The structural properties of these nuclei are shown inFig. 3. The results for 208Pb agree with existing litera-ture [18], indicating a soft energy surface, with all statesbeing nearly degenerate irrespective of their γv, up toβv ≈ 0.1, beyond which the energy rises quickly. Our newresult concerning 129Xe shows instead a minimum aroundβv = 0.2, corresponding to a g2(βv, γv) peaked aroundthe average intrinsic moments (βv, γv) = (0.19, 23.6◦).129Xe appears to be, hence, a rigid triaxial spheroid.

With this knowledge, we perform simulations of208Pb+208Pb and 129Xe+129Xe collisions to assess the

role of γ in high-energy experiments. Following theGlauber Monte Carlo model [23], the colliding nuclei aretreated as batches of independent nucleons sampled froma density, n, usually taken as a Woods-Saxon profile:

n(r, θ, ϕ) ∝(

1 + exp

[1

a

(r −R(θ, ϕ)

)])−1

, (4)

where a is the skin thickness, and R(θ, ϕ) has the sameform as in Eq. (2). The parameters entering Eq. (4) areobtained by fitting the Woods-Saxon profile to the one-body nucleon density returned by a HFB calculation withSLyMR1, constrained to present quadrupole moments βvand γv. The fit yields: a = 0.537 fm, R0 = 6.647 fm,β = 0.062, γ = 27.04◦ for 208Pb, and a = 0.492 fm,R0 = 5.601 fm, β = 0.207, γ = 26.93◦ for 129Xe. Supple-menting the Glauber model with an ansatz [24] for theenergy density of the QGP created in each collision, westudy the impact of γ on the dependence of v2 on 〈pt〉 byevaluating the Pearson correlation coefficient [25]:

ρ(v22 , 〈pt〉) =

⟨δv2

2δ〈pt〉⟩

√⟨(δv2

2

)2⟩⟨(δ〈pt〉

)2⟩ , (5)

where 〈. . .〉 denotes an average over events at fixed cen-trality, and δo = o−〈o〉 for any observable o. This quan-tity is a number between -1 (perfect anti-correlation) and+1 (perfect correlation). In central collisions, one expectsρ(v2

2 , 〈pt〉) > 0 [26]. Figure 2 shows that, for prolate nu-clei, decreasing the value of 〈pt〉 yields maximally ellip-tical overlap geometries, leading to enhanced values ofv2. Therefore, for β > 0 and γ = 0, nuclear deforma-tion yields a negative contribution to ρ(v2

2 , 〈pt〉), whichgradually turns into a positive one towards γ = 60◦ [27].

We evaluate Eq. (5) as a function of the percentage ofoverlap (or impact parameter) of the colliding ions, repre-sented as a percentile, where 20% corresponds roughly toa distance between the colliding ions of 7 (6) femtometersfor 208Pb+208Pb (129Xe+129Xe) events. Our results arein Fig. 4. In central collisions, the role of γ is manifest.Colliding triaxial 129Xe nuclei (red dashed line) enhancesρ(v2

2 , 〈pt〉) compared to the case where the nuclei haveγ = 0 (green dot-dashed line), flipping its sign for thelowest percentiles. This sensitivity of ρ(v2

2 , 〈pt〉) to thetriaxiality implies that dedicated simulation frameworks[28, 29] will be able to perform independent extractionsof γ from high-energy data. Comparing our results withpreliminary LHC measurements by the ATLAS collabo-ration [30], shown in the upper panels of Fig. 4, we seethat they capture the trend of the data, although the ef-fect of the experimental uncertainty on the centrality def-inition, causing the differences between the data pointsdisplayed in the left and in the right panel of the figure,is not included in our calculation. However, such effectsbecome less relevant in the ratio of the two collision sys-tems, shown in the lower panels (purple squares). The

4

centrality (%)0.0

0.1

0.2

0.3

0.4

ρ( v

2 2,〈p t〉)

ATLAS, Pb+Pb Nch-based

ATLAS, Xe+Xe Nch-based

0 5 10 15 20

centrality (%)

0.4

0.5

0.6

0.7

0.8

Xe+

Xe

/P

b+

Pb

Nch-based, 0.5 < pt < 2 GeV

Nch-based, 0.5 < pt < 5 GeV

centrality (%)0.0

0.1

0.2

0.3

0.4

ρ( v

2 2,〈p t〉)

Pb+Pb β = 0.062, γ = 27.04◦

Xe+Xe β = 0.207, γ = 26.93◦

Xe+Xe β = 0.207, γ = 0

ATLAS, Pb+Pb∑ET -based

ATLAS, Xe+Xe∑ET -based

ATLAS, Pb+Pb∑ET -based

ATLAS, Xe+Xe∑ET -based

0 5 10 15 20

centrality (%)

0.4

0.5

0.6

0.7

0.8

Xe+

Xe

/P

b+

Pb

∑ET -based, 0.5 < pt < 2 GeV∑ET -based, 0.5 < pt < 5 GeV

FIG. 4. Theoretical and experimental results on the correlation between elliptic flow and the mean transverse momentum.Symbols are preliminary measurements from the ATLAS collaboration (circles, diamonds, and squares). Error bars on ATLASdata are of the same size as the displayed symbols. Theory results include collisions of 208Pb (black solid line), collisions oftriaxial 129Xe nuclei (red dashed lines), as well as of 129Xe nuclei with γ = 0 (green dot-dashed lines). The upper panels showthe centrality dependence of the correlator ρ(v22 , 〈pt〉) in both 208Pb+208Pb and 129Xe+129Xe collisions. The lower panels showtheir ratio. The reconstruction of the impact parameter in ATLAS data is based on either a raw number of charged hadrons,Nch (left panels), or the energy collected by dedicated calorimeters,

∑ET (right panels). The data points in the upper panels

are obtained from hadrons having 0.5 < pt < 2 GeV, while the ratios in the lower panels are calculated as well for 0.5 < pt < 5GeV [30]. The shaded bands represent the statistical uncertainties on the theoretical results.

pt range of the hadrons used in the analysis, while influ-encing ρ(v2

2 , 〈pt〉) [30], plays also a minor role in Fig. 4.The measured ratios are, hence, very robust. They grantaccess to the triaxiality from data, and confirm our low-energy prediction that 129Xe has β ≈ 0.2 and γ ≈ 27◦. Asimilar analysis can be repeated for any species used incollider experiments, for instance, 197Au nuclei collidedat RHIC [17], which may also present a triaxial groundstate [19]. The possibility of knowing γ from high-energydata represents a stepping stone to a fruitful collabora-tion between the low- and the high-energy nuclear com-munities, to better exploit collider experiments involvingatomic nuclei.

This project has received funding from the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sk lodowska-Curie grant agree-ment No. 839847. M.B. acknowledges support by theFrench Agence Nationale de la Recherche under grant No.19-CE31-0015-01 (NEWFUN). G.G. is supported by theDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under Germany’s Excellence Strat-egy EXC 2181/1 - 390900948 (the Heidelberg STRUC-

TURES Excellence Cluster), SFB 1225 (ISOQUANT)and FL 736/3-1. We acknowledge the computer re-sources and assistance provided by the Centro de Com-putacion Cientıfica-Universidad Autonoma de Madrid(CCC-UAM).

Supplemental material

The calculations we have performed to produce theresults presented in this manuscript are of two kinds.

1. We first evaluate the structure properties of thefirst 1/2+ state of 129Xe and the first 0+ state of208Pb. The goal is to extract the average deforma-tion parameters β and γ of these species, as well asthe parameterization of their radial profiles.

2. Secondly, we use the knowledge of the structuralproperties of these nuclei to perform Monte Carlosimulations of millions of high-energy 129Xe+129Xeand 208Pb+208Pb collisions to produce the resultsshown in Fig. 4.

5

1. Within our nuclear structure model, we build approxi-mations to the eigenstates of the nuclear Hamiltonian Hby solving the variational equation

δ〈Ψ|H|Ψ〉〈Ψ|Ψ〉 = 0 (6)

in a restricted many-body Hilbert space. Here, thenuclear Hamiltonian is taken to be the SLyMR1parametrization of a phenomenological Skyrme-typepseudopotential [21, 22].

In the first step of the calculation, the variationalequation is solved considering Bogoliubov quasiparticlestates as trial wave functions [20]. A series of constrainedminimizations of the energy is performed to explore the(βv, γv) surface and we end up with a set of Bogoliubov-type wave functions {|Φ(βv, γv)〉} that are characterizedby their average dimensionless quadrupole moments

βv =4π

(3R20A)

√q220 + 2q2

22, (7)

γv = arctan

(√2q22

q20

), (8)

where A is the mass number, R0 = 1.2A1/3, and

Qlm ≡ rlYlm(θ, ϕ), (9)

qlm ≡1

2〈Φ(βv, γv)|Qlm + (−1)mQl−m|Φ(βv, γv)〉. (10)

Then, in the second step of the calculation, we consideran enriched variational ansatz that is built as a linearsuperposition of the states in the set {|Φ(βv, γv)〉} pro-jected onto good quantum numbers associated with thesymmetries of the nuclear Hamiltonian [15]. This is thePGCM that was used for example in [31]. The variationalansatz now reads as

|Ψ〉 =∑

(βv,γv)K

f(βv,γv)KPJMKP

NPZ |Φ(βv, γv)〉, (11)

where P JMK , PN , PZ are projection operators onto goodangular momentum J and its projections along the zaxis M , K, neutron number N and proton number Z,respectively [15]. The weights f(βv,γv)K as well as theenergy of the state |Ψ〉 are solutions of the variationalequation, which is in this case is equivalent to a gener-alized eigenvalue problem. In practice, the sets used tobuild the PGCM anzatz contained 15 states for 208Pband 23 states for 129Xe. In particular, for 129Xe, thecalculations were tailored to obtain the best possible de-scription of the lowest 1/2+, which is the experimentalground state but is probably an excited state when per-forming a large-scale PGCM calculation of 129Xe withthe SLyMR1 parametrization.

After obtaining the correlated PGCM wave functions,we extract average quadrupole moments (βv, γv) for the

state |Ψ〉 by computing

βv =∑

(βv,γv)

βv g2(βv, γv), (12)

γv =∑

(βv,γv)

γv g2(βv, γv), (13)

where g(βv, γv) is the so-called collective wave function,whose expression in terms of the f(βv,γv)K of Eq. (11)can be found in [20]. We obtain (βv = 0.06, γv = 25.3◦)for 208Pb and (βv = 0.19, γv = 23.6◦) for 129Xe. To giveconfidence in the fact that our calculations correctly cap-ture the structure of the experimental levels, we mentionin particular that, when including suitable corrections forthe finite size of nucleons [32], our PGCM wave functionsgive charge radii rch = 5.46 fm for 208Pb, and rch = 4.74fm for 129Xe, that reproduce experimental data with anaccuracy better than one percent.

The average multipole moments of the point-nucleondensity are used to inform our high-energy collisionmodel in the following way. First, for a given nucleus, wecompute a new Bogoliubov-type state |Φ(βv, γv)〉 thatis constrained to the appropriate average intrinsic multi-pole moment. Then, we fit a Woods-Saxon density

n(r, θ, ϕ) ∝(

1 + exp

[1

a

(r −R(θ, ϕ)

)])−1

, (14)

with

R(θ, ϕ) = R0

[1 + β

(cos γY20(θ, ϕ) + sin γY22(θ, ϕ)

)],

(15)such that it reproduces at the same time the one-bodydensity and average quadrupole moments of the state|Φ(βv, γv)〉. With this procedure, we obtain the pa-rameters are a = 0.537 fm, R0 = 6.647 fm, β = 0.062,γ = 27.04◦ for 208Pb, and a = 0.492 fm, R0 = 5.601 fm,β = 0.207, γ = 26.93◦ for 129Xe. This Bogoliubov state isnot only an approximation for the lowest state with goodquantum numbers after symmetry restoration, but alsoincorporates effect of shape fluctuations, which for thenuclei discussed here is possible because their wave func-tions have only one peak. The Bogoliubov-type states|Φ(βv, γv)〉 have in general also higher-order multipolemoments [19], whose presence influences the values for βand γ. Including higher-order deformations in the shapeof the Woods-Saxon density of Eq. (15), however, doesnot have a significant impact on the high-energy observ-ables discussed here [33]. But we note that the param-eters β and γ of the fitted Woods-Saxon density are ingeneral different from the average (βv, γv) of the micro-scopic wave function.

2. The simulations of 208Pb+208Pb and 129Xe+129Xe col-lisions are subsequently performed within the frameworkof the Glauber Monte Carlo model [23].

6

The colliding nuclei are treated as batches of A nucle-ons, which, in each realization of the nuclei, are sampledindependently from the distribution of Eq. (14). Beforesampling the nucleons, we consider that the spatial ori-entation of the colliding ions is random at the time ofscattering. We randomly rotate the Woods-Saxon den-sities in space following the so-called Z-X-Z (or 3-1-3)prescription. Consider that the intrinsic nuclear frameand the lab frame, (x, y, z), are initially aligned. We per-form i) a rotation about the z axis by an angle u; ii)a rotation about the x axis by an angle v; iii) an addi-tional rotation about the z axis by an angle w. To ensurethat the rotations of the ellipsoids are sampled uniformlyfrom SO(3), the angle u an the angle w are sampled uni-formly between 0 and 2π, whereas the angle v is sampledsuch that the distribution of cos(v) is uniform between -1and 1. Subsequently, we sample A nucleons for each ionaccording to the rotated Woods-Saxon densities. We ne-glect any effect of short-range correlations in our nuclei,and do not impose, e.g., any minimum distance cutoffamong the sampled nucleon pairs.

For each collision we draw a random impact parame-ter, b, from a distribution dN/db ∝ b. This correspondsto the distance between the centers of the colliding ions.For each ion, we shift, then, the coordinates of the corre-sponding nucleons by +b/2 and −b/2, respectively. Thedirection of this shift, i.e., the direction of the impact pa-rameter, defines the x direction in the transverse plane.Two nucleons, belonging to different parent nuclei, inter-act in the transverse plane if their distance is less than

D =√σNN/π, (16)

where σNN is the inelastic nucleon-nucleon cross section,which at top LHC energy is approximately 7 fm2. A nu-cleon is labeled a participant if it undergoes at least onescattering with a nucleon coming from the target nucleus.The total number of participants is dubbed Npart.

From this point on, we describe the collisions pro-cess and the subsequent QGP formation by means of theTRENTo model of initial conditions [24]. We center ontop of each participant nucleon a two-dimensional Gaus-sian distribution of participant matter of width ω = 0.5fm. The sum of these participant-level distributions givesthe so-called thickness functions of the colliding nuclei.For, say, nucleus A, the thickness reads:

tA(x) =

Npart,A∑

i=1

λi2πω2

e−(x−xi)

2

2ω2 , (17)

where xi is the location of the ith participant inside nu-cleus A, and λi is a normalization drawn independentlyfor each participant from a gamma distribution of unitmean and standard deviation equal to 1/

√2. This par-

ticular tuning of the parameters ω and λ proves powerfulin phenomenological applications at LHC energies [34].

For the collision of nucleus A against nucleus B, the en-tropy density in the transverse plane of the QGP createdin the collision process is then given by:

s(x) ∝√tA(x)tB(x). (18)

From the knowledge of the entropy density, we can obtainall the quantities necessary to draw Fig. 4.

We first need the total entropy in each event:

S ∝∫

x

s(x). (19)

The entropy, S, is used to sort events in centrality classes.As discussed below, experiments are unable to determinethe impact parameter of the collisions, therefore, theyrely on auxiliary variables to reconstruct it [35]. Thesevariables are of more or less direct variants of the numberof particles (N in Eq. (3)) produced in a given event,which is, in turn, in a nearly one-to-one correspondencewith the entropy of the QGP [36]. The distribution ofS provides, hence, a means to sort the simulated eventsinto centrality classes in a way which is consistent withthe methodology of the experimental collaborations. Thecentrality is in general well-approximated by the relation

c =πb2

σinel, (20)

where b is the impact parameter and σinel is the in-elastic nucleus-nucleus cross section, σinel ≈ 770 fm2

in 208Pb+208Pb collisions, and σinel ≈ 570 fm2 in129Xe+129Xe collisions. Expressing c as a percentile frac-tion, one can then relate the centrality values shown inFig. 4 to the impact parameters, irrespective of the spe-cific variable used to define the percentile.

In each centrality class, we evaluate the Pearson cor-relator of Eq. (5), i.e.,

ρ(v22 , 〈pt〉) =

⟨δv2

2δ〈pt〉⟩

√⟨(δv2

2

)2⟩⟨(δ〈pt〉

)2⟩ . (21)

To do so, we need the knowledge of v2 and 〈pt〉 in eachevent. To exhibit results that have a statistical uncer-tainty comparable to that of the experimental ATLASmeasurements, we follow recent theoretical developments[37–39], and evaluate the Pearson coefficient by meansof accurate initial-state predictors. We consider that,in a given centrality class, the value of v2 in the finalstate is linearly correlated with the value of the ellipticanisotropy, ε2, of the initial state of the QGP, correspond-ing to the normalized quadrupole moment of the entropydensity [40],

ε2 =|∫x|x|2ei2Φs(x) |∫x|x|2s(x)

, (22)

where Φ = tan−1(y/x). Further, we consider that thevalue of 〈pt〉 is linearly correlated with the value of the

7

energy, E, carried by the QGP [39] after its formation.The transverse energy density of the system, e(x), is eval-uated from the conformal equation of state of quantumchromodynamics at high temperature:

e(x) ∝ s(x)4/3, (23)

so that the total fluid energy reads:

E ∝∫

x

e(x). (24)

Replacing, then, v2 with ε2 and 〈pt〉 with E in Eq. (21)leads to a very good approximation of the results ob-tained at the end of full hydrodynamic simulations [39].Our curves in Fig. 4 result from 2 × 106 simulationsof 208Pb+208Pb collisions, and 5 × 106 simulations of129Xe+129Xe collisions (for both γ = 0 and γ = 26.93◦).The statistical errors are calculated with the jackkniferesampling method. The simulations are performed bymeans of a code written in Python 3 which we have de-veloped for this application.

In a final note concerning the experimental data points,we emphasize that, since the impact parameter of the col-lisions can not be determined experimentally, the ATLAScollaboration has made use of two different variables tosort their events into centrality classes [41], and performthe measurement of ρ(v2

2 , 〈pt〉) [30]. The first variable isthe raw number of charged particles, Nch, observed inthe central region of the detector, corresponding to thepseudorapidity window |η| < 2.5, where the pseudorapid-ity is defined by η = − ln tan(Θ/2), where Θ is the polarangle in the (y, z) plane of Fig. 1. The second variableis the total transverse energy,

∑ET , deposited by the

products of the collisions in forward calorimeters covering3.2 < |η| < 4.9. Additionally, the measurement has beenperformed for different kinematic ranges of the hadronsused to build the ρ(v2

2 , 〈pt〉) observable. We recall thatthe momentum of a given hadron in the transverse planeis denoted by pt. The observable has been calculatedfor 0.5 < pt < 2 GeV and for 0.5 < pt < 5 GeV, bothconsidered in Fig. 4.

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