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Vol. 39 (2008) ACTA PHYSICA POLONICA B No 4SUPPRESSION OF STATISTICAL BACKGROUNDIN THE EVENT STRUCTUREOF AWAY-SIDE ?φ B.
Vol. 39 (2008) ACTA PHYSICA POLONICA B No 4SUPPRESSION OF STATISTICAL BACKGROUNDIN THE EVENT STRUCTUREOF AWAY-SIDE ?φ DISTRIBUTIONCharles B. ChiuCenter for Particles and Fields and Department of PhysicsUniversity of Texas at Austin, Austin, TX 78712, USARudolph C. HwaInstitute of Theoretical Science and Department of PhysicsUniversity of Oregon, Eugene, OR 97403-5203, USA(Received January 25, 2008)An approach is proposed to analyze the azimuthal distribution of parti-cles produced on the away side in heavy-ion collisions without backgroundsubtraction. Measures in terms of factorial moments are suggested that cansuppress the statistical background, while giving clear distinction betweenone-jet and two-jet event structures on the away side. It is also possi-ble to map the position and strength of the recoil jet to suitably chosenasymmetry moments.PACS numbers: 25.75.–q, 29.90.+r1. IntroductionRecently an important area of investigation in relativistic heavy-ion col-lisions is the azimuthal angular distribution of hadrons produced on theopposite side of particles triggered at high pT [1–3]. That distribution re-veals the properties of the medium effect on a hard recoil parton traversingthe dense system formed by a central collision. Several possibilities havebeen suggested for what may happen to the hard parton and what signa-tures may be registered on the away side [4–8]. Our concern in this paperis not so much on the nature of the possible signatures, but on how best toextract them from a sea of noisy background that is inherent in every nuclearcollision. The conventional approach is to subtract the background. That isaccomplished by first summing over all events subject to specific kinemati-cal cuts. It is meaningless and impossible to make background subtraction(911)912 C.B. Chiu, R.C. Hwaevent by event. However, summing over all events is a process in analysisthat often degrades the signal, since the recoil parton may be absorbed bythe medium, or may emerge on the other side as a reduced minijet, or maygenerate a shock wave. To enhance the signature what is needed is a mea-sure that filters out the statistical fluctuation in every event to the extentpossible, and then let that measure be summed over all events. The aim ofthis paper is to propose such a measure and to demonstrate its effectivenessin the framework of a simple model that simulates events with jets in themidst of a large statistical background.The measure that we propose is factorial moments. Before going intothe details, let us first give some historical background. The factorial mo-ments were first used in the study of intermittency, which quantifies the self-similar behavior of branching processes in multiparticle production [9, 10].It has also been suggested for critical behavior in heavy-ion collisions [11].Although intermittency has been found in elementary collisions, its studyin nuclear collisions has been plagued by a number of difficulties, amongwhich is the problem of statistical fluctuations in large systems despite thetheoretical virtues of the factorial moments. There are several differencesbetween those intermittency studies and our problem here. First, we donot analyze fluctuations in rapidity, and do not consider a wide range ofbin size to search for fractal behavior. Second, we consider only a subsetof rarer events selected by high pT triggers, and examine the away-side ?φdistribution that consists of far lower multiplicities of particles. Third, in in-termittency studies the dynamical fluctuation is convoluted with statisticalfluctuation, whereas our jet signal on the away side is additive relative to thebackground. Having made these remarks to dissociate our work from what-ever vestige there may be from the past, we proceed now to a description ofour measure from the basics.The outline of the remainder of this paper is as follows. Section 2 givesthe definition of the appropriate of factorial moment measures of presentinterest. They are the normalized factorial moments (NFM). Throughoutthis work the term: “FM” approach will be used as a generic term to referto analyses which make use of the NFMs. In that section we will introducea simple model to illustrate the application of the FM approach in analyzingthe away-side azimuthal distribution. In Section 3, we illustrate how thoseevent-by-event fluctuations in the away-side spectrum may be evaluated inthe FM approach. In Section 4, we discuss an approach which reveals theposition and shape information of the 1j case using the FM method. A briefsummary is included in Section 5.Suppression of Statistical Background in the Event Structure of . . . 9132. Factorial moments of a simple model2.1. Definition of FM measuresFor notation convenience, we define ? = ?φ ? π, with the usual ?φvariable defined relative to the trigger momentum in the transverse plane.Consider an interval I around ? = 0, which, for definiteness, may be takento be from ?1.5 to +1.5, although the suitable range is an experimentaldecision. Let I be divided into M equal bins so that the bin size is δ = I/M .In an event let nj denote the number of particles in jth bin. Define thefactorial moment (FM) by? ? Mn! 1 ∑fq = = nj(nj ? 1) · · · (nj ? q + 1) , (1)(n? q)! Mj=1and the normalized factorial moment (NFM) byfqFq = q , (2)f1for each event. It is important to recognize that Eq. (1) is used to determinethe FM event by event, and the summation there may be regarded as thehorizontal average.Let us now consider the hypothetical case where the fluctuation of njfrom bin to bin can be described by a Poisson distribution as an ideal rep-resentation of the backgroundn?nP (n) = e?n?n? , (3)n!where n is the bin multiplicity that fluctuates and n? is the average thatdepends on δ and total event multiplicity N . Then Eq. (1) can be written as∑N n!fq = Pn?(n) . (4)(n? q)!n=qIf N is large compared to n?, it can be approximated by infinity in Eq. (4);(stat) (stat)it then follows from Eq. (3) that fq = n?q, so we have Fq = 1 for any δin the ideal statistical case. When N is finite but large, the corresponding(stat)Fq should be slightly less than 1. Furthermore, it has been pointed outto us that Fq as defined in Eq. (2) is independent of detector efficiency [12].(stat)In reality, Fq is not exactly 1. We expect Fq to fluctuate from eventto event. Let us define in general the event-averaged NFMN1 ∑evt〈Fq〉 (δ) = F(i)q (δ,Ni) , (5)Nevti=1914 C.B. Chiu, R.C. Hwawhere Ni is the multiplicity of the ith event and Nevt is the total numberof events. The central question is whether 〈Fq〉 stays approximately at 1 forpure background in the real data and becomes larger than 1 significantlyenough, when jets exist, so that 〈Fq〉 can be used as an effective measure todistinguish the signal from the noise. We stress that the (vertical) averageover all events in Eq. (5) is performed after the horizontal average is donein Eq. (1), not the other way around. Thus if the answer to the abovecentral question is in the affirmative, then one may regard Fq as effectivelysuppressing the statistical background event-by-event. Since it is impossibleto make background subtraction event-by-event, to suppress the statisticalcontribution to a suitably chosen measure seems to be the best that one cando, and it should be done at the level of each event.2.2. A simple model of away-side distributionTo test the effectiveness of the measure suggested above, let us considera simple model to simulate the ? distributions of events with jets in the pres-ence of significant background. Let Ni particles be distributed randomly inthe interval I with ?1.5 < ? < 1.5. They will be regarded as the back-ground of the ith event. For now, we limit ourselves to the simplest case ofNi being a constant, but later generalize Ni to be Gaussian distributed. Fora 1j event, we add a cluster of particles of multiplicity, m = 5, bunched in asmall interval in ? of extent ? = 0.04 rad. The cluster is randomly located inthe interval ?1 < ? < +1. For a 2j event (to mimic a Mach cone structure)we add two clusters of 5 particles each, and place them symmetrically about? = 0, uniformly distributed in the interval |?| = 0.6–0.8. We label the threecases by bg, bg+1j and bg+2j, respectively, where bg denotes backgroundonly without jets.In Fig. 1 we show the distributions of (a) F2 and (b) F3 for the threecases, for Ni = 60,M = 30 and Nevt = 2000. Although the distributionshave significant overlap, their average values 〈Fq〉 are distinctly separated,as shown by the three lines for q = 2, 3, 4 in Fig. 2, for various bin numbersM . In (a) the average values 〈Fq〉 for the background are all less than,but close to, 1. In (b) for the case including 1j and in (c) for the bg+2jcase, 〈Fq〉 are distinctly higher than 1, especially for q > 2. These resultsrepresent the first indication that the FM method is useful, and that thesuppression of the statistical fluctuation is realized upon event averagingwithout explicit subtraction of the background. That is, with 〈Fq〉 beingaround 1 for background only, any significant enhancement of 〈Fq〉 above 1would indicate the presence of non-statistical signal.In a more realistic situation elliptic flow introduces a ? dependence tothe background. We have accordingly modified the statistical backgroundby a factor 1 + 2v2cos2?, and find that Fig. 2 remains essentially the sameSuppression of Statistical Background in the Event Structure of . . . 915600 600bgbg+1jbg+2j400 400 (b) F3 (a) F2 200 20000 1 2 3 0F2 0 1 F 2 33Fig. 1. Event-by-event distributions of Fq for (a) q = 2 and (b) q = 3, M=30. Thecase shown is for dN/d?|bg = 20.5 5 5q=2q=3q=4 (b) bg + 1j4 4 4 (c) bg + 2j3 (a) bg 3 32 2 21 1 10 0 020 40 20 40 20 40M M MFig. 2. Event-averaged 〈Fq〉 versus bin number M for (a) bg only, (b) bg+1j, and(c) bg+2j. Circles, triangles and squares are for q = 2, 3, 4, respectively. The solidpoints are for v2 = 0, the open points for v2 = 0.1. For dN/d?|bg < 50, the 1j and2j signals quantified by 〈Fq〉 can be distinguished from the background. The caseshown is for dN/d?|bg = 20.for v2 increasing up to 0.1. We do not show the modified Fq distributions inFig. 1 because they overlap with the existing ones (without flow effect) somuch as to cause substantial sacrifice in clarity. However, the modified 〈Fq〉 2dN/dF3916 C.B. Chiu, R.C. Hwacan be shown clearly by the open points in Fig. 2 for v2 = 0.1. They areslightly higher than the corresponding curves for v2 = 0, with the deviationmost noticeable for the bg+1j case at q = 4. Still, the general features ofthe result remain the same.In the test cases presented the background consists of 60 particles, dis-tributed over an interval of I = [?1.5, 1.5], which corresponds to dN/d?|bg≈ 20. We use dN/d? to denote the event-averaged azimuthal distribution,as is conventionally done, although in our simulation here the same notationis used for the distribution in any event, for brevity’s sake. In some experi-mental data the lower bound of passocT has been set as low as 0.15 GeV/c [1],for which the background height is as high as dN/d?|bg ≈ 200. In such casesthe signal of jets on the away side is at the 1% level; it is therefore in thepresence of statistical bin-to-bin fluctuations that can be significantly larger,and our method cannot be expected to be effective. We have found that fordN/d?|bg < 50, the 1j and 2j signals quantified by 〈Fq〉 can be distinguishedfrom the background.It should also be noted that our model for a jet is a cluster of 5 particlesin a small interval ?. Obviously, if a cluster is widely spread out withouta corresponding increase of particle multiplicity in the jet, then the effectof jet becomes indistinguishable from the background fluctuation and onecannot expect any method to be able to extract its properties. Since our pur-pose here is to illustrate a new method of analysis rather than to constructrealistic jet characteristics, the model we use suffices toward that end.3. Event-by-event fluctuations about ? = 0A disadvantage in working with Fq is that one no longer sees visually thepeaks in ?φ distribution associated with jets. Thus one gets a quantitativemeasure of the peaks at the expense of losing information about the locationsof the peaks. However, the loss can be reduced if we elevate the level ofhorizontal analysis of the FM by being more specific about the spatial regionsin ?. To that end let us define± 1∑fq = nj(nj ? 1) · · · (nj ? q + 1) , (6)M±j∈R±where R± stands for the region where ? is>< 0, and M± = M/2 is thenumber of bins in R±. Thus we have fq = (f+q + f?q )/2 for each event. Wefurther define the NFM, as in Eq. (2), byF± = f±/(f )qq q 1 , (7)which implies Fq = (F+ ?q + Fq )/2 for every event.Suppression of Statistical Background in the Event Structure of . . . 917Since event averaging is likely to erase the distinction between F+q andF?q if a jet fluctuates between being on the + and ? sides, it is useful toconsider the difference moments Dq = |F+q ?F?q | and the sum Sq = F+ ?q +Fq .To amplify the effect of the fluctuations let us define the vertical average ofthe p-th moment of Dq by? ? N∑evtDp1 pq (δ) = D(i)q (δ,Ni) , (8)Nevti=1and similarly of Sq. The superscript p is a power of Dq. A plot of 〈Dpq 〉 (δ)pversus 〈Sq 〉 (δ) for the three cases of bg, 1j and 2j for various values of δ canreveal some characteristics of interest. In our simulation we place the twojets always on the opposite sides of ? = 0 in order to contrast the 2j fromthe 1j events. In Fig. 3 we show an array of such plots for p, q = 2, 3, 4. Inall, the background points are clustered together, while the 1j and 2j pointsfan out from the origin, mostly in straight lines. Clearly, two jets on twosides of ? = 0 reduce Dq, resulting in lower 〈Dpq〉 compared to the samefrom 1j. Similar plots of the experimental data would indicate first of allbg0.5 5 500 0 00 5 0 10 20 0 100bg+j 20000.5 2010000 0 00 10 0 100 0 5000bg+2j1 200 100K0.5 100 50K0 0 0 0 20 40 0 500 1000 0 100K p 200Kp p q=4? ? ? ?Fig. 3. Dp versus Sp . The rows are for p = 2, 3 and 4, and the columns forq qq = 2, 3 and 4. Connected points are at M = 20, 30, 40, and 50.p=4 Dp=3 p=2q < q > 918 C.B. Chiu, R.C. Hwawhether the 〈Dpq〉 and 〈Spq 〉 points of the background cluster together at thelower-left corner, and then secondly whether the points arising from 1j and2j are well separated from those from just the background.The implication of Fig. 3 is that the bin size δ is not an essential variable,and that the normalized asymmet?ry ? ? ?A p pqp = Dq / Sq (9)can be a more succinct measure. To see that explicitly we show in Fig. 4the values of Aqp versus δ?1 for the 1j and 2j points. For q = 3 and 4 thedependences on p and δ are not sensitive, and the differentiation between the1j and 2j cases can easily be made, even for q = 2. It seems that this typeof analysis renders more quantitative and distinctive results than 3-particlecorrelation.q=2 q=3 q=40.50.12 1j 0.80.1 p=2 0.4p=3 21j0.08 p=4 0.3 p=20.6p=30.06 p=40.2 0.40.040.1 0.20.020 0 05 10 1 15 5 10 15 5 10 1 15δ? δ?1 δ?Fig. 4. Aqp versus δ?1 for (a) q = 2, (b) q = 3, and (c) q = 4. Thick, medium, thinlines are for p = 2, 3, 4; solid (dashed) lines for bg+1j (bg+2j).We have generalized the value of Ni to be Gaussian distributed, andthen also the jet multiplicity m. We have found no significant effect onthe asymmetry measure Aqp. Since our aim here is only to suggest usefulmeasures, and not their detail numbers, we will not detail the results of theGaussian-distributed multiplicities here.4. A measure reflecting ? dependenceRestricting our attention now to only the 1j case, we consider the ques-tion of how best to reveal the position and shape of a peak on the away sideusing FM. To that end we need to introduce a cut in ?. DefineAqpSuppression of Statistical Background in the Event Structure of . . . 919> 1 ∑f< j∈S> is the set of bins in the range |?|> is the number of< . The overall FM is then given by fq = rq + r>fq , where = M>/M . The corresponding NFM is< <> >F< F>q . The relative magnitude ofF>q versus FqBq = ? . (12)F< + F>q qTo test its usefulness we consider three cases for which the event-averageddistributions above the background are shown in Fig. 5(a); the heights of thebackground, dN/d?|bg , (not shown) are [i] 20, [j] 2, and [k] 0.2. The widthsand locations of the peaks are: [i] (0.4, ±0.8), [j] (0.3, ±0.4), [k] (0.3, 0),respectively. The signal in case [i] is at the 1% level, and corresponds to verylow passocT , whereas case [k] has a sharp peak above a low background thatcan arise from high ptrig and passocT T . In Fig. 5(b) B4(?c) is shown for the threecases, which are well sepa?rate?d. In?all ?cases background alone gives Bq verynearly 0 because either F< =? F>q q for ?c > 0.1 or their sum is muchlarger than their difference. Note that even in case [i] where the doublepeaks are small compared to background, we obtain without backgroundsubtraction nonzero B4(?c), albeit small. In case [j] with dN/?|bg = 2 theaverage bin multiplicity n? of the background is only 0.2, which is much less>than q ≥ 1. Nevertheless, ffluctuations of t?he b?in multiplicities, so Fevent-averaged F<4 are between 0 and 4, resulting in B4 to be around 0.5.In case [k] the strong peak at ? = 0 results in Fq except when ?c is small; thus B4 is large, specially at large ?c. Where itdeviates from 1 is a measure of the narrowness of the peak. Bq for q = 2, 3give similar results, so we do not show them for the sake of clarity in thefigure.920 C.B. Chiu, R.C. Hwa1i0.6 (a) j (b)0.4 k 0.50.20 03 (c)0.8 (d)2 (a)1 0.40?1 0 0 ? 1 0 0.5 ? 1cFig. 5. (a) Event-averaged ? distribution for Nevt = 5000 with M = 30. Thebackground contributions are not shown; their levels are [i] 20, [j] 2, [k] 0.2. Thewidths and locations of the peaks are: [i] (0.4, ±0.8), [j] (0.3, ±0.4), [k] (0.3, 0),respectively. (b) The corresponding B4(?c) distributions. (c) dN/d? for [j] and [k]with bg included. (d) 1?B4 for the distributions in (c).To achieve a better perspective of the mapping between dN/d? andB4, we show in Fig. 5(c) the former for cases [j] and [k] with backgroundincluded. The corresponding plot of 1? B4(?c), which is more intuitive, isgiven in (d). Here [j] exhibits the features of both the broad bump in (a)and the high background in (c), while the peak of [k] reflects the samein (a) and (c). These properties are remarkable due to the drastic differencein the nature of the measures displayed. This demonstrates that withoutbackground subtraction the FM analysis event-by-event results in clear andquantitative description of the ?φ characteristics.We end our presentation of figures here with a general remark on thestatistical errors. By doubling the total number of events used, we havefound that curves shown in all the figures presented are stable with respectthe variations of the total number of events used. In other words, the errorbars due to statistics are not significant.5. SummaryIn this work we have considered the use of factorial moments to analyzethe ?φ distribution of particles produced opposite a trigger. In practice,for a given set of experimental data, one needs to first determine the appli-dN/d?dN/d?1?B4 B4Suppression of Statistical Background in the Event Structure of . . . 921cability of the FM approach by previewing 〈Fq〉’s, to check whether thereare at least some of them which is significantly greater than unity. ForFM-applicable data, the advantage of such a method of analysis is that noexplicit background subtraction is necessary. The FM of each event is sen-sitive to the jet characteristics, while suppressing the effect of statisticalfluctuation of the background; it is also insensitive to the smooth variationdue to elliptic flow. The asymmetry moments Aqp can well separate 1j and 2jrecoil scenarios, and Bq can give a quantitative description of the single-jetcharacteristics. The application of this method to the analysis of the RHICdata on jet correlation may provide a common framework to compare re-sults from widely different experimental conditions and various subtractionschemes.We are grateful to Rene Bellwied, Lanny Ray and Aihong Tang for help-ful comments. We thank Matt Haley for his contribution to the investigationof possible extension of this work that is not reported here. This work wassupported, in part, by the US Department of Energy under Grant No. DE-FG03-96ER40972.REFERENCES[1] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 95, 152301 (2005).[2] H. Beusching, Nucl. Phys. A774, 103 (2006) [nucl-ex/0511044].[3] J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 97, 162301 (2006).[4] J. Casalderrey-Solana, E.V. Shuryak, D. Teaney, J. Phys. Conf. Ser. 27, 22(2005).[5] V. Koch, A. Majumder, X.-N. Wang, Phys. Rev. Lett. 96, 172302 (2006)[nucl-th/0507063].[6] I.M. Dremin, Nucl. Phys. A767, 233 (2006).[7] J.G. Ulery [STAR Collaboration], Nucl. Phys. A783, 511c (2007).[8] C.B. Chiu, R.C. Hwa, Phys. Rev. C74, 064909 (2006).[9] A. Bia?as, R. Peschanski, B273, 703 (1986); Nucl. Phys. B308, 857 (1988).[10] E.A. De Wolf, I.M. Dremin, W. Kittel, Phys. Rep. 270, 1 (1996).[11] A. Bia?as, R.C. Hwa, Phys. Lett. B253, 436 (1991).[12] A. Tang, private communication.
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