Two- and three-pion quantum statistics correlations in Pb-Pb collisions at root S-NN=2.76 TeV at the CERN Large Hadron Collider

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Twoand three-pion quantum statistics correlations in Pb-Pb collisions at root SNN=2.76 TeV at the CERN Large Hadron Collider ALICE Collaboration ALICE Collaboration. (2014). Twoand three-pion quantum statistics correlations in Pb-Pb collisions at root S-NN=2.76 TeV at the CERN Large Hadron Collider. Physical Review C, 89(2), Article 024911. https://doi.org/10.1103/PhysRevC.89.024911 2014 PHYSICAL REVIEW C 89, 024911 (2014) Twoand three-pion quantum statistics correlations in Pb-Pb collisions at √sNN =2.76 TeV at the CERN Large Hadron Collider B. Abelev et al.∗ (ALICE Collaboration) (Received 1 November 2013; revised manuscript received 29 January 2014; published 26 February 2014) Correlations induced by quantum statistics are sensitive to the spatiotemporal extent as well as dynamics of particle-emitting sources in heavy-ion collisions. In addition, such correlations can be used to search for the presence of a coherent component of pion production. Twoand three-pion correlations of same and mixed charge are measured at low relative momentum to estimate the coherent fraction of charged pions in Pb-Pb collisions at √sNN=2.76 TeV at the CERN Large Hadron Collider with ALICE. The genuine three-pion quantum statistics correlation is found to be suppressed relative to the two-pion correlation based on the assumption of fully chaotic pion emission. The suppression is observed to decrease with triplet momentum. The observed suppression at low triplet momentum may correspond to a coherent fraction in charged-pion emission of 23% ±8%. DOI: 10.1103/PhysRevC.89.024911 PACS number(s): 25.75.Gz,05.30.Jp I. INTRODUCTION The techniques of intensity interferometry are often used to extract information of the space-time structure of particleemitting sources [1]. For identical boson correlations, quantum statistics (QS) or Bose-Einstein correlations contribute significantly at low relative momentum. The strength of QS correlations is known to depend on the degree of chaoticity of particle-emitting sources [2,3]. Identical boson QS correlations reach their maximum value for fully chaotic sources (no coherence) and their minimum value for fully coherent sources. The possibility of coherent pion production in high-energy heavy-ion collisions has been considered several times before. In particular, it was proposed that the interior of the high-energy hadron collisions might form a Bose-Einstein condensate [4] with an anomalous chiral order parameter (DCC) [5]. Such a condensate produced in the interior may survive until some time after the relatively hot and chaotic expanding shell decouples and hadronizes. The pion radiation from a condensate is expected to be coherent and thus suppresses Bose-Einstein correlations. Furthermore, initial conditions such as the color-glass condensate (CGC) [6], which invoke the coherent production of partons, might also lead to condensate formation [7]. In this article we present twoand three-pion correlations of same and mixed charge at low relative momentum to estimate the coherent fraction of charged-pion emission in Pb-Pb collisions at √sNN=2.76 TeV at the LHC with ALICE. A number of past experimental efforts have been made to measure the degree of coherence in high-energy heavyion collisions using three-pion Bose-Einstein correlations: NA44, WA98, and STAR [8–10]. The methodology used here ∗Full author list given at the end of the article. Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. represents an improvement over the past efforts, which we summarize in Sec. III. The remainder of this article is organized into six sections. In Sec. II we describe the data-selection procedure. In Sec. III we introduce the methodology used in this analysis. In Sec. IV we describe the treatment of final-state interactions (FSIs). In Sec. Vwe describe the treatment of momentum resolution corrections. In Sec. VII we explain the estimation of systematic uncertainties. In Sec. VIII we present the results of this analysis. We conclude with a possible interpretation of the analysis results in Sec. IX. II. EXPERIMENT AND DATA ANALYSIS Data were taken from the 2011 Pb-Pb run at √sNN= 2.76 TeV at the CERN Large Hadron Collider (LHC) with ALICE [11]. The VZERO detectors [12], located in the forward and backward regions of the detector, were used to form a minimum-bias trigger by requiring a simultaneous signal in both [13]. The charged-particle multiplicity in the VZERO detectors is used to determine the collision centrality. Approximately 34 ×106minimum-bias collisions were used in this analysis. Particle tracking was performed with two azimuthally complete detectors: the inner tracking system (ITS) and the time projection chamber (TPC) [14]. The ITS consists of six layers of silicon detectors: silicon pixel (layers 1–2), silicon strip (layers 3–4), and silicon drift (layers 5–6) detectors. The combined number of readout channels for all six layers is 1.257 ×107. The ITS provides high spatial resolution to the distance of closest approach (DCA) of a particle to the primary vertex. However, it was not used for the momentum determination of particles in this analysis. Cluster sharing within the ITS was found to cause a slight increase in track merging, to which this analysis is especially sensitive. The TPC was used to determine the particle’s momenta and charge via its radius of curvature in the 0.5-T longitudinal magnetic field. The TPC is composed of 159 radially aligned pad rows for each of the 18 azimuthal sectors, totaling 557 568 readout channels. In addition to the tracking capabilities, the ITS and TPC provide particle identification capabilities through the specific ionization energy loss (dE/dx) in the silicon layers and 0556-2813/2014/89(2)/024911(19) 024911-1 ©2014 CERN, for the Alice Collaboration B. ABELEV et al. PHYSICAL REVIEW C 89, 024911 (2014) TPC gas, respectively. We select charged pions within 2 standard deviations (σ) of the expected pion dE/dx value. For momenta greater than 0.6 GeV/c, high pion purity is maintained with the time-of-flight (TOF) detector. The TOF covers the full azimuthal range and the pseudorapidity range |η|<0.9, except for the region 260◦<ϕ<320◦, where no TOF modules were installed to reduce the material budget in front of the photon spectrometer. With TOF we select tracks within 2σof the expected pion TOF values. Tracks which are within 2σof the expected kaon or proton dE/dx or TOF values are rejected. Below 0.45 GeV/c we further reject pion candidates if their dE/dx is within 2σof the expected electron dE/dx value. The pion-pair purity in this analysis is estimated to range from 90% to 94% for the highest and lowest pair momentum, respectively. To ensure uniform tracking in the ITS, TPC, and TOF we require the zcoordinate of the primary vertex to be within a distance of 10 cm from the detector center. We analyze tracks with transverse momenta in the interval 0.16 <p T< 1.0GeV/c and pseudorapidity |η|<0.8. To ensure good momentum resolution, we require a minimum of 70 tracking points in the TPC. Track merging and splitting are known issues for samecharge tracks at very low relative momentum [15]. We minimize the contribution from merged and split pairs through three types of pair cuts. First, we simply reject all pairs whose Lorentz invariant relative momentum, q, is less than 5 MeV/c. Second, we reject all pairs whose angular separation is less than 0.02 and 0.045 rad in the longitudinal and azimuthal direction, respectively. The pair angular separation is evaluated at a radial distance of 1.0 and 1.6 m, where the most pronounced trackmerging and -splitting effects were observed, respectively. Third, we reject pairs that share more than 5% of pad-row tracking points [16]. These three cuts are applied to all terms of the correlation functions (same-event and mixed-event) introduced in the next section. For three-pion correlations we apply these three cuts to each of the three pairs in the triplet. The cuts are only applied to same-charge pairs. Mixed-charge pairs are easily distinguished in the central barrel magnetic field as their trajectories are bent away from each other. III. METHODOLOGY Two-particle correlation functions are binned in narrow intervals of the mean pair transverse momentum, kT= |pT,1+pT,2|/2, and Lorentz invariant relative momenta, q= −(p1−p2)μ(p1−p2)μ. They are defined as the ratio of the inclusive two-particle spectrum, N2(p1,p2) over the product of inclusive single-particle spectra, N1(p1)N1(p2): C2(p1,p2)=N2(p1,p2) N1(p1)N1(p2).(1) The numerator of the correlation function is formed by all pairs of particles from the same event. The denominator is formed by taking one particle from one event and the second particle from another event. The sameand mixed-event two-particle distributions are normalized to each other in the interval 0.15 <q<0.175 GeV/c, sufficiently above the dominant region of low relative momentum correlations and sufficiently narrow to avoid the small influence of background correlations. Only events within the same centrality class are mixed. The centrality classes correspond to the top 0%–5% through 45%–50% of the particle multiplicity distribution estimated with the VZERO detector. Each class has a width of 5%. The isolation of genuine two-pion correlations is complicated by several additional factors. Namely, the resolvable threshold of low relative momentum pairs is limited by track merging and splitting in the ALICE detector. The QS correlation of long-lived resonance decays is largely localized below this threshold and is therefore unobservable. This leads to an apparent decrease of QS correlations and is described by the λor “dilution” parameter in this analysis. Given λ, two-particle correlations can be written as N2(p1,p2)=N(1 −λ)N1(p1)N1(p2) +λK2(q)NQS 2(p1,p2),(2) C2(q)=N(1 −λ)+λK2(q)CQS 2(q),(3) where Nis a residual normalization taking into account the small nonfemtoscopic contributions [17,18]. We allow a different Nfor same and mixed-charge correlations as the nonfemtoscopic contributions can be different. K2(q)isthe FSI correlation. NQS 2and CQS 2(q) are the genuine two-pion QS distribution and correlation, respectively. Here, unlike in most experimental publications on this subject, the λparameter does not include effects of partial coherence. Its deviation below unity can also be attributable to secondary contamination, pion misidentification, and finite qbinning. Same-charge pion QS correlations excluding coherence can be parametrized by CQS,++ 2(q)=1+Ew(Rchq)2e−R2 chq2,(4) Ew(Rchq)=1+∞  n=3 κn n!(√2)nHn(Rchq),(5) where Rch are the characteristic radii of the chaotic component. Ew(Rchq) is the Edgeworth expansion characterizing deviations from Gaussian behavior [19]. Hnare the Hermite polynomials and κnare the Edgeworth coefficients. The first two relevant Edgeworth coefficients (κ3,κ4) are found to be sufficient to describe the non-Gaussian features in this analysis. At the two-pion level we do not include an explicit parametrization of a possible coherent component owing to the large uncertainty of non-Gaussian Bose-Einstein correlations. In this analysis we assume λof mixed-charge pions is identical to that of same-charge pions: λ+− =λ±±. This is a valid assumption at high energies where the secondary contamination from particles and antiparticles are expected to be equal [20]. Three-particle correlation functions are binned in terms of the three invariant relative momenta in the triplet: q12,q31, and q23. The three-particle correlation function is similarly the ratio of the inclusive three-particle spectrum to the product of the inclusive single-particle spectra binned in the pair relative momenta: C3(p1,p2,p3)=N3(p1,p2,p3) N1(p1)N1(p2)N1(p3),(6) Q3=q2 12 +q2 31 +q2 23.(7) 024911-2 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) The numerator of C3is formed by all triplets of particles from the same event. The denominator is formed by taking each of the three particles from different events. We project three-particle correlations against the Lorentz invariant Q3. For three-particle correlations, λ= 1 similarly causes “feed-up” from pure combinatorial distributions and twoparticle correlations as described in Eq. (8) below. The derivation of Eq. (8) is shown in the Appendix. In Eq. (8), N2(pi,pj)N1(pk) terms represent the case where particles i and jare taken from the same event, while particle kis taken from a different event and K3is the three-pion FSI correlation. Isolation of the three-pion QS correlation is done by solving Eq. (8)forNQS 3.UsingNQS 2and NQS 3one can construct a cumulant correlation function, c3,inEq.(9): N3(p1,p2,p3)=f1N1(p1)N1(p2)N1(p3)+f2[N2(p1,p2)N1(p3)+N2(p3,p1)N1(p2)+N2(p2,p3)N1(p1)] +f3K3(q12,q31,q23)NQS 3(p1,p2,p3),(8) c3(p1,p2,p3)=1+2N1(p1)N1(p2)N1(p3)−NQS 2(p1,p2)N1(p3)−NQS 2(p3,p1)N1(p2)−NQS 2(p2,p3)N1(p1) +NQS 3(p1,p2,p3)N1(p1)N1(p2)N1(p3).(9) In Eq. (8), f1,f2, and f3are derived in the Appendix and are given by (1 −λ1/2)3+3λ1/2(1 −λ1/2)2−3(1 −λ1/2)(1 −λ), (1 −λ1/2), λ3/2, respectively. The quantity in square brackets in Eq. (9) represents a three-pion cumulant which has all two-pion correlations removed. Therefore, the three-pion cumulant represents the isolation of genuine three-pion QS correlations. All sameand mixed-event three-particle distributions are normalized to each other in the range where all three pairs satisfy 0.15 <q ij <0.175 GeV/c, sufficiently above the dominant region of low relative momentum correlations and sufficiently narrow to avoid the small influence of background correlations. The novel effects measured with three-particle correlations are isolated with the r3function [21,22]: r3(p1,p2,p3)=c3(p1,p2,p3)−1 CQS 2(p1,p2)−1CQS 2(p3,p1)−1CQS 2(p2,p3)−1 .(10) The r3function isolates the phase of three-pion correlations: r3=Icos()≈I(1 −2/2) [21]. The intercept of r3,I,is expected to be 2 in the case of fully chaotic particle-emitting sources and less than 2 in the case of partially coherent sources. The leading-order contribution to the phase was shown to be quadratic in relative momenta, ≈aμνqμ 12qν 23, which leads to quartic behavior in r3[21]. The antisymmetric tensor aμν characterizes space and momentum source asymmetries related to how the spatial position of maximum pion emission changes with momentum. There are six nonvanishing independent components in aμν . However, owing to limited statistical precision we project r3from three-dimensional invariant relative momenta to one-dimensional Q3. A fit quartic and quadratic in Q3is performed, r3(Q3)=I1−aQ4 3,(11) r3(Q3)=I1−aQ2 3,(12) where Iis the intercept of r3[I=r3(0)], and ais the quartic or quadratic coefficient. The quadratic fit is motivated by previous fit attempts by the STAR collaboration [10]. The coherent fraction (G) can be extracted from the intercept as [21] I=2√1−G1+2G (1 +G)3/2.(13) Equation (13) neglects the effect of the charge constraint on charged coherent states [20,23,24]. In the quantum optics approach to coherent states [25], charged pions can only be in coherent states when positive and negative pions pair together to form a charge neutral state. However, because the charge constraint affects both numerator and denominator of r3in the same direction, its effect on r3for G<30% is expected to increase its intercept by less than 17% [24]. The denominator of r3is measured using the threeparticle combinatorial distribution and two-particle correlation strengths. The two-particle correlation strengths are tabulated from a previous run over the data. They are tabulated in sufficiently narrow intervals or bins of centrality, kT, and three-dimensional (3D) relative momentum to allow reliable interpolation between bins. We bin the two-particle correlations in nine centrality bins (5% wide) and 4 kTbins in the longitudinally comoving system (LCMS). Forty qout, qside, and qlong bins (5 MeV/c wide) are chosen. qout is the projection of the relative momentum along the pair momentum direction. qlong is the projection along the beamline. qside is then perpendicular to the other two (azimuthal projection). The four kTbins are chosen such that they divide the pair distribution into four equally populated intervals. A. Methodology improvement The methodology used here to measure three-pion QS correlations represents an improvement over the past efforts [8–10], which we highlight here. (i) In addition to QS correlations, charged pions also experience a Coulomb repulsion, which reduces the apparent strength of QS correlations. Corrections for the three-body Coulomb interactions are damped in this analysis according to the observed λparameter. 024911-3 B. ABELEV et al. PHYSICAL REVIEW C 89, 024911 (2014) Previously, the Coulomb corrections were undamped and thus overestimated. (ii) The Coulomb corrections are estimated by integrating over an assumed freeze-out distribution of pions. We take into account the effect of resonance decays on the freeze-out distribution. Previously, a Gaussian distribution was assumed. (iii) For the case when λ<1, the measured three-pion correlations contain a feed-up from lower-order correlations, which is now removed. (iv) We apply momentum resolution corrections, which was not universally done in the past efforts. (v) We apply corrections for muon contamination which was not done in the past efforts. (vi) The isolation of the cumulants is done at the pair/triplet distribution level instead of at the correlation function level. (vii) Mixed-charge twoand three-pion correlations are used to help determine the λparameter and to monitor the performance of FSI corrections. IV. FINAL-STATE INTERACTIONS The treatment of FSIs is crucial for this analysis. In addition to QS correlations, identical charged pions also experience FSIs which reduce the apparent strength of QS correlations. The FSIs of charged pions are dominated by the Coulomb interaction. The strong interactions, while small for same-charge pions, are important for mixed-charge pions. Coulomb and strong FSI corrections are included in this analysis for both twoand three-particle sameand mixed-charge correlations. The wave functions for two-pion Coulomb and strong FSIs are known to high precision [26]. Two-pion FSIs are calculated by averaging the modulus square of the two-pion FSI wave functions over an assumed freeze-out particle-emitting source distribution. This is then divided by the corresponding average of plane-wave functions to isolate the pure FSIs. For same-charge pions, the wave functions are symmetrized. Typically, the source distribution is taken to be a spherical Gaussian with a radius matching what is found in the data. Here, we use a more sophisticated approach. All FSIs are calculated directly within THERMINATOR 2 events [27,28]. The pair relative separation at freeze-out in the pair-rest frame, r∗, as well as the space-momentum correlations included in the model are used. THERMINATOR includes all of the known resonance decays. Pions from resonance decays add non-Gaussian features to the freeze-out distribution. Furthermore, they increase the mean value of r∗, which in turn reduces the strength of FSI correlations. The same centrality class and kTrange from the data are used to calculate the FSIs. The freeze-out hypersurfaces in THERMINATOR were calculated within 3D viscous hydrodynamics with an initial and final temperature of 512 and 140 MeV, respectively. The starting time for hydrodynamics was 0.6 fm/c. Three-body FSI wave functions are not known for all regions of phase-space. However, all asymptotic wave functions are known [29]. In particular, the wave-function corresponding to the phase-space region where all three interparticle spacings are large, 0, is given by the product of the three two-body wave functions. It has been shown that the 0wave function is a justified approximation also in the case where the triplet kinetic energy in the triplet rest frame is sufficiently large [30]. It is estimated that triplet energies exceeding ∼7MeVfor6-fm sources justify the use of the 0wave function. The minimum triplet energy considered in this analysis is √3×5≈8.7MeV when all three pair q’s are at their minimum allowed value of 5MeV/c. For the case of same-charge pion FSIs with the 0wave function, the modulus square of the fully symmetrized FSI wave-function is averaged in THERMINATOR events. This is then divided by the corresponding average of fully symmetrized plane waves. The full symmetrization assumes fully chaotic emission. For the case of mixed-charge FSIs, only the samecharge pairs are symmetrized. All Kfactors in this analysis are averaged over the THERMINATOR freeze-out distribution for pairs satisfying r∗<80 fm. For the K3calculation, all three pairs must satisfy this requirement. All three-pion correlations in this analysis are binned in 3D corresponding to the three pair invariant relative momenta: q12,q23,q31. The three-pion FSI correlations are likewise calculated in 3D for the integrated kTrange. Another more commonly used approach to treat three-body FSIs is the Riverside approach [31] for which the three-body FSI correlation, K3, is given by the triple product of Gamov factors (K3=G12G23G31). In the generalized version of this approach, “generalized Riverside” (GRS), each two-body factor is averaged over the assumed source distribution (K3= K12 2K23 2K31 2)[9,10]. In Fig. 1we compare our calculations of three-body FSI correlations using the 0wave function and GRS approach within THERMINATOR events. We observe similar FSI correlations with both methods. V. MOMENTUM RESOLUTION Finite momentum resolution in the ALICE detector generally causes a smearing of the correlation function. We estimate its effect on the correlation functions by assigning a weight to each pair or triplet in HIJING [32] based on the measured correlation strength in real data. The same weight is applied to two versions of each Nn(n=1,2,3) histogram. The first is filled with the nonsmeared ideal qfrom HIJING. The second is filled with the smeared qafter the tracks have been propagated through the simulation of the ALICE detector response. The ratio of the first to the second histogram forms the correction factor for the Nndistributions. The momentum resolution corrections are found to be largestatlowq(Q3), where they increase the raw correlation function by less than 5% (8%) for two-pion (three-pion) correlations. We also observe that the correction factors do not change significantly with kT. After the momentum resolution corrections are applied, we verified that the observed correlation strength and shape matches the assumed values used as a weight in HIJING. VI. MUON CONTAMINATION The pion-pair purity is estimated to be about 93% in HIJING with the simulated ALICE detector response. The 024911-4 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) 3 K 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 Ωsame-charge, same-charge, GRS 0 Ωmixed-charge, mixed-charge, GRS 0 0.02 0.04 0.06 0.08 0.1 )-1) 0 Ω( 3 / (K 3 KΔ -0.1 0 0.1 same-charge mixed-charge )c (GeV/ 3 Q FIG. 1. (Color online) Comparison of sameand mixed-charge three-pion FSI correlations. 0wave function and generalized Riverside (GRS) method are shown. The calculation was performed in THERMINATOR (0%–5%). The bottom panel shows the difference between the two methods, K3=K3(0)−K3(GRS), divided by K3(0)−1. leading order misidentified pair is the muon pion combination. The rest of the misidentified combinations taken together contribute less than 1% to the total pairs. We estimate that about 93% of the muons contaminating our sample originate from primary-pion decays. The primary parent pion is expected to interact with the other primary pions via QS +FSI. We therefore expect that the muon pion pairs contaminating our sample will contain a residual pion pion correlation. For the three-pion case the muon pion pion combination dominants the misidentified triplets. We form a correction factor for all two-pion (three-pion) terms by assigning a QS +FSI weight to the parent pions in the pair (triplet) which subsequently decayed into muons. A smeared correlation is obtained when the assigned correlation is binned in relative momentum using the muon momentum. The ratio of the assigned correlation to the smeared correlation forms our correction factor. The correction is applied to same and mixed-charge correlations and is found to increase λby about 5% while having a negligible effect on the extracted radii. The correction increases the two-pion correlation by about 1.5% at low q and rapidly decreases for larger q. The correction increases the three-pion correlation by about 3% at low Q3and by about 1% for high Q3. VII. SYSTEMATIC UNCERTAINTIES The dominant systematic uncertainty in this analysis pertains to the unknown spatiotemporal pion distribution at freeze-out on which the fitting of the correlation functions and FSI calculations depends. Typically, a Gaussian profile is assumed in most femtoscopic analyses. However, the known resonances taken all together will generally give rise to non-Gaussian features in the freeze-out distribution. The systematic uncertainty of the freeze-out distribution is twofold in this analysis. First, it creates an uncertainty in the wave-function integration for the FSI calculation. However, the qdependence of FSI correlations is largely invariant to reasonable variations of the assumed freeze-out distribution and radius. A possible mismatch of the freeze-out distribution and radius in THERMINATOR as compared to the data is largely absorbed by the λparameter of the global fits to sameand mixed-charge two-pion correlations presented in the Results section. We assign a 2% uncertainty on the two-pion FSI correlations based on the maximum observed difference between FSIs calculated in THERMINATOR and Gaussian particleemitting source profiles after “rescaling” by an effective λ parameter. We also assign a 2% uncertainty on the r∗- dependent part of the FSI wave functions [26]. Second, the freeze-out distribution uncertainty creates an uncertainty in the fitting of the same-charge correlation functions. A convenient account of sufficiently small deviations from Gaussian behavior in the QS correlation functions can be obtained through an Edgeworth expansion [19]. Deviations from Gaussian behavior are also expected from a finite coherent component [20]. Non-Gaussian features in the QS correlation functions can also occur in more trivial ways. Spherical Gaussian freeze-out distributions create Gaussian QS correlation functions as a function of q. Non-Gaussian features in 1D correlation functions can arise simply from nonequal 3D radii in the LCMS frame. However, we note that Rout ≈Rside and Rlong is only ∼20% larger than Rout and Rside [15]. Also, kTand centrality bins whose widths are not sufficiently narrow will create a mix of different radii and therefore will not be described by a single Gaussian function. However, our chosen centrality bin width (5%) and kTbin width (100 MeV/c for two-particle correlations) are sufficiently narrow to mostly avoid this feature given the known kTdependencies of the radii [15]. More non-Gaussian features are expected for our three-particle correlations as the kTbin is much wider (1 GeV/c). The momentum resolution of low-momentum particles (pT<1GeV/c) is dominated by multiple scatterings within the ALICE detector. The ALICE material budget uncertainty is conservatively estimated to be ±10%. Our studies suggest a near one-to-one correspondence of the material budget uncertainty with the momentum resolution uncertainty. We apply a 10% uncertainty on all the momentum resolution corrections. For r3the momentum resolution correction uncertainty is 024911-5 B. ABELEV et al. PHYSICAL REVIEW C 89, 024911 (2014) found to be 1%. It is not the dominant uncertainty because both numerator and denominator are affected in the same direction. We study the uncertainties associated with tracking in the ALICE detector in several ways. We study the effect of different magnetic-field orientations in the TPC. The pion particle identification (PID) cuts are tightened by 10%. The angular separation cuts for same-charge pairs are increased by 50%. Positive pions are compared to negative pions. All the uncertainties in this category except for PID were found to be negligible. A 0.3% and 1% systematic uncertainty owing to PID were assigned for three-pion correlation functions and r3, respectively. Concerning r3, additional systematics are included. Imperfect isolation of the three-pion QS cumulant (FSI corrected) is the dominant uncertainty for r3which mostly affects the larger values of Q3where the cumulant is smallest. The chosen λ parameter (λ=0.7) used in extracting the QS correlations in both the numerator and the denominator, while largely canceling in the ratio, is varied by 0.1. Mixed-charge threepion cumulant correlations (c±±∓ 3) reveal a slight residual correlation of about 1.005 for all centralities. The residual cumulant correlation in the mixed-charge channel is used as a systematic uncertainty in the same-charge channel. Also, small variations of the powers mand nin Eq. (8) which brought c±±∓ 3 closer to unity resulted in similar systematic variations for r3. This procedure is valid if the true FSI-corrected mixed-charge cumulant correlation is expected to be near unity. The GRS approach to Coulomb corrections is found to give a better description of the mixed-charge correlations than the 0wave function. For this reason we choose the GRS approach as our principal method and use the 0wave function as a systematic variation for all three-pion correlations. Finally, nonfemtoscopic background correlations associated with minijets [33], while negligible for the highest multiplicity collisions, create a small uncertainty in the extraction of twopion QS correlation strengths. A linear fit to the background is made in the interval 0.2<q<0.4GeV/c and extrapolated into the femtoscopic region, q<0.15 GeV/c. The correction only has a non-negligible effect on r3for large Q3and above ∼40% centrality. VIII. RESULTS A. Two pions We first present the two-pion correlation functions. Figures 2and 3show the sameand mixed-charge correlation functions versus qin 6 kTbins for 0%–5% and 45%–50% centrality, respectively. Global fits for same and mixed-charge correlations are performed for each kTbin separately. Two types of global fits are shown. The dotted lines correspond to Gaussian fits (Ew=1), while the solid lines correspond to non-Gaussian fits with Edgeworth coefficients (Ew= 1). Our strict pair cuts cause a lack of data for same-charge correlations at low qat high kT, where a larger fraction of the pairs moves collinearly and thus is more susceptible to track merging and splitting. Concerning the purely Gaussian fits in Figs. 2and 3,the average χ2per degree of freedom (NDF) is 39. It is clear that a spherical Gaussian fully chaotic source can be ruled out. The global fits underestimate mixed-charge correlations for each 1 1.1 1.2 1.3 =2.76 TeV NN sALICE Pb-Pb 0-5% c<0.3 GeV/ T k0.2< 1 1.1 1.2 1.3 Gauss Edgeworth c<0.5 GeV/ T k0.4< 0 0.05 1 1.1 1.2 1.3 c<0.7 GeV/ T k0.6< same-charge mixed-charge c<0.4 GeV/ T k0.3< c<0.6 GeV/ T k0.5< 0 0.05 c<0.8 GeV/ T k0.7< 2 C )c (GeV/q FIG. 2. (Color online) C2for same-charge (solid red circles) and mixed-charge pions (open blue squares) for 0%–5% centrality. The global fits with dotted lines correspond to Gaussian same-charge fits (Ew=1). The global fits with solid lines correspond to non-Gaussian fits with Edgeworth coefficients (Ew= 1). Shaded boxes represent the momentum resolution correction uncertainty. FSI uncertainties are smaller than the symbol sizes. kTand centrality bin. The fits indicate the possibility of significant non-Gaussian features in the same-charge correlation functions and/or the possibility of two separate suppression parameters. An individual fit to mixed-charge correlations suggests λ∼0.7. An individual fit to same-charge correlations with a Gaussian function suggests λ∼0.4. Concerning the Edgeworth fits in Figs. 2and 3,the average χ2/NDF is 1.5. Sameand mixed-charge correlations are simultaneously well described with an Edgeworth fit. A common λparameter is now able to describe both sameand mixed-charge correlations. This may demonstrate the significance of non-Gaussian same-charge correlations and/or the presence of a coherent component. Fits including coherence with and without the charge constraint were also attempted. The charge constraint on coherent states in the quantum optics [25] approach leads to a slight modification of both same-charge and mixed-charge correlations [20]. It leads to a slight decrease of the suppression 024911-6 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) 1 1.1 1.2 1.3 1.4 1.5 =2.76 TeV NN sALICE Pb-Pb 45-50% c<0.3 GeV/ T k0.2< 1 1.1 1.2 1.3 1.4 1.5 Gauss Edgeworth c<0.5 GeV/ T k0.4< 0 0.05 1 1.1 1.2 1.3 1.4 1.5 c<0.7 GeV/ T k0.6< same-charge mixed-charge c<0.4 GeV/ T k0.3< c<0.6 GeV/ T k0.5< 0 0.05 c<0.8 GeV/ T k0.7< 2 C )c (GeV/q FIG. 3. (Color online) C2for same-charge (solid red circles) and mixed-charge pions (open blue squares) for 45%–50% centrality. Same details as for Fig. 2. of same-charge correlations (1 5G2) and also an enhancement of mixed-charge correlations (1 5G2)[20]. Coherence may also explain the observation of separate suppression parameters as it only suppresses same-charge correlations. However, given the uncertainty of non-Gaussian same-charge correlations, we find that two-pion correlations alone are inconclusive in determining the presence of coherence. The λand radii fit parameters for both global fit types are shown in Fig. 4. The Edgeworth coefficients from ALICE data are shown in Table I. The corresponding Edgeworth coefficients from THERMINATOR are shown in Table II.The Edgeworth coefficients presented in Tables Iand II quantify the non-Gaussian structure of the same-charge correlation functions. They may also be influenced by a coherent component. The comparison of Table Ito Table II demonstrates a discrepancy in the shape of QS correlations between THERMINATOR and ALICE data. The values for the overall normalization, N, are typically within 0.005 from unity. We observe that λ∼0.7 and is largely kTindependent for the Edgeworth fits. The pion-pair purity and the primary-pair purity in this analysis are estimated to be about 93% and 84%, respectively. The correction for muon λ 0.5 0.6 0.7 0.8 =2.76 TeV NN sALICE Pb-Pb )c (GeV/ T k 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (fm) ch R 6 8 10 12 Gauss, 0-5% Gauss, 45-50% Edgeworth, 0-5% Edgeworth, 45-50% FIG. 4. (Color online) Fit parameters versus kTfor Gaussian and Edgeworth global fits in Figs. 2and 3. (Top) λvalues. (Bottom) Rch values. Shaded bands represent systematic uncertainties. contamination accounts for pion misidentification. We therefore expect λ<0.84. The Gaussian radii are larger than what is typically reported [15] owing to the global fit procedure which incorporates mixed-charge correlations to better constrain the λparameter. The Edgeworth radii for the chaotic component are observed to be larger than the purely Gaussian radii by ∼10%. We note that it has also been shown that the presence of a finite coherent component can influence the width (∝1/Rch) of same-charge correlations [2,3,20]. In particular, for the TABLE I. κ3and κ4Edgeworth coefficients from ALICE data corresponding to global fits in Figs. 2and 3.kT1and kT6represent our lowest and highest kTintervals, respectively. kT1kT2kT3kT4kT5kT6 κ3 0%–5% 0.14 0.13 0.12 0.12 0.1 0.094 45%–50% 0.23 0.22 0.23 0.25 0.25 0.24 κ4 0%–5% 0.29 0.33 0.37 0.38 0.43 0.46 45%–50% 0.19 0.22 0.22 0.24 0.25 0.31 024911-7 B. ABELEV et al. PHYSICAL REVIEW C 89, 024911 (2014) TABLE II. κ3and κ4Edgeworth coefficients from THERMINATOR. kT1and kT6represent our lowest and highest kTintervals, respectively. kT1kT2kT3kT4kT5kT6 κ3 0%–5% 0.18 0.22 0.27 0.31 0.35 0.4 45%–50% 0.25 0.27 0.3 0.34 0.36 0.42 κ4 0%–5% 0.076 0.12 0.17 0.18 0.22 0.23 45%–50% 0.034 0.061 0.081 0.085 0.11 0.084 case when the radius of a coherent component is smaller than the chaotic component, same-charge correlations appear broader than expected by the chaotic component alone. This can incorrectly give the impression of a smaller chaotic source. This may also arise from a momentum dependence of a coherent component (not considered in our fits). For all cases, we observe Rch to decrease with increasing kT. A comparison of the kTevolution of sameand mixedcharge correlations in Figs. 2and 3reveals that same-charge correlations change rapidly with increasing kTwhile mixedcharge correlations change very little. The widening of samecharge correlations with increasing kTis potentially caused by radial flow [34,35]. In an expanding source, pairs with large kTare preferentially formed from particles within the same space-time interval. Thus, larger values of kTmeasure smaller lengths of homogeneity. In QS correlations, this will demonstrate itself as a widening of the correlation function with increasing kT. Similarly, mixed-charge pairs of larger kTmayalsomeasure smaller lengths of homogeneity owing to radial flow. Mixed-charge correlation strengths may therefore increase with increasing kTbecause FSI correlations are larger for smaller sources. In Fig. 5we present mixed-charge correlations in the form of a ratio, C+− 2(kT6)/C+− 2(kT1), where kT6 and kT1represent our highest (sixth) and lowest (first) kT bins, respectively. Comparing the ALICE data to the diluted THERMINATOR calculation in Fig. 5, it is clear that the observed mixed-charge correlations evolve less rapidly in real data as compared to the THERMINATOR expectation. This may be caused by a discrepancy of λor the freeze-out size in THERMINATOR as compared to the data. To distinguish between them, we also compare the ALICE data to the undiluted THERMINATOR calculation in Fig. 5, where only “interacting” pairs with r∗<80 fm are used. Such a procedure can help remove the effect of the λparameter from the comparison. The kTevolution of mixed-charge correlations is better described with the undiluted THERMINATOR expectation, which indicates a discrepancy of the kTevolution of the λparameter in THERMINATOR as compared to the data. B. Three pions We now present the three-pion sameand mixed-charge correlation functions in two KT,3=|pT,1+pT,2+pT,3|/3 bins. Two KT,3intervals were chosen such that they divide the number of triplets into two roughly equal halves. The same-charge three-pion correlations in six centrality bins and )c (GeV/q 0 0.02 0.04 0.06 0.08 0.99 1 1.01 1.02 1.03 1.04 =2.76 TeV NN sALICE Pb-Pb 0-5% , ALICE T,1 k/ T,6 k , Therminator (diluted) T,1 k/ T,6 k , Therminator (undiluted) T,1 k/ T,6 k ) T,1 k( +- 2 C) / T,6 k( +- 2 C FIG. 5. (Color online) Ratio C+− 2(kT6)/C+− 2(kT1), comparing mixed-charge correlations between the highest (sixth) and lowest (first) kTbins. Open circles represent the THERMINATOR comparison using all pion pairs (diluted). Open squares represent the THERMINATOR calculation only using pion pairs with r∗<80 fm (undiluted). Error bars include statistical and systematic uncertainties. two KT,3bins are shown in Figs. 6and 7. Also shown are the cumulant correlation functions, c3, for which the two-pion correlations and FSIs are removed. The dilution of correlations caused by λ<1 is also removed when we consider c3. Extraction of the cumulant correlation function, c3, requires an assumption on the λparameter. We use the λparameter obtained from two-pion global fits excluding coherence and incorporating an Edgeworth expansion to the full kTrange (0 <k T<1.0). From central to peripheral collisions, λranges from 0.65 to 0.70. In Figs. 6and 7we observe that the raw same-charge three-pion correlations are suppressed far below the expected value for fully chaotic emission [C±±± 3(Q3= 0) <6] as was similarly seen for C±± 2. The same-charge cumulant correlation also appears to be suppressed below its maximum [c3(Q3=0) <3] although a reliable extrapolation to Q3=0 is needed to be sure. The mixed-charge three-pion correlations and cumulant correlations in six centrality bins and two KT,3bins are shown in Figs. 8and 9. For mixed-charge correlations, c±±∓ 3 is expected to be equal to unity in the presence of only QS and FSIs. The construction of the cumulant correlation function removes FSI effects and the dilution when λ<1. The mixed-charge cumulant correlation is largely consistent with unity for both KT,3bins although the positive residue for the highest KT,3bin is about 2 times larger than for the lowest bin. This demonstrates the validity of asymptotic three-body FSI wave functions for Pb-Pb collisions at the LHC for Q3>10 MeV/c. We note that it may also be possible for a residue to exist for c±±∓ 3with charge-constrained coherent states [20]. The cumulant correlation functions in Figs. 8 024911-8 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) H. Buesching,35 S. Bufalino,7P. Buncic,5O. Busch,30 Z. Buthelezi,71 D. Caffarri,77 X. Cai,78 H. Caines,15 A. Caliva,60 E. Calvo Villar,79 P. Camerini,80 V. Canoa Roman,5F. Carena,5W. Carena,5F. Carminati,5A. Casanova D´ ıaz,63 J. Castillo Castellanos,46 E. A. R. Casula,81 V. Catanescu,28 C. Cavicchioli,5C. Ceballos Sanchez,82 J. 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S´ anchez Rodr´ ıguez,126 L. ˇ S´ andor,47 A. Sandoval,55 M. Sano,62 G. Santagati,49 D. Sarkar,11 E. Scapparone,20 F. Scarlassara,77 R. P. Scharenberg,74 C. Schiaua,28 R. Schicker,30 C. Schmidt,29 H. R. Schmidt,111 S. Schuchmann,35 J. Schukraft,5M. Schulc,2T. Schuster,15 Y. Schutz,5,34 K. Schwarz,29 K. Schweda,29 G. Scioli,9 E. Scomparin,7P. A. Scott,18 R. Scott,128 G. Segato,77 J. E. Seger,85 I. Selyuzhenkov,29 J. Seo,69 E. Serradilla,55,64 A. Sevcenco,95 A. Shabetai,34 G. Shabratova,51 R. Shahoyan,5A. Shangaraev,67 N. Sharma,48,128 S. Sharma,45 K. Shigaki,139 K. Shtejer,56 Y. Sibiriak,17 S. Siddhanta,89 T. Siemiarczuk,97 D. Silvermyr,39 C. Silvestre,36 G. Simatovic,136 R. Singaraju,11 R. Singh,45 S. Singha,11,132 V. Singhal,11 B. C. Sinha,11 T. Sinha,84 B. Sitar,75 M. Sitta,92 T. B. Skaali,38 K. Skjerdal,25 R. Smakal,2N. Smirnov,15 R. J. M. Snellings,60 C. Søgaard,88 R. Soltz,1J. Song,69 M. Song,114 F. Soramel,77 S. Sorensen,128 M. Spacek,2I. Sputowska,50 M. Spyropoulou-Stassinaki,103 B. K. Srivastava,74 J. Stachel,30 I. Stan,95 G. Stefanek,97 M. Steinpreis,105 E. Stenlund,88 G. Steyn,71 J. H. Stiller,30 D. Stocco,34 M. Stolpovskiy,67 P. Strmen,75 A. A. P. Suaide,27 M. A. Subieta Vasquez,56 T. Sugitate,139 C. Suire,90 M. Suleymanov,13 R. Sultanov,16 M. ˇ Sumbera,3T. Susa,32 T. J. M. Symons,65 A. Szanto de Toledo,27 I. Szarka,75 A. Szczepankiewicz,5M. Szymanski,108 J. Takahashi,87 M. A. Tangaro,24 J. D. Tapia Takaki,90,140 A. Tarantola Peloni,35 A. Tarazona Martinez,5A. Tauro,5G. Tejeda Mu˜ noz,93 A. Telesca,5C. Terrevoli,24 A. Ter Minasyan,17,66 J. Th¨ ader,29 D. Thomas,60 R. Tieulent,86 A. R. Timmins,54 A. Toia,33,35 H. Torii,110 V. Trubnikov,21 W. H. Trzaska,83 T. Tsuji,110 A. Tumkin,76 R. Turrisi,33 T. S. Tveter,38 J. Ulery,35 K. Ullaland,25 J. Ulrich,73 A. Uras,86 G. L. Usai,81 M. Vajzer,3M. Vala,47,51 L. Valencia Palomo,43,90 S. Vallero,30,56 P. Vande Vyvre,5 L. Vannucci,138 J. W. Van Hoorne,5M. van Leeuwen,60 A. Vargas,93 R. Varma,10 M. Vasileiou,103 A. Vasiliev,17 V. Vechernin,26 M. Veldhoen,60 M. Venaruzzo,80 E. Vercellin,56 S. Vergara Lim´ on,93 R. Vernet,141 M. Verweij,70 L. Vickovic,107 G. Viesti,77 J. Viinikainen,83 Z. Vilakazi,71 O. Villalobos Baillie,18 A. Vinogradov,17 L. Vinogradov,26 Y. Vinogradov,76 T. Virgili,98 Y. P. Viyogi,11 A. Vodopyanov,51 M. A. V¨ olkl,30 K. Voloshin,16 S. A. Voloshin,70 G. Volpe,5B. von Haller,5I. Vorobyev,26 D. Vranic,5,29 J. Vrl´ akov´ a,68 B. Vulpescu,43 A. Vyushin,76 B. Wagner,25 J. Wagner,29 V. Wagner,2M. Wang,34,78 Y. Wang,30 D. Watanabe,62 M. Weber,54 J. P. Wessels,31 U. Westerhoff,31 J. Wiechula,111 J. Wikne,38 M. Wilde,31 G. Wilk,97 J. Wilkinson,30 M. C. S. Williams,20 B. Windelband,30 M. Winn,30 C. Xiang,78 C. G. Yaldo,70 Y. Yamaguchi,110 H. Yang,46,60 P. Yang,78 S. Yang,25 S. Yano,139 S. Yasnopolskiy,17 J. Yi,69 Z. Yin,78 I.-K. Yoo,69 I. Yushmanov,17 V. Zaccolo,52 C. Zach,2 A. Zaman,13 C. Zampolli,20 S. Zaporozhets,51 A. Zarochentsev,26 P. Z ´ avada,112 N. Zaviyalov,76 H. Zbroszczyk,108 I. S. Zgura,95 M. Zhalov,57 F. Zhang,78 H. Zhang,78 X. Zhang,43,65,78 Y. Zhang,78 C. Zhao,38 D. Zhou,78 F. Zhou,78 Y. Zhou,60 H. Zhu,78 J. Zhu,78 J. Zhu,78 X. Zhu,78 A. Zichichi,9,19 A. Zimmermann,30 M. B. Zimmermann,5,31 G. Zinovjev,21 Y. Zoccarato,86 M. Zynovyev,21 and M. Zyzak35 (ALICE Collaboration) 1Lawrence Livermore National Laboratory, Livermore, California 94550, USA 2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic 3Nuclear Physics Institute, Academy of Sciences of the Czech Republic, ˇ Reˇ z u Prahy, Czech Republic 4Physics Department, Panjab University, Chandigarh, India 5European Organization for Nuclear Research (CERN), Geneva, Switzerland 6Politecnico di Torino, Turin, Italy 7Sezione INFN, Turin, Italy 8Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary 9Dipartimento di Fisica e Astronomia dell’Universit` a and Sezione INFN, Bologna, Italy 024911-16 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) 10Indian Institute of Technology Bombay (IIT), Mumbai, India 11Variable Energy Cyclotron Centre, Kolkata, India 12Department of Physics, Aligarh Muslim University, Aligarh, India 13COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan 14Korea Institute of Science and Technology Information, Daejeon, South Korea 15Yale University, New Haven, Connecticut 06520, USA 16Institute for Theoretical and Experimental Physics, Moscow, Russia 17Russian Research Centre Kurchatov Institute, Moscow, Russia 18School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom 19Centro Fermi-Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi,” Rome, Italy 20Sezione INFN, Bologna, Italy 21Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine 22Faculty of Engineering, Bergen University College, Bergen, Norway 23Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit¨ at Frankfurt, Frankfurt, Germany 24Dipartimento Interateneo di Fisica “M. Merlin” and Sezione INFN, Bari, Italy 25Department of Physics and Technology, University of Bergen, Bergen, Norway 26V. Fock Institute for Physics, Saint Petersburg State University, St. Petersburg, Russia 27Universidade de S˜ ao Paulo (USP), S˜ ao Paulo, Brazil 28National Institute for Physics and Nuclear Engineering, Bucharest, Romania 29Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨ ur Schwerionenforschung, Darmstadt, Germany 30Physikalisches Institut, Ruprecht-Karls-Universit¨ at Heidelberg, Heidelberg, Germany 31Institut f¨ ur Kernphysik, Westf¨ alische Wilhelms-Universit¨ at M¨ unster, M¨ unster, Germany 32Rudjer Boˇ skovi´ c Institute, Zagreb, Croatia 33Sezione INFN, Padova, Italy 34SUBATECH, Ecole des Mines de Nantes, Universit´ e de Nantes, CNRS-IN2P3, Nantes, France 35Institut f¨ ur Kernphysik, Johann Wolfgang Goethe-Universit¨ at Frankfurt, Frankfurt, Germany 36Laboratoire de Physique Subatomique et de Cosmologie (LPSC), Universit´ e Joseph Fourier, CNRS-IN2P3, Institut Polytechnique de Grenoble, Grenoble, France 37Departamento de F´ ısica de Part´ ıculas and IGFAE, Universidad de Santiago de Compostela, Santiago de Compostela, Spain 38Department of Physics, University of Oslo, Oslo, Norway 39Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA 40Physics Department, University of Cape Town, Cape Town, South Africa 41Sezione INFN, Catania, Italy 42Gangneung-Wonju National University, Gangneung, South Korea 43Laboratoire de Physique Corpusculaire (LPC), Clermont Universit´ e, Universit´ e Blaise Pascal, CNRS-IN2P3, Clermont-Ferrand, France 44Physics Department, University of Rajasthan, Jaipur, India 45Physics Department, University of Jammu, Jammu, India 46Commissariat ` a l’Energie Atomique, IRFU, Saclay, France 47Institute of Experimental Physics, Slovak Academy of Sciences, Koˇ sice, Slovakia 48Institute of Physics, Bhubaneswar, India 49Dipartimento di Fisica e Astronomia dell’Universit` a and Sezione INFN, Catania, Italy 50The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 51Joint Institute for Nuclear Research (JINR), Dubna, Russia 52Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 53Institut Pluridisciplinaire Hubert Curien (IPHC), Universit´ e de Strasbourg, CNRS-IN2P3, Strasbourg, France 54University of Houston, Houston, Texas 77004, USA 55Instituto de F´ ısica, Universidad Nacional Aut´ onoma de M´ exico, Mexico City, Mexico 56Dipartimento di Fisica dell’Universit` a and Sezione INFN, Turin, Italy 57Petersburg Nuclear Physics Institute, Gatchina, Russia 58Saint Petersburg State Polytechnical University, St. Petersburg, Russia 59Technische Universit¨ at M¨ unchen, Munich, Germany 60Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands 61Gauhati University, Department of Physics, Guwahati, India 62University of Tsukuba, Tsukuba, Japan 63Laboratori Nazionali di Frascati, INFN, Frascati, Italy 64Centro de Investigaciones Energ´ eticas Medioambientales y Tecnol´ ogicas (CIEMAT), Madrid, Spain 65Lawrence Berkeley National Laboratory, Berkeley, California 94704, USA 024911-17 B. ABELEV et al. PHYSICAL REVIEW C 89, 024911 (2014) 66Moscow Engineering Physics Institute, Moscow, Russia 67Institute for High Energy Physics, Protvino, Russia 68Faculty of Science, P. J. ˇ Saf´ arik University, Koˇ sice, Slovakia 69Pusan National University, Pusan, South Korea 70Wayne State University, Detroit, Michigan 48202, USA 71iThemba LABS, National Research Foundation, Somerset West, South Africa 72Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands 73Institut f¨ ur Informatik, Johann Wolfgang Goethe-Universit¨ at Frankfurt, Frankfurt, Germany 74Purdue University, West Lafayette, Indiana 47907, USA 75Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia 76Russian Federal Nuclear Center (VNIIEF), Sarov, Russia 77Dipartimento di Fisica e Astronomia dell’Universit` a and Sezione INFN, Padova, Italy 78Central China Normal University, Wuhan, China 79Secci´ on F´ ısica, Departamento de Ciencias, Pontificia Universidad Cat´ olica del Per´ u, Lima, Peru 80Dipartimento di Fisica dell’Universit` a and Sezione INFN, Trieste, Italy 81Dipartimento di Fisica dell’Universit` a and Sezione INFN, Cagliari, Italy 82Centro de Aplicaciones Tecnol´ ogicas y Desarrollo Nuclear (CEADEN), Havana, Cuba 83University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, Finland 84Saha Institute of Nuclear Physics, Kolkata, India 85Physics Department, Creighton University, Omaha, Nebraska 68102, USA 86Universit´ e de Lyon, Universit´ e Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne, France 87Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 88Division of Experimental High Energy Physics, University of Lund, Lund, Sweden 89Sezione INFN, Cagliari, Italy 90Institut de Physique Nucleaire d’Orsay (IPNO), Universite Paris-Sud, CNRS-IN2P3, Orsay, France 91Centro de Investigaci´ on y de Estudios Avanzados (CINVESTAV), Mexico City and M´ erida, Mexico 92Dipartimento di Scienze e Innovazione Tecnologica dell’Universit` a del Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy 93Benem´ erita Universidad Aut´ onoma de Puebla, Puebla, Mexico 94Instituto de Ciencias Nucleares, Universidad Nacional Aut´ onoma de M´ exico, Mexico City, Mexico 95Institute of Space Science (ISS), Bucharest, Romania 96Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India 97National Centre for Nuclear Studies, Warsaw, Poland 98Dipartimento di Fisica “E.R. Caianiello” dell’Universit` a and Gruppo Collegato INFN, Salerno, Italy 99Sezione INFN, Bari, Italy 100Sezione INFN, Rome, Italy 101University of Liverpool, Liverpool, United Kingdom 102Institute for Nuclear Research, Academy of Sciences, Moscow, Russia 103Physics Department, University of Athens, Athens, Greece 104Sezione INFN, Trieste, Italy 105Department of Physics, Ohio State University, Columbus, Ohio 43210, USA 106Chicago State University, Chicago, Illinois 60628, USA 107Technical University of Split FESB, Split, Croatia 108Warsaw University of Technology, Warsaw, Poland 109A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia 110University of Tokyo, Tokyo, Japan 111Eberhard Karls Universit¨ at T¨ ubingen, T¨ ubingen, Germany 112Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic 113Department of Physics, Sejong University, Seoul, South Korea 114Yonsei University, Seoul, South Korea 115KTO Karatay University, Konya, Turkey 116Zentrum f¨ ur Technologietransfer und Telekommunikation (ZTT), Fachhochschule Worms, Worms, Germany 117Excellence Cluster Universe, Technische Universit¨ at M¨ unchen, Munich, Germany 118Department of Applied Physics, Aligarh Muslim University, Aligarh, India 119California Polytechnic State University, San Luis Obispo, California 93407, USA 120The University of Texas at Austin, Physics Department, Austin, Texas 78712, USA 121Suranaree University of Technology, Nakhon Ratchasima, Thailand 122Helsinki Institute of Physics (HIP), Helsinki, Finland 123Inha University, College of Natural Sciences 024911-18 TWOAND THREE-PION QUANTUM STATISTICS . . . PHYSICAL REVIEW C 89, 024911 (2014) 124Vestfold University College, Tonsberg, Norway 125Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom 126Universidad Aut´ onoma de Sinaloa, Culiac´ an, Mexico 127M. V. Lomonosov Moscow State University, D. V. Skobeltsyn Institute of Nuclear Physics, Moscow, Russia 128University of Tennessee, Knoxville, Tennessee 37996, USA 129Indian Institute of Technology Indore, Indore (IITI), India 130Dipartimento di Fisica dell’Universit` a “La Sapienza” and Sezione INFN Rome 131University of Belgrade, Faculty of Physics and “Vinˇ ca” Institute of Nuclear Sciences, Belgrade, Serbia 132National Institute of Science Education and Research, Bhubaneswar, India 133Institut f¨ ur Kernphysik, Technische Universit¨ at Darmstadt, Darmstadt, Germany 134Konkuk University, Seoul, South Korea 135Budker Institute for Nuclear Physics, Novosibirsk, Russia 136University of Zagreb, Zagreb, Croatia 137Institute of Theoretical Physics, University of Wroclaw, Wroclaw, Poland 138Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy 139Hiroshima University, Hiroshima, Japan 140The University of Kansas, Lawrence, Kansas 66045, USA 141Centre de Calcul de l’IN2P3, Villeurbanne, France *Deceased. 024911-19