Symmetric cumulant $sc_{2,4} \left \{ 4 \right \}$ and asymmetric cumulant $ac_{2} \left \{ 3 \right \}$ from transverse momentum conservation and flow

Multiparticle cumulants method can be used to reveal long-range collectivity in small and large colliding systems. The four-particle symmetric cumulant $sc_{2,4} \left \{ 4 \right \}$, three-particle asymmetric cumulant $ac_{2} \left \{ 3 \right \}$, and the normalized cumulants $nsc_{2,4} \left \{ 4 \right \}$ and $nac_{2} \left \{ 3 \right \}$ from the transverse momentum conservation and flow are calculated. The interplay between the two effects is also investigated. Our results are in a good agreement with the recent ATLAS measurements of multiparticle azimuthal correlations with the subevent cumulant method, which provides insight into the origin of collective flow in small systems.


I. INTRODUCTION
High-energy nucleus-nucleus (A+A) collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) can create an extremely dense and hot environment in which confined quarks and gluons are released into a deconfined state of matter called the quark-gluon plasma (QGP) [1][2][3][4][5].One of the most significant experimental signatures of the QGP properties is the collective flow due to its sensitivity to the dynamical evolution of the QGP, which can transfer the asymmetries in the initial geometry space into the anisotropies in the final momentum space [6][7][8][9][10].The magnitude of the azimuthal anisotropy in the transverse plane of the final momentum space can be quantified in terms of the Fourier expansion coefficient, dN  dφ ∝ 1 + n v n cos[n(φ − Ψ n )] [11][12][13], where the anisotropic flow coefficients v 1 , v 2 , v 3 , and v 4 are directed, elliptic, triangular, and quadrangular flows, respectively.Studies of collective flow have shown that the QGP is a nearly perfect fluid with strong coupling, i.e., the ratio of the shear viscosity to the entropy density η/s is close to the minimum value of 1/4π [14][15][16].In addition to the collective flow, anisotropic flow in measurements contains nonflow, which includes the short-range correlation such as jets, resonance decays, and Bose-Einstein correlation, and long-range correlation such as the transverse momentum conservation (TMC) [17][18][19][20][21][22].
Previous studies have found that there is a linear relationship between the flow v n and the corresponding eccentricity ε n , i.e., v n ∝ ε n [71,72].However, it has been argued that the set of flow coefficients {v n } and the set of eccentricities {ε n } can be linked by a response matrix, which implies that there is a nonlinear correlation between v m and v n [73][74][75].The symmetric cumulants m can be inscribed to carry a correlation between v n and v m capable of responding to the geometrical shape eccentricities ε n and ε m of the initial state phase during the evolution of the QGP, in addition to information about the interactions of the final state [76][77][78].Moreover, ac n {3} = v 2 n v 2n cos 2n(Ψ n − Ψ 2n ) involves not only the correlation between flow harmonics v n and v 2n but also the correlation between event planes Ψ n and Ψ 2n .The generalized symmetric cumulants sc k,l,m {6} have been proposed to explore the collectivity in large systems such as Pb+Pb collisions at LHC energies [79].The ALICE experiment measured sc 4,2 {4} and sc 3,2 {4} and found that there is a positive correlation between v 2 and v 4 , and a negative correlation between v 2 and v 3 [80], which has shown that sc n,m {4} is very sensitive to the temperature dependence of η/s in noncentral collisions [81].To suppress the nonflow contribution to sc n,m {4} and ac 2 {3}, a subevent method has been proposed, in which particles in different pseudorapidity intervals are divided into two or more subevents.The subevent method has been performed in p+p and p+Pb collisions using PYTHIA and HIJING models, which shows that the subevent method can indeed suppress nonflow contributions [82,83].The recent ATLAS experimental results have demonstrated that the signal of four-particle symmetric cumulant sc 2,4 {4} and three-particle asymmetric cumulant ac 2 {3} gradually decreases from the standard method to the subevent method, as a result of the effective suppression of nonflow contribution from jets [84].
In this paper, we calculate the four-particle symmetric cumulant sc 2,4 {4}, three-particle asymmetric cumulant ac 2 {3}, and their normalized cumulants nsc 2,4 {4} and nac 2 {3} based on transverse momentum conservation and collective flow.Compared to the recent ATLAS experimental measurements with the subevent method, we aim to understand and explore the origin of the collectivity in small systems.
II. sc2,4 {4} AND ac2 {3} FROM TRANSVERSE MOMENTUM CONSERVATION First, we summarize the calculation method of the TMC, which is assumed to be the only effect of correlations between final particles.The k-particle probability distribution f ( p 1 , . . ., p k ) for the N -particle system with imposed transverse momentum conservation is given by [85][86][87][88] where p 2 F denotes the mean value of p 2 over the full space F , Our goal is to calculate the four-particle symmetric cumulant sc 2,4 {4} and three-particle asymmetric cumulant ac 2 {3}.The four-particle symmetric cumulant and and three-particle asymmetric cumulant are defined as follows: A. sc2,4 {4} For four particles, we have where and To calculate e i2(φ1−φ2)+i4(φ3−φ4) at given transverse momenta p 1 , p 2 , p 3 , and p 4 , we expand exp(−Φ) in Φ.In the numerator, the first nonzero term is given by Φ 6 /720 and we neglect all higher terms.In the denominator, it is enough to take the first term, exp(−Φ) ≈ 1, since the next terms are suppressed by the power of 1/N .To simplify our calculation, we assume that all the transverse momentum p i are equal.In this case, we obtain Performing analogous calculations we obtain and Using Eq. ( 3) we find B. ac2 {3} For three particles, we have Using Eq. ( 4) we find III. sc2,4 {4} AND ac2 {3} FROM TRANSVERSE MOMENTUM CONSERVATION AND FLOW Next, we calculate the contribution of the TMC and the collective flow to the four-particle symmetric cumulant sc 2,4 {4} and three-particle asymmetric cumulant ac 2 {3}.The particle emission azimuthal angle distribution measured with respect to the reaction plane is characterized by a Fourier expansion, where v n and Ψ n denote the nth-order flow coefficient and the reaction plane angle.In our calculations we consider v 2 , v 3 , and v 4 only.
A. sc2,4 {4} The four-particle probability distribution with TMC can be written as [87] where Using and including all the terms up to the one containing the pure TMC effect, e where and Note that the above items are approximate results, where we kept the terms up to v 4 n .The full results and the ratios relative to the full results are shown in Eq. (A.1) and the left panel of Fig. 5 in the Appendix, respectively.
Similarly, we have where and Finally where and B. ac2 {3} The three-particle probability distribution with TMC can be written as We have where and Note that the above items are approximate results, where we kept the terms up to v 3 n .The full results and the ratios relative to the full results are shown in Eq. (A.2) and the right panel of Fig. 5 in the Appendix, respectively.

C. nsc2,4 {4} and nac2 {3}
The normalized cumulants are defined as follows: where v 2 {2} and v 4 {2} are from Eqs. ( 25) and (28).The normalized cumulants only reflect the strength of the correlation between v 2 and v 4 , whereas the unnormalized cumulants have contributions from both the correlations between the two different flow harmonics and the individual harmonics.
IV. RESULTS Based on Eqs. ( 11) and ( 14), we present the four-particle symmetric cumulants sc 2,4 {4} and three-particle asymmetric cumulants ac 2 {3} from transverse momentum conservation only as a function of the number of particles N for various values of transverse momenta p = 0.6, 0.9, 1.2 GeV in Fig. 1.In our calculation, p 2 F = 0.25 GeV 2 .It can be seen that the values of sc 2,4 {4} and ac 2 {3} from transverse momentum conservation decrease and tend to zero as N increases, and that sc 2,4 {4} and ac 2 {3} also increase with transverse momenta p, which are consistent with the trends found in Refs.[82,83] using the PYTHIA model.This is a manifestation of the property of the TMC that it is more effective at smaller N and rather negligible at larger N .According to Eqs. ( 22), ( 25), (28), and (32), Fig. 2 shows sc 2,4 {4} and ac 2 {3} from transverse momentum conservation and flow as a function of the number of particles N for various values of transverse momenta p = 0.6, 0.9, 1.2 GeV.In our calculation, we set v 2 = 0.08, v 3 = 0.0175, v 4 = 0.08 2 , p 2 F = 0.5 2 , v 2F = 0.025, Ψ 2 = 0, cos(4(Ψ 4 −Ψ 2 )) = 0.8, and cos(2Ψ 2 −6Ψ 3 +4Ψ 4 ) = −0.15.The values of the correlations among different combinations of event planes here are from Ref. [89].We observe that both sc 2,4 {4} and ac 2 {3} decrease with the increase of multiplicity and their magnitudes are consistent with the data.Note that since the multiplicity N refers to the number of particles under the influence of the TMC rather than the number of experimentally detected charged particles, we multiply the experimental number of charged particles by 1.5 to obtain the total number of particles N in the experimental data in all figures.In comparison with Fig. 1, for larger N , sc 2,4 {4} and ac 2 {3} do not converge to zero, which is caused by the existence of flow due to hydrodynamics.When N is relatively small, sc 2,4 {4} and ac 2 {3} increase with increasing momentum p, whereas when N is large, sc 2,4 {4} and ac 2 {3} hardly change with momentum.3: sc2,4 {4} and ac2 {3} from the TMC, the TMC and collective flow, and plus interplay as a function of the number of particles N for momentum p = 0.9 GeV.The ATLAS data for 0.3 < pT < 3 GeV in p+p collisions at 13 TeV using the four-subevent cumulant method or three-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively [84].
To expound how the TMC and collective flow affect sc 2,4 {4} and ac 2 {3}, Fig. 3 presents the respective contributions from the TMC only (denoted as "TMC"), the TMC and collective flow (denoted as "TMC+flow"), and plus interplay (denoted as "TMC+flow+interplay") for p = 0.9 GeV.Here "TMC" refers to the terms that depend only on N and p, "flow" refers to the terms that depend only on v n and Ψ n , and "interplay" refers to terms that depend on both N , p, v n , and Ψ n in Eqs. ( 22), ( 25), (28), and (32).In Fig. 3, "TMC+flow" means the sum of "TMC" and "flow", and "TMC+flow+interplay" means the combination of all three of the above.We see that collective flow makes the curve higher and the contribution from interplay is present when N is small, but almost negligible when N is large.It can be understood as when N is small, the TMC dominates, and when N is large, the contribution from collective flow becomes significant.Figure 4 shows that the normalized cumulants nsc 2,4 {4} and nac 2 {3} from the TMC and flow decrease with the increase of multiplicity, which can basically describe the experimental data.Since Fig. 2 has shown that sc 2,4 {4} and ac 2 {3} can describe the ATLAS data, it indicates that our results on two-particle v 2 {2} and v 4 {2} should also be consistent with the experimental data.In the left plot of Fig. 4, because the TMC contribution is small when N is large, our results close to 2.0 for nsc 2,4 {4} at very large N reflect the correlation between v 2 {2} and v 4 {2} produced by hydrodynamics.In the right plot of Fig. 4, we see that nac 2 {3} is close to 1, suggesting that the event planes Ψ 2 and Ψ 4 gradually converge in the same direction at large N , consistent with the hydrodynamic expectation.The increase in nsc 2,4 {4} and nac 2 {3} with decreasing N and the increase in nsc 2,4 {4} and nac 2 {3} with increasing p are both due to the TMC effect.

V. CONCLUSIONS
In this paper, we calculated the four-particle symmetric cumulants sc 2,4 {4}, three-particle asymmetric cumulants ac 2 {3}, and the normalized cumulants nsc 2,4 {4} and nac 2 {3}, originating from the transverse momentum conservation and flow.As expected, when the number of particles is small, the correlation comes from the TMC, and when the number of particles is large, the collective flow is dominant.Our results are consistent with the ATLAS data using the subevent cumulant method and therefore allow for a better understanding of collectivity in small systems.In the future, we can calculate the higher order symmetric cumulants sc k,l,m {6} in the same way to understand how the TMC and collective flow affects the coupling between v k , v l , and v m in small systems.

APPENDIX
The full results of the Eq. ( 24) are as follows:   Based on Fig. 5, the ratios of the approximate results of Eqs. ( 24) and ( 34) to the full results of Eqs.(A.1) and (A.2) in this appendix are both close to 1, with the worst approximations of 99.244% and 99.378% respectively, suggesting that Eqs. ( 24) and ( 34) can be good proxies for the results of Eqs.(A.1) and (A.2) in this appendix.

FIG. 1 :
FIG.1:The four-particle symmetric cumulants sc2,4 {4} and three-particle asymmetric cumulants ac2 {3} from transverse momentum conservation only as a function of the number of particles N for various values of transverse momenta p.

FIG. 2 :
FIG.2: sc2,4 {4} and ac2 {3} from transverse momentum conservation and flow as a function of the number of particles N for various values of transverse momenta p.The ATLAS data for 0.3 < pT < 3 GeV in p+p collisions at 13 TeV using the four-subevent cumulant method or three-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively[84].
FIG.3: sc2,4 {4} and ac2 {3} from the TMC, the TMC and collective flow, and plus interplay as a function of the number of particles N for momentum p = 0.9 GeV.The ATLAS data for 0.3 < pT < 3 GeV in p+p collisions at 13 TeV using the four-subevent cumulant method or three-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively[84].

FIG. 4 :
FIG.4: nsc2,4 {4} and nac2 {3} from the TMC and flow as a function of the number of particles N for various values of transverse momenta p.The ATLAS data for 0.3 < pT < 3 GeV in p+p collisions at 13 TeV using the three-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively[84].

FIG. 5 :
FIG. 5:The ratios of the approximate result to the full result, i.e., Eq. (24) / Eq. (A.1) (left panel) and Eq.(34) / Eq. (A.2) (right panel), as a function of the number of particles N , for different values of the transverse momenta p .