Identification of flow-background to subtract in jet-like azimuthal correlation
aa r X i v : . [ nu c l - e x ] J a n Identification of flow-background to subtract in jet-like azimuthal correlation
Quan Wang and Fuqiang Wang Department of Physics, Purdue University, 525 Northwestern Ave., West Lafayette, IN 47907
We derive an analytical form for flow-background to jet-like azimuthal correlation in a clusterapproach. We argue that the elliptic flow parameter to use in jet-correlation background is that fromtwo-particle method excluding non-flow correlation unrelated to the reaction plane, but includingcross-terms between cluster correlation and cluster flow. We verify our result with Monte Carlosimulations. We discuss implications of our finding in the context of jet-like correlations from STARand PHENIX.
PACS numbers: 25.75.-q, 25.75.Gz, 25.75.Ld
I. INTRODUCTION
Jet-like angular correlation studies with high transverse momentum ( p T ) trigger particles have provided valuable in-formation on the properties of the medium created in relativistic heavy-ion collisions [1, 2]. In such studies, correlationfunctions are formed in azimuthal angle difference between an associated particle and a high p T trigger particle, whichpreferentially selects (di-)jet. One important aspect of these studies is the subtraction of combinatorial backgroundwhich itself is non-uniform due to anisotropic particle distribution with respect to the reaction plane– both the triggerparticle and the associated particles are correlated with the common reaction plane in an event. One critical part isto determine flow parameters, mainly elliptic flow ( v ), to use in constructing background.There are many v measurements [3, 4]. They contain various degrees of non-flow contributions, such as those fromresonance decays and jet correlations. Those non-flow effects should not be included in the background to subtractfrom jet-like correlations. We shall refer to this jet-correlation background as flow-background. The anisotropic flow tobe used for flow-background should be ideally that from two-particle method, v { } [5, 6], because jet-like correlationis analyzed by two-particle correlation method. Moreover, two-particle anisotropic flow contains fluctuations whichshould be included in jet-correlation flow-background [5, 6].Non-flow is due to azimuthal correlations unrelated to the reaction plane, such as resonances, (mini)jets, or generally,clusters. Non-flow in two-particle v { } was studied in a cluster approach [7]; analytical form was derived for each non-flow component. In this paper, we shall demonstrate that the flow to be used in jet-correlation background subtractionshould be the two-particle v { } excluding cluster correlations unrelated to the reaction plane, but including cross-terms between cluster correlations and cluster flow. We verify our result with Monte Carlo simulations. We thenexamine available jet-like correlation data from STAR [1] and PHENIX [2] with our refined flow-background. Finallywe discuss implications of our result on the observed conical emission signal [8] and on medium modification of jetsin general. II. ELLIPTIC FLOW FOR JET-CORRELATION BACKGROUND
In this section, we derive an analytical form for flow-background to jet-correlation in the cluster approach, as usedin our non-flow study [7]. We suppose a relativistic heavy-ion collision event is composed of hydrodynamic mediumparticles, jet-correlated particles, and particles correlated via clusters. Hydro-particles, high p T trigger particles, andclusters are distributed relative to reaction plane ( ψ ) by dNdφ = N π [1 + 2 v cos 2( φ − ψ )] (1)with the corresponding elliptic flow parameter v and multiplicity N . Particle azimuthal distribution with respect toa trigger particle is 1 N t dNd ∆ φ = dN hy d ∆ φ + X k = jet ∈ clus dN d,k d ∆ φ + X k ∈ jet dN d,k d ∆ φ + dN d,jet d ∆ φ (2)where ∆ φ = φ − φ t . In Eq. (2), dN d,jet /d ∆ φ is jet-correlation signal of interest. All other terms are backgrounds; N d,k is the number of daughter particles in cluster k .If trigger particle multiplicity is Poisson and effects due to interplay between collision centrality selection (usuallyvia multiplicity) and trigger bias are negligible, then the background event of the triggered (di-)jet should be identicalto any inclusive event, without requiring a high p T trigger particle, but with all other event selection requirements asfor the triggered event [9]. Therefore, experimentally one can use inclusive events to obtain flow-background:1 N t dNd ∆ φ = a dN hy d ∆ φ + X k = jet ∈ clus dN d,k d ∆ φ + X k ∈ jet dN d,k d ∆ φ inc + dN d,jet d ∆ φ (3)where a is a normalization factor, often determined by the assumption of ZYAM or ZYA1 (zero jet-correlated yieldat minimum or at ∆ φ = 1) [10, 11], and is approximately unity.In this paper we are interested in the anisotropic flow to be used in jet-correlation background. So we will notconcern ourselves with the background normalization, but only the background shape. We rewrite the background inEq. (2) as dN bg d ∆ φ = dN hy d ∆ φ + X k = jet ∈ clus dN d,k d ∆ φ + X k ∈ jet dN d,k d ∆ φ = dN hy d ∆ φ + X cl N cl dN d d ∆ φ (4)where we have eliminated subscript ‘ inc ’ to lighten notation. We have summed over all cluster types ‘ cl ’ including jet-correlation, where N cl is the number of clusters of type ‘ cl ’. Different cluster types include jet and minijet correlations,resonance decays, etc.The hydro-background is simply dN hy d ∆ φ = N hy π (1 + 2 v ,t v ,hy cos 2∆ φ ) (5)where v ,t is elliptic flow parameter of trigger particles and v ,hy is that of hydro-medium particles.The cluster particles background is given by dN d d ∆ φ = Z π d ˜ φ t ρ t ( ˜ φ t ) Z π d ˜ φ k ρ cl ( ˜ φ k ) Z π d ∆ φ i f d (∆ φ i , ˜ φ k ) × π δ (∆ φ i + ˜ φ k − ∆ φ − ˜ φ t ) (6)where ˜ φ t = φ t − ψ , ˜ φ k = φ k − ψ , ∆ φ i = φ i − φ k , and ρ t ( ˜ φ t ) = π (cid:16) v ,t cos 2 ˜ φ t (cid:17) and ρ cl ( ˜ φ k ) = π (cid:16) v ,cl cos 2 ˜ φ k (cid:17) are density profiles (i.e., v -modulated distributions) of trigger particles and clusters rela-tive to the reaction plane, respectively. We have assumed that the cluster axis (or cluster parent) distribution is alsoanisotropic with respect to the reaction plane. In Eq. (6), f d (∆ φ i , ˜ φ k ) = dN d,k d ∆ φ i ≡ dN d ( ˜ φ k ) d ∆ φ i is distribution of daughterparticles in cluster relative to cluster axis (cluster correlation function), which may depend on the cluster axis relativeto the reaction plane ˜ φ k [12]. Decomposing ρ t ( ˜ φ t ), we obtain dN d d ∆ φ = 12 π Z π d ˜ φ k ρ cl ( ˜ φ k ) Z π d ∆ φ i f d ( d ∆ φ i , ˜ φ k )+ 2 v ,t π Z π d ˜ φ k ρ cl ( ˜ φ k ) Z π d ∆ φ i f d (∆ φ i , ˜ φ k ) cos 2(∆ φ i + ˜ φ k − ∆ φ ) . (7)Because of symmetry, f d (∆ φ i , ˜ φ k ) = f d ( − ∆ φ i , − ˜ φ k ) and ρ cl ( ˜ φ k ) = ρ cl ( − ˜ φ k ), we have R π d ˜ φ k ρ cl ( ˜ φ k ) R π d ∆ φ i f d (∆ φ i , ˜ φ k ) sin 2(∆ φ i + ˜ φ k ) = 0. Therefore dN d d ∆ φ = 12 π Z π d ˜ φ k ρ cl ( ˜ φ k ) N d ( ˜ φ k ) + 2 v ,t π cos 2∆ φ Z π d ˜ φ k ρ cl ( ˜ φ k ) Z π d ∆ φ i f d (∆ φ i , ˜ φ k ) cos 2(∆ φ i + ˜ φ k ) . (8)Realizing that elliptic flow parameter of particles from clusters is given by v ,d ≡ h cos 2( φ − ψ ) i cl = 1 N d Z π d ˜ φ k ρ cl ( ˜ φ k ) Z π d ∆ φ i N d ( ˜ φ k ) f d (∆ φ i , ˜ φ k ) cos 2(∆ φ i + ˜ φ k ) . (9)we rewrite Eq. (8) into dN d d ∆ φ = N d π (1 + 2 v ,t v ,d cos 2∆ φ ) . (10)From Eq. (5) and (10) we obtain the total background as given by dN bg d ∆ φ = N bg π " v ,t N hy N bg v ,hy + X cl N cl N d N bg v ,d ! cos 2∆ φ , (11)where N bg = N hy + X cl N cl N d . (12)The v ’s in Eqs. (5), (10), and (11) include fluctuations, so they should be replaced by p h v i . The hydro-particles p h v i is equivalent to two-particle v { } because there is no non-flow effect between hydro-particle pairs; same forthe cluster p h v i because there is no non-flow effect between different clusters (we consider sub-clusters to be partof their parent cluster). Thus Eq. (11) should be N bg d ∆ φ = N bg π (1 + 2 v ,t v ,bg cos 2∆ φ ) (13)where v ,bg = N hy N bg v { } hy + X cl N cl N d N bg v { } d . (14)We note that here cluster includes single-particle (within a give p T range) cluster, which generally is part of a parentcluster including particles of all p T . Those single-particle clusters do not contribute to non-flow in v { } d , but theydiffer from single hydro-particles because they may possess different v values.In principle, v ,t should have a similar expression as Eq. (14) out of symmetry reason: v ,t = N t,hy N t,tot v { } t , hy + X cl t N cl t N t , cl t N t , tot v { } t , cl t . (15)where N t,hy is number of high p T trigger particles from hydro-medium (i.e., background trigger particles), v { } t , hy is the elliptic anisotropy of those background trigger particles, N cl t is number of clusters of type ‘ cl t ’ containingat least one trigger particle, N t,cl t is number of trigger particles per cluster, v { } t , cl t is elliptic flow parameterof trigger particles from clusters, and N t,tot = N t,hy + X cl t N cl t N t,cl t . The only difference is that trigger particlesare dominated by clusters (mostly jets), and those clusters are dominated by single-trigger-particle clusters; hydro-medium contribution to trigger particle population should be small. We note that jet-correlation functions are usuallynormalized by total number of trigger particles including those from hydro-medium background.If particle correlation in clusters does not vary with cluster axis relative to the reaction plane, v { } d ≡ v { } cl h cos 2∆ φ i cl , (16)and v ,bg = N hy N bg v { } hy + X cl N cl N d N bg v { } cl h cos 2∆ φ i cl . (17) III. TWO-PARTICLE v IN CLUSTER MODEL
Obviously, the elliptic flow in Eq. (14) or (17) contains not only the two-particle anisotropy relative to the reactionplane, but also non-flow related to angular spread of clusters. How to obtain the elliptic flow as in Eq. (14) or (17)?In [7] we have derived two-particle v { } in a general hydro+cluster approach: v { } = N hy N bg v { } hy + X cl N cl N d N bg v { } d ! + X cl N cl N d N bg (cid:0) h cos 2∆ φ ij i cl − v { } d (cid:1) . (18)The quantity in the first pair of parentheses in r.h.s. of Eq. (18) is elliptic flow due to correlation with respect tothe reaction plane. The second term in the r.h.s. arises from cluster correlation [7]; the small correction v { } d is dueto assumptions of Poisson statistics for number of clusters and particle multiplicity in clusters, but not the productof the two [7]. Since elliptic flow is formally defined to be relative to the reaction plane, the first term in r.h.s. ofEq. (18) may be considered as “true” elliptic flow (except flow fluctuation effect), v , flow . We note, however, it is notnecessarily as same as hydro-flow because of contamination from clusters due to coupling between cluster correlationand cluster flow. The second term in r.h.s. of Eq. (18) can be considered as non-flow, v , non − flow ; non-flow is due tocorrelations between particles from the same dijet or the same cluster. Eq. (18) can be expressed into v { } = v , flow + v , non − flow . (19)Comparing Eq. (18) with Eq. (14), we see that v ,bg = v , flow , (20)i.e., the quantity in the first pair of parentheses in r.h.s. of Eq. (18) is the v parameter in Eq. (14) that is needed inconstructing jet-correlation background. In other words, elliptic flow parameter that should be used in jet-correlationflow background is the “true” two-particle elliptic flow (i.e., due to the reaction plane and including fluctuation). IV. MONTE CARLO CHECKS
In this section, we verify our analytical result by Monte Carlo simulations. We generate events consisting ofthree components. One component is hydro-medium particles according to Eq. (1), given hydro-particles ellipticflow parameter v ,hy and Poisson distributed number of hydro-particles with average multiplicity N hy . The secondcomponent is clusters, given cluster elliptic flow parameter v ,cl and Poisson distributed number of clusters withaverage N cl ; each cluster is made of particles with Poisson multiplicity distribution with average N d and Gaussianazimuth spread around cluster axis with σ d . The third component is trigger particles with accompanying associatedparticles; the trigger particle multiplicity is Poisson with average N t , and the elliptic flow parameter is v ,t . Theassociated particles are generated for each trigger particle by correlation function: f (∆ φ, ˜ φ t ) = C ( ˜ φ t ) + N ns ( ˜ φ t ) √ πσ ns ( ˜ φ t ) exp (cid:20) − (∆ φ ) σ ns ( ˜ φ t ) (cid:21) + N as ( ˜ φ t ) √ πσ as ( ˜ φ t ) exp − (cid:16) ∆ φ − π + θ ( ˜ φ t ) (cid:17) σ as ( ˜ φ t ) + exp − (cid:16) ∆ φ − π − θ ( ˜ φ t ) (cid:17) σ as ( ˜ φ t ) , (21)where the near- and away-side associated particle multiplicities are Poisson with averages N ns ( ˜ φ t ) and N as ( ˜ φ t ),respectively. The Gaussian widths of the near- and away-side peaks are fixed, and the two away-side symmetric peaksare set equal and their separation is fixed. All parameters in the jet-correlation function of Eq. (21) can be dependenton the trigger particle azimuth relative to the reaction plane, ˜ φ t .We first verify Eq. (18) by generating events with hydro-particles and jet-correlated particles. (We do not includeother clusters except jet-correlations.) We use N hy = 150, v ,hy = 0 .
05, and N t = 2, v ,t = 0 .
5. We use the largetrigger particle v in order to maximize the effect of non-flow. For jet-correlation function, we generate back-to-backdijet with N ns = 0 . N as = 1 . σ ns = 0 . σ as = 0 .
7, and θ = 0 (referred to as dijet model). We fix v in thesimulation, i.e., v fluctuation is not included. We simulate 10 events and calculate v { } = h cos 2∆ φ ij i . Includingonly hydro-particles, we obtain v { } hy = 0 . ± . v { } inc = 0 . ± . v { } trig evt = 0 . ± . v { } bg = 0 . ± . v isas same as that obtained from inclusive events, v { } bg = v { } inc , and both are smaller than that from triggeredevents only.We can in fact predict the inclusive event v by Eq. (17) using the “hydro + dijet” model. The average p h cos 2∆ φ ij i of jet-correlated particle pairs within the same dijet is p h cos 2∆ φ ij i jet = h cos 2∆ φ i jet = 0 . ± . h cos 2∆ φ i jet = N ns N ns + N as exp (cid:0) − σ ns (cid:1) + N as N ns + N as exp (cid:0) − σ as (cid:1) cos 2 θ = 0 . θ = 0. The average p h cos 2∆ φ ij i of pairs of particles from different dijets is 0 . ± . v { } d , jet = v , t h cos 2∆ φ i jet = 0 . × . . p h cos 2∆ φ ij i for cross-talkpairs of background particle and jet-correlated particle is 0 . ± . p v { } hy v , t h cos 2∆ φ i jet = √ . × . × . . v { } = q(cid:0) . × .
05 + . . × . (cid:1) + × . . (0 . − . ) = 0 . v { } inc or v { } bg obtained from simulation.We now verify Eq. (14) or (17) as the correct v to be used for jet-correlation background subtraction. We generatePoisson distributed hydro-particles with average multiplicity N hy = 150 and fixed elliptic flow parameter v ,hy = 0 . N t = 2 .
0; we use differentjet-correlation functions (discussed below). We also include clusters that do not have trigger particles (referred toas minijet clusters); the particle multiplicity per minijet cluster is Poisson distributed with average N d = 5, and thenumber of minijet clusters is also Poisson distributed but we vary the average number of clusters N cl ; we fix thecluster shape to be Gaussian with width σ d = 0 . h cos 2∆ φ i cl = exp (cid:0) − σ d (cid:1) = 0 . v ,cl = 0 .
20. We simulate 10 events and form raw correlation functionsnormalized by the number of trigger particles. In order to extract the real background v from the simulations,we subtract the input jet-correlation function. If the jet-correlation function varies with the trigger particle anglerelative to the reaction plane, the trigger multiplicity weighted average jet-correlation function is subtracted. We fitthe resultant background function to B (1 + 2 v ,t v , fit cos 2∆ φ ) where B and v , fit are fit parameters. We treat theinput v ,t as known; we did not include any complication into v ,t . We compare the fit v , fit to the calculated one byEq. (14) or (17). We study several cases with different shapes for jet-correlation function, as well as varying valuesfor some of the input parameters:(i) “hydro + dijet” model: we generate back-to-back dijets accompanying trigger particles, without other clusters.The calculated v ,bg by Eq. (17) is v ,bg = . × .
05 + × . . × . × . . v ,bg by Eq. (17)is v ,bg = . × .
05 + × . × . × . × . . × . × . . θ = 1, thus h cos 2∆ φ i jet = 0 . v ,bg by Eq. (17) is v ,bg = . × .
05 + × . × . × . × . . × . × . . v ,bg , which gives v ,bg = . × .
05 + × . × . × . × . . × . . v , fit is supposed to be the real background v ,bg . The fit errors are dueto statistical fluctuations in the simulation. As can be seen, the calculated v ,bg reproduces the real background v ,bg in every case. The v ,bg values differ from the hydro-background v due to contributions from cross-talks betweencluster correlation and cluster flow. Also shown in Table I are the two-particle v { } from all pairs in inclusive events.The v { } values differ from v ,bg due to non-flow contributions between particles from the same dijet or the samecluster.Figure 1(a) shows the raw correlation function for case (iii) and flow background using the calculated v ,bg byEq. 14 and normalized by ZYA1. Figure 1(b) shows the ZYA1-background subtracted jet-correlation function, usingthe calculated v ,bg for flow background. The background-subtracted jet-correlation is compared to the input signal.As shown in Fig. 1(b), the shapes of the input signal and extracted signal are the same, which is not surprising becausethe calculated v ,bg is the correct value to use in flow background subtraction. The roughly constant offset is due toZYA1-normalization. V. HOW TO “MEASURE” JET-CORRELATION BACKGROUND (REACTION PLANE) v Two-particle angular correlation is analyzed by STAR and is decomposed into two components [15]: one is theazimuth quadrupole, v { } , that is due to correlations of particles to a common source, the reaction plane; theother is minijet correlation that is due to angular correlation between particles from the same minijet or the samecluster. Of course, any such decomposition is model-dependent; because the functional form for minijet (or cluster) TABLE I: Monte Carlo verification of analytical results of elliptic flow parameter to be used in jet-correlation background.Hydro-particle multiplicity, trigger particle multiplicity, jet-correlated near- and away-side multiplicities, number of minijetclusters, and particle multiplicity per minijet cluster are all generated with Poisson distributions, with averages N hy , N t , N ns , N as , N cl , and N d , respectively. The jet-correlation function is given by Eq. (21), with near- and away-side Gaussian widthfixed to be σ ns = 0 . σ as = 0 .
7, respectively. The minijet cluster Gaussian width is fixed to σ d = 0 .
5. The elliptic flowparameters for hydro-particles, trigger particles, and clusters are v ,hy , v ,t , and v ,cl , respectively, and are fixed over all eventswithout fluctuation. We use N hy = 150, N t = 2, N d = 5, v ,hy = 0 .
05, and v ,cl = 0 . v { } v , fit Calculated v ,bg (i) hydro + dijet N cl = 0, v ,t = 0 . C = 0, N ns = 0 . N as = 1 .
2, 0.05557(8) 0.05505(8) 0.05500 σ ns = 0 . σ as = 0 . θ = 0(ii) hydro + minijet + dijet N cl = 10, v ,t = 0 . C = 0, N ns = 0 . N as = 1 .
2, 0.08465(6) 0.07115(8) 0.07126 σ ns = 0 . σ as = 0 . θ = 0(iii) hydro + minijet + near-side + N cl = 10, v ,t = 0 . C = 0, N ns = 0 . N as = 1 .
2, 0.08172(6) 0.06815(8) 0.06813 σ ns = 0 . σ as = 0 . θ = 1(iv) hydro + minijet + near-side + N cl = 10, v ,t = 0 . C = 0, N ns = 0 . N as = 1 .
2, 0.08279 0.06883(35) 0.06910+ clusters σ ns = 0 . σ as = 0 . θ = 1 φ∆ -1 0 1 2 3 4 5 φ ∆ dd N × t r i g N (a) φ∆ -1 0 1 2 3 4 5 φ ∆ dd N × t r i g N (b) φ∆ -1 0 1 2 3 4 5 φ ∆ dd N × t r i g N (c) FIG. 1: (a) Simulated raw correlation from the “hydro + minijet + near-side + away-side double-peak” model (Case iii inTable I). ZYA1-normalized flow-background using the calculated v ,bg is shown as the curve. Note the input v in the simulationis purposely made much larger than real data to exaggerate non-flow effect. (b) Background-subtracted jet-correlation (datapoints) compared to the input correlation signal (upper curve). The background uses the calculated v ,bg and is normalizedto signal by ZYA1. The input signal shifted down by a constant is shown in the lower curve. (c) As same as (b) except thesubtracted background uses the decomposed v { } . correlation is unknown a priori, one has to make assumptions about its functional form. Modulo this caveat, if clustercorrelation and flow correlation is properly decomposed, the azimuth quadrupole should correspond to the first termin r.h.s. of Eq. (18), v { } = N hy N bg v { } hy + X cl N cl N d N bg v { } d . (22)This is identical to Eq. (14). That is, the elliptic flow parameter from a proper
2D quadrupole-minijet decompositionis exactly what is needed for jet-correlation background calculation.A 2D quadrupole-minijet decomposition, with an assumption of the minijet correlation structure, has been carriedout experimentally by STAR as a function of centrality but including all p T [15]. One may restrict to narrow p T windows to obtain v { } as a function of p T , however, statistics can quickly run out with increasing p T because the2D decomposition method requires particle pairs.We perform a decomposition of flow and cluster correlation using our simulation data where the shape of clustercorrelation is known from the simulation input. We form two-particle correlation between all particles (untriggeredcorrelation). We fit the two-particle correlation with the sum of cluster correlation and flow to extract v { } from thesimulation data. Figure 1(c) shows the ZYA1-background subtracted jet-correlation function, using the decomposed v { } for flow background. The background-subtracted jet-correlation is compared to the input signal. The shapesof the input signal and extracted signal are the same, which demonstrates that the decomposed v { } is close tothe real elliptic flow value. Again, the roughly constant offset is due to ZYA1-normalization.One natural question to ask is why not to decompose jet-correlation and jet-background directly from high- p T triggered correlation function. One obvious reason is, again, that jet-correlation shape is unknown a priori, thusone cannot simply fit triggered correlation to a given functional form. This is the same caveat mentioned above inparticle pair correlation without a special trigger particle, where the minijet shape function has to be assumed in thedecomposition of minijet correlation and flow. The situation in jet-like correlation is direr because the main interestof jet-like correlation studies is the investigation of medium modification to jet-correlation structure. Furthermore,even when the functional form of jet-correlation signal is known, as is the case in our simulation, we found that thedecomposed jet-correlation signal shape deviates significantly from the input one. This is because the jet-correlationsignal is not orthogonal to flow background, but rather entangled, both with near- and away-side peaks, and henceone can get false minimum χ in decomposing the two components with limited statistics. VI. DISCUSSIONS AND SUMMARY
In experimental analysis, v values from various methods have been used for jet-correlation background. STAR usedthe average of the event plane v { EP } and the four-particle v { } and used the range between them (or between v { } and v { } ) as systematic uncertainties [8, 11]. The event plane v { EP } and two-particle v { } contain significantnon-flow contributions, while the non-flow contributions are significantly reduced in the four particle v { } [16]. Onthe other hand, effect of flow fluctuation is positive in v { EP } and v { } but is negative in v { } . This ensures thatthe true v is smaller than v { EP } and v { } , and is most likely larger than v { } . It is worth to note that theflow parameter to be used in jet-correlation background subtraction should include flow fluctuation effect as in v { } ,which makes the v { } parameter as the lower limit rather conservative.The recently measured v { } magnitudes from STAR are larger than v { } in peripheral and medium centralcollisions, confirming the validity to use v { } as the lower systematic limit of v . In central collisions, however, theextracted v { } is smaller than v { } although the difference is significantly smaller than the difference between v { EP } and v { } . This would suggest, assuming that the decomposed v { } reflects the real flow background (i.e.,the minijet shapes used in the decomposition is close to reality), that the used v values for background calculationin dihadron correlation analysis in STAR would be too large by about 1 σ systematic uncertainty. In three-particlecorrelation analysis [8], the v { } was included in the v systematic uncertainty assignment.STAR has also measured elliptic flow at mid-rapidity in the main Time Projection Chamber (TPC) using event-plane constructed by particles at forward and backward rapidities in the forward TPCs, v { FTPC } . The obtained v { FTPC } is smaller than v { EP } using particles from the main TPC only, however, it is still significantly larger than v { } . This suggests that some but not all non-flow effects are removed from v { FTPC } . The remaining non-flow maybe dominated by the long range ∆ η correlation (ridge) observed in non-peripheral heavy-ion collisions [11, 17, 18].PHENIX used v { BBC } results from the event plane method where the event plane is determined by particles in theBeam-Beam Counter several units of pseudo-rapidity away from particles used in jet-correlation analysis [19]. Therapidity gap in the PHENIX measurement of v { BBC } is larger than that in the STAR measurement of v { FTPC } ,so non-flow effect should be smaller in the PHENIX measurement. However, it is possible that the v { BBC } valuesused by PHENIX for background calculation can be also too large if non-flow ridge correlation persists to very largepseudo-rapidity gap.In summary, we have derived an analytical form for jet-correlation flow-background in a cluster approach. Weargue that the elliptic flow v parameter to be used in jet-correlation background is that from two-particle methodexcluding non-flow correlation unrelated to the reaction plane, but including cross-terms between cluster correlationand cluster flow. We have verified our result by Monte Carlo simulation for various jet-correlation signal shapes as wellas varying other input parameters to the simulation. We demonstrate that the v parameter to use in jet-correlationflow background is as same as the v { } from a proper 2D quadrupole-minijet decomposition of two-particle angularcorrelation. However, we note that 2D quadrupole-minijet decomposition requires a model for minijet correlationshape, which gives rise to systematic uncertainty on the extracted v { } which require further studies. Acknowledgment
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