Seventh and eighth-order cumulants of net-proton multiplicity distributions in heavy-ion collisions at RHIC-STAR

. We report the ﬁrst measurements of seventh and eighth-order cumu-lants of net-proton distributions in Au + Au collisions at √ s NN = 27, 54.4, and 200 GeV. The measurements are performed at mid-rapidity | y | < 0 . 5 within 0.4 < p T < 2 . 0 GeV / c using the Time Projection Chamber and Time-of-Flight detector. Motivation for the measurements comes from lattice-QCD and QCD based model calculations that predict their negative signs for a crossover quark-hadron transition. While 0-40% centrality measurements at √ s NN = 54.4 and 200 GeV are consistent with zero within large uncertainties, at √ s NN = 27 GeV, they are negative with ≤ 1 . 4 σ signiﬁcance. The peripheral 70-80% measurements are either positive or consistent with zero for the three energies.


Introduction
The phase diagram of strongly interacting matter is called as the Quantum-Chromo-Dynamics (QCD) phase diagram [1].It has at least two distinct phases: hadronic phase where quarks and gluons are confined within hadrons and the quark-gluon plasma (QGP) phase where they are deconfined.The transition between the two phases at vanishing baryonic chemical potential (µ B ) is shown to be a smooth crossover by first-principle lattice QCD calculations [2].At large µ B , several QCD-based model calculations indicate a first-order phase transition terminating at a QCD critical point [3,4].Experimental exploration of the QCD phase diagram forms one of the primary goals of the heavy-ion collision experiments.Higher-order cumulants of event-by-event net-particle distributions have been suggested as sensitive observables in this regard [5,6].Ratio of the measured cumulants are constructed to eliminate the system volume dependence and facilitate a direct comparison with ratio of susceptibilities (χ n ) calculated from lattice QCD, QCD-based models, and thermal models [7,8].
The STAR experiment at RHIC has measured cumulants (C n ) of net-proton (as a proxy for net-baryon) distributions up to sixth-order in Au+Au collisions from √ s NN = 3 − 200 GeV [9][10][11][12][13].A non-monotonic collision energy dependence of net-proton cumulant ratio C 4 /C 2 was observed, which is consistent with a model calculation that includes a critical point [9].Furthermore, the net-proton C 6 /C 2 shows an increasingly negative sign with decreasing collision energy in the range of √ s NN = 7.7 − 200 GeV [13].The negative values, albeit with large uncertainties, and the energy dependence are consistent with lattice QCD calculation (µ B < 110 MeV) that includes a crossover quark-hadron transition [6].
The study presented in these proceedings extends the net-proton cumulant measurements to even higher orders, i.e.C 7 /C 1 and C 8 /C 2 .The lattice-QCD and various QCDbased models predict negative seventh and eighth-order net-baryon susceptibilities for a crossover quark-hadron transition at vanishing baryonic chemical potential near the transition temperature T = T pc with larger magnitude compared to those at fifth-and sixthorders [6,14,15].Calculations from a model, called the Polyakov-loop extended quarkmeson (PQM) model [15], presented in Fig 1 shows that χ B 8 is an order more negative than χ B 6 at T = T pc for vanishing chemical potential (µ q ∼ 0).In addition to search for crossover in QCD phase diagram, net-proton cumulant ratios C 7 /C 1 and C 8 /C 2 can also be used to study the validity of hadron resonance gas (HRG) models: Canonical Ensemble (CE) vs. the Grand Canonical Ensemble (GCE) frameworks.In the CE approach [16], the cumulant ratios show a departure from unity and a strong collision energy dependence as compared to GCE [6], where they are unity across all collision energies.8 .This confirms that in the transition region, two derivatives with respect to µq/T are indeed equivalent to one derivative with respect to T .
From these calculations, as well as from calculations of the lower order cumulants χ B 2 and χ B 4 , we obtain the ratios R B n,m of the n-th and m-th cumulants.Results obtained for µ q/T = 0 and µq/T > 0 are shown in Figs. 5  and 6, respectively.We note that these ratios approach unity at low temperatures, as it is the case also in the hadron resonance gas model.In the transition region, they reflect the expected O(4) scaling properties; they have a shallow maximum close to the transition region before they drop sharply.In particular, they show pronounced minima with R B n,2 < 0 in the vicinity of the chiral crossover temperature.The exact location of these minima and 8 .This confirms that in the transition region, two derivatives with respect to µ q /T are indeed equivalent to one derivative with respect to T .
From these calculations, as well as from calculations of the lower order cumulants χ B 2 and χ B 4 , we obtain the ratios R B n,m of the n-th and m-th cumulants.Results obtained for µ q /T = 0 and µ q /T > 0 are shown in Figs. 5  and 6, respectively.We note that these ratios approach unity at low temperatures, as it is the case also in the hadron resonance gas model.In the transition region, they reflect the expected O(4) scaling properties; they have a shallow maximum close to the transition region before they drop sharply.In particular, they show pronounced minima with R B n,2 < 0 in the vicinity of the chiral crossover temperature.The exact location of these minima and 13 q Sign of ; 2 and ; 7 together sensitive to hadronic phase, QGP phase and U VW .
Hadronic Phase GeV/c, only the TPC was used to select (anti-)protons whereas both TPC and TOF were required for (anti-)proton identification in the higher momentum region.To suppress the effect of initial system volume fluctuations on cumulants, Centrality-Bin-Width-Correction (CBWC) was applied [17].To correct the cumulants for finite detection efficiency, an analytical correction was performed where the detector response was assumed to follow binomial distribution [18].For estimating statistical uncertainties, bootstrap method was used [19,20].Systematic uncertainties on the measurements were estimated varying tracking efficiency, track selection, and particle identification criteria.Centrality Dependence of Net-Proton % 5 /% and % 7 /% - q Central 0-40% measurements consistent with zero within uncertainties for 54.4 and 200 GeV.Measurement at √i jj = 27 GeV negative with ~1.4Ä significance.q Peripheral data close to zero for the three energies.

tFigure 4 :
Figure 4: The sixth and eighth order cumulants of the net baryon number fluctuations at µ q/T = 0 in the PQM model.The temperature is given in units of the pseudo-critical temperature Tpc(mπ) corresponding to a maximum of the the chiral susceptibility.The shaded area indicates the chiral crossover region.these derivatives have been implemented directly into the analysis of the flow equations (see Appendix).In Fig.4we show the sixth and eighth order cumulants of the net baryon number fluctuations computed at µq/T = 0 within the PQM model for physical values of the pion mass.The basic features dictated by O(4) symmetry restoration, as discussed in the previous sections, are readily identified in the figure.Moreover, the positions of the two extrema of χ B 6 correspond approximately to the zeros of χ B 8 .This confirms that in the transition region, two derivatives with respect to µq/T are indeed equivalent to one derivative with respect to T .From these calculations, as well as from calculations of the lower order cumulants χ B 2 and χ B 4 , we obtain the ratios R B n,m of the n-th and m-th cumulants.Results obtained for µ q/T = 0 and µq/T > 0 are shown in Figs.5 and 6, respectively.We note that these ratios approach unity at low temperatures, as it is the case also in the hadron resonance gas model.In the transition region, they reflect the expected O(4) scaling properties; they have a shallow maximum close to the transition region before they drop sharply.In particular, they show pronounced minima with R B n,2 < 0 in the vicinity of the chiral crossover temperature.The exact location of these minima and

Figure 4 :
Figure 4: The sixth and eighth order cumulants of the net baryon number fluctuations at µ q /T = 0 in the PQM model.The temperature is given in units of the pseudo-critical temperature T pc (m π ) corresponding to a maximum of the the chiral susceptibility.The shaded area indicates the chiral crossover region.these derivatives have been implemented directly into the analysis of the flow equations (see Appendix).In Fig.4we show the sixth and eighth order cumulants of the net baryon number fluctuations computed at µ q /T = 0 within the PQM model for physical values of the pion mass.The basic features dictated by O(4) symmetry restoration, as discussed in the previous sections, are readily identified in the figure.Moreover, the positions of the two extrema of χ B 6 correspond approximately to the zeros of χ B 8 .This confirms that in the transition region, two derivatives with respect to µ q /T are indeed equivalent to one derivative with respect to T .From these calculations, as well as from calculations of the lower order cumulants χ B 2 and χ B 4 , we obtain the ratios R B n,m of the n-th and m-th cumulants.Results obtained for µ q /T = 0 and µ q /T > 0 are shown in Figs.5 and 6, respectively.We note that these ratios approach unity at low temperatures, as it is the case also in the hadron resonance gas model.In the transition region, they reflect the expected O(4) scaling properties; they have a shallow maximum close to the transition region before they drop sharply.In particular, they show pronounced minima with R B n,2 < 0 in the vicinity of the chiral crossover temperature.The exact location of these minima and
Figure 1.χ B 6 and χ B 8 from the PQM model as a function of T/T pc [15].
! " and !# at STAR-RHIC −Ashish Pandav for the STAR Collaboration

Figure 3 .
Figure 3. C 7 /C 1 (a) and C 8 /C 2 (b) of net-proton distributions from Au+Au collisions as a function of collision energy.The results for two centrality bins, 0-40% (filled squares) and 70-80% (open diamonds), are presented.The bars represent the statistical uncertainties.The shaded bands on 0-40% data points and caps on the 70-80% data represent systematic uncertainties.Insets are added in both panel containing peripheral 70-80% data for better visibility.The HRG CE calculations for C 8 /C 2 [16] are also shown.