Method for the Analysis of Forward-Backward Multiplicity Correlations in Heavy-Ion Collisions
In heavy-ion (A-A) collisions, the correlations among the particles produced across wide range in rapidity, probe the early stages of the reaction. The analyses of forward-backward multiplicity correlations in these collisions are complicated by several effects, which are absent or minimized in hadron-hadron collisions. This includes effects, such as the centrality selection in the A-A collisions, which interfere with the measurement of the dynamical correlations. A method, which takes into account the fluctuations in centrality selection, has been utilized to determine the forward-backward correlation strength in A-A collisions. This method has been validated by using the HIJING event generator in case of Au-Au collisions at = 200 GeV and Pb-Pb collisions at = 2.76 TeV. It is shown that the effect of impact parameter fluctuations is to be considered properly in order to obtain meaningful results.
The major goals of colliding heavy-ions at relativistic energies are to create a new form of matter, called quark-gluon plasma (QGP), and to study its properties. The QGP matter is formed very early in the reaction and it is a major challenge to experimentally probe this initial stage as majority of the detected particles are emitted at freeze-out. Correlations, that are produced across a wide range in rapidity are thought to reflect the earliest stages of the heavy-ions collisions, free from final state effects larry (). The study of correlations among particles produced in different rapidity regions may provide an understanding of the elementary (partonic) interactions which lead to the hadronization. Several experiments involving collisions of electrons, muons and protons show strong short-range correlations (SRC) over a region of about 1 unit in rapidity alner (); SRC (); aexopoulos (). In high-energy nucleon-nucleon collisions ( 100 GeV) the non-single differactive inelastic cross section increases significantly with energy, as does the magnitude of the long-range forward-backward multiplicity correlations. The component involving the long range correlations in these collisions has been shown to increase with the energy aexopoulos (). These effects can be understood in terms of multiparton interactions walker2004 (). In case of heavy-ion collisions, it has been predicted that multiple parton interactions would produce long-range forward-backward multiplicity correlations that extend beyond one unit in rapidity, compared to hadron-hadron scattering at the same energy. The model based on multipomeron exchanges (Dual Parton Model) predicts the existence of long range correlations capela1 (); capela2 (). In the Color Glass Condensate (CGC) framework, the correlations of the particles created at early stages of the collision can spread over large rapidity intervals, unlike the particles produced at the later stage larry (). Thus the measurement of the long range rapidity correlations of the produced particle could give information about the space-time dynamics of the early stages of the collisions.
The measurement of forward-backward (FB) multiplicity correlations in nucleus-nucleus collisions has been studied by the STAR experiment at RHIC star_lrc (); tjt2 (); tjt3 (); bks1 (); bks2 (). These results have generated a great deal of theoretical interest LRCCGC (); HSD (); pajares1 (); Bzdak (); Yan1 (); Gelis (); Yan2 (); Lappi (); Bialas (); Wit (); Broniowski ().
Forward-backward correlations have been characterized by the correlation strength, , the slope extracted from a linear relationship between the average multiplicity measured in the backward rapidity hemisphere () and the multiplicity in the forward rapidity hemisphere, . This relationship can be expressed as alner ():
In this definition, can be positive or negative with a range of . This maximum (minimum) represents total correlation (anti-correlation) of the produced particles separated in rapidity. is the limiting case of entirely uncorrelated particle production. Experimentally, the slope of in hadron-hadron experiments is found to be positive alner (). In Eq.(1), the intercept, , is related to the number of uncorrelated particles.
The correlation strength can also be expressed in terms of the ratio of the covariance of the forward-backward multiplicity and the variance of the forward multiplicity. This is done by performing a linear regression of Eq.(1) and minimizing . Thus, Eq.(1) can be expressed in terms of the following calculable average quantities,
where and are the forward-forward and backward-forward dispersions.
The correlations obtained from above expressions can be a combination of both short and long-ranges. The short-range correlations (SRC) normally extend over a small range of pseudorapidity () and are due to various short-range order effects alner (). These correlations can arise from various effects, such as particles produced from cluster decay, resonance decay, or jet correlations. The particles produced in a single inelastic collision are known to only exhibit SRC SRC (). Long-range correlations (LRC) are correlations that extend over a wide range in pseudorapidity, beyond . The presence of LRC is a violation of short-range order. Short-range order is expected to hold as long as unitarity constraints are neglected alner (). In the approximation of short-range order, only single scattering can be considered. Therefore, quantum mechanical probability is not conserved, since it is possible to have multiple scattering terms.
Recently, FB correlations have been studied extensively with different model simulations, particularly the Color Glass Condensate (CGC) Lappi () model and the Color String Percolation model (CSPM) pajares2 (). The CGC provides a QCD based description and predicts the growth of LRC with collision centrality. It is argued that long-range rapidity correlations are due to the fluctuations of the number of gluons and can only be created in early times shortly after the collision LRCCGC (); Lappi (). In CGC the long range component has the form:
where is coupling constant and is related to the saturation momentum and is a constant. From the above expression, it is observed that as the centrality increases the FB correlation also increases because decreases pajares2 (). A similar behavior is also obtained in the CSPM approach. In the CSPM, is expressed in terms of the string density , which is related to the number of strings formed in the collision:
which vanishes at low string density and at high density grows to become , where is a constant, independent of the density and energy pajares2 (). The experimental data for Au+Au collision at = 200 GeV star_lrc () show similar trends as predicted by CGC and CSPM.
FB correlation strength has also been studied in the framework of wounded nucleon model Bzdak (); wounded (). The results are compared to the STAR data star_lrc () in Au+Au collisions at = 200 GeV. It has been concluded that FB correlation strength for central collisions are due to the fluctuations of wounded nucleons at a given centrality bin. Thus it is essential to control the centrality of the collisions while reporting the experimental results on correlations.
In the data analysis adopted for the STAR experiment star_lrc (), the centrality was defined using the charged particle multiplicity in the mid rapidity region. To avoid a self-correlation of the results with the window used for centrality definition, a profile method was used. In this paper we investigate the profile method to extract the LRC strength in heavy-ion collisions and demonstrate its applicability in Pb-Pb collisions at = 2.76 TeV using HIJING event generator hijing ().
Ii Analysis Method
In a center-of-mass coordinate system, the forward and backward hemispheres have been conventionally defined to be opposite to each other, as shown in the schematic diagram of figure 1. and are the charged particle multiplicities within the forward and backward measurement intervals within a width of . In our analysis, a value of has been chosen. The FB correlations are measured symmetrically around with varying rapidity gaps, designated as , measured from the center of each bin. Thus depending on the available window, the values of are possible.
In this analysis, data from the HIJING event generator have been used, in which the particles are produced based on perturbative QCD processes hijing (). Nearly one million minimum bias Au-Au events at =200 GeV and Pb-Pb events at =2.76 TeV have been generated and used for the analysis. The centrality of the collision is normally designated interms of the impact parameter of the collision. In the experiments, it is not possible to determine the impact parameter directly, hence one uses charged particle multiplicity within a range of , which is not overlapping with the range where the analysis is performed. This is called reference multiplicity (). The use of non-overlapping pseudo-rapidity regions, one for the centrality determination and other ones for FB analysis, avoid bias on the correlation measurements. In the experiments, it is ideal to obtain reference multiplicity from very forward measurement of charged particles. But if this is not available, then the centrality can be defined from the central windows as well. For example, in the present study, for determining FB correlations in = 0.2, 0.4 and 0.6, reference multiplicity has been obtained within , while for = 0.8 and 1.0 the sum of the multiplicities from and used for centrality determination. For = 1.2, 1.4, 1.6 and 1.8,…. the centrality is taken from . In the correlation analysis, the centrality windows are normally selected over a range of cross section, which correspond to a range in reference multiplicity. Within a given centrality window, the FB multiplicity correlations can be affected by the fluctuations in impact parameter and number of participants. In order to extract true correlation, it is desirable to control the centrality and minimize the effect of centrality fluctuations.
To calculate the correlation strength as a function of -gap and as a function of centrality, two different method have been discussed. In the first method, the quantities such as, , , and , have been obtained by averaging over the events within a centrality bin, and thereby calculating the dispersions, and . This method of event averaging does not take the fluctuation within a centrality window into account. This method is termed as method.
In order to eliminate or reduce the effect of the centrality window on the correlation analysis, a second method, called the profile method () has been introduced. In this method, the distributions of , , and , have been plotted as a function of the reference multiplicities. Linear fits to , and second order polynomial fits to and have been made. These distributions, along with the fits are shown in Fig. 2. These fit parameters are used to extract the and , binned by centrality, and normalized by the total number of events in each bin. This is shown in Fig. 3. This method removes the dependence of the FB correlation strength on the width of the centrality bin. In the next section, results from both the average and profile methods will be presented and compared.
Iii Results and Discussion
The FB correlations have been studied for Pb-Pb collisions at = 2.76 TeV using the HIJING event generator. The forward-forward and backward-forward dispersions are calculated as a function of centrality, within a pseudo-rapidity gap extending up to 2.2 units, using both average and profile methods. Figure 4 shows and as a function of for two overlapping centralities, 0-5% and 0-10% of total cross sections. The dispersions remain approximately constant over the rapidity ranges covered. It is observed that yields higher values of both and compared to . This is true for both the centrality windows.
It is expected that the correlation strength increases with the increase of the centrality of the collision. The correlation strengths, , are calculated from the ratios of the dispersions for six different centrality windows, 0-2.5%, 0-5%, 0-10%, 10-20%, 20-30%, and 30-40% of the cross section. These centralities are determined from the reference multiplicities as discussed above. Results from both the methods are presented in Fig. 5, where the upper panel shows the values of using method and the lower panel gives the results for method. We observe that for method does not follow any regular pattern in terms of centrality selection. For example, the is seen to be higher for the 0-10% centrality bin compared to 0-2.5% and 0-5% centrality bin, which is counter intuitive to our expectation. This shows that the impact parameter fluctuations are not completely removed when method is used. On the other hand, it can be seen that using the method, the values of , has an increasing trend with the increase of centrality of the collisions. The correlation strength is highest for 0-2.5% centrality, as expected.
In order to confirm the above observation, a study using the impact parameter window for centrality selection, rather than the reference multiplicity, has been made. Results for for various impact parameter selections are shown in Figs. 6(a) and (b), for the average and profile methods, respectively. The average method arrives at improper results. In this example, the larger centrality window yields highest correlation strength, which should not be the case. On the other hand, the method gives similar results whether centrality selection is made using impact parameter or the reference multiplicity.
A similar study has been made for Au-Au collisions at = 200 GeV using HIJING event generator for top 10% in central collisions. The upper panel of Fig. 7 show , and the lower panel shows , respectively, for both the average and profile methods. The results yield higher values for compared to . The profile results are similar to what had been reported by the STAR Collaboration at RHIC star_lrc (); tjt3 ().
Finally, a comparison of the correlation strengths have been made for Au-Au collisions at = 200 GeV and Pb-Pb collisions at = 2.76 TeV using the results from HIJING event generator, and following the method. The results of the study for two centrality windows (0-10% and 30-40%) are shown in Fig. 8. It is observed that, for the non-central collisions of 30-40% cross section, the correlation strengths are very similar. For central collisions, a decreasing trend is observed for Au-Au collision at = 200 GeV, whereas for Pb-Pb collisions at = 2.76 TeV, a flatter distribution is observed. This implies a much stronger correlation over a broad range in pseudorapidity at the LHC energy compared to those at RHIC.
Study of forward-backward multiplicity correlation strengths in hadron-hadron and heavy-ion collisions provide crucial information towards understanding particle production mechanisms and represent useful tool for differentiating different types of reactions and their energy dependence. It has been observed that the correlations show strong short range correlations and also extend to much wider separation in rapidity. In heavy-ion collisions, the correlation strengths are expected to increase with increase of the beam energy as well as centrality of the collision. Within a given centrality window, the fluctuations in the impact parameter or the number of participants lead to multiplicity fluctuations which affect the accurate determination of correlation strength. It is therefore needed to control the centrality of the collisions while performing the correlation analysis.
In this manuscript, two different methods, the average method and the profile method, have been presented to study the forward-backward multiplicity correlations in heavy-ion collisions as a function centrality. It is observed that in the method, the correlation strength does not follow any pattern as a function of centrality window. This reflects the impact parameter fluctuation due to finite centrality bin width. The second method, , has been introduced, which properly takes care of the effects due to finite centrality bin width. Appropriate centrality dependence has been observed in going from peripheral to central method. A comparison of the correlation strengths have been made for Au-Au collision at = 200 GeV and Pb-Pb collisions at = 2.76 TeV using the data from HIJING event generator. It has been observed that the correlation strengths are higher for higher energy collision. The correlation strengths decrease as a function of the rapidity gap. This decrease is much slower at LHC energy compared to that of the RHIC energies. The method can be used to study the FB correlation strength as a function of centrality in the Pb-Pb collisions at LHC. As is shown in this work, along with the correlation strength (), it is essential to show the behavior of both the forward-forward () and backward-forward () dispersions as a function of pseudo-rapidity gap () for different centrality classes. This will allow to make a direct comparison of experimental data with theoretical models, such as, CGC and CSPM.
This research was supported by the Office of Nuclear Physics within the U.S. Department of Energy Office of Science under Grant No. DE-FG02-88ER40412. S.D and T.N were supported by the Department of Atomic Energy, Government of India.
- (1) Y. V. Kovchegov, E. Levin, and L. McLerran, Phys. Rev. C63, 024903 (2001).
- (2) G. J. Alner et al., Phys. Rep. 154, 247 (1987).
- (3) A. Capella, and J. Tran Thanh van, Z. Phys. C18, 85 (1983).
- (4) T. Alexopoulos et al., Phys. Lett. B353, 155 (1995).
- (5) W.D. Walker, Phys. Rev. D69, 034007 (2004).
- (6) A. Capella and A. Krzywicki, Phys. Rev. D18, 120 (1978).
- (7) A. Capella et al., Phys. Rep. 236, 225 (1994).
- (8) B. I. Abelev et al. (STAR Collaboration), Phys. Rev. Lett. 103, 172301 (2009).
- (9) T. Tarnowsky (STAR Collaboration), Int. J. Mod. Phys. E 16, 3363 (2007).
- (10) T. Tarnowsky (STAR Collaboration), Proceedings of Science (CPOD07) 019,2007, nucl-ex/0711.1175.
- (11) B. K. Srivastava (STAR Collaboration), Int. J. Mod. Phys. E16, 3371 (2007).
- (12) B. K. Srivastava (STAR Collaboration), Eur. Phys. J A31, 3862 (2007).
- (13) N. Armesto, L. McLerran, and C. Pajares, Nucl. Phys. A781, 201 (2007).
- (14) V. P. Konchakovski, M. Hauer, G. Torrieri, M. I. Gorenstein, and E. L. Bratkovskaya, Phys. Rev. C79, 034910 (2009).
- (15) P. Brogueira, J. Dias de Deus, and C. Pajares, Phys. Lett. B675, 308 (2009).
- (16) A. Bzdak, Phys. Rev. C80, 024906 (2009).
- (17) Yu-Liang Yan et al., Phys. Rev. C79, 054902 (2009).
- (18) F. Gellis, T. Lappi, and L. McLerran, Nucl. Phys. A828, 149 (2009).
- (19) Yu-Liang Yan et al., Phys. Rev. C81, 044914 (2010).
- (20) T. Lappi, and L. McLerran, Nucl. Phys. A832, 330 (2010).
- (21) A. Bialas, and K. Zalewski, Phys. Lett. B698, 416 (2011).
- (22) K. Fialkowski, and R. Wit, nucl-th/1203.3671.
- (23) A. Olszewski, and W. Broniowski, nucl-th/1303.5280.
- (24) C. Pajares, Nucl. Phys. B854, 125 (2011).
- (25) A. Bialas, M. Bleszynski, and W. Czyz, Nucl. Phys. B111, 461 (1976).
- (26) X. -N. Wang, and M. Gyulassy, Phys. Rev. D44, 3501 (1991).