Detection of source inhomogeneity through event-by-event two-pion Bose-Einstein correlations

We develop a method for detecting the inhomogeneity of the pion-emitting sources produced in ultra-relativistic heavy ion collisions, through event-by-event two-pion Bose-Einstein correlations. The root-mean-square of the error-inverse-weighted fluctuations between the two-pion correlation functions of single and mixed events are useful observables for the detection. By investigating the root-mean-square of the weighted fluctuations for different impact parameter regions people may hopefully determine the inhomogeneity of the particle-emitting in the coming Large Hadron Collider (LHC) heavy ion experiments.

space-time structure of the particle-emitting sources produced in high energy heavy ion collisions [1,2,3,4]. Because of the limitation of data statistics, the usual HBT investigations are performed for mixed events and the HBT radii are obtained by fitting the two-pion HBT correlation functions from the mixed events to Gaussian parametrized formulas.
On event-by-event basis, the density distribution of the source may be not a Gaussian distribution. An inhomogeneous particle-emitting source on event-by-event basis may be a more general case because of the fluctuating initial matter distribution in high energy heavy ion collisions [5,6,7,8]. In Ref. [9], a granular source model was used to explain the Relativistic Heavy Ion Collider (RHIC) HBT results, R out /R side ≈ 1 [10,11,12,13]. The main idea of the explanation is that the evolution time for the granular sources is short and the short evolution time, which can not be averaged out after event-mixing, may lead to the HBT results R out ∼ R side [9,14,15]. Recent source imaging researches for the collisions of √ s NN = 200 GeV Au+Au indicate that the pion-emitting source with the selections 40 < centrality < 90% and 0.20 < k T < 0.36 GeV/c is far from a Gaussian distribution [16,17,18]. Although the long tail of the two-pion source function at large separation r is believed mainly the contribution of long-lived resonances [16,17], the enhancement of the source function at small r may possibly arise from the source inhomogeneity [18].
For inhomogeneous particle-emitting sources, the single-event two-pion HBT correlation functions may exhibit fluctuations relative to the HBT correlation functions of mixed events [19,20]. Detecting and investigating this event-by-event fluctuations is important for finally determining the source inhomogeneity and understanding the initial conditions and evolution of the system in high energy heavy ion collisions.
Hydrodynamics may provide a direct link between the early state of the system and final observables and has been extensively used in high energy heavy ion collisions. In hydrodynamical calculations the system evolution is determined by the initial conditions and the equation of state (EOS) of the system. Smoothed Particle Hydrodynamics (SPH) is a suitable candidate that can be used to treat the system evolution with large fluctuating initial conditions for investigating event-by-event attributes [7,21]. It has been used in high energy heavy ion collisions for a wide range of problems [7,8,21,22,23,24,25]. In the present letter we use SPH to describe the system evolution. The system initial states are given by the NEXUS event generator [6] at τ 0 = 1 fm/c for √ s NN = 200 GeV Au+Au collisions at RHIC and τ 0 = 0.5 fm/c for √ s NN = 5500 GeV Pb+Pb collisions at LHC. The EOS is obtained with the entropy density suggested by QCD lattice results [14,26,27,28].
In the EOS, the QGP phase is considered as an ideal gas of massless quarks (u, d, s) and gluons [22,23]. The hadronic gas is composed of the resonances with mass below 2.5 GeV/c 2 , where volume correction is taken into account [22,23]. The transition temperature between the QGP and hadronic phases is taken to be T c = 160 MeV, and the width of the transition is taken to be 0.1T c [28].
However, in order to investigate the whole space-time structure of the system an nonlocal coordinate frame is needed and we work in the center-of-mass frame of the system. Figure   1   Assuming that final identical pions are emitted at the space-time configuration characterized by a freeze-out temperature T f , we may generate the pion momenta according to Bose-Einstein distribution and construct the single-event and mixed-event two-pion correlation functions [9,20]. Figure 2 shows the two-pion correlation functions C(q side , q out , q long ) for the single and mixed events for √ s NN = 200 GeV Au+Au with impact parameters b = 10 fm (up panels) and b =5 fm (down panels). Here q side , q out , and q long are the components of "side", "out", and "long" of relative momentum of pion pair [29,30]. In each panel of initial rapidity of the "smoothed particles", η 0 > 0 or η 0 < 0.
We have seen that the two-pion correlation functions of the single events exhibit eventby-event fluctuations. However, in the usual mixed-event HBT measurements, these fluctuations are smoothed out. In order to observe the event-by-event fluctuations, we investigate the distribution dN/df of the fluctuations between the correlation functions of single and mixed events, |C s (q i ) − C m (q i )|, with their error-inverses as weights [20], In calculations we take the width of the relative momentum q i bin to be 10 related to the energy √ s NN of the collisions. For a finite N ππ we have to reduce variable numbers in analysis although it will lose some details. In Fig. 4 we show the distributions of f for the variables of transverse relative momentum q trans and relative momentum q of the pion pairs for the 40 simulated events with b = 5 fm. One can see that for N ππ = 5 × 10 6 , the distributions for FIC are much wider than those for SIC both for q trans and q. Even for N ππ = 5 × 10 5 , the widths for FIC are visibly larger than those for SIC. In order to examine the distributions quantitatively, we calculate the root-mean-square (RMS) of f . Figure 5 shows  hundreds for central collisions. The order of N ππ is about 10 5 (∼ M 2 π /2). However, at the higher energy of LHC, M π will be about two thousands and the order of N ππ will be 10 6 . In this case the large differences between the RMS of f for inhomogeneous and homogeneous sources provide a great opportunity to detect the source inhomogeneity.
Another problem in experimental data analysis is that the impact parameter b is hardly to be held at a fixed value and usually be limited in a region. In such a case the large f rms values may arise from the source inhomogeneity as well as the variation of b in the region.
So one should also consider the effect of the b variation in inhomogeneity detections. In summary, on event-by-event basis the initial density distribution of matter in high energy heavy ion collisions is highly fluctuating. The fluctuating initial conditions lead to event-by-event inhomogeneous particle-emitting sources. In this letter we developed a method for detecting the source inhomogeneity through event-by-event two-pion correlations. We find that the RMS of the error-inverse-weighted fluctuations f are useful observables for detecting the inhomogeneity of the sources. The high identical pion multiplicity in the coming LHC heavy ion collisions provides a great opportunity to do the detections. By investigating the RMS values of f for different impact parameter regions people may hopefully determine the inhomogeneity of the particle-emitting produced in the coming LHC experiments.
We thank Dr. C. Y. Wong and Dr. D. C. Zhou for helpful discussion. This research was supported by the National Natural Science Foundation of China under Contracts No.