Probing nuclear structure in relativistic p–O and O–O collisions at the LHC
through the measurement of anisotropic flow coefficients

Abstract

Report issue for preceding element

RHIC and LHC plan to inject ${}^{16}\rm O$ nuclei with a focus to investigate collectivity and the origin of quark-gluon plasma signatures in small collision systems. ${}^{16}\rm O$ nuclei is known to possess clusters of $\alpha$-particles (${}^{4}\rm He$) inside the nucleus. In this paper, we study the anisotropic flow coefficients such as elliptic flow ($v_{2}$) and triangular flow ($v_{3}$), which are sensitive to the nuclear geometry of colliding nuclei, for p–O and O–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and 7 TeV respectively. The study is performed employing a hybrid model encompassing IPGlasma + MUSIC + iSS + UrQMD. The results of the clustered nuclear geometry are compared with those of the Woods – Saxon nuclear profile. Both initial and final state anisotropies are explored. This study is thus one of its first kind, where the study of anisotropic flow coefficients for p–O and O–O collisions is presented using a hybrid hydrodynamics model. We find a small effect of $\alpha$-clustering in p–O, while a significant one for O–O collisions. It is also observed that the magnitude of the effect correlates with the size of the ⁴He.

Report issue for preceding element

I INTRODUCTION

Report issue for preceding element

The heavy-ion collisions at the Large Hadron Collider (LHC), CERN, and Relativistic Heavy-Ion Collider (RHIC), BNL, aim to recreate a deconfined matter of partons, known to have existed shortly after the Big Bang. This deconfined and thermalised medium of partons, known as Quark-Gluon Plasma (QGP), is transient in nature and is shown to possess collective phenomena similar to fluids ALICE:2022wpn ; Heinz:2013th . Due to the short lifetime of QGP, its existence in heavy-ion collisions is often inferred by studying indirect signatures, which include strangeness enhancement Rafelski:1982pu , collectivity Voloshin:1994mz , quarkonia suppression Matsui:1986dk , jet-quenching Gyulassy:2000fs ; Gyulassy:2000er ; Levai:2001dc , to name a few. Small systems like proton-proton (pp) and p–Pb are often considered as baselines to study QGP and cold nuclear matter effects in heavy-ion collisions. Thanks to the observation of these collective and other heavy-ion-like phenomena in high-multiplicity pp and p–Pb collisions at the LHC energies ALICE:2024vzv ; ALICE:2016fzo , the small collision systems are of great interest, where the existence of a QGP medium is hinted at. Thus, both RHIC and LHC prepare to inject small nuclei, like e.g., Oxygen (${}^{16}\rm O$) ions into the beam pipe to perform O–O and p–O collisions Brewer:2021kiv ; Katz:2019qwv . These collision systems are important as they bridge the multiplicity gap between high multiplicity pp, p–Pb, and peripheral Pb–Pb collisions. Studying them ought to provide insights into the nature of the QCD medium in small systems where some of the signatures– like jet quenching and quarkonia suppression– are absent. Further p–O collisions are specifically important as their measurements at the LHC would be useful to tune the cosmic ray air shower models, leading us to solve the existing puzzles Scaria:2023coa .

Report issue for preceding element

One of the key aspects to investigate in p–O and O–O collisions is rooted in low-energy nuclear physics, which suggests the presence of $\alpha$-clustered nuclear structure in light nuclei having $4n$ number of nucleons. Oxygen (${}^{16}\rm O$) is one such nuclei where the $\alpha$-particles (${}^{4}\rm He$ nucleus) arrange themselves at the corners of a regular tetrahedron gamow ; Wheeler:1937zza ; Bijker:2014tka ; Wang:2019dpl ; He:2014iqa ; He:2021uko ; Otsuka:2022bcf . In heavy-ion collisions, several studies argue for a modification in the final state particle production due to a change in the initial state nuclear structure of the colliding nuclei Behera:2023nwj ; Giacalone:2021udy ; Haque:2019vgi ; Behera:2023oxe ; ALICE:2021ibz ; ALICE:2018lao ; ATLAS:2019dct ; CMS:2019cyz ; STAR:2015mki ; Giacalone:2021udy ; Haque:2019vgi ; PHENIX:2018lia . The geometry scan program at RHIC is dedicated to understanding the particle production due to the specific geometry of the colliding nuclei PHENIX:2018lia ; PHENIX:2021ubk ; STAR:2022pfn ; STAR:2023wmd . A number of studies at the RHIC and LHC energies have been performed to study the impact of clustered structure in the final state particle production and flow in O–O collisions Li:2020vrg ; Rybczynski:2019adt ; Sievert:2019zjr ; Huang:2019tgz ; Behera:2023nwj ; Behera:2021zhi ; Lim:2018huo ; Summerfield:2021oex ; Schenke:2020mbo ; Rybczynski:2019adt ; Sievert:2019zjr ; Huang:2019tgz ; Huss:2020whe ; Zakharov:2021uza ; Giacalone:2024ixe ; Zhang:2024vkh ; R:2024eni ; Prasad:2024ahm ; Ding:2023ibq ; Wang:2021ghq ; Rybczynski:2017nrx ; Svetlichnyi:2023nim . It is observed that due to a compact nuclear geometry, O–O collisions with clustered nuclear geometry yield a higher particle multiplicity and energy density in the central collisions as compared to a traditional Woods-Saxon nuclear profile Behera:2021zhi . Additionally, the clustered O–O collisions lead to significantly large values of triangular flow Behera:2023nwj ; Prasad:2024ahm . One of our earlier studies concluded that the anisotropic flow fluctuations are sensitive to the clustered nuclear geometry of the colliding nuclei Prasad:2024ahm . In several studies, the authors have performed O–Au, C–Au, O–Pb, and Ne–Pb collisions at the RHIC and LHC energies to investigate the clustered structure of light nuclei Bozek:2014cva ; Broniowski:2013dia ; Giacalone:2024luz ; Lim:2018huo . However, similar studies are underperformed in p–O collisions, which are necessary to understand the effect of the clustered structure of ${}^{16}\rm O$ in asymmetric small collisions so as to identify the effects coming from the initial nuclear profile and small system dynamics. Additionally, anisotropic flow measurements in p–O and O–O collisions are capable of making remarkable additions to the present understanding of partonic collectivity recently observed in pp and p–Pb collisions by the ALICE collaboration ALICE:2024vzv . Interestingly, one recent study based on AMPT investigates the effect of clustered nuclear geometry in p–O collisions R:2024eni , which shows that the effects of $\alpha$-clustering are significantly different from those of the unclustered nuclear profile, as far as the flow-observables under study are concerned. The uniqueness of $\alpha$-cluster results is found to be consistent with that of the AMPT results on O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV as well R:2024eni ; Behera:2021zhi .

Report issue for preceding element

In this paper, we simulate p–O and O–O collisions at at $\sqrt{s_{\rm NN}}=9.61$ TeV and 7 TeV, respectively, where the measurements with $\alpha$-clustered structure of ${}^{16}\rm O$ nuclei are compared with those of the Woods-Saxon profile. We measure initial state eccentricity ($\epsilon_{2}$), triangularity ($\epsilon_{3}$) and compare them with final state anisotropic flow coefficients such as elliptic flow ($v_{2}$) and triangular flow ($v_{3}$) for both $\alpha$-clustered and Woods-Saxon nuclear profiles for both p–O and O–O collisions. The anisotropic flow coefficients are calculated using the two-particle Q-cumulant method. We use a hybrid model referred to as IPGlasma + MUSIC + iSS + UrQMD in the work.

Report issue for preceding element

The paper is organised as follows. We start with a brief introduction in Section I. Section II discusses the event generation using the hybrid model and the two-particle Q-cumulant method. In Section III, we present our results with necessary discussions. Section IV summarises the study with a brief outlook.

Report issue for preceding element

II Event Generation and Methodology

Report issue for preceding element

II.1 IP-Glasma + MUSIC + iSS + UrQMD

Report issue for preceding element

In this work, we employ a hybrid framework – IP-Glasma+MUSIC+iSS+UrQMD model to simulate the evolution of the ultra-relativistic p–O and O–O collisions, as it gives a good description of particle production and flow across small to large collisions McDonald:2016vlt . Here, the initial conditions of the collisions are described by the impact-parameter-dependent Glasma (IP-Glasma) model, MUSIC handles the hydrodynamic evolution of the fireball, the iSS package does the particlization from the MUSIC hypersurface, and the UrQMD transport model carries out the hadronic interactions. These four stages are briefly discussed below, along with the details of the parameters/settings used in our study.

Report issue for preceding element

II.1.1 IP-Glasma: Pre-equilibrium

Report issue for preceding element

IP-Glasma is a theoretical framework based on Color Glass Condensate (CGC) effective field theory, which simulates the early stage dynamics of gluon fields during relativistic collisions Schenke:2012wb ; Schenke:2012hg ; McLerran:1993ni ; McLerran:1993ka . It is an integration of two frameworks McDonald:2016vlt : the impact-parameter-dependent dipole saturation model (IP-Sat) model Bartels:2002cj ; Kowalski:2003hm , which handles the initial nuclear configurations, and the Glasma framework Krasnitz:1998ns ; Krasnitz:1999wc ; Krasnitz:2001qu ; Krasnitz:2000gz , which evolves these configurations post-collision till thermalization. The two nuclei, described as two sheets of CGC fields, are boosted and made to collide with each other at $\tau=0$. Further, the classical Yang-Mills equations, which model the glasma evolution, are solved numerically using a lattice approach (lattice size, L = 14, lattice spacing, a = 0.02 fm) Schenke:2020mbo , which generates the initial energy-momentum tensor ($T_{YM}^{\mu\nu}$) of the system. The model accounts for the two essential sources of event-by-event fluctuations: 1) the random spatial distribution of nucleons inside the colliding nuclei 2) the fluctuating color charge densities within each nucleon. The color charge fluctuations are described by the IP-Sat model, where the sub-nucleonic fluctuations are modelled as three Gaussian hotspots per nucleon. The nucleonic positions are randomly sampled for each event, according to the Woods-Saxon distribution with the parameter settings being the same as those in Ref. Prasad:2024ahm .

Report issue for preceding element

In addition, since we aim to understand the effect of $\alpha$-clustering on the flow coefficients, we also sample nucleons into an $\alpha$-clustered geometry such that the four $\alpha$-clusters of the ${}^{16}\rm O$ nucleus are at the corners of a regular tetrahedron, while the nucleons inside each cluster are distributed according to Woods-Saxon profile. The values of the parameters chosen for these distributions can be found in Ref. Prasad:2024ahm . Finally, the output of IP-Glasma is a fluctuating energy-momentum tensor computed at $\tau_{\rm switch}=0.4~{}\rm fm$, which is fed to MUSIC simulations for hydrodynamic evolution.

Report issue for preceding element

II.1.2 MUSIC: Hydrodynamics

Report issue for preceding element

Using viscous relativistic hydrodynamics, MUSIC evolves the energy-momentum tensor obtained from IP-Glasma at the thermalization time $\tau_{0}=0.4$ fm, under the assumption of local thermal equilibrium. The evolution incorporates shear and bulk viscous effects, solving the conservation law $\partial_{\mu}T^{\mu\nu}$ Schenke:2010nt ; Schenke:2010rr ; Paquet:2015lta . The simulation is performed in a (2+1)D boost-invariant set-up, using an equation-of-state parametrization "s95p-v1.2" that interpolates between lattice-QCD and hadron resonance gas Huovinen:2009yb . A constant shear viscosity to entropy ratio of $\eta/s=0.12$ is used along with a temperature-dependent bulk viscosity $\zeta/s(T)$ as described in Ref. Schenke:2020mbo . Further, the Kurganov-Tadmor numerical algorithm solves the hydrodynamic equations Kurganov:2000ovy ; Jeon:2015dfa . The evolution continues until the local energy density falls to a switching energy density, $e_{\rm switch}=0.18~{}\rm GeV/fm^{3}$ Schenke:2020mbo , at which a freeze-out hypersurface is constructed, marking the end of the hydrodynamic description.

Report issue for preceding element

II.1.3 iSS: Particlization

Report issue for preceding element

The hydrodynamic freeze-out hypersurface data produced by MUSIC acts as the input for the iSS (iSpectraSampler) Shen:2014vra ; Denicol:2018wdp , which converts the fluid-like medium into hadrons, using the Cooper-Frye formula Cooper:1974mv ; Dusling:2009df ; Schenke:2020mbo . Particle sampling is done based on the local flow velocity, temperature, and viscous corrections, reflecting the momentum and spatial distributions of particles emitted at freeze-out. iSS allows oversampling; hence, to increase statistical precision without re-running hydrodynamics, a finite number of events can be sampled from each MUSIC hypersurface for p–O(O–O) collisions at $\sqrt{s_{\rm NN}}=9.61(7)$ TeV. The total number of hadronic freezeout performed per IPGlasma + MUSIC event i.e. $N_{\rm sample}=$ 200 in our study.

Report issue for preceding element

II.1.4 UrQMD: Hadronic cascade

Report issue for preceding element

Particles sampled from iSS are propagated using Ultra-relativistic Quantum Molecular Dynamics (UrQMD) microscopic transport model, with its default settings, which provides a realistic simulation of final-stage hadronic interactions such as elastic and inelastic hadronic scatterings, resonance decays, strong decays, and baryon-antibaryon annihilations Bass:1998ca ; Bleicher:1999xi . This model, which functions by solving the Boltzmann transport equation using Monte Carlo techniques, handles hadrons up to 2.25 GeV in mass and performs a dynamical freeze-out for different particle species Schenke:2020mbo . The model outputs the final-state four momenta and PIDs, which can be stored for subsequent examination and analysis.

Report issue for preceding element

II.2 Two-particle Q-Cumulant method

Report issue for preceding element

One of the important signatures of the hydrodynamic behavior of the system formed in relativistic nuclear collisions is given by collectivity. The studies of anisotropic flow coefficients are thus crucial to understand the collective behaviour of the system formed in nuclear collisions. Anisotropic flow is quantified using the coefficients of the Fourier expansion of the azimuthal distribution of the particles in the final state, given as follows Voloshin:1994mz ,

Report issue for preceding element

$$\frac{dN}{d\phi}\propto 1+\sum_{n=1}^{\infty}2v_{n}\cos[n(\phi-\psi_{n})]\ .$$ (1)

Here, $\phi$ is the azimuthal angle, $\psi_{n}$ is the $n$^th harmonic symmetry plane angle, and $v_{n}$ are the coefficients of the Fourier expansion, also known as the anisotropic flow coefficients. $v_{2}$, $v_{3}$, etc., are respectively called as elliptic flow, triangular flow, etc, and can be calculated as, $v_{n}=\langle\cos[n(\phi-\psi_{n})]\rangle$. The estimation of $\psi_{n}$ is not trivial in experiments, thus, we use the two-particle Q-cumulant method to estimate the flow coefficients Bilandzic:2010jr ; Zhou:2015iba . The estimation of $n$^th order flow coefficient with Q-cumulant method requires the $n$^th order flow vectors ($Q_{n}$), defined as follows,

Report issue for preceding element

$$Q_{n}=\sum_{j=1}^{M}e^{in\phi_{j}}\ ,$$ (2)

where $M$ is the number of charged particles in an event. As mentioned in Section II, in this study, we have used a hybrid of IPGlasma + MUSIC + iSS + UrQMD models, where each IPGlasma + MUSIC event is passed 200 times through iSS + UrQMD. To reduce the random fluctuations arising due to sampling of a finite number of particles, we define the flow vector of a super event ($Q^{\rm SE}_{n}$) as follows McDonald:2016vlt ,

Report issue for preceding element

$$Q^{\rm SE}_{n}=\sum_{j=1}^{N_{\rm sample}}\sum_{k=1}^{M_{j}}e^{in\phi_{jk}}\ .$$ (3)

Here, $M_{j}$ is the multiplicity of the $j$^th hadronic sample, and $\phi_{jk}$ is the azimuthal angle of the $k$^th particle in this sample. To reduce the contribution of nonflow, one can make two super-sub events, $A$ and $B$ respectively, separated by a pseudorapidity gap, $|\Delta\eta|>1$. Corresponding flow vector can be denoted as, $Q^{A}_{n}$ and $Q^{B}_{n}$ with multiplicities $M_{A}$ and $M_{B}$ respectively. Consequently, the two-particle correlation is given by,

Report issue for preceding element

$$\langle 2\rangle_{\Delta\eta}=\frac{Q_{n}^{A}\cdot Q_{n}^{B*}}{M_{A}\cdot M_{B}}\ .$$ (4)

The corresponding two-particle Q-cumulant can be calculated using the following expression.

Report issue for preceding element

$$c_{n}\{2,|\Delta\eta|\}=\langle\langle 2\rangle\rangle_{\Delta\eta}\ ,$$ (5)

where, $\langle\langle\dots\rangle\rangle$ denotes the average over the super events. Finally, the anisotropic flow coefficients can be calculated using the following expression.

Report issue for preceding element

$$v_{n}\{2,|\Delta\eta|\}=\sqrt{c_{n}\{2,|\Delta\eta|\}}\ .$$ (6)

II.3 Estimation of Spatial Geometry from IP-Glasma

Report issue for preceding element

One can quantify the initial state spatial anisotropy of the collision overlap region using eccentricity ($\epsilon_{2}$), triangularity ($\epsilon_{3}$), etc., as follows Petersen:2010cw ; Prasad:2022zbr ,

Report issue for preceding element

$$\epsilon_{n}=\frac{\sqrt{\langle{r^{n}\cos(n\phi_{\text{part}})}\rangle^{2}+\langle{r^{n}\sin(n\phi_{\text{part}})}\rangle^{2}}}{\langle{r^{n}}\rangle}\ ,$$ (7)

where $r$ and $\phi_{\rm part}$ denote the radial distance and azimuthal angle of the participant nucleons from the center in polar coordinates. However, with IP-Glasma, we can estimate the spatial anisotropy of the fluid just before the start of the MUSIC hydrodynamics. Here we have the energy density, $\varepsilon(x,y)$, in each fluid cell of dimension (${\textrm{d}}x,{\textrm{d}}y$) at ($x,y$) from the center ($0,0$). Thus, Eq. (7) can be modified as follows Schenke:2013aza .

Report issue for preceding element

$$\epsilon_{n}=\frac{\sqrt{\big{(}\iint_{A}{r^{n}\varepsilon(x,y)\cos(n\phi_{\rm part})}{\textrm{d}}x\;{\textrm{d}}y\big{)}^{2}+\big{(}\iint_{A}{r^{n}\varepsilon(x,y)\sin(n\phi_{\rm part})}{\textrm{d}}x\;{\textrm{d}}y\big{)}^{2}}}{\iint_{A}{r^{n}\varepsilon(x,y){\textrm{d}}x\;{\textrm{d}}y}}$$ (8)

Here, the integral runs over all the fluid cell elements in the transverse area $A$. $r=\sqrt{x^{2}+y^{2}}$ is the radial distance of the fluid cell and $\phi_{\rm part}=\tan^{-1}\big{(}\frac{y}{x}\big{)}$ is the corresponding azimuthal angle. From here onwards, we shall show the results of eccentricity and triangularity from IP-Glasma using Eq. (8).

Report issue for preceding element

III Results and Discussions

Report issue for preceding element

In this section, the generated initial eccentricities via $\epsilon_{n}$, and measured final state azimuthal anisotropies via $v_{n}$ ($n=1,~{}2$) for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV simulated using the IP-Glasma+MUSIC+iSS+UrQMD model, are presented. The centrality selection is performed via geometrical slicing using the impact-parameter distribution obtained from IP-Glasma. Using the conventional definition of centrality, the most central collisions (e.g. 0–10%) are the almost head-on collisions for which the impact parameter values are close to zero, and the number of participant nucleons, $N_{\rm part}$, is close to the total number of nucleons in both the colliding nuclei.

Report issue for preceding element

Figure 1: Charged particle multiplicity density, $\langle dN_{\rm ch}/d\eta\rangle_{|\eta|<0.5}$, normalized to the average number of participating nucleon pairs, $\langle N_{\rm part}\rangle/2$, plotted as a function of $\langle N_{\rm part}\rangle$ in p–O, and O–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and $7$ TeV, respectively.

Report issue for preceding element

Figure 1 shows the charged particle multiplicity density, $\langle dN_{\rm ch}/d\eta\rangle$, at midrapidity, $|\eta|<0.5$, normalized to the average number of participating nucleon pairs, $\langle N_{\rm part}\rangle/2$, plotted as a function of $\langle N_{\rm part}\rangle$ in p–O and O–O collisions at $\sqrt{s_{\rm NN}}=9.61$ and $7$ TeV, respectively, using the hybrid model used in this work. The presence of $\alpha$-clusters in ¹⁶O yields more charged particles towards high-$N_{\rm part}$, and significantly lower yield in low-$N_{\rm part}$ as compared to the same numbers from the collisions of Woods-Saxon type ¹⁶O nuclei in O–O collisions. This is understood as a fact that the $\alpha$-cluster density profile samples the nucleons inside a tight radius, making the nuclear geometry more compact than the Woods-Saxon type sampling, which has an elongated, smooth tail in its probability distribution that can go up to large radial distances. Therefore, more matter is squeezed up at a shorter distance from the center of the nucleus for the $\alpha$-clustered density profile than the Woods-Saxon type profile. This possibly leads to a higher $\langle N_{\rm coll}\rangle$ towards the central collisions (high-$N_{\rm part}$) and explains the observed higher yield than the Woods-Saxon case. In p–O collisions, we see a slightly different trend. Here, $\langle dN_{\rm ch}/d\eta\rangle/(\langle N_{\rm part}\rangle/2)$ for $\alpha$-cluster case is higher for the intermediate-$\langle N_{\rm part}\rangle$ (midcentral collisions). This is because, in the $\alpha$-cluster profile, most of the matter is concentrated near the RMS radius of the $\alpha$-particle, while the center is mostly empty, unlike the Woods-Saxon profile, where the nuclear density is higher at the center. Thus, p–O collisions near mid-central class lead to a higher $N_{\rm coll}$ in the clustered nuclear profile, while the central p–O collisions with clustered structure of Oxygen (high-$\langle N_{\rm part}\rangle$) have relatively smaller yield compared to the Woods-Saxon profile. Interestingly, the $\langle dN_{\rm ch}/d\eta\rangle/(\langle N_{\rm part}\rangle/2)$ of O–O for both nuclear profiles shows a qualitative sharp rise towards $N_{\rm part}\lesssim 2A$, where $A$ is the mass number of the colliding nuclei. In Ref. ALICE:2018cpu , it is shown that for a specific $N_{\rm part}$, the value of $\langle dN_{\rm ch}/d\eta\rangle/(\langle N_{\rm part}\rangle/2)$ in Xe–Xe is higher as compared to Pb–Pb collisions. This indicates that the number of binary collisions in Xe–Xe is higher than Pb–Pb for similar values of $N_{\rm part}$ ALICE:2018cpu ; ALICE:2015juo . Similarly, in Fig. 1, a higher value of $\langle dN_{\rm ch}/d\eta\rangle/(\langle N_{\rm part}\rangle/2)$ in p–O collisions than O-O near $\langle N_{\rm part}\rangle\simeq 6$ for both the nuclear density profiles is also attributed to a higher $\langle N_{\rm coll}\rangle$ in p–O collisions as compared to O–O collisions.

Report issue for preceding element

Figure 2: Upper panel shows the average eccentricity and triangularity as a function of impact-parameter based centrality class (%) for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV (left) and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV (right) for Woods-Saxon and $\alpha-$clustered nuclear density profiles, using IPGlasma. The ratios of $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$ from Woods-Saxon to $\alpha-$clustered nuclear density profile are shown in the lower panel.

Report issue for preceding element

III.1 Initial spatial anisotropy

Report issue for preceding element

Figure 2 shows the collision centrality dependence of IP-Glasma generated initial geometrical eccentricities ($\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$) in p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV (left) and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV (right), with both Woods-Saxon and $\alpha$-clustered type nuclear density for ¹⁶O. The bottom panels show the ratio of eccentricities of the same order between Woods-Saxon and $\alpha$-clustered density profiles.

Report issue for preceding element

In case of p–O collisions, $\langle\epsilon_{2}\rangle$ increases from central to peripheral collisions, for both Woods-Saxon and $\alpha$-cluster type density profiles. The value of $\langle\epsilon_{2}\rangle$ towards the peripheral collisions is around $25\%$ higher than the most central p–O collisions for both types of nuclear density profiles. Although both the density profiles show similar trends for the centrality dependence of $\langle\epsilon_{2}\rangle$, the values of $\langle\epsilon_{2}\rangle$ from $\alpha$-cluster type density profile dominate over the Woods-Saxon density profile across all the centrality bins. A similar centrality dependence is also observed for $\langle\epsilon_{3}\rangle$, with $\alpha$-cluster type density profile dominating over the Woods-Saxon density profile across all the centrality bins, except towards the most peripheral case, where the values of $\langle\epsilon_{3}\rangle$ from both type of density profiles converge. This behavior is also represented in the bottom ratio plot of the left panel, where the ratio is less than one for both $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$. Clearly, $\langle\epsilon_{2}\rangle$ is quantitatively greater than $\langle\epsilon_{3}\rangle$ for p–O collisions, consistent with the observations from heavy-ion collisions such as p–Pb and Pb–Pb collisions ALICE:2019zfl ; ALICE:2016ccg .

Report issue for preceding element

In the right panel of Figure 2 O–O collisions, involving Woods-Saxon density profile, $\langle\epsilon_{2}\rangle$ shows an increasing trend from central to peripheral collisions; however, this increase is steadier than in p–O collisions and it leads to a rise of more than $40\%$ of its value towards the peripheral collisions compared to the most central case. On the other hand, $\langle\epsilon_{2}\rangle$ for the $\alpha$-cluster case dominates over the Woods-Saxon profile from central to mid-central collisions, yet approaches similar values towards the peripheral case. Thus, a noticeable nuclear density profile dependence of $\langle\epsilon_{2}\rangle$ is observed, with the eccentricity in the 0–70% centrality class being clearly higher for $\alpha$-cluster profile than that for the Woods-Saxon nuclear density profile Prasad:2024ahm . Further, $\langle\epsilon_{3}\rangle$ for O–O collisions increases from central to peripheral collisions for the Woods-Saxon profile as in p–O collisions, while for the case of $\alpha$-cluster profile, it shows a decreasing-and-saturating trend with the centrality. In the most central O–O collisions, $\langle\epsilon_{3}\rangle$ from $\alpha$-clustered nuclear density profile is observed to be $\approx$ $30\%$ higher than that of the Woods-Saxon case. This feature is also visible in the AMPT results for O–O collisions at $\sqrt{s_{\rm NN}}$ = 7 TeV Behera:2023nwj , and should arise as a consequence of the overlap of two triangular sides of the colliding tetrahedral oxygen nuclei with the $\alpha$-clustered nuclear density profile. The value of $\langle\epsilon_{3}\rangle$ from Woods-Saxon dominates over the $\alpha$-clustered profile in the off-central O–O collisions, and this can be quantitatively analyzed from the bottom panel of the right plot of Fig. 2.

Report issue for preceding element

III.2 Anisotropic flow coefficients

Report issue for preceding element

Figure 3: Elliptic and triangular flow calculated using the two-particle Q-cumulant method as a function of collision centrality for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV (left) and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV (right) from the IP-Glasma + MUSIC + iSS + UrQMD framework. Results include both Woods-Saxon and $\alpha-$clustered type ${}^{16}\rm O$ nuclei. The ratio of $v_{\rm 2}\{2,|\Delta\eta|>1.0\}$ and $v_{\rm 3}\{2,|\Delta\eta|>1.0\}$ from Woods-Saxon to $\alpha-$cluster density profile is shown in the lower panels.

Report issue for preceding element

Using the two-particle Q-cumulant method, we calculated the elliptic and triangular flow for all charged particles in $|\eta|<$ 2.5, in p–O and O–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and $\sqrt{s_{\rm NN}}=7$ TeV respectively. A pseudorapidity gap of $|\Delta\eta|>1.0$ is imposed between the particle pairs to remove short-range jet-like correlations, if any. Note: In addition to the pseudorapidity gap, $|\Delta\eta|>1$, estimation with $|\Delta\eta|>2$ i.e. $v_{\rm n}\{2,|\Delta\eta|>2\}$ was also performed, so as to ensure maximum reduction of non-flow contributions. Since the results showed little or no difference from the estimated $v_{\rm n}\{2,|\Delta\eta|>1\}$, we resort to estimation with $|\Delta\eta|>1$.

Report issue for preceding element

In Fig. 3, both $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ as a function of collision centrality are shown, for both Woods-Saxon and $\alpha$-cluster type ${}^{16}\rm O$ nuclei in p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV, from the IP-Glasma + MUSIC + iSS + UrQMD framework. As one can see from the left plot of Fig. 3, the nuclear density profile dependence for $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ is almost negligible in p–O collisions. For both nuclear density profiles, $v_{2}\{2,|\Delta\eta|>1.0\}$ drops gradually from the most central to peripheral p–O collisions, with the $\alpha$-cluster profile showing slightly higher values than the Woods-Saxon profile in the 10–20% centrality class, appearing as a small bump-like structure. On the other hand, $v_{3}\{2,|\Delta\eta|>1.0\}$ shows no significant centrality dependence, and the trends for both Woods-Saxon and $\alpha$-cluster profile cases almost overlap with one another.

Report issue for preceding element

On the contrary, for O–O collisions, both $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ have strong dependence on the choice of nuclear density profiles. $v_{2}\{2,|\Delta\eta|>1.0\}$ for the Woods-Saxon case has a smooth increase from central to midcentral collisions and then a relatively gradual decrease towards the peripheral collisions. For the O–O collisions with $\alpha$-clustered profile for ${}^{16}\rm O$ nuclei, the initial rise until midcentral collisions and the following drop towards peripheral collisions are much steeper compared to the Woods-Saxon case. In fact, we see in the $\alpha$-cluster case that the $v_{2}\{2,|\Delta\eta|>1.0\}$ attains a peak at 10–20% centrality class ($\approx$ 30% higher than the corresponding $v_{2}$ from Woods-Saxon profile). This is also similar to the small bump observed in the case of p–O collisions for the same centrality of collisions. However, as one moves from mid-central to peripheral O–O collisions, the magnitude of $v_{2}\{2,|\Delta\eta|>1.0\}$ steeply falls to almost half of its peak value, with $v_{2}\{2,|\Delta\eta|>1.0\}$ from Woods-Saxon case starting to dominate over the elliptic flow from the $\alpha$-cluster profile case.

Report issue for preceding element

In addition, $v_{3}\{2,|\Delta\eta|>1.0\}$ has a smooth decrease from the central to peripheral O–O collisions for both types of nuclear density profiles studied in this work; the $\alpha$-cluster case has a higher value of $v_{3}\{2,|\Delta\eta|>1.0\}$ for the most central collisions compared to the same from the Woods-Saxon case. However, for the rest of the centrality bins, the Woods-Saxon type profile leads the value of $v_{3}\{2,|\Delta\eta|>1.0\}$.

Report issue for preceding element

The major contribution to the nuclear density profile dependence of $v_{\rm n}$ arises from the initial state eccentricities. For instance, $v_{2}$ in the (10–20%) centrality and $v_{3}$ in the (0–10%) centrality class for the $\alpha$-clustered profile in O–O collisions are reflections of the corresponding $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$. However, the effect of the centrality dependence of $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$ is not very well carried forward to the final state, beyond mid-central collisions, owing to the smaller size and lifetime of the system formed. However, for the case of O–O collisions, the differences in the initial spatial anisotropy between the Woods-Saxon and $\alpha$-cluster profile propagate nicely to the final state azimuthal anisotropy, as visible from the bottom ratio panels of Fig. 2 and Fig. 3.

Report issue for preceding element

In a nutshell, the distinction between the $\alpha$-clustered and Woods-Saxon nuclear density profiles in p–O collisions is not as prominent as it is for the O–O collisions in the measurements of anisotropic flow coefficients, i.e., $v_{2}$ and $v_{3}$. These differences might arise due to the difference in their system size, since p–O collisions are asymmetric, and form a smaller system as compared to O–O collisions. In addition, proton is not an ideal probe to capture the differences in the nuclear density profiles of ¹⁶O, which one should generally expect from a much bigger system in O–O collisions. This is interesting, since our expectation was that the smaller-sized bombarding proton would provide better resolution.

Report issue for preceding element

III.3 Multiplicity dependence of $v_{\rm n}$ and $v_{\rm n}$/$\langle\epsilon_{\rm n}\rangle$

Report issue for preceding element

Figure 4: $v_{\rm n}\{2,|\Delta\eta|>1.0\}$ (left panel) and $v_{\rm n}\{2,|\Delta\eta|>1.0\}$/$\langle\epsilon_{n}\rangle$ (right panel) as a function of average charged-particle multiplicity ($\langle N_{\rm ch}\rangle_{|\eta|<2.5}$) measured in $|\eta|<2.5$ for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model, for Woods-Saxon and $\alpha-$cluster density profiles of ${}^{16}\rm O$ nuclei.

Report issue for preceding element

One of the major motivations for the future p–O and O–O collisions at the LHC comes from the fact that these collision systems form a perfect system size to fill the multiplicity gap between the collision systems, pp, p–Pb, and Pb–Pb. Or in other words, studying p–O and O–O collisions (where produced multiplicity is expected to be intermediate to that of other systems mentioned above) helps us in understanding how the signatures of hydrodynamic behaviour (or any other signatures of QGP) undergo a transition as we move down from high to low multiplicity, analogously, large to small systems Brewer:2021kiv . It is already quite well known that the particle multiplicity plays a crucial role in transforming the initial eccentricities to the final-state flow. Compared to a low multiplicity event, a collision achieving a larger multiplicity would naturally imply that the hydrodynamic phase during its evolution is longer, which results in an efficient (complete) transformation of initial spatial anisotropies to final-state momentum anisotropies. As we observed in Fig. 3, O–O collisions show significant dependence of $v_{\rm n}$ on the initial clustering, and the initial-state effects are reflected better in O–O than in p–O collisions. In this context, studying $\langle\epsilon_{\rm n}\rangle$ and $v_{\rm n}$ for their charged particle multiplicity dependence would help us discern where the system behaviour changes, in turn enabling us to quantify the hydrodynamic limit.

Report issue for preceding element

Shown in the left plot of Fig. 4 is the elliptic and triangular flow as a function of average charged-particle multiplicity in the pseudorapidity range $|\eta|<2.5$ and $p_{\rm T}>0.3$ GeV/$c$ ($\langle N_{\rm ch}\rangle_{|\eta|<2.5}$), for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV, generated using IP-Glasma + MUSIC + iSS + UrQMD framework. Since the $\langle N_{\rm ch}\rangle$ is calculated based on the impact parameter-based event classes, Fig. 4 is visually the mirror image of Fig. 3. However, the purpose of Fig. 4 is to understand the multiplicity dependence of $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ for two different collision systems i.e. p–O and O–O, for both $\alpha$-clustered and Woods-Saxon nuclear density profiles of colliding ${}^{16}\rm O$, to be in line with the experimental results which are presented as a function of multiplicity.

Report issue for preceding element

As can be seen, $v_{2}\{2,|\Delta\eta|>1.0\}$ has the maximum value at $\langle N_{\rm ch}\rangle\simeq$ 600 for O–O collisions involving $\alpha$-clustered nuclear density profile, while this sharp peak is not observed in case of Woods-Saxon profile for O–O, as already discussed earlier. If compared with the results in Ref. ALICE:2019zfl , $v_{2}\{2,|\Delta\eta|>1.0\}$ achieved by the (10-20%) central O–O collisions involving compact $\alpha$-clusters, is comparable to the elliptic flow attained in mid-central Pb–Pb collisions. Now, as the particle multiplicity in the desired pseudorapidity ($\eta$) range falls down by an order for O–O collisions ($\langle N_{\rm ch}\rangle\simeq$ 75), the $v_{2}\{2,|\Delta\eta|>1.0\}$ is reduced by 40%. Interestingly, even in the highest multiplicity p–O collisions with $\alpha$-clustered ${}^{16}\rm O$ nuclei (where $\langle N_{\rm ch}\rangle\simeq$ 100), we see a tiny bump like structure in the trend for $v_{2}\{2,|\Delta\eta|>1.0\}$, similar to the observation in O–O collisions with $\alpha$-cluster profiles. Additionally, in the case of the Woods-Saxon profile, the observation of a gradual decrease of $v_{2}\{2,|\Delta\eta|>1.0\}$ from high to low multiplicity classes, with no abrupt jumps, is common for both O–O and p–O collisions. In the case of triangular flow, $v_{3}\{2,|\Delta\eta|>1.0\}$ obtained in high to intermediate multiplicity p–O collisions is very similar to that obtained in the lowest multiplicity O–O collisions, which is expected from Ref. ALICE:2019zfl . Though we see a moderate dependence on nuclear density profiles for $v_{3}\{2,|\Delta\eta|>1.0\}$ in O–O collisions, there is no significant influence of $\alpha$-clustering on $v_{3}\{2,|\Delta\eta|>1.0\}$ in the case of p–O collisions.

Report issue for preceding element

Scaling $v_{\rm n}$ by $\langle\epsilon_{\rm n}\rangle$ is a quantified way to understand how much of the initial spatial anisotropy is converted to the final state momentum anisotropy and hence is believed to probe the transport properties of the medium, and the applicability of hydrodynamic models to the system under study. In the right plot of Fig. 4, we thus present $v_{2}$/$\langle\epsilon_{2}\rangle$ and $v_{3}$/$\langle\epsilon_{3}\rangle$ as a function of $\langle N_{\rm ch}\rangle_{|\eta|<2.5}$ for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV, generated using IP-Glasma + MUSIC + iSS + UrQMD framework, for Woods-Saxon and $\alpha$-cluster density profiles of ${}^{16}\rm O$ nucleus. The scaled ratios now reveal an interesting trend: the increase of $v_{3}$/$\langle\epsilon_{3}\rangle$ with $\langle N_{\rm ch}\rangle_{|\eta|<2.5}$ experiences a smooth transition from p–O to O–O collisions where the rate of increment seems to be alike for both collision systems. Similarly, $v_{2}$/$\langle\epsilon_{2}\rangle$ also increases with $\langle N_{\rm ch}\rangle_{|\eta|<2.5}$; the slight bump-like structure observed in the trends of $v_{2}$ in p–O collisions still persists in the ratio $v_{2}$/$\langle\epsilon_{2}\rangle$, while it is smoothened out in the case of O–O collisions. However, the nuclear density profile dependence is no longer seen to remain once scaling of $v_{n}$ with $\langle\epsilon_{\rm n}\rangle$ is performed, except for the $v_{2}$/$\langle\epsilon_{2}\rangle$ case of p–O collisions. This is a testimony that, in this work, the choice of nuclear density profile has little to no effect on the medium formed.

Report issue for preceding element

III.4 Impact parameter dependence of $v_{n}$

Report issue for preceding element

Figure 5: $v_{\rm 2}\{2,|\Delta\eta|>1.0\}$ and $v_{\rm 3}\{2,|\Delta\eta|>1.0\}$ as a function of $\rm b/\rm r_{\alpha}$ (where, $\rm r_{\alpha}=1.676$ fm) for p–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IP-Glasma + MUSIC + iSS + UrQMD model, for Woods-Saxon and $\alpha-$cluster density profiles of colliding ${}^{16}\rm O$ nucleus.

Report issue for preceding element

In Figs. 3 and 4, we observed that the effect of clustered structure is mostly highlighted in the high-multiplicity or most central regions. Thus, it is important to quantify the regions as a function of impact parameter ($b$) that is capable of highlighting the effects in the final state flow coefficients for the choice of nuclear profile ($\alpha$-clustered versus Woods-Saxon). This would help us understand whether the size of the $\alpha$-particle (radius of $\alpha$-particle, $\rm r_{\alpha}=1.676$ fm) has any role to play. In Fig. 5, we show $v_{n}\{2,|\Delta\eta|>1.0\}$ as a function of impact parameter scaled with the RMS radius of $\alpha$-particle (${}^{4}\text{He}$ nucleus) ($b/r_{\alpha}$) for both p–O and O–O collisions at $\sqrt{s_{\rm NN}}=9.61$ TeV and $7$ TeV, respectively, using IP-Glasma + MUSIC + iSS + UrQMD. Figure 5 would aid in quantitatively conceiving the effect of the clustered structure in the final state flow observables as a function of $b/r_{\alpha}$. Interestingly, in O–O collisions, one finds that the dependence of elliptic and triangular flow on nuclear density profile is maximised near $b/r_{\alpha}\simeq 1$. Here, $v_{2}\{2,|\Delta\eta|>1.0\}$ near $b=r_{\alpha}$ shows a peak like structure for the $\alpha$-cluster case and is absent for the Woods-Saxon profile, the effect of which originates from the initial eccentricity, as shown in Fig. 2. Similarly, a higher $v_{3}\{2,|\Delta\eta|>1.0\}$ for $b/r_{\alpha}<1$ is also attributed to the $\alpha$-clustered case which is absent for the Woods-Saxon profile. Additionally, in Fig. 5 we observe that, for O–O collisions towards the higher impact parameter values, the Woods-Saxon nuclear profile have larger value of $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ as compared to the $\alpha$-clustered structure, also reflected in the bottom panel where the ratio of Woods-Saxon to $\alpha$-clustered structure is larger than one. This is similar to the observations made in the right panel of Fig. 3 and 4, where $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ in mid-central and peripheral O–O collisions deviate significantly from that of $\langle\epsilon_{2}\rangle$ and $\langle\epsilon_{3}\rangle$, respectively, in Fig. 2. This could be attributed to the combined effects due to the smaller lifetime of the fireball towards the mid-central or peripheral collisions and the difference in the medium response to the evolution of $v_{2}$ from $v_{3}$. Additional contributions could arrive from the large fluctuations of anisotropic flow coefficients due to a smaller number of charged particles. However, due to small particle multiplicity in p–O collisions, no such explicit effects of clustered nuclear structure are observed throughout $b/r_{\alpha}$ regions. The variations of $v_{n}\{2,|\Delta\eta|>1.0\}$ with the choice of nuclear density profile are well reflected in the bottom ratio panel. Therefore, in brief from Fig. 5, it is clear that the effects of clustered density profile are well observed in the region $b/r_{\alpha}\lesssim 1$. The sought-after effects are not visible when the particle multiplicity is small, like in the p–O collisions.

Report issue for preceding element

IV SUMMARY

Report issue for preceding element

In this study, we have investigated the effect of the clustered nuclear structure of ${}^{16}\rm O$ nuclei through p–O and O–O collisions at the LHC energies using IP-Glasma + MUSIC + iSS + UrQMD. The final state anisotropic flow coefficients are studied and are compared with the corresponding spatial anisotropies of the initial collision overlap geometry. Our findings are summarized as follows:

Report issue for preceding element

1. For both p–O and O–O collision systems, we find that the $\langle\epsilon_{2}\rangle$ rises from central to peripheral collisions with a clear dominance of $\alpha$-cluster over the Woods-Saxon profile case. Similar is the trend for $\langle\epsilon_{3}\rangle$ except for O–O collisions with $\alpha$-clustering. Here, the $\langle\epsilon_{3}\rangle$ is found to be maximum in the 0–10% centrality class and falls thereafter in mid-central collisions.

   Report issue for preceding element

2. The centrality dependence of initial spatial anisotropies ($\langle\epsilon_{\rm n}\rangle$) is not effectively converted to final-state elliptic and triangular flow, especially in the mid-central to peripheral p–O and O–O collisions, owing to the less dense systems formed in the off-central collisions.

   Report issue for preceding element

3. The nuclear density profile seems to have little influence on $v_{2}$ and $v_{3}$ in p–O collisions, while it has a noticeable effect in the case of O–O collisions. In spite of this, $v_{2}\{2,|\Delta\eta|>1.0\}$ is found to achieve its maximum in the (10–20%) centrality class, for both p–O and O–O collisions involving $\alpha$-clustered ${}^{16}\rm O$.

   Report issue for preceding element

4. Once $v_{\rm n}$ is scaled with the corresponding $\langle\epsilon_{\rm n}\rangle$, the nuclear profile dependence vanishes completely for O–O collisions. This scaling also brings the multiplicity-dependent curve of $v_{\rm n}$/$\langle\epsilon_{\rm n}\rangle$ for p–O and O–O collisions to a single continuous line.

   Report issue for preceding element

5. The effects of $\alpha$-clustering in O–O collisions are observed to manifest well in the region $b/r_{\rm\alpha}\lesssim 1$, therefore it is comparable with the size of the ⁴He.

   Report issue for preceding element

In this work, we present a systematic study of how the $\alpha$-clustered nuclear structure of the colliding ${}^{16}\rm O$ nucleus in collision systems ranging from p–O to O–O collisions impacts the final-state medium anisotropy. This study can further be useful as a hydrodynamic prediction for future experimental investigations of p–O and O–O collisions at the LHC. A comparison with experimental data would help constrain the nuclear density profile for the oxygen nuclei and aid in tuning other models.

Report issue for preceding element