Multiparticle azimuthal cumulants in p+Pb collisions from a multiphase transport model

A new subevent cumulant method was recently developed, which can significantly reduce the non-flow contributions in long-range correlations for small systems compared to the standard cumulant method. In this work, we study multi-particle cumulants in $p$+Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 5.02$ TeV with a multiphase transport model (AMPT), including two- and four-particle cumulants ($c_{2}\{2\}$ and $c_{2}\{4\}$) and symmetric cumulants [SC(2, 3) and SC(2, 4)]. Our numerical results show that $v_{2}\{2\}$ is consistent with the experimental data, while the magnitude of $c_{2}\{4\}$ is smaller than the experimental data, which may indicate either the collectivity is underestimated or some dynamical fluctuations are absent in the AMPT model. For the symmetric cumulants, we find that the results from the standard cumulant method are consistent with the experimental data, but those from the subevent cumulant method show different behaviors. The results indicate that the measurements from the standard cumulant method are contaminated by non-flow effects, especially when the number of produced particles is small. The subevent cumulant method is a better tool to explore the $real$ collectivity in small systems.


I. INTRODUCTION
One experimental signature suggesting the formation of nearly perfect fluid in ultrarelativistic nucleus-nucleus (A+A) collisions is the azimuthal anisotropy of produced particles.
The measured anisotropies provide strong evidence of collective flow, which is commonly believed to be related to the hot QCD medium that expands collectively and transfers asymmetries in the initial geometry space into azimuthal anisotropies of produced particles in the final momentum space [1][2][3][4][5][6]. The feature of collectivity appears in the form of "ridge": enhanced pair production in a small azimuthal angle interval, ∆φ ∼ 0, extended over a wide range of pseudorapidity intervals ∆η [7][8][9][10]. The azimuthal structure of the ridge is typically analyzed via a Fourier decomposition, dN pairs /d∆φ ∼ 1 + 2 v 2 n cos(n∆φ). The second (elliptic; v 2 ) and third (triangular; v 3 ) Fourier harmonics are under intensive studies, because they are assumed to directly reflect the medium response to the initial geometry. For a small collision system, such as proton-proton (p + p) or proton-nucleus (p+A) collisions, it was assumed that the transverse size of the produced system is too small compared to the mean free path of constituents. Thus, it was expected that the collective flow in small systems should be much weaker than that in A+A collisions. However, recent observations of large long-range ridge-like correlations and v n coefficients in small systems [11][12][13][14][15][16][17] challenges the above paradigm of collective flow.
Since hydrodynamic flow implies a global collectivity involving all particles in the event, k-particle azimuthal cumulants, c n {k}, are often used to measure the true v n [18,19]. The standard cumulant method, known as the Q-cumulant [19], use all k-particle multiplets in the entire detector acceptance to calculate c n {k}. But this method can be contaminated by non-flow effects, like jet-like correlation, especially when the multiplicity is small. Recently an improved cumulant method, referred to as the "subevent cumulant," in which particles are divided into different subevents separated in the pseudorapidity η direction, was developed [20]. Compared to the standard cumulant method, the new method can more effectively suppress intra-jet (single-jet) and inter-jet (di-jet) correlations. Recent ATLAS measurements have shown that the subevent method provides a more precise determination of c n {4} associated with long-range collectivity in small systems [21].
Multi-particle correlation between different orders of flow harmonics is another complementary observable which provides additional constraints on the medium properties [22,23].
Such mixed-harmonic correlations are measured through the so-called symmetric cumulant, SC(n, m), with n = m. The CMS Collaboration recently obtained results for SC (2,3) and SC (2,4) in p+p and p+Pb collisions based on the standard cumulant method [24]. However, Huo et al. argued that the measurements of SC(n, m) in small systems are not trustworthy due to dominating non-flow effects, unless the subevent method is utilized [25]. But their argument is based on the PYTHIA and HIJING models, which have no collective flow.
Therefore it is necessary to verify this assertion with models that contain both collective flow and non-flow.
However, the collectivity from the AMPT model has been interpreted as a parton escape mechanism where the azimuthal anisotropy is mainly generated by the anisotropic parton escape instead of hydro-like interactions [45,46]. The controversy surrounding the origin of collectivity in small systems needs to be further tested in more experimental and theoretical efforts.
In this work, we adopt the newly developed subevent cumulant method to suppress nonflow effects to investigate the flow in the AMPT model for p+Pb collisions at √ s N N = 5.02 TeV. The two-and four-particle azimuthal cumulants, (c 2 {2} and c 2 {4}), and multi-particle azimuthal correlations between v 2 and v 3 and between v 2 and v 4 , [SC(2, 3) and SC(2, 4)], are calculated using both standard and subevent cumulant methods. We find that the AMPT model can well describe the two-particle v 2 {2} data, but with a magnitude of c 2 {4} smaller than the experimental data. To further shed light on the origin and evolution of multiparticle correlations, the evolution of c 2 {k} values is traced at different phases in the AMPT model. Significant differences in symmetric cumulants, SC(2, 3) and SC (2,4), between the standard and the subevent cumulant methods are also observed. Our results suggest that either the collectivity is underestimated or some non-Gaussian dynamical fluctuations are missing in the AMPT model. We find that the subevent cumulant method is a better probe to investigate the real collectivity in small systems.

II. THE AMPT MODEL
A multiphase transport model [47], which is a hybrid dynamical transport model, is utilized in this work. We use the string melting AMPT version to simulate p+Pb collisions at √ s NN = 5.02 TeV. The string melting version consists of four main components: fluctuating initial conditions from the HIJING model [48], elastic parton cascade simulated by the ZPC model [49] for all partons from the melting of hadronic strings, a quark coalescence model for hadronization, and hadron rescatterings described by the ART model [50]. For details, see the review [47]. For the setting of parameter values, we follow the recent AMPT study with a modest elastic parton-parton cross section σ = 3 mb, which has been shown to be capable of reproducing the long-range correlation and two-particle v n coefficients in p+Pb collisions at √ s NN = 5.02 TeV [31,32].

III. MULTIPARTICLE CUMULANTS
The cumulant method has been developed to characterize multi-particle correlations related to the collective expansion of system, while reducing non-flow contributions order by order [18,51]. A 2k-particle azimuthal correlator 2k is obtained by averaging over all unique combinations in one event then over all events, where the first two terms are . For a given harmonic n, the twoand four-particle cumulants can be determined: The flow coefficients v n can be analytically obtained from the two-and four-particle The framework of the standard cumulant [19] expresses multi-particle correlations in terms of powers of the flow vector Q n = e inφ . The multi-particle correlations and cumulants can be calculated through a single loop over all events. In the standard cumulant method, the particles are chosen from the entire detector acceptance. In small systems, the non-flow correlations, especially the jet and dijet, dominate the azimuthal correlations.
Hence, the standard cumulant may be strongly biased by these non-flow correlations, while the subevent cumulant method is designed to further suppress these non-flow correlations.
In the subevent cumulant method, the entire event is divided into two subevents or three subevents. Specifically, in the two-subevent method, the event is divided into two (labelled as a and b) according to −η max < η a < 0 and 0 < η b < η max ; in the three-subevent method, the event is divided into three (labelled as a, b and c) according to −η max < η a < −η max /3, −η max /3 < η b < η max /3 and η max /3 < η c < η max . Then one can get the 2k-particle azimuthal correlators as follows: The symmetric cumulant is also based on the multi-particle cumulants, which measures the correlation between different flow harmonics on the basis of event-by-event fluctuations.
The SC(n, m) is defined below: Similarly, we can easily get the SC(n, m) with the subevent methods, SC(n, m) three−sub = e in(φ a More details can be found in Ref. [20]. from the published AMPT results [32] and CMS data [52], respectively.
In order to make our results directly comparable to the experimental measurements, we choose η max = 2.5 in our analysis to mimic the ATLAS detector acceptance for charged particles. The event selection is based on N ch , the number of charged particles in |η| < 2.5 and p T > 0.4 GeV. The cumulant calculations are carried out using the charge particles in |η| < 2.5 and a certain p T selection, 0.3 < p T < 3 GeV, and the number of charged particle in this p T range, N sel ch . We need to point out N sel ch and N ch are not the same due to different p T ranges. Then 2k is averaged over events with the same N sel ch to obtain the 2k and SC(n, m). Finally, the cumulant results are obtained by mapping N sel ch to N ch , where we follow the ATLAS procedure exactly with the same kinematic cuts [21]. filled circles represent the ATLAS data using three-subevent cumulant method [21]. Figure 1 shows the v 2 {2} results with the subevent cumulant method, and compares them with the two-particle v 2 {2, |∆η| > 2} from the published AMPT results [32] and the CMS data [52]. The three results are in good agreement. conservation [57]. After parton cascade, c 2 {2} is enhanced and c 2 {4} changes sign from positive to negative at a certain value of N ch , which maybe due to the interplay between transverse momentum conservation and an anisotropic flow generated by parton cascade [58]. is in good agreement with the CMS data. However, we find the SC(2, 3) from the subevent methods stays negative for the whole range of multiplicity. Our results strongly suggest that the measurements using the standard cumulant method are contaminated by the non-flow effects. On the other hand, the SC(2, 4) from the standard cumulant method is comparable with the experimental data, but it is much larger than those with subevent methods. It also suggests that non-flow contributions need to be removed to obtain a clean signal of collectivity, especially in the low-multiplicity region in small systems.

V. CONCLUSIONS
The subevent cumulant method is utilized to study multi-particle correlations in p+Pb collisions within the AMPT model. The two-and four-particle cumulants, (c 2 {2} and c 2 {4}),