Anisotropic flow fluctuation as a possible signature of clustered nuclear geometry
in O–O collisions at the Large Hadron Collider

The primary goal of relativistic heavy-ion collisions at the RHIC and the LHC is to create a deconfined thermalized medium of partons, also known as Quark Gluon Plasma (QGP), which is believed to have existed a few microseconds after the Big Bang. However, due to the transient nature of QGP, it can not be observed directly in experiments. Instead, several indirect signatures from experimentally measurable observables can signify the presence of such a deconfined strongly interacting matter in heavy-ion collisions. One such observable is azimuthal anisotropy, which characterizes the collective behavior of the medium formed in heavy-ion collisions. It is quantified in terms of the Fourier expansion coefficients of azimuthal distribution of final state charged hadrons, as follows [1]:

Here, $\phi$ is the azimuthal angle, $\psi_{n}$ stands for the $n$^th order symmetry plane angle and $v_{n}$ being the $n$^th order flow coefficients. Here, $v_{2}$ quantifies the elliptic flow, $v_{3}$ gives the estimate for triangular flow, and so on. Both $v_{2}$ and $v_{3}$ depend upon the transport properties of the QCD medium formed during heavy-ion collisions. In addition, studies based on hydrodynamic calculations show that the flow coefficients are affected strongly by a change in the ratio of shear viscosity to entropy density ($\eta/s$) [2, 3]. Moreover, the sensitivity of flow coefficients to $\eta/s$ is large for higher-order harmonics, i.e., $v_{3}$ is more sensitive to $\eta/s$ as compared to $v_{2}$ [3]. In several studies, $v_{2}$ and $v_{3}$ are found to be associated with the initial eccentricities of the participating nucleons, where the values of anisotropic flow coefficients can change when the nuclear distribution is modified [6, 4, 5]. In addition, in Xe–Xe collisions, an observation of a larger value of $v_{2}$ in the most central collisions, as compared to that observed in Pb–Pb collisions, is anticipated due to a deformed nuclear structure of Xe nucleus [7, 8, 10, 9]. In a similar line, the nuclear deformation of Uranium, being a prolate shape nucleus, is reflected in a higher $v_{2}$, when compared to that observed in the most central collisions of Au nuclei, which is close to a spherical nucleus [11]. Similarly, one notices that the values of $v_{2}$ and $v_{3}$ vary with the change in the deformation parameters of Uranium in U–U collisions, and the effect is enhanced for $v_{2}$, as compared to $v_{3}$ [4, 5, 12].

Elliptic flow is one of the observables sensitive to the initial state effects in heavy-ion collisions [13]. However, the value of elliptic flow fluctuates from one event to another, which is a reflection of the fluctuation in the initial energy density and event-by-event participant nucleon distribution before the formation of QGP [14]. These fluctuations in the initial density profile of the participating nucleons can generate odd flow harmonics and lead to fluctuations in the harmonic flow coefficients [14]. Further, the fluctuations in the initial energy density can develop symmetry planes of different orders in various kinematic regions [15, 16, 17]. In other words, the initial density fluctuations can develop transverse momentum ($p_{\rm T}$) and pseudorapidity ($\eta$) dependence of $\psi_{n}$. Therefore, to comprehend the event-by-event initial density fluctuations and their impact on medium evolution, the study of anisotropic flow fluctuations becomes crucial. Anisotropic flow fluctuations are well explored in both experimental and theoretical frontiers of heavy-ion collisions [18, 19, 20, 21, 22, 23, 24]. The measurements by the ATLAS experiment hint at the non-Gaussian behavior of the flow fluctuations in Pb–Pb collisions [25]. Consequently, this observation has led to studies constraining their probability distribution functions [26]. However, the anisotropic flow fluctuations in small collision systems are not yet well understood. Moreover, one may conceive that for a small number of participants, the contribution to fluctuations can increase. Finally, if the evolution of anisotropic flow fluctuations has more than one contributor, then the results can vary significantly from the expectations.

In addition to the collective behavior of the QGP, several other signatures for the presence of a QGP medium have been observed in heavy-ion collisions. For the measurement of such signatures, the baseline measurements are obtained from proton-proton ($pp$) collisions, where the formation of a QCD medium is not anticipated. However, recent observations of strangeness enhancement [27], ridge-like structure [29, 28], radial flow effects [30, 31, 32], etc., in high multiplicity $pp$ collisions have highlighted the significance of studying the small collision systems from different aspects. In addition, recent measurements with ALICE confirm the observation of partonic collectivity in $pp$ and p–Pb collisions, where a characteristic grouping (within $\sim 1\sigma$) and splitting (within $\sim 5\sigma$) of elliptic flow for mesons and baryons at intermediate $p_{\rm T}$ is observed [33]. The LHC and RHIC have plans to inject Oxygen (¹⁶O) nuclei to perform p–O and O–O collisions [34, 35], which can bridge the multiplicity gap between $pp$, p–Pb and Pb–Pb collisions and may provide insight into the possible formation of QGP droplets in small systems. As already mentioned, the presence of collective transverse expansion of the medium (also known as collectivity) in ultrarelativistic collisions of nuclear matter is regarded as one of the key signatures of QGP. This leads to the appearance of long-range multi-particle azimuthal correlations in the final state. Through many extensive studies, it has been understood that this collective expansion can be modeled via relativistic hydrodynamics with dissipative effects, which is responsible for these long-range correlations observed in heavy-ion collisions. By comparing the hydrodynamics model predictions with experimentally observed azimuthal anisotropy, it confirms that the initial-state effects such as eccentricity, nuclear deformation, and density fluctuations in the colliding species get embedded in the final-state multi-particle correlations through this collective medium evolution and manifest as finite measurable flow coefficients. Based on these observations from heavy-ion collisions, the existence of long-range multi-particle correlations in high multiplicity events in small collision systems can strongly be associated with the formation of an early deconfined and locally thermalized partonic phase in small collision systems such as O–O collisions.

Furthermore, ¹⁶O is a stable nucleus having a double magic number and is theorized to possess clusters of $\alpha$ (⁴He) particles, where the $\alpha$-particles arrange themselves in the corners of a tetrahedron [36, 37, 38, 39, 40, 41, 42]. Figure 1 shows a pictorial representation of the nucleons inside an ¹⁶O nucleus for an $\alpha$-clustered structure (left) and non-clustered distribution of nucleons (right). With the RHIC geometry scan program, ¹⁶O collisions would be an interesting study which may unveil the inner geometry of ¹⁶O nuclei [12, 43, 44, 45]. In recent years, there have been several studies performed to understand the O–O collisions both at RHIC and LHC energies [51, 52, 50, 46, 47, 48, 6, 49, 53]. A few of them investigate the use of hydrodynamical models [54, 55, 56], and Glauber model calculations [46, 47, 48]. Some intend to study the parton energy loss and jet quenching effects in O–O collisions [57, 58]. Interestingly, there have been a few phenomenological studies that aim to establish the effect of the presence of an $\alpha$-cluster nucleus in the final state particle production [51, 52, 50, 60, 61, 62, 63, 6, 59, 64, 65, 66]. A few of these studies employ O–O collisions, while others employ O–Au, C–Au, O–Pb, and Ne–Pb collisions. Nevertheless, the presence of a clustered nuclear structure of ¹⁶O is reflected in the final state azimuthal anisotropy when compared to a Woods-Saxon nuclear density profile. The lattice QCD estimate of a minimum initial energy density requirement of 1 GeV/$\rm fm^{3}$ to form a deconfined partonic medium is conjectured to be fulfilled in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV across all centrality classes as shown in Ref. [49]. In Ref. [59], using a multi-phase transport model (AMPT) [67], authors have shown that an away side broadening is observed in the two-particle correlation function for an $\alpha$-clustered nucleus. In addition, an increased value of $v_{3}$, and $v_{3}/v_{2}$ for most central O–O collisions are also reported for an $\alpha$-clustered structure. A similar observation was reported in Ref. [51], where authors study the anisotropic flow in C–Au collisions with three $\alpha$-clusters of ¹²C nucleus. However, so far, no studies have shown the effect of change in nuclear density profile on event-by-event flow fluctuations. In this paper, for the first time, we report the behavior of azimuthal anisotropy and flow fluctuations in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using a hydrodynamically expanding system. For this study, we use a hybrid of IPGlasma initial condition, MUSIC hydrodynamics, iSS as a particlisation, and UrQMD as an afterburner for the hadronic cascade. Here, we compare results from an $\alpha$-clustered nuclear distribution with a Woods-Saxon nuclear distribution.

This paper is organized as follows. We start with a brief introduction and motivation to this study in Section I. We briefly discuss the event generation and methodology in Section II. Then, we present our findings and discuss the results in Section III. Finally, we summarise our findings with a brief outlook in Section IV.

In this section, we briefly introduce the hybrid simulation framework used to generate events, including a discussion on the multi-particle cumulant method for the anisotropic flow estimation.

In this section, we start with a discussion of the multiplicity selection, followed by discussions on the results for anisotropic flow coefficients. Finally, we discuss the multiplicity dependence of flow fluctuations.

Figure 4 shows the distribution of charged particles produced in the pseudorapidity region $-3.7<\eta<-1.7$ and $2.8<\eta<5.1$ (V0M) in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model for the Woods-Saxon (upper) and $\alpha$-cluster (lower) nuclear density profiles. The lower panel shows the ratio of normalized yields of V0M for Woods-Saxon and $\alpha$-cluster nuclear distributions. Also, different multiplicity classes are defined based on the percentile slices on the V0M distribution for the Woods-Saxon nuclear density profile. Here, for a fair comparison, we use the same cuts of V0M distribution for all the multiplicity classes for both the nuclear density profiles. The first multiplicity class, e.g., 0–10%, has the highest particle multiplicity, while the last multiplicity class, 50–90%, has the events having the lowest final state charged particle multiplicity, as shown in Fig. 4. Thus, one may infer that the 0–10% class refers to the most central collisions, while the last multiplicity class, 50–90% in this study, implies a peripheral collision. Further, one can refer to Fig. 8 in the Appendix .1 for the average value of the impact parameter corresponding to each multiplicity class for both Woods-Saxon and $\alpha$-cluster nuclear density profiles. In Fig. 4, one finds a higher yield for V0M for the O–O collisions having an $\alpha$-clustered nucleus. In addition, for a similar value of V0M, one finds a higher yield for the $\alpha$-clustered distribution of nucleons inside the ¹⁶O nuclei for V0M $>300$, which can be seen in the bottom ratio plot. In Ref. [49], the probability distribution of $\alpha$-clustered nuclear density profile of ¹⁶O nuclei is compared with Woods-Saxon and Harmonic-Oscillator nuclear density profiles. Here, one finds that the probability of finding nucleons in Harmonic-Oscillator and Woods-Saxon nuclear density profiles spreads over long distances from the center of ¹⁶O nuclei. In contrast, the probability of finding nucleons in the $\alpha$-cluster profile is limited to some finite distance from the center of the nucleus. This demonstrates the compactness of the $\alpha$-clustered structure of ¹⁶O nuclei, which may lead to increased energy density, as shown in Ref. [49]. A large energy density can lead to an increase in final state particle yield, as can be seen in Fig. 4.

Figure 5 shows $v_{2}\{2,|\Delta\eta|>1.0\}$ and $v_{3}\{2,|\Delta\eta|>1.0\}$ with a pseudorapidity gap, $|\Delta\eta|>1.0$, as a function of multiplicity class for Woods-Saxon and $\alpha$-cluster nuclear density profiles of ¹⁶O in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One notices a moderate multiplicity dependence on both elliptic and triangular flow in both Woods-Saxon and $\alpha$-cluster nuclear density profiles. The values of $v_{2}\{2,|\Delta\eta|>1.0\}$ for the $\alpha$-cluster nuclear density profile are higher than the Woods-Saxon nuclear density profiles throughout the multiplicity classes except for the 50–90% case, where the trend is reversed. This nuclear density profile dependence of $v_{2}\{2,|\Delta\eta|>1.0\}$ has significantly large effects from the initial state eccentricities, as shown in Fig. 9. Further, the value of elliptic flow is the largest for the 0–10% multiplicity class and decreases as the final state particle multiplicity decreases. In the 0–10% multiplicity class, the observed high values of elliptic flow can be attributed to the initial eccentricity fluctuations. In contrast, towards the low multiplicity regions, although the value of eccentricity increases, the continued decrease in the system size and lifetime takes over, restricting the complete transfer of initial eccentricity to the final state azimuthal anisotropy. This results in a smaller value of anisotropic flow coefficients towards the low multiplicity regions than for the high multiplicity events. However, the observed behavior of elliptic flow as a function of multiplicity class and nuclear density profile assuming a hydrodynamical expansion vary significantly from that observed in a kinetic theory-based model, such as AMPT [6]. In contrast, we observe an interesting behavior for $v_{3}\{2,|\Delta\eta|>1.0\}$ as a function of multiplicity class for both the nuclear density profiles. One finds that $v_{3}\{2,|\Delta\eta|>1.0\}$ decreases from 0–10% to 50–90% multiplicity classes, having similar slopes for the Woods-Saxon and $\alpha$-clustered nuclear density profiles within 20–90% multiplicity classes. This consistency in the triangular flow between two nuclear density profiles for the mentioned multiplicity class is consistent with the AMPT predictions, as shown in Ref. [6]. Interestingly, it can be observed from the lower panel of Figure 5, a sudden rise of $v_{3}\{2,|\Delta\eta|>1.0\}$ in the 0–10% multiplicity class for the $\alpha$-clustered in contrast to a Woods-Saxon density profile. These results are similar to the AMPT model calculations in Ref. [6]. We believe that the sudden rise in $v_{3}\{2,|\Delta\eta|>1.0\}$ for 0–10% multiplicity class of ¹⁶O nuclei having an $\alpha$-clustered nuclear density profile can be attributed to the presence of an initial tetrahedral arrangement of four $\alpha$-particles inside the ¹⁶O nucleus.

The upper panel of Fig. 6 shows the ratio $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ as a function of multiplicity class in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model for Woods-Saxon and $\alpha$-cluster nuclear density profiles. The bottom panel shows the ratio of $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ for Woods-Saxon to an $\alpha$-cluster nuclear density profile. Interestingly, when the ¹⁶O nuclei having the Woods-Saxon nuclear distributions collide, their elliptic and triangular flow values follow a modest dependence on final state multiplicity, as one can see in Fig. 5. Thus, the ratio $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ also exhibits a modest decrease with a decrease in particle multiplicity. However, one finds a notably stronger increase in $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ for 0–10% multiplicity class for the $\alpha$-cluster case, followed by a modest decrease in $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ as a function of multiplicity class. Since these effects originate from the initial state, as shown in Fig. 10, the stronger increase in the value of $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ in the 0–10% multiplicity class reflects the presence of a tetrahedral arrangement of four $\alpha$-clusters inside the ¹⁶O nuclei, giving rise to a large intrinsic triangular flow. Although one finds $v_{2}\{2,|\Delta\eta|>1.0\}$ for $\alpha$-cluster density profile is larger compared to that of Woods-Saxon nuclear density profile, this tetrahedral structure raises $v_{3}\{2,|\Delta\eta|>1.0\}$ even higher for the high-multiplicity events, which is also consistent with transport models such as AMPT, shown in Ref. [6]. Thus, the observation of a sudden rise of $v_{3}\{2,|\Delta\eta|>1.0\}$ in tetrahedral $\alpha$-clustered case can lead to an observation of a spike in $v_{3}\{2,|\Delta\eta|>1.0\}/v_{2}\{2,|\Delta\eta|>1.0\}$ for the highest charged particle multiplicity events and consequently, the observation of which in O–O collisions may signify the presence of a $\alpha$-clustered nuclear geometry of ¹⁶O nuclei.

Figure 7 (a) shows the evolution of elliptic flow using two- and four-particle Q-cumulant method as a function of multiplicity class for Woods-Saxon and $\alpha$-cluster nuclear density profiles in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One finds a small multiplicity dependence for $v_{2}\{2,|\Delta\eta|>1.0\}$ for both the nuclear density profiles within 0–30% multiplicity class. In contrast, using the four-particle cumulants method, we observe a significant charged particle multiplicity dependence of elliptic flow. The value of $v_{2}\{4\}$ for the $\alpha$-cluster case remains almost unchanged as a function of multiplicity class, the same trend as of $v_{2}\{2\}$. However, for the Woods-Saxon case, $v_{2}\{4\}$ maintains a consistent decreasing feature with the decrease in the particle multiplicity. Figure 7 (b) shows the mean value of elliptic flow ($\langle v_{2}\rangle$), evaluated using Eq. (21), for Woods-Saxon and $\alpha$-cluster nuclear density profiles in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. $\langle v_{2}\rangle$ is found to be larger for O–O collisions having an $\alpha$-cluster nuclear density profile as compared to a Woods-Saxon distribution. The multiplicity class dependence of $\langle v_{2}\rangle$ for both the nuclear density profiles is similar to $v_{2}\{4\}$, as $v_{2}\{2,|\Delta\eta|>1.0\}$ is nearly independent of multiplicity class in the 0–30% region. Figure 7 (c) shows a multiplicity class dependence of elliptic flow fluctuation ($\sigma_{v_{2}}$) for Woods-Saxon and $\alpha$-cluster nuclear density profiles in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using IPGlasma + MUSIC + iSS + UrQMD model. One finds that $\sigma_{v_{2}}$ for Woods-Saxon and $\alpha$-cluster nuclear density profiles have complementing multiplicity dependence, i.e., when going from the 0–10% to 20–30% multiplicity classes, $\sigma_{v_{2}}$ for Woods-Saxon increases while for an $\alpha$-cluster nuclei, it shows a decreasing trend. This makes one of the interesting distinctions between the two nuclear density profiles with the study of flow fluctuations. In Pb–Pb collisions, due to the absence of the elliptic geometry in the initial state, a larger contribution to the elliptic flow in the most central collisions comes from elliptic flow fluctuations; however, towards the mid-central collisions, the contribution of elliptic flow-fluctuations to elliptic flow decreases, as the elliptic geometry of the nuclear participants leads the elliptic flow contribution [98, 102, 103]. Interestingly, the $\alpha$-cluster case of O–O collisions shows a similarity in the evolution of elliptic flow fluctuations with the Pb–Pb collisions, which is clearly different in O–O collisions with a Woods-Saxon nuclear density profile. The source of this opposite behavior of $\sigma_{v_{2}}$ for both the nuclear density profiles can thus intrinsically be attributed to the difference in the nuclear geometry of the ¹⁶O nucleus. Interestingly, the multiplicity class dependence of $\langle v_{2}\rangle$ and $\sigma_{v_{2}}$ are complementary to each other for the Woods-Saxon case in O–O collisions. However, this correlation between $\langle v_{2}\rangle$ and $\sigma_{v_{2}}$ is not completely applicable for O–O collisions with an $\alpha$-cluster nuclear density profile. Figure 7 (d) shows the evolution of relative elliptic flow fluctuation ($F(v_{2})$) as a function of multiplicity class in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV. Interestingly, we find a similar multiplicity dependence of $F(v_{2})$ as found for $\sigma_{v_{2}}$. The multiplicity class dependence of $F(v_{2})$ in O–O collisions with Woods-Saxon nuclear distribution resembles a p–Pb collision [98]. In contrast, for $\alpha$-cluster nuclear density profiles, $F(v_{2})$ is higher for 0–10% multiplicity class and decreases with a decrease in particle multiplicity. This is an important observation where we find that a change in the initial nucleon distribution (Woods-Saxon versus $\alpha$-cluster) can cause a difference in the multiplicity class or collision centrality dependence of relative elliptic flow fluctuations for a hydrodynamically evolving system.

In this paper, we study the anisotropic flow coefficients in O–O collisions at $\sqrt{s_{\rm NN}}=7$ TeV using a hybrid of IPGlasma + MUSIC + iSS + UrQMD model and try to probe the final state effects of the presence of an exotic $\alpha$-cluster nuclear geometry in contrast to a uniform Woods-Saxon nuclear distribution inside a ¹⁶O nucleus. Here, we study the multiplicity class dependence of anisotropic flow coefficients. In addition, for the first time, we report studies of elliptic flow fluctuations with a motivation to distinguish O–O collisions having $\alpha$-cluster geometry from a regular Woods-Saxon distribution. The observation of large triangular flow in the highest multiplicity O–O collisions having an $\alpha$-clustered nuclei compared to a Woods-Saxon nuclear distribution is consistent with transport model predictions. In addition, we observe a significant multiplicity class dependence of relative elliptic flow fluctuations in both the nuclear distributions. With this study, it is clear that the presence of an $\alpha$-cluster nuclear geometry has a significant effect on both elliptic and triangular flow. However, we note a distinctive feature appears when observables related to fluctuations are studied, i.e., triangular flow and elliptic flow fluctuations. While probing a nuclear density profile is a subject matter of low-energy nuclear physics, it will be very interesting to find an appropriate observable that can be measured in TeV nuclear collisions to do the job. This makes the present study more interesting.

S. P. acknowledges the doctoral fellowship from UGC, Government of India. R. S. sincerely acknowledges the DAE-DST, Government of India funding under the mega-science project – “Indian participation in the ALICE experiment at CERN” bearing Project No. SR/MF/PS-02/2021-IITI (E-37123). G. G. B. gratefully acknowledges the Hungarian National Research, Development and Innovation Office (NKFIH) under Contracts No. OTKA K135515, No. NKFIH NEMZ_KI-2022-00031, “2024-1.2.5-TET-2024-00022” and Wigner Scientific Computing Laboratory (WSCLAB, the former Wigner GPU Laboratory). The authors gratefully acknowledge the MoU between IIT Indore and Wigner Research Centre for Physics (WRCP), Hungary, for the techno-scientific international cooperation.