A Review of Elliptic Flow of Light Nuclei in Heavy-Ion Collisions at RHIC and LHC Energies

We present a review of the measurements of elliptic flow ($v_{2}$) of light nuclei ($d$,$\bar{d}$, $t$, $^{3}\rm He$, and $^{3}\overline{\rm He}$) from the RHIC and LHC experiments. Light (anti)nuclei $v_{2}$ have been compared with that of (anti)proton. We observed a similar trend in light nuclei $v_{2}$ to that in identified hadron $v_{2}$ with respect to the general observations such as ($p_{\rm T}$) dependence, low $p_{\rm T}$ mass ordering, and centrality dependence. We also compared the difference of nuclei and antinuclei $v_{2}$ with the corresponding difference of $v_{2}$ of proton and antiproton at various collision energies. Qualitatively they depict similar behavior. We also compare the data on light nuclei $v_{2}$ to various theoretical models such as blast-wave and coalescence. We then present a prediction of $v_{2}$ for $^{3}\rm He$ and $^{4}\rm He$ using coalescence and blast-wave models.


I. INTRODUCTION
The main goals of high energy heavy-ion collision experiments have primarily been to study the properties of Quark Gluon Plasma (QGP) and the other phase structures in the QCD phase diagram [1][2][3][4]. The energy densities created in these high energy collisions are similar to that found in the universe, microseconds after the Big Bang [2,5,6]. Subsequently, the universe cooled down to form nuclei. It is expected that high energy heavy-ion collisions will allow to study the production of light nuclei such as d, t, 3 He and their corresponding anti-nuclei. There are two possible production mechanisms for light (anti-)nuclei. The first mechanism is thermal production of nucleus-antinucleus pairs in elementary nucleonnucleon or parton-parton interactions [7][8][9]. However, due to their small (∼few MeV) binding energies, the directly produced nuclei or anti-nuclei are likely to breakup in the medium before escaping. The second mechanism is via final state coalescence of produced (anti-)nucleons or from transported nucleons [10][11][12][13][14][15][16][17]. The quark coalescence as a mechanism of hadron production at intermediate transverse momentum has been well established by studying the number of constituent quark (NCQ) scaling for v 2 of identified hadrons measured at RHIC [18][19][20][22][23][24][25]. Light nuclei may also be produced via coalescence of quarks similar to the hadrons. But the nuclei formed via quark coalescence is highly unlikely to survive in the high temperature environment due to their small binding energies. In case of hadron formation by quark coalescence, the momentum space distribution of quarks are not directly measurable in experiments. However, in case of nucleon coalescence, momentum space distributions of both the constituents (nucleons) and the products (nuclei) are measurable in heavy-ion collision experiments. Therefore, measurements of v 2 of light nuclei provides a tool to understand the production mechanism of light nuclei and freeze-out properties at a later stage of the evolution. It also provides an excellent opportunity to understand the mechanism of coalescence at work in high energy heavy-ion collisions.
The production of light (anti-)nuclei has been studied extensively at lower energies in Bevelac at LBNL [26,27], AGS at RHIC [28,29] and SPS at CERN [30,31]. In the AGS experiments, it was found that the coalescence parameter (B 2 ) is of similar magnitude for both d and d indicating similar freeze-out hypersurface of nucleons and anti-nucleons. Furthermore, the dependence of B 2 on collision energy and p T indicated that light nuclei production is strongly influenced by the source volume and transverse expansion profile of the system [31,32]. In this paper, we review the results of elliptic flow of light nuclei measured at RHIC and LHC and discuss the possible mechanisms for the light nuclei production.
The paper is organized as follows. Sec. II briefly describes the definition of v 2 , identification of light (anti-)nuclei in the experiments and measurement of v 2 of light (anti-)nuclei. In Sec. III, we present the v 2 results for minimum bias collisions from various experiments. We also discuss the centrality dependence, difference between nuclei and anti-nuclei v 2 as well as the energy dependence of deuteron v 2 . In the same section, we present the atomic mass number scaling and also compare the experimental results with various theoretical models. Finally in Sec. IV, we summarize our observations and discuss the main conclusions of this review.

A. Elliptic flow (v2)
The azimuthal distribution of produced particles in heavy-ion collision can be expressed in terms of a Fourier where φ is the azimuthal angle of produced particle, Ψ r is called the reaction plane angle and the Fourier coefficients v 1 , v 2 and so on are called flow co-efficients [33]. Ψ r is defined as the angle between the impact parameter vector and the x-axis of the reference detector in the laboratory frame. Since it is impossible to measure the direction of impact parameter in heavy-ion collisions, a proxy of Ψ r namely the event plane angle Ψ n is used for the flow analysis in heavy-ion collisions [34]. The v 2 is measured with respect to the 2 nd order event plane angle Ψ 2 [34]. Ψ 2 is calculated using the azimuthal distribution of the produced particles. In an event with N particles, the event plane angle Ψ 2 is defined as [34]: X 2 and Y 2 are defined as where w i are weights given to each particle to optimise the event plane resolution [34,35]. Usually the magnitude of particle transverse momentum p T is used as weights as the v 2 increases with p T . Special techniques are followed while calculating the event plane angle so that it does not contain the particle of interest whose v 2 is to be calculated (self-correlation) and also the nonflow effects (e.g., jets and short range correlations) are removed as much as possible [22,34,38]. Heavy-ion experiments use the η-sub event plane method to calculate the elliptic flow of identified hadrons as well as for light nuclei. In this method, each event is divided into two sub-events in two different η-windows (e.g., positive and negative η). Then two sub-event plane angles are calculated with the particles in each sub-event. Each particle with a particular η is then correlated with the sub event plane of the opposite η. This ensures that the particle of interest is not included in the calculation of event plane angle. A finite η gap is applied between the two subevents to reduce short range correlations which does not originate from flow. The distribution of the event plane angles should be isotropic in the laboratory frame for a azimuthally isotropic detector. If the distribution of the event plane angles is not flat in the laboratory frame (due to detector acceptance and/or detector inefficiency) then special techniques are applied to make the distribution uniform. The popular methods to make the Ψ 2 distribution uniform is the φ-weight and recentering [36,37]. In the φ-weight method, one takes the actual azimuthal distribution of the produced particle, averaged over large sample of events, and then uses inverse of this distribution as weights while calculating the correlation of the particles with the event plane angle [36,37]. In the recentering method, one subtracts X n and Y n from the eventby-event X n and Y n , respectively, where X n and Y n denotes the average of X n and Y n over a large sample of similar events. The main disadvantage of applying one of these methods is that it does not remove the contribution from higher flow harmonics. Therefore, another correction method known as the shift correction is used to remove the effects coming from higher flow harmonics. In this method, one fits the Ψ 2 distribution (after applying φ-weight and/or recentering method) averaged over all events, with a Fourier function. The Fourier co-efficients from this fit (obtained as fit parameters) are used to shift the Ψ 2 of each event, to make the distribution uniform in the laboratory frame [36,37].
Since the number of particles produced in heavy-ion collisions are finite, the calculated event plane angle Ψ 2 may not coincide with Ψ r . For this reason, the measured v obs 2 with respect to Ψ 2 is corrected with the event plane resolution factor R 2 , where In order to calculate the event plane resolution, one calculates two sub-event plane angles Ψ a 2 and Ψ b 2 , where a and b corresponds to two independent sub-events. If the multiplicities of each sub-event are approximately half of the full event plane, then the resolution of each of subevent plane can be calculated as [33,34], However, the full event plane resolution can be expressed as, where, χ 2 = v 2 /σ and I 0 , I 2 are modified Bessel functions [33,34]. The parameter σ is inversely proportional to the square-root of N , the number of particles used to determine the event plane [33,34]. To calculate the resolution for full event plane (Ψ 2 ), one has to solve the Eq. (6) iteratively for the value of χ 2 using the subevent plane resolution ( cos[2(Ψ a 2 − Ψ r )] ) which is calculated experimentally using Eq. (5). The χ 2 value is then multiplied with √ 2 as χ 2 is propotional to √ N , and re-used in Eq. (6) to calculate the resolution of the full event plane. In a case of very low magnitudes, the full event plane resolution can be approximately given as [33,34], The procedure for calculating full and sub-event plane resolutions using sub-event plane angles and various approximations are discussed in detail in [33,34].

B. Data on light nuclei
For this review, we have collected light nuclei v 2 data from the STAR [38] and PHENIX [39] experiments at RHIC and ALICE experiment at LHC [40]. The table I summarises the measurement of light nuclei v 2 available till date.
In the STAR experiment, to identify light nuclei using TPC, a variable Z [38] is defined as Then the light nuclei yields are extracted from these Zdistributions in differential p T and (φ−Ψ 2 ) bins for either minimum bias collisions or in selected centrality classes. The (φ − Ψ 2 ) distribution is then fitted with a 2 nd order Fourier function namely, The Fourier co-efficient v 2 is called elliptic flow and is extracted from the fit. As we discussed in the previous subsection this measured v 2 is then corrected with the event plane resolution factor (R 2 ) [22,38].
In the ALICE experiment, light nuclei in the low p T region (< 1.0 GeV/c for d, d) are identified by comparing the variance (σ dE dx ) of the measured dE/dx in the TPC with the corresponding theoretical estimate (dE/dx| theo ) [40,44]. Light nuclei are considered identified if the measured dE/dx lies within ±3σ dE dx of the dE/dx| theo . On the other hand, the light nuclei yield are extracted from the mass squared (m 2 TOF ) distribution using the TOF detector. The mass of each particle is calculated using the time-of-flight (t) from the TOF detector and the momentum (p) information from the TPC [39,40,44]. Both the ALICE and PHENIX experiments use the TOF detector to identify light nuclei at high p T (> 1.0 GeV/c). The mass of a particle can be calculated using the TOF detector as, where the track length L and momentum p are determined with the tracking detectors placed inside magnetic field [24,39,40,44]. After getting the m 2 for each particle, a selection cut is implemented to reject tracks which have their m 2 several σ away from the true m 2 value of the light nuclei, as done in the STAR experiment [38]. The ALICE experiment, on the other hand defines a quantity ∆m such that, ∆m = m TOF − m nucl , where m nucl is the mass of the light nuclei under study. The distribution of ∆m is then fitted with an Gaussian + exponential function for signal and an exponential function for the background [40]. Then v 2 of light nuclei is calculated by fitting the v 2 (∆m) with the weighted function, where the total measured v Tot 2 is the weighted sum of that from the signal (v Sig 2 ) and background (v Bkg   2 ). The v Tot 2 of the candidate particles are calculated using the scalar product method and corrected for the event plane resolution [40].
The PHENIX experiment calculates charged average v 2 of (anti)-deuterons as, The quantity R 2 = cos(2(Ψ 2 − Ψ r )) can readily be identified as the resolution of the event plane angle [39]. The resolution of full event plane Ψ 2 is calculated with subevent planes (Ψ a 2 , Ψ b 2 ) estimated using two Beam Beam Counter (BBC) detectors [24,39]. The detailed procedure of calculating the full event plane resolution from sub-events are already menioned in the previous subsection. The large η gap between the central TOF and the BBCs (∆η > 2.75) reduces the effects of non-flow significantly [24,39]. The nuclei v 2 calculated in PHENIX is also corrected for the contribution coming from backgrounds, mainly consisting of mis-identification of other particles (e.g., protons) as nuclei. A p T dependent correction factor was applied on the total v 2 (referred as v Sig+Bkg where v Sig+Bkg  30-40% centrality in ALICE. Since PHENIX and ALICE do not have measurements in the minimum bias collisions, we only show the results for mid-central collisions. The data points of PHENIX and ALICE correspond to inclusive d+d v 2 . The general trend of nuclei v 2 of all species is the same− it increases with increasing p T . The slight difference of v 2 between STAR and PHENIX is due to the difference in centrality ranges. The centrality range for PHENIX is 20-60% and that for STAR is 0-80%. From the trend in Fig. 1 it seems that light nuclei v 2 shows mass ordering, i.e. heavier particles have smaller v 2 value compared to lighter ones, similar to v 2 of identified particles [22,24,47]. In order to see the mass ordering effect more clearly, we restrict the x−axis range to 2.5 GeV/c and compare the v 2 of d with the v 2 of identified particles such as π + , K 0 s (K in Pb+Pb) and p as shown in Fig. 2. We see that d v 2 at all collision energies is lower than the v 2 of the identified hadrons at a fixed value of p T . Although mass ordering is a theoretical expectation from the hydrodynamical approach to heavy-ion collisions [49], coalescence formalism for light nuclei can also give rise to this effect. Recent studies using AMPT and VISHNU hybrid model suggest that mass ordering is also expected from transport approach to heavy-ion collisions [50,51]. The v 2 of light nuclei is negative for some collision energies as shown in Fig. 1. This negative v 2 is expected to be the outcome of strong radial flow in heavy-ion collisions [52].
In order to study the energy dependence of light nuclei v 2 , we compare the deuteron v 2 from collision energy √ s N N = 7.7 GeV to 2760 GeV as shown in Fig. 3. creases with decreasing collision energy [22].
The STAR experiment has measured the difference of nuclei (d) and anti-nuclei (d) v 2 for collision energies √ s N N = 19.6, 27, 39, 62.4 and 200 GeV [38]. Figure 4 shows the difference of d and d v 2 as a function of collision energy. For comparison, the difference of proton and anti-proton v 2 is also shown [22]. We observe that the difference of d and d v 2 remains positive for √ s N N = 7.7 − 39 GeV. However, for √ s N N ≥ 62.4 GeV the difference of d and d v 2 is almost zero. The difference of d and d v 2 qualitatively follows the same trend as seen for difference of p and p v 2 [22]. It is easy to infer from simple coalescence model that light (anti-)nuclei formed via coalescence of (anti-)nucleons, will also reflect similar difference in v 2 as the constituent nucleon and anti-nucleon. The difference in v 2 between particles and their antiparticles has been attributed to the chiral magnetic effect in finite baryon-density matter [53], different v 2 of produced and transported particles [54], different rapidity distributions for quarks and antiquarks [55], the conservation of baryon number, strangeness, and isospin [56], and different mean-field potentials acting on particles and their antiparticles [57].
The centrality dependence of light nuclei v 2 measured by the STAR and ALICE is shown in Fig. 5. STAR has measured d and d v 2 in two different centrality ranges namely 0-30% and 30-80% for collision energies below √ s N N = 200 GeV. In case of √ s N N = 200 GeV, the light nuclei v 2 is measured in three different centrality ranges namely 0-10% (central), 10-40% (mid-central) and 40-80% (peripheral) as high statistics data were available. ALICE has measured inclusive d + d v 2 in 6 different centrality ranges namely, 0-5%, 5-10%, 10-20%, 20-30%, 30-40% and 40-50%. We only present the results from 0-5%, 20-30% and 40-50% centrality from ALICE as shown in Fig. 5. The v 2 of d shows strong centrality dependence for all collision energies studied in the STAR experiment. We observe that more central events has lower v 2 compared to peripheral events. d shows the same trend as d for collision energies down to √ s N N = 27 GeV. The STAR experiment could not study centrality dependence of d below √ s N N = 27 GeV due to limited event statistics [38]. Comparing the centrality dependence of d(d) v 2 from STAR and ALICE we can see that both experiments show strong centrality dependence of light nuclei v 2 . The centrality dependence of light nuclei v 2 is analogous to the centrality dependence observed for identified nucleon (p, p) v 2 [58,59].   [21]. Therefore, we expect that the light (anti-)nuclei v 2 divided by A, should scale with p(p) v 2 . Here, we essentially assume that the v 2 of (anti-)proton and (anti-)neutron are same as expected from the observed NCQ scaling of identified particle v 2 [22]. Figure 6 shows the atomic mass number scaling of light nuclei v 2 from STAR, PHENIX and AL-ICE experiments. Since ALICE does not have results in minimum bias events, so we used both p+p and d+d v 2 from 30-40% centrality range. We observe that light nuclei v 2 from STAR and PHENIX show atomic mass   The pT spectra are used from [25,64]. Markers for PHENIX data corresponds to 20-60% and markers for ALICE data corresponds to 30-40% central events.
number scaling up to p T /A ∼1.5 GeV/c. However, deviation of the scaling of the order of 20% is obesrved for d+d v 2 from ALICE. The scaling of light (anit-)nuclei v 2 with (anti-)proton v 2 suggests that light (anit-)nuclei might have formed via coalescence of (anti-)nucleons at a later stage of the evolution at RHIC energies for p T /A up to 1.5 GeV/c [10][11][12][13][14]. However, this simple picture of coalescence may not be holding for ALICE experiment at LHC energies. On the contrary, there is another method to produce light nuclei, for example by thermal production in which it is assumed that light nuclei are produced thermally like any other primary particles [8,9]. Various thermal model studies have successfully reproduced the different ratios of produced particles as well as light nuclei in heavy-ion collisions [8,9].
In order to investigate the success of these models, both STAR and ALICE has compared the elliptic flow of light nuclei with the predictions from blast-wave models [38,40]. Figure 7 shows the v 2 of light nuclei predicted from blast-wave model using the parameters obtained from fits to the identified particles v 2 [40,60]. We observe that blast-wave model reproduces v 2 of light nuclei from STAR with moderate success except for low p T (< 1.0 GeV/c) where v 2 of d(d) are under-predicted for all collision energies. However, the blast-wave model seems to successfully reproduce the d+d v 2 from ALICE. The low relative production of light nuclei compared to identified nucleons at RHIC collisions energies supports the procedure of light nuclei production via coalescence mechanism [10][11][12][13][14]. However, the success of blast-wave model in reproducing the nuclei v 2 at LHC and moderate success at RHIC suggest that the light nuclei production is also supported by thermal process [8,9]. The light nuclei production in general might be a more complicated coalescence process, e.g., coalescence of nucleons in the local rest frame of the fluid cell. This scenario might give rise to deviations from simple A scaling [38].
At RHIC energies the light nuclei v 2 have been compared with results from a hybrid AMPT+coalescence model [38]. A Multi Phase Transport (AMPT) model is an event generator with Glauber Monte Carlo initial state [61]. The AMPT model includes Zhang's Partonic Cascade (ZPC) model for initial partonic interactions and A Relativistic Transport (ART) model for later hadronic interactions [61]. The nucleon phase-space information from the AMPT model is fed to the coalescence model to generate light nuclei [38,62]. Figure 8 shows the light nuclei v 2 from the coalescence model and compared to the data. The coalescence model prediction for d+d in Pb+Pb collisions at √ s N N = 2760 GeV is taken from [63]. The coalescence model fairly reproduces the measurement from data for all collision energies except for the lowest energy √ s N N = 7.7 GeV. The AMPT model generates nucleon v 2 from both partonic and hadronic interactions for all the collision energies presented. However, increased hadronic interactions compared to partonic, at lowest collision energies, is not implemented in the AMPT+coalescence model. This could be the reason behind the deviation of the data from the model predictions at lowest collision energy [22]. We have performed simultaneous fit to the v 2 and p T spectra of identified hadrons + light nuclei using the same blast-wave model as used in [40,47]. The simultaneous fit of v 2 and p T spectra for measurements from the PHENIX and the ALICE experiment are shown in Fig. 9. We find that the inclusion of light nuclei results to the fit does not change the fit results compared to the blast-wave fit performed only on identified hadron v 2 and p T spectra. This indicates that the light nuclei v 2 and p T spectra is well described by the blast-wave model.

C. Model prediction of 3 He and 4 He v2
We have predicted the v 2 of 3 He and 4 He using the simple coalescence and blast-wave model. Since protons and neutrons have similar masses and same number of constituent quarks, they should exhibit similar collective behavior and hence, similar magnitude of v 2 . Therefore, we parametrize the elliptic flow of p + p v 2 using the fit formula [65], where a, b, c, and d are fit parameters and n is the constituent quark number of the particle [65]. The fit to p+p v 2 (solid lines) from the PHENIX and ALICE experiment is shown in Fig. 10(a) and (b), respectively. Assuming similar magnitude of neutron v 2 as that of proton, we then predict the v 2 of 3 He and 4 He as, This simplified coalescence model prediction of 3 He and 4 He v 2 are shown in Fig. 10(a) and (b) as blue (thindotted) lines. For comparison, the blast-wave model predicted v 2 of 3 He and 4 He from the fit parameters obtained in Fig. 9 are also shown in red (thick-dotted) lines. We observe characteristic difference is observed in the prediction of 3 He and 4 He v 2 from the coalescence and the blast-wave model. As one expects from the mass ordering effect of blast-wave model, the in future, would significantly improve our knowledge on the mechanisms of light nuclei formation in heavy-ion collisions [44,[66][67][68].

IV. SUMMARY AND CONCLUSIONS
We have presented a review of elliptic flow v 2 of light nuclei (d, t and 3 He) and anti-nuclei (d and 3 He) from STAR experiment, and inclusive d+d v 2 from PHENIX at RHIC and ALICE at LHC. Similar to identified hadrons, the light nuclei v 2 show a monotonic rise with increasing p T and mass ordering at low p T for all measured collision energies. The beam energy dependence of d v 2 is small at intermediate p T and only prominent at high p T , which is similar to the the trend as observed for the charged hadron v 2 . The v 2 of nuclei and antinuclei are of similar magnitude for top collision energies at RHIC but at lower collision energies, the difference in v 2 between nuclei and anti-nuclei qualitatively follow the difference in proton and anti-proton v 2 . The centrality dependence of light (anti-)nuclei v 2 (p T ) is similar to that of identified hadrons v 2 (p T ).
Light (anti-)nuclei v 2 is found to follow the atomic mass number (A) scaling for almost all collision energies at RHIC suggesting coalescence as the underlying process for the light nuclei production in heavy-ion collisions. However, a deviation from mass number scaling at the level of 20% is observed at LHC. This indicates that a simple coalescence process may not be the only underlying mechanism for light nuclei production. Furthermore, a transport-plus-coalescence model study is found to approximately reproduce the light nuclei v 2 measured at RHIC and LHC. The agreement of coalescence model with the data from PHENIX and STAR are imperceptibly better than the blast-wave model. However, at the LHC energy, the light nuclei v 2 is better described by blast-wave model rather than the simple coalescence model. The coalescence mehcanism, intuitively, should be the prominent process of light nuclei production. However, the breaking of mass scaling at LHC energy and success of blast-wave model prevent us to draw any definitive conclusion on the light nuclei production mechanism.
We observed orders of magnitude difference in 3 He and 4 He v 2 as predicted by blast-wave and coalescence model. The blast-wave model predicts almost zero v 2 for 3 He and 4 He up to p T = 2.5 GeV/c, whereas the coalescence model predicts significant v 2 for 3 He and 4 He at same p T range. Hence, the precise measurements of 3 He and 4 He v 2 in future can significantly improve the knowledge of the light nuclei production mechanism in heavy-ion collisions.