Global baryon number conservation encoded in net-proton fluctuations measured in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV

Experimental results are presented on event-by-event net-proton fluctuation measurements in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV, recorded by the ALICE detector at the CERN LHC. These measurements have as their ultimate goal an experimental test of Lattice QCD (LQCD) predictions on second and higher order cumulants of net-baryon distributions to search for critical behavior near the QCD phase boundary. Before confronting them with LQCD predictions, account has to be taken of correlations stemming from baryon number conservation as well as fluctuations of participating nucleons. Both effects influence the experimental measurements and are usually not considered in theoretical calculations. For the first time, it is shown that event-by-event baryon number conservation leads to subtle long-range correlations arising from very early interactions in the collisions.

Recently the effects of limited acceptance were studied [15]. There, it was investigated under which conditions net-baryon fluctuations depend on the size of the acceptance. An obvious case is fluctuations caused by correlations due to baryon number conservation. To identify these and other long-range correlations it is interesting to perform the experimental analysis as a function of the acceptance size.
The analysis is set up by providing the necessary definitions. The first and second cumulants of the net-baryon probability distribution P(∆n B ), with ∆n B = n B − n B , are defined as The second cumulant can be represented as a sum of the corresponding cumulants for single baryons and antibaryons plus the correlation term for the joint probability distributions of baryons and antibaryons P(n B , n B ) κ 2 (∆n B ) = κ 2 (n B ) + κ 2 (n B )− where the mixed moment n B n B is defined as n B n B P(n B , n B ).
Equation 3 shows that, for vanishing correlations between baryons and antibaryons ( n B n B = n B n B ), the second cumulant of net-baryons is exactly equal to the sum of the corresponding second cumulants for baryons and antibaryons.
The data presented below were obtained by analyzing about 13 × 10 6 minimum-bias (cf. [16] for definition) Pb-Pb events at a center-of-mass energy per nucleon-nucleon pair of √ s NN = 2.76 TeV recorded by the ALICE detector [17] in the year 2010. Two forward scintillator arrays (V0) are located on either side of the interaction point and cover the pseudorapidity intervals 2.8 < η < 5.1 and −3.7 < η < −1.7 [18]. A minimum-bias trigger condition is defined by requiring a combination of hits in the two innermost layers of the ITS and coincidence in both V0 detectors. The [20]. In order to keep the tracking efficiency as high as possible, only the TPC detector was used for particle identification, while precise vertex determination was performed with the ITS detector. The track selection criteria used are described in Section 3.1 of [21]. Charged particle tracks with at least 80 out of maximum of 159 specific energy loss (dE/dx) samples in the TPC were used in this analysis to ensure best particle identification. Moreover, in order to suppress contributions of secondary particles from weak decays, the distance-of-closest-approach (DCA) of the extrapolated track to the primary collision vertex was taken to be less than 2 cm along the beam direction. In the transverse plane, a more restrictive and transverse momentum (p T ) dependent DCA cut of less than (0.018 cm + 0.035p −1.01 T ) with p T in GeV/c, was imposed [22].
Since the energy loss of charged particles in the gas volume of the TPC depends explicitly on the particle momentum (p), the analysis was performed in momentum space. Correspondingly, the particle identification (PID) procedure consists of correlating particle momentum with the specific energy loss dE/dx. This allows the event-by-event fluctuation analysis to be performed with high overall reconstruction efficiency of about 80% for protons, almost independent of the collision centrality. The latter is important because small efficiencies induce Poisson fluctuations. To ensure the best possible dE/dx resolution, the phase space coverage was restricted to 0.6 < p < 1.5 GeV/c and |η| < 0.8 for the present analysis. The corresponding p T range at |η| = 0.8 is about 0.45 < p T < 1.12 GeV/c, which gradually approaches the used momentum range towards midrapidity. Moreover, a differential analysis is provided as function of ∆η in the range ∆η = 0.2 to 1.6.
The cumulants of net-protons were then reconstructed using the Identity Method (IM) [21,[23][24][25][26][27]. This method is designed to deal efficiently with the overlapping dE/dx distributions of protons, kaons, pions and electrons considered in the present analysis. Their specific probability distributions were obtained by unfolding the moments of the measured multiplicity distributions for each particle species. The IM counts proxies of particle multiplicities W j event-by-event where j stands for a particle type, x i denotes the measured values of dE/dx for a given track i and ρ j (x) is the inclusive dE/dx distribution of particle type j within a specified phase space bin. The summation in Eq. 5 runs over all selected n tracks in the given event. The pure and mixed moments of the W j distributions were calculated by averaging over all events, leading to the moments of the true multiplicity distributions.
The IM is based on input of moments of W j distributions and inclusive dE/dx fits in bins of momentum and pseudorapidity. The dE/dx distributions were fit with generalized Gaussian functions, taking into account non-Gaussian components of the experimental dE/dx distributions. The fits of inclusive distributions of dE/dx were performed separately for positively and negatively charged particles in 20 MeV/c momentum and 0.1 units of η bins.
In the upper panel of Fig. 1 the centrality dependence of the efficiency-corrected second cumulants of netprotons is compared with the sum of the mean multiplicities (first cumulants) of protons and antiprotons. Also included are the first and second cumulants of protons and antiprotons. The efficiency correction for the second moments was performed by transporting the particle tracks, simulated with the HIJING [28] and AMPT [29] event generators, through the ALICE apparatus. The procedure was cross-checked by assuming binomial response functions as outlined in [30,31]. The accuracy of the correction procedure was estimated to be on the percent level and is included in the systematic uncertainties. We note that possible corrections for volume fluctuations such as discussed in [32,33] were not applied to the data since, at LHC energies, second cumulants of net protons, our main observable, are protected from such effects [34]. The subsample approach was chosen to estimate the statistical uncertainties of the reconstructed cumulants [21,35]. In order to evaluate systematic uncertainties stemming from track selection criteria, the applied selection ranges were varied around their nominal values. Other sources of systematic uncer-tainties originate from the parameters of the ρ j (x i ) functions, entering Eq. 5. The latter was estimated by hypothesis testing using the Kolmogorov-Smirnov (K-S) statistics. For this purpose the parameters of the ρ j (x i ) functions were varied by testing the null hypothesis that measured dE/dx distributions and fit functions are similar within a given significance level of 10% (cf. [21,35] for details). The final systematic uncertainties on cumulants were computed by adding in quadrature the maximum systematic variations from individual contributions.
By their definition, cumulants are extensive quantities, i.e., are proportional to the system volume. This also explains the centrality dependence of all cumulants, presented in the upper panel of Fig. 1. To remove the system size dependence, normalized cumulants R 1 and R 2 are introduced as In the middle and bottom panels of Fig. 1, deviations from unity are visible for both R 1 and R 2 . Moreover, the amount of deviation for R 2 is about twice as large compared to that of R 1 .
In order to shed light on these observations, the results are compared with predictions from a model constructed recently [34], in which participant fluctuations are included following the analysis of the ALICE centrality selection [18]. Within uncertainties, the model predictions are fully consistent with the measured R 2 values, lending support to the interpretation that volume fluctuations are at the origin of the observed deviation. This is also supported by the observation of a small structure observed in the 10-20% centrality class, where, compared to the first two centrality classes, the centrality bin width is doubled.
On the other hand, by construction, for vanishing net-proton numbers, R 1 should not contain any contributions from volume fluctuations, i.e., the values of R 1 obtained from the model should be consistent with unity [34]. The origin for the deviation of the measured R 1 values from unity must therefore be beyond the volume fluctuations scenario. To further understand these differences, the acceptance dependence is studied.
The analysis is performed in eight different pseudorapidity intervals from |η| < 0.1 up to |η| < 0.8 in steps of 0.1. The obtained normalized second cumulants R 1 of net-protons are presented in Fig. 2. Again the data are below unity, with the deviation linearly increasing with increasing acceptance.
Such a behavior was predicted based on the assumption of global baryon number conservation, which induces correlations between protons and antiprotons leading to the following dependence on the acceptance factor α [34, 37] where α = n p / N 4π B with n p and N 4π B referring to the mean number of protons inside the acceptance and the mean number of baryons in full phase space. It should be further noted that, for non-central collisions, baryon transport to midrapidity has to be taken into account, which is rather model dependent. In order to avoid the model dependence, the comparison is performed only for the central events and in the estimate of the alpha parameter only produced baryons are used. In doing so, the number of baryons are used in the pseudorapidity range of |η| < 0.5 as reported in [16, 38-40]. Next, using HIJING and AMPT simulations, estimates were obtained for the total average number of baryons in full phase space. The average number of protons n p entering into the definition of α (cf. Eq. 7) was taken from the current analysis for each pseudorapidity interval. Finally, using these values of α, the pink band in Fig. 2 is calculated with Eq. 7. The finite width of the band reflects the difference between predictions of the two event generators.  Inspection of Fig. 2 shows that, for small pseudorapidity ranges of |η| < 0.4 corresponding to ∆η < 0.8, the experimentally measured net-proton distributions closely follow a Skellam distribution. This agreement is expected because of the small acceptance window as discussed above. For ∆η > 0.8, deviations from the Skellam distribution are observed. The amount of deviation is small but significant and in good agreement with the prediction assuming global baryon number conservation. The observed deviation is therefore consistent with the assumption of global baryon number conservation, i.e. conservation within the full phase space.
On the other hand, local baryon number conservation may induce additional correlations between protons and antiprotons, which would lead to a further reduction of the measured κ 2 (n p − n p ) [36]. In Fig. 2 the data are compared to the predictions from an analysis of effects of local baryon number conservation for different values of correlation width ∆y corr between protons and antiprotons. Within the experimental uncertainties the data are best described with the assumption of global baryon number conservation, which corresponds to the correlation width ∆y corr > 2|y beam | but the data are also consistent with a large correlation width of ∆y corr = 5 [36]. The results from the HIJING event generator (cf. blue solid line in Fig. 2), which can be described with ∆y corr = 2, and from a recent study reported in [41] would imply much stronger correlations between protons and antiprotons, not consistent with the experimental data. We note here that correlations arising from baryon or charge conservation have also been analyzed in the framework of balance functions [42,43]. Such an analysis could also shed interesting light on global vs. local baryon conservation.
From the present results it is concluded that effects due to local baryon number conservation are not large, if present at all in second cumulants of net-protons. The large correlation length observed in the data implies that the normalized second cumulant R 1 is determined by collisions in the very early phase of the Pb-Pb interaction [44]. This interesting result would be important to follow up with measurements over a larger rapidity range, such as proposed in [45]. We note that long range rapidity correlations were investigated in other contexts in [46,47].
After accounting for baryon number conservation, the data are in agreement with the corresponding second cumulants of the Skellam distribution. As mentioned in the introduction, LQCD predicts a Skellam behavior for the second cumulants of net-baryon distributions at a pseudo-critical temperature of about 156 MeV [11], which is very close to the freeze-out temperature from the HRG model applied to the ALICE data [7]. Critical behavior is predicted by LQCD for higher cumulants of net-baryon distributions [48], which will be the topic of future investigations.

Acknowledgements
The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The       [45] D. Adamová et al., "A next-generation LHC heavy-ion experiment", arXiv:1902.01211 [physics.ins-det].