Critical fluctuations of the proton density in A+A collisions at $158A$ GeV

We look for fluctuations expected for the QCD critical point using an intermittency analysis in the transverse momentum phase space of protons produced around midrapidity in the 12.5\% most central C+C, Si+Si and Pb+Pb collisions at the maximum SPS energy of 158$A$~GeV. We find evidence of power-law fluctuations for the Si+Si data. The fitted power-law exponent $\phi_{2} = 0.96^{+0.38}_{-0.25}\text{ (stat.)}$ $\pm 0.16\text{ (syst.)}$ is consistent with the va\-lue expected for critical fluctuations. Power-law fluctuations had previously also been observed in low-mass $\pi^+ \pi^-$ pairs in the same Si+Si collisions.

Studies of QCD suggest the existence of a critical point in the phase diagram of strongly interacting matter. Close to this point, according to recent theoretical investigations, the net-proton density carries the critical fluctuations of the chiral order parameter. Using intermittency analysis in the transverse momentum phase space of protons produced around midrapidity in the 12.5% most central C+C, Si+Si and Pb+Pb collisions at the maximum SPS energy of 158A GeV we find evidence of power-law fluctuations for the Si+Si and Pb+Pb data. The fitted power-law exponent approaches the value expected for critical fluctuations. This suggests that the freeze-out states of these two systems are located in the phase diagram in the neighbourhood of the chiral critical point. The current understanding of the phase diagram of strongly interacting matter, as supported by several lattice QCD calculations [1,2], suggests the existence of a critical point (CP) at finite baryochemical potential and temperature. This CP is the endpoint of a line of first order transitions associated with the partial restoration of chiral symmetry when the temperature T , for given baryochemical potential µ B , increases beyond a critical value T c . For present experiments with heavy-ion collisions the search for the hypothetical "chiral" CP is of high priority. Many theoretical studies have been performed to find suitable observables [3][4][5][6]. To this end it is necessary to consider the fluctuations of the chiral condensate qq which is the order parameter of the chiral transition (q is the quark field). The quantum state which carries the quantum numbers as well as the critical properties of the chiral condensate is the sigma-field σ( x). In a heavy-ion collision the excitation of the vacuum leading to the formation of the chiral condensate is likely to occur. However, the condensate will be unsta-ble, decaying mainly into pions, at time scales characteristic of the strong interaction. The critical properties of the condensate are transferred to detectable π + π − pairs with invariant mass close to their production threshold [5]. In a finite-density medium there is a mixing between the chiral condensate and the baryon density inducing critical fluctuations in the baryon density [7,8] as well. Furthermore, critical fluctuations of the chiral condensate are also directly transferred to the net-proton density through the coupling of the protons with the isospin zero σ-field [9]. Thus detecting the QCD CP through fluctuations of the net-proton density is a promising strategy.
One of the main goals of the NA49 experiment [10] and its successor NA61 [11] is an extensive search for the CP by changing energy and size of the colliding nuclei in order to scan the phase diagram with a dense set of freezeout states at different T and µ B . The observables of interest are either event-by-event fluctuations of integrated quantities like multiplicity [12] and average transverse momentum [13] or local power-law fluctuations related to density-density correlations detectable through the measurement of factorial moments in momentum space [14] within the framework of intermittency analysis [15]. Our approach differs from the intermittency studies in the past [16] in analysing quantities directly related to the order parameter of the phase transition (sigma condensate or net-baryon density). The advantage is that for a critical system the fluctuations of these quantities are expected to obey power laws with exponents determined by the corresponding critical exponents [4,6].
In the present work we calculate the second factorial moment of the proton density as a function of the number M of subdivisions in each transverse momentum space direction (-1.5 GeV/c < p T,x , p T,y < 1.5 GeV/c), where n i is the number of protons in the cell i. The analysed data were recorded by the NA49 experiment in A+A collisions at maximum CERN SPS energy of 158A GeV ( √ s N N = 17.3 GeV). We can restrict the analysis to protons because the antiprotons are clearly far fewer for all considered systems. Theoretical work [6] points out the emergence of intermittencyi.e. power-law behaviour of F 2 (M ) as a function of M 2with expected exponent (intermittency index) φ 2,cr = 5 6 , when the system freezes out exactly at the chiral CP. The critical intermittency index is determined from universality class arguments. In the case of the two-dimensional system of interest (transverse momentum components) the above index can be revealed, if one studies protons in a small region (window) around midrapidity where their density reaches a plateau value. When moving the position of the window towards beam rapidity, the critical behaviour is gradually masked [6]. Also, a weakening of the intermittent behaviour is expected as the distance of the freeze-out state from the CP increases, leading to a decrease of the value of φ 2 and/or a modification of the power-law dependence [5,14]. Our aim is to explore the systems C+C, Si+Si and Pb+Pb looking for power-law behaviour of the second factorial moment of the proton density in transverse momentum space as a signature for the approach to the CP.
For the analysis we used data on the 12.5% most central collisions of "C", "Si" and Pb nuclei on C (2.4% interaction length), Si (4.4%) and Pb (1%) targets, respectively, collected by the NA49 experiment at √ s N N = 17.3 GeV. The "C" beam as defined by the online trigger and offline selection was a mixture of ions with charge Z = 6 and 7 (intensity ratio 69:31); the "Si" beam of ions with Z = 13, 14 and 15 (intensity ratio 35:41:24) [17]. The event statistics amounted to 210k events for "C"+C, 176k events for "Si"+Si and 1480k events for Pb+Pb. The standard event and track selection cuts of the NA49 experiment were applied as described in [18]. Specifically, to avoid double counting split tracks we only accepted tracks for which the ratio of number of measured points to estimated maximum number of points in the TPCs exceeds 55%. Proton identification [18] used the measurements of particle energy loss dE/dx in the gas of the time projection chambers. The inclusive dE/dx distribution for positively charged particles in each reaction was fitted in 10 bands of momentum p to a sum of contributions f α (dE/dx, p) from different particle species α with α = π, K, p, e. The probability P for a track with energy loss x i and momentum p i of being a proton is then given . The value of P for proton candidates had to exceed 80% for the "C"+C and "Si"+Si systems and 90% for Pb+Pb. Moreover the candidates had to fulfill the criterion |y CM | < y beam − 0.5 (y beam ≈ 2.9 in the cms) to avoid contamination by spectator protons [19].
In order to reveal a potentially underlying power-law behaviour of the factorial moments calculated from the data, we have to subtract the background due to uncorrelated protons (e.g. from hyperon decays) and/or the contamination by particles misidentified as protons. The background contribution is simulated by mixed events which, by construction, contain all the background effects but not the correlation signal. We then search for powerlaw behaviour of the corrected correlator ∆F 2 (M ) where the superscripts (d), (m) refer to data and mixed events respectively [5].
In Fig. 1 we show the second factorial moments F 2 (M ) versus M 2 for the three analysed systems (red circles). In the same figure we also plot the second factorial moments for the corresponding mixed events (black triangles). For the "Si"+Si and Pb+Pb systems we observe that for large M values the factorial moments of the data are clearly larger than those of the mixed events. The difference between the two moments increases with increasing num-ber of cells M 2 , a typical characteristic of intermittent behaviour. This observation is an indication for sizeable correlations among the produced protons. However, due to small statistics the errorbars of the F 2 values are quite large for "Si"+Si. In the "C"+C case the factorial moment values of the data overlap with those of the mixed events suggesting the absence of an intermittency effect in this system. The background corrected correlator ∆F 2 as a function of M 2 is shown in Fig. 2. The panels (a),(b),(c) illustrate, in log-linear scale, ∆F 2 (M 2 ) for "C"+C, "Si"+Si and Pb+Pb respectively. Only the region of large number of cells (M 2 > 2000) is displayed, where the intermittency effect, if present, is expected to show up [15,20]. In order to make our analysis more quantitative we fitted the power-law function ∆F 2 (M ) ∼ (M 2 ) φ2 shown by the lines in Fig. 2. A complication for the fitting procedure is the fact that the factorial moment values at consecutive M 2 are correlated. Several approaches like the correlated χ 2 (Cχ 2 ) [21], the independent bin (IB) [22] and the sparse binning (SB) [23] method were employed, which are designed to reduce the effect of these data point correlations. We found that for the Pb + Pb dataset the different methods give identical results fully compatible with the direct power-law fit (DPF). For "Si" + Si the results for φ 2 have some spread (φ 2,Cχ 2 = 0.76 ± 0.28, φ 2,IB = 0.91±0.25 and φ 2,SB = 0.88±0.21), presumably due to small statistics. However, they are still compatible with each other within errors. The variation of these values is used to obtain a systematic error for "Si" + Si leading to: δφ (Si) 2,sys = 0.12. Finally for the "C" + C sys-tem only the direct power-law fit is applicable. We also estimated the systematic error of φ 2 for the Pb + Pb system using different proton purity requirements (70%, 80 % and 90%). Decreasing purity the power-law behaviour of the correlator is pushed to larger values of M 2 as expected. Despite this, the associated power-law fits give (for the datasets with different purities) φ 2 values which are compatible within errors confirming the efficiency in the subtraction of the background. Restricting the fit region to the domain M 2 ≥ 10000 we obtain (for the 90% purity case) φ 2 = 0.36 ± 0.06 where we estimated the systematic error δφ (P b) 2,sys = 0.06 as half the range of the three fit results. Due to low statistics it is not possible to increase purity for the "C" + C and "Si" + Si, but the combinatorial background is much smaller in these systems.
For all performed fits the χ 2 value per degree of freedom was found in the range [0.4, 0.9]. The quality with which the fit describes the variation of the correlator with the number of cells M 2 is measured in terms of the coefficient of determination R 2 [24]. Using the DPF we obtained the intermittency index φ 2 = −0.16 ± 0.44 for the "C"+C system with R 2 ≈ 0 indicating that the "C" + C-correlator does not follow a power law in M 2 . The presence of many negative ∆F 2 values leads to a negative φ 2 with a large error. For the "Si"+Si system the DPF gave φ 2 = 0.88 ± 0.10 with R 2 = 0.60, well above the the minimum acceptable value of 0.5, supporting that the correlator in this case follows to a good approximation a power law. Finally for the Pb+Pb system the DPF gave φ 2 = 0.37 ± 0.01 with R 2 = 0.96 which is very close to the optimal value R 2 = 1 valid for an exact power-law dependence.
Thus we observe significant power-law fluctuations of the proton density in transverse momentum space for the "Si"+Si and Pb+Pb systems at √ s N N = 17.3 GeV with exponents approaching the QCD prediction (5/6). Both systems appear to freeze out near the CP in the phase diagram. It is illustrative to plot the fitted intermittency indices φ 2 as a function of the size A of the nuclei (see Fig. 3). To guide the eye we include in the plot also the QCD prediction for the CP φ 2,cr = 5/6 (red solid line) [6]. Note that the φ 2 value of the "C"+C system is added in the plot for completeness, although it is not associated with critical fluctuations since it is negative and the power-law behavior is questionable (R 2 ≈ 0). An analogous intermittency effect was found recently [14] in central Si+Si collisions at √ s N N = 17.3 GeV for π + π − pairs with invariant mass close to their production threshold (σ-field fluctuations). The corresponding analysis of Pb+Pb data was not possible due to limitations of the experimental resolution.
Two remarks are in order before going to our conclusions. First, in Fig. 4 we show that the intermittency effect observed for the Pb+Pb system is confined, as the-  oretically expected, to protons at midrapidity (the same is true also for "Si"+Si) where the density of the produced fireball is approximately constant in rapidity [6]. We compared the correlator ∆F 2 for the Pb+Pb system calculated for protons with 1 < y CM < 1.4 ( Fig. 4a) with that for −0.75 < y CM < 0.75 (Fig. 4b). Clearly the effect gets suppressed for protons away from the midrapidity region. We repeated the analysis for protons with −0.5 < y CM < 0.5 and found φ 2 = 0.46 ± 0.04 with R 2 = 0.78. Thus within the errors there is no observ-able change of φ 2 by further decreasing the size of the window around midrapidity. However, the R 2 value decreases since the statistics becomes worse. Second, we calculated the correlator ∆F 2 for protons at midrapidity in the 12.5% most central Pb+Pb collisions at √ s N N = 8.8 GeV. In Fig. 5 we compare this result with that found at √ s N N = 17.3 GeV. We clearly observe that the critical fluctuations disappear at lower energy. Thus, our analysis suggests that the critical baryochemical potential is closer to 240 MeV, the freeze-out baryochemical potential at √ s N N = 17.3 GeV, than to µ B ≈ 380 MeV, the value at √ s N N = 8.8 GeV [25]. A similar analysis cannot be performed for the "Si"+Si system at √ s N N = 8.8 GeV due to insufficient statistics. In summary, we performed a search for critical fluctuations in NA49 data using factorial moments of the proton density in transverse momentum space. We calculated the background subtracted second factorial moment ∆F 2 in the 12.5% most central collisions of "C"+C, "Si"+Si and Pb+Pb at √ s N N = 17.3 GeV. We found a strong intermittency effect for the "Si"+Si and the Pb+Pb system at midrapidity. This effect is expressed through the power-law dependence: The intermittency index φ 2 for the "Si"+Si system approaches in size the QCD prediction for the CP.