General balance functions of identified charged hadron pairs of $(\pi,{\rm K},{\rm p})$ in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV

First measurements of balance functions (BFs) of all combinations of identified charged hadron $(\pi,\rm K,\rm p)$ pairs in Pb$-$Pb collisions at $\sqrt{s_{\rm NN}} = 2.76$ TeV recorded by the ALICE detector are presented. The BF measurements are carried out as two-dimensional differential correlators versus the relative rapidity ($\Delta y$) and azimuthal angle ($\Delta\varphi$) of hadron pairs, and studied as a function of collision centrality. The $\Delta\varphi$ dependence of BFs is expected to be sensitive to the light quark diffusivity in the quark$-$gluon plasma. While the BF azimuthal widths of all pairs substantially decrease from peripheral to central collisions, the longitudinal widths exhibit mixed behaviors: BFs of $\pi\pi$ and cross-species pairs narrow significantly in more central collisions, whereas those of $\rm KK$ and $\rm pp$ are found to be independent of collision centrality. This dichotomy is qualitatively consistent with the presence of strong radial flow effects and the existence of two stages of quark production in relativistic heavy-ion collisions. Finally, the first measurements of the collision centrality evolution of BF integrals are presented, with the observation that charge balancing fractions are nearly independent of collision centrality in Pb$-$Pb collisions. Overall, the results presented provide new and challenging constraints for theoretical models of hadron production and transport in relativistic heavy-ion collisions.

Convincing evidence for the production of strongly interacting quark-gluon plasma (QGP) in heavyion (AA) collisions has been reported from a variety of measurements at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) [1][2][3][4], including observations of strong elliptic flow [5][6][7], suppression of high transverse momentum (p T ) hadron production [8][9][10][11][12][13], suppression of quarkonium states [14][15][16][17][18][19], as well as dihadron correlation functions [20,21]. Many of these findings are quantitatively explained by hydrodynamic calculations in which the QGP matter undergoes radial and azimuthally anisotropic collective motion. The existence of the latter is well established based on measurements of flow coefficients with finite pseudorapidity (η) gap and multi-particle cumulants, whereas the presence of the former is inferred in part from the increase of average transverse momenta with the mass of hadrons [22], the centrality dependence of event-by-event p T fluctuations [23,24], as well as the observed narrowing of the near-side peak of balance functions (BFs) in central collisions relative to that observed in peripheral collisions [25][26][27][28][29][30]. Balance functions essentially amount to differences of correlation functions of like-sign and unlike-sign charges. They are measured, typically, as functions of particle pair separation in azimuth angle and rapidity. They indicate the degree to which the production of a positive charge is accompanied by the production of a negative charge somewhere in phase space. As such, BFs probe the balancing of charge distributions in momentum space and theoretical studies show they are sensitive to the details of the time (i.e., whether particles are produced early or late), production mechanisms, and transport of balancing charges.
Measurements of BFs were originally proposed as a tool to investigate the delayed hadronization and two stages of quark production in the QGP formed in AA collisions [31]. These terms refer to the notion that quark production occurs in two distinct stages, the first at the onset, and the second at the very end (just before hadronization and freeze-out) of AA collisions. The two stages are posited to be separated by a period of isentropic expansion whose duration depends on the multiplicity of produced quarks and gluons and thus the collision impact parameter. Hadron pairs produced at the onset of collisions feature large longitudinal separation (i.e., rapidity differences ∆y) whereas pairs produced after the expansion have smaller ∆y separations determined by the smaller temperature of the system at that time. AA collisions with smaller impact parameters are expected to produce larger systems with a longer isentropic stage in which late particle production dominates. The longitudinal and azimuthal widths of BFs are thus expected to progressively decrease from peripheral to central collisions as the fraction of late particle production increases. BFs could also provide a precise probe of balancing particle production [32-35], the hadrochemistry of particle production [34,36], as well as the collision dynamics [37,38]. Recent studies also indicate that the BF dependence on pair separation in azimuth is sensitive to the diffusivity of light quarks, a measure of the diffusion and scattering of quarks within the QGP, which has thus far received only limited attention [36,39]. Finally, BFs also provide a tool to calibrate measurements of the Chiral Magnetic Effect [40,41] and net charge/baryon fluctuations deemed essential for the determination of QGP susceptibilities [42,43].
Few measurements of BFs of identified hadrons have been reported to date. At RHIC, these include BF measurements of charged hadrons, pion pairs, kaon pairs, as well as proton/antiproton pairs [25][26][27], whereas at the LHC, only charged hadron BFs have been reported [28,29]. Of these, only the results published by ALICE were fully corrected for detector acceptance and particle losses (efficiency). Integrals of measured BFs have not been considered and no cross-species BFs have been published. Theoretical analyses of measured BFs have consequently focused mainly on the interpretation of the narrowing with collision centrality of charged hadron BFs. The full potential of BFs as a probe of the evolution dynamics and chemistry of the QGP has thus so far been underexploited. In this paper, general balance functions of identified charged hadron species (π, K, p) are reported for the first time. These general BFs are corrected for efficiency and non uniform acceptance effects and it becomes possible to study the effects of two-stage quark production, light quark diffusivity, and relative balancing fractions using BFs of nine distinct identified pairs of charged hadron species.
The BF of a species of interest, α, and an associated species, β , was originally defined in terms of conditional densities [31] but it is convenient to compute BFs in terms of normalized cumulants R 2 according to with where ρ α 1 ( p α ) ≡ dN/d p α and ρ αβ 2 ( p α , p β ) ≡ dN pair /d p α d p β are single-and particle-pair densities of species α and β measured at momenta p α and p β , respectively, while labels + and − stand for positive and negative charges. Normalized cumulants R 2 are robust observables, i.e., independent to first order of measurement efficiencies. They are sensitive to the strength of correlation between species α and β . Their properties were described in several publications [44][45][46][47]. The combination of R 2 correlation functions, normalized by single particle densities, as per Eq. (1), is strictly equivalent to the balance function introduced in Ref. [31,32] and measures the correlation between positive and negative particles of species α and β constrained by charge conservation. Integrals of inclusive charge balance functions, I +− B (Ω) ≡ Ω B +− d∆η, are expected to lie within the range 0 < I +− B (Ω) ≤ 1 for limited acceptances Ω. However, they converge to unity for full acceptance coverage. Furthermore, fractions I αβ B (Ω)/I α B (Ω) are determined by the hadrochemistry of the QGP and transport properties of the medium. In the full acceptance coverage limit, the denominator of this fraction must satisfy (Ω) → 1 [43]. In this paper, the identified particle BFs of nine pairs of charged hadrons (π ± , K ± and p/p) ⊗ (π ± , K ± , and p/p) are reported as joint functions of the relative rapidity (∆y) and azimuthal angle (∆ϕ) and studied as a function of collision centrality. Measurements of R αβ 2 ( p α , p β ) are carried out in terms of the rapidity and azimuthal angle y α , ϕ α , y β , and ϕ β for fixed p T ranges, and averaged across the pair acceptance to yield correlation functions R αβ 2 (∆y, ∆ϕ) with ∆y = y α − y β and ∆ϕ = ϕ α − ϕ β following the procedure used in Ref. [44]. The densities of associated particles, ρ β 1 , used in Eq. (1), are integrated from p T -dependent densities reported in prior ALICE measurements [22] to match the p T ranges used in measurements of the normalized cumulants R 2 . The correlators R αβ 2 and densities ρ β 1 are corrected for p T -dependent particle losses and non uniform acceptance. Densities ρ β 1 were additionally corrected for minor contamination effects as per the procedure described in [22]. The measured BFs thus feature absolute normalization which enables meaningful determination of their integrals and collision centrality dependence.
As already mentioned, the shape of the BFs vs. ∆y and ∆ϕ is sensitive to the timescales at which particles are produced during the system evolution. Early emission occurs at large effective collisional energy √ s and is thus expected to yield broad BFs in ∆y and ∆ϕ, whereas late emission, at collisional energy commensurate with the system temperature, is expected to produce much narrower near side peak correlations vs. ∆y and ∆ϕ [31]. Additionally, the integral of the BFs shall also provide increased sensitivity to the hadrochemistry of the collisions. Indeed whereas contributions to single-particle spectra from hadronic resonance decays must be inferred from models, integrals of the BFs are directly sensitive to the magnitude of (hadronic) feeddown contributions. For instance, by comparing the integrals of π + π − and π ± K ∓ BFs, sensitivity to the relative strengths of processes that lead to such correlated pairs of particles is acquired. It becomes possible to better probe the role of hadronic resonance decay contributions and increased sensitivity to the hadrochemistry of the QGP and its susceptibilities is gained [36].
The BFs presented are based on 1 × 10 7 minimum bias (MB) Pb-Pb collisions at √ s NN = 2.76 TeV collected in 2010 by the ALICE collaboration. Descriptions of the ALICE detector and its performance have been reported elsewhere [48,49]. The minimum bias trigger required a combination of hits in the V0 detectors and layers of the SPD detector. The V0 detectors, which cover the full azimuth and the pseudorapidity ranges −3.7 < η < −1.7 and 2.8 < η < 5.1, also provided a measurement of the charged particle multiplicity used to classify collisions into centrality classes corresponding to 0-5% (most central) to 80-90% (most peripheral) of the Pb-Pb hadronic cross section [50]. Some centrality classes have been combined to optimize the statistical accuracy of the BFs reported. Particle momenta were determined based on Kalman fits of charged particle tracks reconstructed in the Time Projection Chamber (TPC). The particle identification (PID) of charged hadrons was performed based on specific energy loss (dE/dx) measured in the TPC and particle velocities measured in the Time-of-Flight detector (TOF). Track quality criteria based on the number of space points, the distance of closest approach to the collision primary vertex, and the χ 2 of the Kalman fits were used to restrict the measurements to primary particles produced by the Pb-Pb collisions and suppress contamination from tracks resulting from weak decays and interactions of particles with the apparatus. Additionally, PID selection criteria based on deviations of dE/dx and TOF from their respective expectation values, at a given momentum, and for each species of interest, were used to optimize the species purity. These and other selection criteria are reported in detail below in the context of a discussion of systematic uncertainties. The analysis focused on the low p T range, commonly known as the "bulk" physics regime. Slightly different p T ranges were used for each species to optimize yields and species purity. Charged pions and kaons were selected in the range 0.2 ≤ p T ≤ 2.0 GeV/c, whereas (anti-)protons are within 0.5 ≤ p T ≤ 2.5 GeV/c. The selected rapidity range, largely determined by the TOF coverage, was set to |y π | ≤ 0.8 and |y p | ≤ 0.6 for measurements of B ππ and B pp , respectively, and set to |y| ≤ 0.7 for all other BFs reported.
Track reconstruction efficiencies and PID purity were studied with Monte Carlo simulations of Pb-Pb collisions produced with the HIJING generator [51] and propagated through a model of the ALICE detector with GEANT3 [52]. Selected track quality and PID criteria yield purities of 97%, 95%, and 94% for π ± , K ± , and p/p, respectively, thereby minimizing species contamination and its impact on correlation functions. Corrections for track losses were carried out using a weighting technique [46]. Weights are calculated independently for positive and negative tracks of each species considered, for each centrality range, both magnetic field polarities used in the measurements, versus y, ϕ, p T , as well as the longitudinal position of the primary vertex (PV) of each event, z vtx . Various selection criteria were applied to minimize residual instrumental effects while optimizing particle yields. The PV is required to be in the range |z vtx | ≤ 6 cm of the nominal interaction point. Tracks are required to have a minimum of 70 reconstructed TPC space points (hits), out of a maximum of 159, and a track fit with χ 2 value per degree of freedom smaller than 2.0 to ensure good track quality. Contamination of BFs by secondary particles (i.e., weak decays or particles scattered within the detector) is suppressed by requiring distances of closest approach (DCA) to PV chosen as DCA z ≤ 2.0 cm in the longitudinal direction and DCA xy ≤ 0.04, 0.04, 2.0 cm in the transverse plane for π ± , p/p, and K ± , respectively. Contamination by e + e − pairs from photon conversion is suppressed by removing tracks closer than 1σ dE/dx to the TPC Bethe-Bloch median, at a given momentum, for electrons.
Systematic uncertainties on the amplitudes of B α,β and their integrals were calculated as quadratic sums of systematic uncertainties of the correlation function R α,β 2 and the systematic uncertainties on the published single particle densities [22] used in the computation of the BFs. Uncertainties on R α,β 2 were assessed based on variations of conditions and selection parameters employed in the analysis. A statistical test [53] was used to identify potential biases introduced by those variations and determine their statistical significance. Systematic uncertainties, corresponding to a relative deviation at the maximum of B α,β associated with operation with two solenoidal magnetic field polarities, are smaller than 4%. Potential biases associated with track selection criteria are up to 3%, whereas the presence of misidentified and secondary particles contribute up to 4%, while kinematic dependencies of the detection efficiency are estimated to be 1%. Systematic uncertainties on the single particle densities [22] are species and collision centrality dependent and typically range from 5 to 10%.
In order to obtain BF for all nine combinations of π ± , K ± , and p/p species pairs, R αβ 2 (∆y, ∆ϕ) correlators were first measured, in each centrality class, for all 36 αβ permutations of positive and negative π, K, and p. These correlators were then combined according to Eq. (1) and multiplied by the single particle densities ρ β 1 in the |y| ≤ 0.5 rapidity range [22]. Figure 1 shows the B αβ (∆y, ∆ϕ) of ππ, KK, and pp pairs in semicentral collisions for illustrative purposes. The nine measured BFs exhibit common features, including prominent near-side peaks centered at (∆y, ∆ϕ) = (0, 0) and relatively flat and featureless awaysides. The flat away-side arises from the fact that positive and negative particles of a given species feature essentially equal azimuthal anisotropy relative to the collision symmetry plane. It is also an indicator of the fast radial flow profile of the emitting sources, which manifests as strong focusing on the near-side peak [37], although the various species pairs demonstrate different centrality-dependent near-side peak shapes, widths, and magnitudes that indicate that they are subject to different charge balancing pair production and transport mechanisms, as well as final state effects. For instance, B ππ exhibits a deep and narrow dip at (∆y, ∆ϕ) = (0, 0), within the near-side correlation peak, resulting in part from the Hanbury Brown-Twiss (HBT) effect, with a depth and width that vary with the source size and thus the centrality [32]. B KK exhibits much weaker HBT effects, whereas B pp also features a narrow dip centered at (∆y, ∆ϕ) = (0, 0) within a somewhat elongated near-side peak that may reflect the annihilation of pp pairs. Pairs of protons and antiprotons emitted at small relative ∆η and ∆ϕ (as well as small relative p T ) are more likely to interact, and thus annihilate, than pairs produced at large separation, thereby leading to a depletion of pairs near ∆y = 0 and ∆ϕ = 0.
The evolution with collision centrality of B αβ , for all nine combinations α, β = π, K, p, is examined by considering their projections onto the ∆y and ∆ϕ axes in Figs. 2 and 3, respectively. The shape and amplitude of B ππ projections onto ∆y exhibit the strongest centrality dependence, whereas those of B πK , B πp , B Kπ and B pπ display significantly smaller dependence on centrality. Uncertainties on the rest of the ∆y projections do not make it possible to claim any centrality dependence albeit some hints are visible in the cases of B KK and B pp . The evolution with collision centrality of the measured BFs is further characterized in terms of their longitudinal and azimuthal standard deviation (σ ) widths, noted σ ∆y and σ ∆ϕ , respectively, as well as their integral, I αβ B , as shown in Fig. 4. In the longitudinal direction, the widths σ ∆y of all species pairs, except those of KK and pp pairs, exhibit a significant narrowing from peripheral to central collisions. In contrast, B KK is essentially independent in both shape and width σ ∆y with changing collision centrality, whereas the width σ ∆y of B pp features little centrality dependence even though this balance function exhibits some shape dependence on centrality.
Differences in the evolution of the longitudinal σ of pions and kaons BFs were already observed in Au-Au collisions at RHIC [26] and were then interpreted as resulting in part from strong radial flow profiles Figure 2: Balance function of species pairs (π, K, p) ⊗ (π, K, p) projected onto the ∆y axis for particle pairs within the full range |∆ϕ| ≤ π. Vertical bars and open boxes represent statistical and systematic uncertainties, respectively. and two-stage emission [31,32]. The independence of the width σ ∆y of the B KK relative to the narrowing BFs of all other pairs observed in this work suggests two-stage quark production might also be at play at the TeV collision scale. Indeed, pions might be predominantly formed from the light u,ū, d, andd quarks most abundantly produced in the second quark production stage, whereas kaon production would largely result from ss pairs predominantly created during the early stages of collisions [31, 32]. Several distinct models have had success in describing the yield of produced hadrons, and more specifically baryons. Such models invoke a range of production mechanisms including parton fragmentation, effective mostly at high-p T , as well as parton coalescence and recombination, playing a predominant role at low and intermediate p T [54][55][56]. Statistical thermal models and production models involving color transparency [57] and baryon junctions [58] have also had a good measure of success. Single particle spectra of baryons thus do not provide sufficiently discriminating constraints to fully identify baryon production mechanisms. The added information provided by cross-species BFs shall thus contribute by adding new constraints for models of particle production and transport. In particular, given that neutrons, protons, and their excited states are composed of light u and d quarks, believed to be copiously produced in late stage emission (within the context of the two-stage quark production model), it is conceivable that these baryons are predominantly produced by coalescence (recombination) of light quarks in the late stage of the collisions. However, baryons (B) and antibaryons (B) have a relatively large mass and carry a conserved baryonic charge. The question then arises as to whether BB correlated pairs might originate before the formation of thermalized QGP, during the early stages of AA collisions. Late BB production is expected to be characterized by narrow longitudinal BFs while early stage emissions would produce pairs with a much wider ∆y range [31, 32]. It is clear from Fig. 2    : Longitudinal (∆y) σ widths (left), azimuthal (∆ϕ) σ widths (center), and integrals (right) of balance functions of the full species matrix of π ± , K ± , and p/p with centrality. For ∆y and ∆ϕ widths, Kπ, pπ, and pK have the same values with πK, πp, and Kp, respectively. For the longitudinal widths, the relative azimuthal angle range for all the species pairs is the full azimuth range |∆ϕ| ≤ π. For the azimuthal widths, the relative rapidity range used for all species pairs is |∆y| ≤ 1.2, with the exception of |∆y| ≤ 1.4 for ππ and |∆y| ≤ 1.0 for pp. Vertical bars represent statistical uncertainties while systematic uncertainties are displayed as dash line bands.
balance functions that might result from early BB pair separation. Detailed models of BB production and transport that account for (strong) decays from resonant states are required, evidently, to firmly establish this conclusion. Figure 4 shows that the σ ∆ϕ widths of the nine BFs exhibit narrowing trends from peripheral to central collisions. The widths σ ∆ϕ feature a wide spread of values at a given collision centrality, with those of KK pairs being the largest and those of πK the smallest. The widths also exhibit similar reductions with increasing collision centrality. These observations are in agreement with azimuthal BFs already reported from observations at RHIC for unidentified charged particle and identified ππ, KK pairs [25,26], as well as unidentified charged particle BFs in collisions at the LHC [28,29]. This narrowing can be qualitatively understood as resulting from the larger estimated transverse expansion velocity present in more central AA collisions [59]. It competes with an opposing trend associated with light quark diffusivity, expected to broaden and smear out the long range tails of the ∆ϕ BFs for systems featuring increasingly large lifespans [39]. Given the radial boost profile and contributions from resonance decays can be largely calibrated based on the shape of single particle p T spectra, the BF projections presented in Fig. 2, 3 and the evolution of their widths σ ∆y and σ ∆ϕ , shown in Fig. 4, then provide the first comprehensive set of azimuthal BFs to estimate the diffusivity of light quarks at the LHC [36,39]. The above discussion neglects possible contributions from the fragmentation of jets but these are anticipated to be small in the p T range of this measurement. Quantitative estimates of such contributions would need to be accounted for in theoretical modeling of balance functions reported in this work for the purpose of determining the diffusivity of light quarks.
Contributions of φ → K + + K − decays to B KK were studied using simulated events from the HIJING generator [51]. The amplitude of the near-side peak of B KK is reduced by about 30% when contributions from φ -meson decays are explicitly excluded, while the correlator ∆y and ∆ϕ widths increase by about 7-8%. Effects associated with radial flow, not present in HIJING, could reduce this broadening effect and possibly induce a narrowing of the ∆y width of B KK in more central collisions. However, no such narrowing is observed thereby signaling a more intricate production and transport evolution with competing contributions from φ produced at hadronization of the QGP and by coalescence of kaons within a hadron phase.
The evolution with the collision centrality of the integrals I αβ B of the nine species-pairs B αβ (∆y, ∆ϕ) shown in the right panel of Fig. 4 is also of considerable interest. By definition, a balance function B αβ (∆y, ∆ϕ, ∆p T ) measures the "likelihood" of finding a charge balancing particle of a type β , e.g., π + , with a pair separation ∆y, ∆ϕ, ∆p T away from a reference particle of type α, e.g., π − . But charge balancing can be accomplished, on average, by distinct species, e.g., p, K + , and more rarely produced heavier particles, in additions to π + . The integral, I αβ B (4π), of B αβ (∆y, ∆ϕ, ∆p T ) over the full phase space is thus proportional to the average fraction (and probability in the full phase space limit) of balancing partners of species β . Indeed, neglecting contributions from species other than pions, kaons, and protons, one expects the sum, I α B (4π) ≡ I απ B (4π) + I αK B (4π) + I αp B (4π) to converge to unity, I α B (4π) ≈ 1, in the full acceptance limit [43]. Integrals I (Ω < 4π) cannot, strictly speaking, be considered charge balancing probabilities. They nonetheless provide useful indicators of the hadrochemistry as well as the flavor and baryon number transport in AA collisions. As such, integrals I αβ B shown in Fig. 4 as a function of collision centrality are surprising on two accounts. First, they show that the balance fractions are all, but one, approximately independent of collision centrality. The notable exception is the ππ integral which increases by about 20% from peripheral to central collisions. Second, close examination of these pairing fractions shows they are rather different than inclusive probabilities of observing π, K, and p/p in Pb-Pb collisions. For instance, I Kπ B is not larger than I KK B by the π/K ∼ 6.7 ratio of inclusive single particle yields and I pp B is larger than I pK B also in contrast to observed K/p ∼ 3 yield ratios [22]. Hadron species charge balancing pairing fractions are thus indeed very different than the relative probabilities of single hadrons, and as such, provide new and useful information to further probe the hadronization of the QGP. This difference arises because the set of processes P 2 that lead to a specific balancing pair αβ (e.g., P 2 :→ α ± + β ∓ + X) is, by construction, far smaller than the set of processes P 1 leading to a given particle species α or β (e.g., P 1 :→ α ± + X or P 1 :→ β ∓ +Y ). It is remarkable, nonetheless, that the pairing fractions I αβ B exhibit essentially no collision centrality dependence while single particle yield ratios are known to exhibit a weak dependence on collision centrality [9,60]. Note that the observed rise of I ππ B in more central collisions may artificially result from increased kinematic focusing of pions with centrality in the p T and ∆y acceptance of this measurement. The higher velocity flow fields encountered in more central Pb-Pb collisions could indeed shift and focus the yield of associated pions. Why such a shift is not as important for other charge balancing pairs remains to be elucidated with a comprehensive model accounting for the flow velocity profile and appropriate sets of charge conserving processes yielding balancing charges in the final state of collisions. Recent deployments of hydrodynamic models feature the former but lack the latter [61][62][63]. Further theoretical work is thus required to interpret the observed collision centrality dependence of the pairing probabilities displayed in Fig. 4. As such calculations become available, the data reported in this work, and specifically the integral I αβ B shown in Fig. 4, shall provide increased sensitivity to the hadrochemistry of the QGP and its susceptibilities.
In summary, this paper presents the first measurements of the collision centrality evolution of same and cross-species balance functions of identified π ± , K ± and p/p at the LHC. Measured as functions of particle pair separation in rapidity (∆y) and azimuth (∆ϕ), the BFs exhibit prominent near-side peaks centered at (∆y, ∆ϕ) = (0, 0) which feature different shapes, amplitudes, and widths, and varied dependencies on collision centrality. The BFs of species-pairs measured in this work feature narrowing ∆ϕ widths in more central collisions, owing to the strong radial flow field present in central Pb-Pb collisions. Theoretical studies beyond the scope of this work shall use this data to put upper limits on the diffusivity coefficients of light quarks. In the longitudinal direction, the σ widths of BFs of all species pairs decrease with centrality except for those of KK and pp pairs. The shape and width of KK BFs are independent of collision centrality, while the pp BFs peak shapes depend only minimally on centrality. The observed centrality independence of the KK and narrowing σ of other species in the longitudinal direction are qualitatively consistent with effects associated with radial flow and the two-stage quark production scenario, which posits that quark production occurs predominantly in early and late stages separated by a period of isentropic expansion. Integrals I             [35] A. Bialas, "Balance functions in coalescence model", Phys. Lett. B 579 (2004) 31-38, arXiv:hep-ph/0308245.