Unfolding of event-by-event net-charge distributions in heavy-ion collisions
P. Garg, D. K. Mishra, P. K. Netrakanti, A. K. Mohanty, B. Mohanty
aa r X i v : . [ h e p - ph ] J un Unfolding of event-by-event net-charge distributionsin heavy-ion collisions
P. Garg ∗ Department of Physics, Banaras Hindu University, Varanasi-221005, IndiaE-mail: [email protected]
D. K. Mishra, P. K. Netrakanti, A. K. Mohanty
Nuclear Physics Division, Bhabha Atomic Research Center, Mumbai 400094, India
B. Mohanty
School of Physical Sciences, National Institute of Science Education and Research,Bhubaneswar 751005, IndiaE-mail: [email protected]
An unfolding method, based on Bayes theorem is presented to obtain true event-by-event net-charge multiplicity distribution from a corresponding measured distribution, which is subjectedto detector artifacts. The unfolding is demonstrated to work for widely varying particle productionmechanism, beam energy and collision centrality. Further the necessity of taking into account thedetector effects is emphasized before comparing the experimental measurements to the theoreticalcalculations, particularly in case of higher moments. The advantage of this approach being thatone need not construct new observable to cancel out detector effects which loose their ability tobe connected to physical quantities calculable in standard theories.8 th International Workshop on Critical Point and Onset of DeconfinementMarch 11 - 15, 2013Napa, California, USA ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ nfolding of event-by-event net-charge distributions in heavy-ion collisions
P. Garg
1. Introduction
It has been suggested that the higher moments of fluctuations are very sensitive to the proxim-ity of critical point as they have strong dependence on correlation length [1] and they are related tothe susceptibility of the system [2, 3]. The higher moments of event-by-event distribution of con-served quantities like net-charge, net-baryon and net-strangeness have been important observablesto characterize the system formed in heavy ion collision experiments [4]. Further suggestions aremade to explore the QCD phase transition and freeze out conditions in heavy ion collisions, usingthe higher moments [5, 6].Most of the experimental measurements depend on the detector acceptance, particle countingefficiency and other background effects [10]. Some of them are very difficult to exclude for anobservable which is based on an event-by-event analysis. Therefore the experimentally measuredevent-by-event distributions are shown without taking care of these corrections [1, 7, 8, 9]. Thesecorrections are applied on an average basis to correct the particle yields in heavy ion collisionexperiments [10], but to apply these corrections on an event-by-event observable is not trivial.Hence, comparing uncorrected event-by-event observables with the theoretical observables shouldbe done carefully as it may lead to different physics conclusions.Although, some observables have been constructed in order to cancel out the detector effectsto first order [11, 12, 13, 14], but while making these constructs one may loose the ability tocompare them to the theoretically calculated quantities. Therefore, to compare higher moments ofmultiplicity distributions with the theoretical results, one should consider the experimental artifacts.In the present work, an approach based on bayesian theorem of probability is demonstrated towork successfully to remove the detector artifacts on an event-by-event basis. Such a method hassome constraints in terms of proper detector modeling and a large event-by-event multiplicities.
2. Event generators
Two different event generators are used to explore our proposal. For the present study, HIJING[15] (version 1.37) and THERMINATOR [16] (version 2.0) event generators provide the facilityto incorporate different particle production mechanism. Using these event generators, net-chargedistributions are obtained within the pseudo-rapidity of -0.5 to 0.5 and transverse momentum rangeof 0 . < p T < . GeV / c with in full azimuthal coverage. The average charge particle countingefficiency is taken to be 65%, derived from the charged pion efficiency, as is given in Ref. [10].Further to demonstrate our proposal at different energies, HIJING is also used at √ s NN = 27, 39,62.4, 130 and 200 GeV for most central Au+Au events.
3. Bayes method for the unfolding of distributions
The RooUnfoldBayes [17] algorithm of RooUnfold package [18] is used to demonstrate thepresent proposal. The algorithm based on Bayes theorem of probability is used to show that the truedistributions can be reconstructed from the distributions which are affected by systematic biasesand detection efficiency.To demonstrate it, 5M Au+Au collision events are generated for each centrality bin using HI-JING for 19.6 GeV and THERMINATOR for 200 GeV. For each event positive ( N + ) and negative2 nfolding of event-by-event net-charge distributions in heavy-ion collisions P. Garg ( N − ) charge particles are selected and a net-charge distribution ( D N = N + − N − ) is constructed onan event-by-event basis.Now, to mimic the experimental situation, individual N + and N − are smeared with a Gaus-sian function of width 10%, and the mean value corresponding to the average efficiency of 65%as obtained from the parametrization of the p T dependent efficiency for charged pions from STARexperiment [10]. Afterwards, these smeared distributions are used to construct the net-charge dis-tribution, we will call it the measured distribution . E v en t F r a c t i on -4 -3 -2 -1 = 1.7 fm AuAu 19.6 GeV
TrueMeasuredBayes(a) ) + Positively charged particles (N0 50 100 150 200 250 300 R a t i o = B a y e s / T r ue (b) HIJING ) - Negatively charged particles (N0 50 100 150 200 250 3000.51.01.52.02.53.03.5 -100 -50 0 50 100-4-3-2-1 (c) ) - - N + Net charged particles (N-100 -50 0 50 1000.51.01.52.02.53.03.5
Figure 1: (Color online) Top panel: Event-by-event distribution of positive, negative and net-charge (de-noted as “True”, solid line) in Au+Au collisions for impact parameter b = 1.7 fm at √ s NN = 19.6 GeV fromHIJING event generator. Also shown are the corresponding distributions after applying acceptance and ef-ficiency effects as discussed in the text (denoted as “Measured”, open circles). The unfolded distributionsare shown as red stars and denoted as “Bayes”. Bottom panel: Shows the ratio of the unfolded to the Truedistributions. The measured distribution of net-charge is unfolded with response matrix obtained from thetraining procedure using iterative Bayes theorem. The present study uses the optimal value of 4for the regularization. True, measured and unfolding are performed in a way to eliminate the finitecentrality bin-width effect. The moments of net-charge distributions are derived using cumulantmethod [19].
4. Results and Discussions
The true, measured and unfolded distributions for positive charge (panel a), negative charge(panel b) and net charge (panel c) are shown in Figure 1. These distributions are for most centralevents corresponding to an average impact parameter of 1.7 fm of Au+Au collisions from HIJINGat 19.6 GeV on an event by event basis. Solid lines, open circles and red stars represent the truedistributions, measured distributions and the unfolded distributions respectively, for all the cases.It is evident that for all the cases, the true distributions are reproduced from the measured distribu-tions, using the unfolding technique. Also the ratios, presented in the bottom panel suggest that theunfolding procedure is able to get back the true distribution.3 nfolding of event-by-event net-charge distributions in heavy-ion collisions
P. Garg
We have seen the variation of mean, sigma, skewness and kurtosis as a function of centrality( N part ), obtained from the net-charge distributions in Au+Au collisions at √ s NN = 19.6 GeV fortrue, measured and unfolded distributions [20]. The moments computed from unfolded distribu-tions and true distributions were found in good agreement. Further, the ratios of unfolded to truemoments were close to unity. It suggests that the unfolding method reproduced the results of truedistribution from the measured distributions. Same conclusions can be drawn from Fig. 2 where s / M , S s and ks are drawn as a function of N part . part N / M s AuAu 19.6 GeV
TrueMeasuredBayes ) - - N + (N (a) part N s S (b)HIJING part N s k (c) Figure 2: (Color online) Ratio (panel a) and product of moments (panel (b) and (c)) of net-charge distribu-tions in Au+Au collisions at √ s NN = 19.6 GeV from HIJING event generator. The results are for the True,measured and Bayes unfolded distributions as a function of N part . It is observed (Fig. 3) that the dependence of ks and S s is very different for true and smeareddistributions, as a function of N part . It implies that any physics conclusion associated with the vari-ation of S s and ks with N part for net-charge distributions could be highly misleading. However,the results for measured distributions can be unfolded nicely to get back the results of true distri-butions. part N
100 200 300 400 / M s AuAu 200 GeV
TrueMeasuredBayes ) - - N + (N (a) part N
100 200 300 400 s S -0.10.00.10.2 THERMINATOR (b) part N
100 200 300 400 s k -50510152025 (c) Figure 3: (Color online) Product of moments of net-charge distributions in Au+Au collisions at √ s NN = 200GeV from THERMINATOR event generator. The results are for the True, measured and Bayes unfoldeddistributions as a function of N part . Another event generator, THERMINATOR is also used to check the validity of proposedbayesian approach. Figure 3 shows the s / M , S s and ks of net-charge distributions from thetrue, measured and unfolded distributions at √ s NN = 200 GeV, as a function of N part . Here also theratio and products of moments from unfolded distributions are reproduced as true distributions up4 nfolding of event-by-event net-charge distributions in heavy-ion collisions P. Garg to a good extent, suggesting that the method proposed in this paper works equally well for parentdistributions produced from very different particle production mechanisms as well as over a widerange of beam energies. Besides 19.6 GeV, HIJING is used for √ s NN = 27, 39, 62.4, 130 and 200GeV with the same procedure. In this energy dependence study only 0-5% central Au+Au eventsare used.In Figure 4, s / M , S s and ks are drawn as a function of √ s NN for true, measured, andunfolded distributions. Here also a good agreement is found between True and Unfolded moments.This demonstrate that the proposed method works over a wide range of energies as well. (GeV) NN s / M s (a) AuAu (0-5%)TruthMeasuredBayes ) - - N + (N (GeV) NN s s S (b) HIJING (GeV) NN s s k (c) Figure 4: (Color online) Product of moments of net-charge distribution in 0-5% Au+Au collisions as afunction of √ s NN from HIJING event generator. The results are for the True, measured and Bayes unfoldeddistributions.
5. Summary
The Bayesian unfolding method is successfully demonstrated to unfold back the measureddistributions, which are subjected to detector effects like finite particle counting efficiencies. Thecentrality dependent study for moments and their product and ratios is carried with HIJING andTHERMINATOR at 19.6 GeV and 200 GeV respectively. It is observed that the detector effects canmodify the results significantly and these effects can be removed by bayesian unfolding method.Also, for wide range of energies ( √ s NN =19.6, 27,39, 62.4, 130 and 200 GeV), a good agreementis found between true and unfolded moments, s / M , S s and ks . However, there are limitationsin terms of proper modeling of the detector response. Also it requires high multiplicity and largeevent statistics for building better response matrix. Although main advantage of Bayesian approachis that, one don’t have to construct new observables to cancel out the detector effects. Further, moredetails of the present work can be found in Ref. [20].
6. Acknowledgements
Financial assistance from the Department of Atomic Energy, Government of India is gratefullyacknowledged. BM is supported by the DST Swarna Jayanti project fellowship. PG acknowledgesfinancial support from CSIR, New Delhi, India. This work is carried out using the NPD-BARCcluster facility at HBNI. 5 nfolding of event-by-event net-charge distributions in heavy-ion collisions
P. Garg
References [1] M. A. Stephanov, Phys. Rev. Lett. , 032301 (2009) [arXiv:0809.3450 [hep-ph]].[2] A. Bazavov et al. [HotQCD Collaboration], Phys. Rev. D , 034509 (2012) [arXiv:1203.0784[hep-lat]].[3] M. Cheng, P. Hendge, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann, R. D. Mawhinney andC. Miao et al. , Phys. Rev. D , 074505 (2009) [arXiv:0811.1006 [hep-lat]].[4] M. M. Aggarwal et al. [STAR Collaboration], Phys. Rev. Lett. , 022302 (2010) [arXiv:1004.4959[nucl-ex]].[5] F. Karsch and K. Redlich, Phys. Lett. B , 136 (2011) [arXiv:1007.2581 [hep-ph]].[6] B. Friman, F. Karsch, K. Redlich and V. Skokov, Eur. Phys. J. C , 1694 (2011) [arXiv:1103.3511[hep-ph]].[7] M. M. Aggarwal et al. [WA98 Collaboration], Phys. Rev. C (2002) 054912 [nucl-ex/0108029].[8] K. Adcox et al. [PHENIX Collaboration], Phys. Rev. Lett. (2002) 082301 [nucl-ex/0203014].[9] P. Braun-Munzinger, B. Friman, F. Karsch, K. Redlich and V. Skokov, Phys. Rev. C , 064911(2011) [arXiv:1107.4267 [hep-ph]].[10] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C , 034909 (2009) [arXiv:0808.2041[nucl-ex]].[11] S. Mrowczynski, Phys. Lett. B (1999) 8 [nucl-th/9905021].[12] S. A. Voloshin, V. Koch and H. G. Ritter, Phys. Rev. C , 024901 (1999) [nucl-th/9903060].[13] A. Bialas, Phys. Rev. C , 024904 (2007) [hep-ph/0701074].[14] C. Pruneau, S. Gavin and S. Voloshin, Phys. Rev. C , 044904 (2002) [nucl-ex/0204011].[15] M. Gyulassy and X. -N. Wang, Comput. Phys. Commun. , 307 (1994) [nucl-th/9502021].[16] A. Kisiel, T. Taluc, W. Broniowski and W. Florkowski, Comput. Phys. Commun. , 669 (2006)[nucl-th/0504047].[17] G. D’ Agostini, Nucl. Instrum. Meth. A , 487 (1995).[18] T. Adye, [arXiv:1105.1160 [physics.data-an]]. RooUnfold package[19] X. Luo, J. Phys. G , 025008 (2012) [arXiv:1109.0593 [nucl-ex]].[20] P. Garg, D. K. Mishra, P. K. Netrakanti, A. K. Mohanty and B. Mohanty, J. Phys. G , 055103(2013) [arXiv:1211.2074 [nucl-ex]]., 055103(2013) [arXiv:1211.2074 [nucl-ex]].