Non-flow effects in correlation between harmonic flow and transverse momentum in nuclear collisions
Chunjian Zhang, Arabinda Behera, Somadutta Bhatta, Jiangyong Jia
NNon-flow effects in correlation between harmonic flow and transverse momentum innuclear collisions
Chunjian Zhang, Arabinda Behera, Somadutta Bhatta, and Jiangyong Jia
1, 2, ∗ Department of Chemistry, Stony Brook University, Stony Brook, NY 11794, USA Physics Department, Brookhaven National Laboratory, Upton, NY 11976, USA (Dated: February 11, 2021)A large anti-correlation signal between elliptic flow v and average transverse momentum [ p T ] wasrecently measured in small collision systems, consistent with a final-state hydrodynamic responseto the initial geometry. This negative v – [ p T ] correlation was predicted to change to positivecorrelation for events with very small charged particle multiplicity N ch due to initial-state momentumanisotropies of the gluon saturation effects. However, the role of non-flow correlations is expectedto be important in these systems, which is not yet studied. We estimate the non-flow effects in pp , p Pb and peripheral PbPb collisions using
Pythia and
Hijing models, and compare them withthe experimental data. We show that the non-flow effects are largely suppressed using the rapidity-separated subevent cumulant method (details of the cumulant framework are also provided). Themagnitude of the residual non-flow is much less than the experimental observation in the higher N ch region, supporting the final-state response interpretation. In the very low N ch region, however, thesign and magnitude of the residual non-flow depend on the model details. Therefore, it is unclear atthis moment whether the sign change of v – [ p T ] can serve as evidence for initial state momentumanisotropies predicted by the gluon saturation. PACS numbers: 25.75.Gz, 25.75.Ld, 25.75.-1
I. INTRODUCTION
In high-energy hadronic collisions, particle correlations are an important tool to study the multi-parton dynamics ofQCD in the strongly coupled non-perturbative regime [1]. Measurements of azimuthal correlations in small collisionsystems, such as pp and p +A collisions [2–6], have revealed a strong harmonic modulation of particle densitiesd N / d φ ∝ + ∑ ∞ n = v n cos n ( φ − Φ n ) . Measurement of v n and their event-by-event fluctuations have been performedas a function of charged particle multiplicity N ch in pp and p +A collisions. It is found that the azimuthal correlationsinvolve all particles over a wide pseudorapidity range. A key question is whether this multi-particle collectivity reflectsinitial momentum correlation from gluon saturation effects (ISM) [7], or a final-state hydrodynamic response to theinitial transverse collision geometry (FSM) [8].Recently, the correlation between v n and [ p T ] , the average transverse momentum of particles in each event, wasproposed to be a sensitive observable to distinguish between the initial-state and final-state effects [9]. The lowestorder of such correlation is characterized by the covariance cov ( v n , [ p T ]) ≡ ⟨ v n [ p T ]⟩ − ⟨ v n ⟩ ⟨[ p T ]⟩ [10] with the averagecarried over events, which have been measured at the LHC [11, 12]. In the final-state dominated scenario, the flowharmonics are diven by the initial spatial eccentricity ε n , v n ∝ ε n , while the [ p T ] is related to the transverse size of theoverlap region: events with similar total energy but smaller transverse size in the initial state are expected to have astronger radial expansion and therefore larger [ p T ] [13]. In p +A and A+A collisions, hydrodynamic model calculationspredict a positive cov ( v n , [ p T ]) at large N ch which changes to negative cov ( v n , [ p T ]) towards small N ch region[14–16]. In the initial-state dominated scenario, however, the initial momentum correlations from gluon saturation areexpected to give a positive contribution to cov ( v n , [ p T ]) at small N ch region [9]. Therefore, the N ch dependence ofcov ( v , [ p T ]) , after considering both initial and final-state effects, is predicted to exhibit a double sign change as afunction N ch . The experimental observation of such sign change was further argued to provide a strong evidence forthe gluon saturation physics [9].On the other hand, momentum correlations could also arise from “non-flow” effects from resonance-decays, jetsand dijets [17]. Such non-flow correlations usually involve a few particles from one or two localized pseudorapidityregions, in contrast to the initial momentum correlation from gluon saturation, which spans continuously over a largerapidity range similar to hydrodynamic flow. The non-flow effects are often suppressed by correlating particles fromtwo or more subevents separated in pseudorapidity. This so-called subevent cumulant method [18] has been validated ∗ Correspond to [email protected] a r X i v : . [ nu c l - t h ] F e b for several multi-particle correlators involving flow harmonics of same or different orders[18–20], such as four-particlecumulants c n { } = ⟨ v n ⟩ − ⟨ v n ⟩ , four-particle symmetric cumulants ⟨ v n v m ⟩ − ⟨ v n ⟩ ⟨ v m ⟩ and three particle asymmetriccumulants ⟨ v n v m v n + m cos ( n Φ n + m Φ m − ( n + m ) Φ n + m )⟩ . It is found that results from the standard cumulant methodare contaminated by non-flow correlations in pp , p A and peripheral AA collisions, while they are largely suppressedin the subevent method that requires three or more subevents [21, 21, 22]. Since covariance cov ( v n , [ p T ]) is a three-particle correlator, it can be measured with two subevent or three-subevent methods, which suppress the non-flowwhile keeping the genuine long range multi-particle correlations associated with ISM and FSM.In this paper, we study the influence of the non-flow correlations to covariance cov ( v n , [ p T ]) in pp , p Pb and PbPbcollisions using
Pythia8 [23] A2 tune and
Hijing v1.37 [24] models in the standard and subevent methods. Wefind that the non-flow correlations give a positive contribution to cov ( v , [ p T ]) , which are strongly suppressed in thethree-subevent method, but not completely eliminated. The sign and magnitude of the residual non-flow are modeldependent. Therefore, the mere observation of change of cov ( v , [ p T ]) from negative to positive towards low N ch inthe experimental results may not serve as evidence for the presence of gluon saturation. II. METHODOLOGY AND MODEL SETUP
The covariance cov ( v n , [ p T ]) is a three-particle correlator, which is obtained by averaging over unique triplets ineach event, and then over all events in an event class [10, 14]:cov ( v n , [ p T ]) = ⟨ ∑ i,j,k,i ≠ j ≠ k w i w j w k e in ( φ i − φ j ) ( p T ,k − ⟨[ p T ]⟩)∑ i,j,k,i ≠ j ≠ k w i w j w k ⟩ (1)where the indices i , j and k loop over distinct charged particles to account for all unique triplets, the particle weight w i is constructed to correct for detector effects, and the ⟨⟩ denotes average over events. In order to reduce short-range“non-flow” correlations, pseudorapidity gaps are often explicitly required between the particles in each triplet. Thisanalysis uses the so-called standard, two-subevent and three-subevent methods [18] to explore the influence of non-flowcorrelations as detailed below.The choices of η ranges for the subevents are identical to those used by the ATLAS experiment [11, 12]. In thestandard method, all charged particles within ∣ η ∣ < . a and c with a ∆ η gap in between to reduce non-flow effects: − . < η a < − . , . < η c < .
5. The two particles contributing to the flow vector are chosen as one particle eachfrom a and c , while the third particle providing the p T weight is taken from either a or c . In the three-subeventmethod, three non-overlapping subevents a , b and c are chosen: − . < η a < − . , ∣ η b ∣ < . , . < η c < .
5. Theparticles contributing to flow are chosen from subevents a and c while the third particle is taken from subevent b .A direct calculation of the nested-loop in Eq. (1) is computationally expensive. Instead, it can be expandedalgebraically within the multi-particle cumulant framework [18, 25] into a polynomial function of vectors and scalars: q n ; k = ∑ i w ki e inφ i ∑ i w ki , o n ; k = ∑ i w ki e inφ i ( p T ,i − ⟨[ p T ]⟩)∑ i w ki , p m ; k = ∑ i w ki ( p T ,i − ⟨[ p T ]⟩) m ∑ i w ki , τ k = ∑ i w k + i /(∑ i w i ) k + (2)where the sum runs over particles in a given event or subevent and “ k ” and “ m ” are natural integer powers. It isstraightforward to show that expansion of Eq. (1) in the three methods gives:cov ( v n , [ p T ]) std = ⟨ (∣ q n ;1 ∣ − τ ) p − τ R ( o n ;2 q ∗ n ;1 ) + τ p − τ + τ ⟩ (3)cov ( v n , [ p T ]) = ⟨ R [( q n ;1 p − τ o n ;2 ) a ( q ∗ n ;1 ) c + ( q n ;1 p − τ o n ;2 ) c ( q ∗ n ;1 ) a ] − ( τ ) a + − ( τ ) c ⟩ (4)cov ( v n , [ p T ]) = ⟨ R [( q n ;1 ) a ( q ∗ n ;1 ) c ]( p ) b ⟩ (5)where the R denotes the real component of the complex number.Experimentally, the v n - [ p T ] correlation is aften presented in normalized form known as Pearson’s correlationcoefficient [10], ρ ( v n , [ p T ]) = cov ( v n , [ p T ])√ var ( v n )√ var ([ p T ]) , (6)where the var ([ p T ]) and var ( v n ) are variances of p T fluctuations and v n fluctuations, respectively. The var ([ p T ]) isobtained using all the pairs in the full event ∣ η ∣ < . ([ p T ]) = ⟨ ∑ i,j,i ≠ j w i w j p T ,i − ⟨[ p T ]⟩)( p T ,j − ⟨[ p T ]⟩)∑ i,j,i ≠ j w i w j ⟩ = ⟨ p − p − τ ⟩ (7)The dynamical variance var ( v n ) are calculated in terms of two-particle cumulant c n { } and four particle cumulants c n { } following Ref. [12]: var ( v n ) ≡ ⟨ v n ⟩ − ⟨ v n ⟩ = c n { } std + c n { } . (8)The c n { } , being a four-particle correlator, is known to be relatively insensitive to non-flow correlations but usuallyhas poor statistical precision. Therefore it is obtained from the standard cumulant method using the full event. On theother hand, the two particle cumulants c n { } is more susceptible to non-flow correlations and therefore is calculatedfrom the two-subevent method with the η choices discussed above. This definition is mostly free of non-flow in largecollision systems. But in small systems, this definition could still be biased by non-flow effects as we discussed inAppendix A.To evaluate the influence of non-flow correlations to cov ( v n , [ p T ]) and ρ ( v n , [ p T ]) , the Pythia8
A2 tune [23]and
Hijing v1.37 [24] models are used to generate pp events at √ s =
13 GeV, p Pb and peripheral PbPb events at √ s NN = .
02 TeV, respectively. These models contain significant non-flow correlations from jets, dijets, and resonancedecays and can be used to quantify the efficacy of non-flow suppression in these methods. In these simulations, theparticle weight are set to be unity, w i = N ch , the number of charged particles in ∣ η ∣ < . p T > . ( v n , [ p T ]) are calculated in three p T ranges using the standard and subevent methods:0 . < p T < . < p T < . < p T < dN ch / dη , which is assumed to be 1/5 of N ch , N ch ≈ dN ch / dη . III. RESULTS
Figure 1 compares the results of cov ( v n , [ p T ]) from the standard and subevent methods in pp collisions from Pythia8 model. The values from the standard method are positive for all harmonics. This is because the correlationsare dominated by the jet fragmentations, which produce clusters of particles with larger p T and enhanced azimuthalcorrelations at ∆ φ ∼
0, and therefore tend to simultaneously increase the v n and [ p T ] . The values from the two-subevent method are positive for even harmonics and negative for odd harmonics, consistent with the dominance ofcorrelations from away-side jet fragments: the away-side correlations are expected to give a more negative v andlarger [ p T ] , and therefore a negative value of cov ( v , [ p T ]) . For the three-subevent method, the values of cov ( v , [ p T ]) are positive at dN ch / dη ≲
10 and are slightly negative for dN ch / dη >
10. The magnitudes of cov ( v n , [ p T ]) are largestfor the standard method, and smallest for the three-subevent method. Similar ordering among the three methods areobserved in all three collision systems and all p T selections, and the magnitudes of signal from three-subevent methodare always the smallest, suggesting that this method is least affected by non-flow. For the remaining discussion, wefocus on discussing results from the three-subevent method. h /d ch dN ] ) T ,[ p c o v ( v -3 · standardtwo-subeventthree-subevent <2 GeV T h | pp 13TeV, PYTHIA h /d ch dN ] ) T ,[ p c o v ( v -0.1-0.0500.050.1 -3 · <2 GeV T h | pp 13TeV, PYTHIA h /d ch dN ] ) T ,[ p c o v ( v -3 · <2 GeV T h | pp 13TeV, PYTHIA FIG. 1: The cov ( v n , [ p T ]) as a function of dN ch / dη for n = . < p T < pp Pythia8 . Figure 2 compares the results of cov ( v n , [ p T ]) from three p T ranges. The overall magnitudes of cov ( v n , [ p T ]) arelarger in the higher p T range, reflecting a larger non-flow correlation at higher p T . The values of cov ( v , [ p T ]) exhibitqualitatively a similar sign change behavior at dN ch / dη ∼ −
10 for all p T ranges. The values of cov ( v , [ p T ]) aremostly positive, and the values of cov ( v , [ p T ]) seem to be systematically below zero. h /d ch dN ] ) T ,[ p c o v ( v -0.100.1 -3 · <2 GeV T T T pp 13TeV, PYTHIA|<2.5 h | three-subevent h /d ch dN ] ) T ,[ p c o v ( v -0.0500.05 -3 · pp 13TeV, PYTHIA|<2.5 h | three-subevent h /d ch dN ] ) T ,[ p c o v ( v -0.02-0.0100.010.02 -3 · pp 13TeV, PYTHIA|<2.5 h | three-subevent FIG. 2: The cov ( v n , [ p T ]) as a function of dN ch / dη from the three-subevent method for n = p T ranges in 13 TeV pp collisions. To further investigate the origin of the sign change of cov ( v , [ p T ]) in the low dN ch / dη region, Figure 3 compares the pp results from Pythia8 with those obtained from the
Hijing model. The results are in good quantitative agreementfor dN ch / dη >
20. In the dN ch / dη <
30 range and towards lower dN ch / dη , the Hijing results show a stronger decreasecompared to the
Pythia8 results. The
Hijing results start to increase at dN ch / dη <
10 similar to
Pythia8 , but exceptfor the lowest p T range of 0 . < p T < ( v , [ p T ]) to change sign. Theresults from pp collisions at √ s = √ s =
13 TeV results for dN ch / dη <
20, suggesting that residual non-flow is larger at lower √ s at the same dN ch / dη . h /d ch dN ] ) T ,[ p c o v ( v -0.2-0.100.1 -3 · pp 13TeV, PYTHIA pp 13TeV, HIJINGpp 5TeV, HIJING <2 GeV T h /d ch dN ] ) T ,[ p c o v ( v -0.2-0.100.1 -3 · <2 GeV T h /d ch dN ] ) T ,[ p c o v ( v -0.2-0.100.1 -3 · <5 GeV T FIG. 3: The cov ( v , [ p T ]) as a function of dN ch / dη from the three-subevent method compared between three pp collisionsystems for 0 . < p T < . < p T < . < p T < Figure 4 compares the results of cov ( v , [ p T ]) between pp , p Pb and PbPb collisions, separately in three p T ranges.The p Pb and PbPb values are negative at low dN ch / dη region, whose magnitudes increase with p T . This is differentfrom the pp results, which are positive at dN ch / dη ≲ dN ch / dη >
10 region, the pp values are negativeand lower than those for the p Pb and PbPb collisions. The values for p Pb collisions are close to but consistently lowerthan those in PbPb collisions, suggesting a slightly larger residual non-flow in p Pb collisions.In order to estimate the non-flow effects on the ρ ( v n , [ p T ]) , we need to choose an appropriate normalization inEq. (6). The var ( v n ) directly obtained from these models should not be used, because they only contain non-flow.Instead, we estimate var ( v n ) from the previous published measurements of v n { } and v n { } in these three collisionsystems [6, 21, 26] as: var ( v n ) = ⟨ v n ⟩ − ⟨ v n ⟩ = v n, tmp { } ⎛⎝ − [ v n { } v n { } ] ⎞⎠ (9)The v n, tmp { } were measured using the two-particle correlation and improved template method from Ref. [26] thatexplicitly subtracts the non-flow correlations. The p T dependence of the v n, tmp { } are taken from Ref. [26]. Thevalues of v { }/ v { } are taken from Ref. [21] for pp and p Pb and from Ref. [6] for PbPb, which are found to be inthe range of 0.71–0.74 as a function of dN ch / dη , and they are assumed to be independent of p T . The v { } term h /d ch dN ] ) T ,[ p c o v ( v -0.1-0.0500.050.1 -3 · pp 13TeV, PYTHIApPb 5TeV, HIJINGPbPb 5TeV, HIJING <2 GeV T h | three-subevent h /d ch dN ] ) T ,[ p c o v ( v -0.1-0.0500.050.1 -3 · <2 GeV T h | three-subevent h /d ch dN ] ) T ,[ p c o v ( v -0.1-0.0500.050.1 -3 · <5 GeV T h | three-subevent FIG. 4: The cov ( v , [ p T ]) as a function of dN ch / dη from the three-subevent method compared between three collision systemsfor 0 . < p T < . < p T < . < p T < leads to a 28% reduction to var ( v ) . For third-order harmonics, the values of v { }/ v { } have been found to be verysmall Ref. [22] and therefore is neglected in this study, i.e. we assume var ( v ) = v , tmp { } . Examples of the dN ch / dη dependence of var ( v ) and var ( v ) are given in Figure 7 of Appendix A. h /d ch dN ] ) T ,[ p ( v r -0.500.5 <2 GeV T h | pp 13TeV PYTHIA pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING h /d ch dN ] ) T ,[ p ( v r -0.500.5 <2 GeV T h | pp 13TeV PYTHIA pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING h /d ch dN ] ) T ,[ p ( v r -0.500.5 <5 GeV T h | pp 13TeV PYTHIA pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING h /d ch dN ] ) T ,[ p ( v r -20246 <2 GeV T h | · pp 13TeV PYTHIA 1/3 · pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING h /d ch dN ] ) T ,[ p ( v r -20246 <2 GeV T h | · pp 13TeV PYTHIA 1/3 · pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING h /d ch dN ] ) T ,[ p ( v r -20246 <5 GeV T h | · pp 13TeV PYTHIA 1/3 · pp 13TeV HIJING pPb 5TeV HIJINGPbPb 5TeV HIJING FIG. 5: The ρ ( v , [ p T ]) (top) and ρ ( v , [ p T ]) (bottom) estimated via as a function of dN ch / dη from the three-subevent methodcompared between three collision systems for 0 . < p T < . < p T < . < p T < ρ ( v , [ p T ]) from pp collisions have been scaled down by a factor of 3. The results of ρ ( v , [ p T ]) and ρ ( v , [ p T ]) are shown in Figure 5 for the three collision systems. They provide anestimate of the expected non-flow contributions to the experimentally measured ρ ( v n , [ p T ]) . In the 0 . < p T < dN ch / dη >
12 region, the values of ∣ ρ ( v , [ p T ])∣ are < .
02 in p Pb and PbPb collisions and are < .
05 in the pp collisions. An experimental observation of a signal much larger than these values could be a clear indication ofnon-trivial initial- and final-state correlations unrelated to non-flow. In the higher p T and dN ch / dη >
20 region, thevalues of ∣ ρ ( v , [ p T ])∣ are ≲ .
02 in the PbPb and ≲ .
06 in the p Pb collisions, but are significantly larger in the pp collisions ( ∼ . − . ρ ( v , [ p T ]) , current statistical uncertainties do not provide a precise lower limit for thenon-flow contributions in p Pb and PbPb collisions. But in pp collisions and higher p T , the non-flow effects could leadto ρ ( v , [ p T ]) values significantly larger than one.Equipped with these detailed knowledge of non-flow, we are ready to discuss its impact on the interpretation ofthe v n – [ p T ] correlation in terms of ISM and FSM. The top panels of Figure 6 compare the non-flow expectationof cov ( v , [ p T ]) with the ATLAS data [11]. The strength of the non-flow correlations is much smaller than theexperimental data in the PbPb collisions (which covers dN ch / dη >
20 region), but could be significant in p Pb collisionsin 0 . < p T < dN ch / dη ∼
20. The results arealso compared to the CGC-hydro model for p Pb [9] that includes both ISM and FSM but without non-flow. In the dN ch / dη >
20 region where the FSM dominates, the model over-predicts the experimental data. In the dN ch / dη < . < p T < . < p T < dN ch / dη ∼ −
10, but would still remain negative at dN ch / dη ∼ h /d ch dN ] ) T ,[ p c o v ( v -0.0200.02 -3 · · <2 GeV T h /d ch dN ] ) T ,[ p c o v ( v -0.050 -3 · <2 GeV T pPb HIJING 3subPbPb HIJING 3sub pPb Data 1907.05176PbPb Data 1907.05176 pPb CGC-hydro 2006.15721 h /d ch dN ] ) T ,[ p ( v r -0.4-0.200.2 · <2 GeV T h /d ch dN ] ) T ,[ p ( v r -0.4-0.200.2 <2 GeV T pPb HIJING 3subPbPb HIJING 3sub pPb Data 1907.05176PbPb Data 1907.05176 pPb CGC-hydro 2006.15721 FIG. 6: The cov ( v , [ p T ]) (top) and ρ ( v , [ p T ]) (bottom) as a function of dN ch / dη in p Pb and PbPb collisions for 0 . < p T < . < p T < dN ch / dη <
10 in the left panels have been rescaled by the factors in order to fit into the y-ranges.
The bottom panels of Figure 6 show the same comparison in terms of ρ ( v , [ p T ]) . The qualitative behaviors arelargely the same, with a few important quantitative differences from cov ( v , [ p T ]) . The non-flow contributions relativeto the experimental data are larger, especially in the p Pb collisions, reaching more than 50% of the experimental valuesat dN ch / dη ∼
20 in 0 . < p T < ([ p T ]) in HIJING are about a factorof 2 smaller than the experimental values, leading to a more negative ρ ( v , [ p T ]) closer to the data. We also cautionthat the ATLAS var ( v n ) data, calculated via Eq. (8), are still biased by non-flow contributions (see Appendix A),which reduce ρ ( v , [ p T ]) slightly further. The main message of Figure 6 is that the interpretation of the cov ( v , [ p T ]) at low dN ch / dη region is rather complicated. Firstly, the non-flow contributions from our model studies are negativeand could account for some of the observed negative signal in the low dN ch / dη range that are also associated with theFSM. Secondly, the negative non-flow contributions compete with the ISM and may eliminate the sign-change in theactual measurement. Thirdly, the fact that Pythia8 model shows a positive ρ ( v , [ p T ]) at dN ch / dη <
10 (Figure 5)suggests that the sign of non-flow contributions is model-dependent and could also be positive. In the latter case,even if experiments observe a positive ρ ( v , [ p T ]) , one could not easily interpret this signal as generated by the ISM. IV. SUMMARY
The influences of non-flow effects to the three-particle correlation between harmonic flow v n and event-by-eventaverage transverse momentum [ p T ] , cov ( v n , [ p T ]) , are studied in pp , p Pb and peripheral PbPb collisions for n = − Pythia8 and
Hijing event generators, which contain only non-flow correlations suchas fragmentation of jet and dijets and resonance decays, but have no genuine long-range multi-particle correlationsfrom the initial-state or the final-state evolution. The efficacy of non-flow suppression via the rapidity separatedthree-subevent method has been tested, and is observed to give smallest ∣ cov ( v n , [ p T ])∣ values in comparison to thestandard and two-subevent methods for all harmonics, collision systems and p T ranges investigated in this paper. Thevalues of cov ( v , [ p T ]) from the three-subevent method are negative in the region dN ch / dη >
20 and approach zerotowards higher dN ch / dη . The magnitudes of the cov ( v , [ p T ]) are much smaller than the experimentally measuredvalues in the p Pb and PbPb collisions, suggesting that the measured cov ( v , [ p T ]) values in dN ch / dη >
20 reflectgenuine correlations arising from the final-state interactions. In the region dN ch / dη <
20, the values of cov ( v , [ p T ]) decrease toward more negative values in Hijing simulations of p Pb and PbPb collisions, but increases in
Hijing and
Pythia8 simulations of pp collisions. They reach a maximum (positive for Pythia8 but is negative in
Hijing ) ataround dN ch / dη ∼
20 before decreasing again for dN ch / dη <
5. The differences between
Hijing and
Pythia8 suggestthat the non-flow contributions in dN ch / dη <
20 region are highly model-dependent. The predicted sign change of dN ch / dη from initial-state momentum correlation due to gluon saturation physics may not be observed if the non-flowcontributions are negative, or unambiguous if the non-flow contributions are positive. Further detailed quantitativemodel investigation of these different sources are required.We appreciate comments from G. Giacalone, S. Huang, J. Nagle, and B. Schenke. We thank B. Schenke for sharingthe CGC-hydro model calculation, and J. Nagle’suggestion to compare pp collisions between Pythia8 and
Hijing .This work is supported by DEFG0287ER40331 and PHY-1913138.
Appendix A: Influence of non-flow on var ( v n ) In the ATLAS measurement [12], the var ( v n ) was calculated using Eq. (8). In the low dN ch / dη region, the c n { } and the resulting ρ ( v n , [ p T ]) could be strongly biased by the non-flow correlations. Figure 7 compares var ( v n ) fromEq. (8) with those estimated via Eq. (9) based on published v n data in three collision systems. They are presentedin terms of √ var ( v n ) in order to be shown in the familiar scale as the single-particle v n values. In p Pb and PbPbcollisions, the non-flow is sub-dominant for dN ch / dη >
20 but can be larger than the genuine flow signal at lower dN ch / dη values. In the pp collisions, the non-flow contribution is comparable or larger than the genuine flow signalover the full dN ch / dη range.To estimate the possible bias of the non-flow, we add the var ( v n ) from flow and non-flow of Figure 7 in quadraturesum: var ( v n ) mod = √ var ( v n ) + var ( v n ) − flow . The var ( v n ) mod is then used to obtain a modified form of Pearsoncoefficient ρ ( v n , [ p T ]) mod . The results are shown in Figure 8. Comparing to the original unbiased results in Figure 5,the magnitudes of the ρ ( v n , [ p T ]) mod are much reduced in the low dN ch / dη region due to the large non-flow bias tovar ( v n ) . The differences between the three systems are also artificially reduced. Therefore, it is important to use avar ( v n ) that is free of non-flow effects by following the procedure given in Eq. (9). [1] E. Shuryak, Rev. Mod. Phys. , 035001 (2017), arXiv:1412.8393 [hep-ph] .[2] S. Chatrchyan et al. (CMS), Phys. Lett. B , 795 (2013), arXiv:1210.5482 [nucl-ex] .[3] ALICE Collaboration, Phys. Lett. B , 29 (2013), arXiv:1212.2001 [nucl-ex] .[4] G. Aad et al. (ATLAS), Phys. Rev. Lett. , 182302 (2013), arXiv:1212.5198 [hep-ex] .[5] G. Aad et al. (ATLAS), Phys. Rev. C , 044906 (2014), arXiv:1409.1792 [hep-ex] .[6] V. Khachatryan et al. (CMS), Phys. Rev. Lett. , 012301 (2015), arXiv:1502.05382 [nucl-ex] .[7] K. Dusling and R. Venugopalan, Phys. Rev. D , 094034 (2013), arXiv:1302.7018 [hep-ph] .[8] P. Bozek and W. Broniowski, Phys. Rev. C , 014903 (2013), arXiv:1304.3044 [nucl-th] . h /d ch dN ) v a r( v <2 GeV T h | pp 13TeV, PYTHIApPb 5TeV, HIJINGPbPb 5TeV, HIJINGpp 13TeV, DatapPb 5TeV, DataPbPb 5TeV, Data h /d ch dN ) v a r( v <2 GeV T h | h /d ch dN ) v a r( v <5 GeV T h | h /d ch dN ) v a r( v <2 GeV T h | pp 13TeV, PYTHIApPb 5TeV, HIJINGPbPb 5TeV, HIJINGpp 13TeV, DatapPb 5TeV, DataPbPb 5TeV, Data h /d ch dN ) v a r( v <2 GeV T h | h /d ch dN ) v a r( v <5 GeV T h | FIG. 7: The √ var ( v ) (top) and √ var ( v ) (bottom) as a function of dN ch / dη compared between three collision systems for0 . < p T < . < p T < . < p T < Pythia8 and
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