QCD Critical Point and Net-Proton Number Fluctuations at RHIC-STAR

In heavy-ion collision physics it is predicted that under a very high temperature and baryon density a deconfined quark-gluon plasma (QGP) phase can be created and studying the QCD phase structure is one of the main goals. A phase diagram in terms of temperature and baryon chemical potential ($\mu_{\mathrm{B}}$) is usually used to explore the QCD phase structure. Regarding the phase transition between the QGP phase and hadronic phase, first principle Lattice QCD calculation nature_Aoki at $\mu_{\mathrm{B}}$ = 0 MeV suggests a smooth crossover transition. While at large $\mu_{\mathrm{B}}$, various QCD-based models predict first order phase transition 1st_1 . Thermodynamically there should be an end point of the first order phase boundary which is called QCD critical point. The possible QCD critical point and the first order phase boundary have been investigated both experimentally PhysRevLett.112.032302 ; PhysRevLett.113.092301 ; PhysLettB785.551 and theoretically.

Higher-order cumulants of conserved quantities like net-baryon ($B$), net-charge ($Q$) and net-strangeness ($S$) are proposed as promising observables to search for the QCD critical point and the first order phase boundary. Higher-order cumulants are sensitive to the correlation length ($\xi$) PhysRevLett.103.262301 and are directly related to susceptibility ($\chi$) of the system susceptibility . It is predicted that the fourth-order fluctuations will exhibit a non-monotonic energy dependence PhysRevLett.107.052301 ; PhysRevD.85.034027 ; PhysRevD.95.014038 when passing through the critical region. For $5^{th}$- and $6^{th}$-order cumulants recent calculations from Lattice QCD PhysRevD.101.074502 and the functional renormalisation group approach (FRG) fu2021hyperorder show that they will be negative due to the crossover transition between QGP and hadronic phase. At high baryon density region, on the other hand, they are also sensitive to the first order phase boundary volker2 .

The data of Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 7.7 - 200 GeV and $p+p$ collisions at $\sqrt{\mathrm{s}}$ = 200 GeV are collected in RHIC Beam Energy Scan program phase I. Protons and antiprotons are identified by the Time Projection Chamber (TPC) and Time of flight (TOF) detectors at rapidity window $-0.5<y<0.5$ and transverse momentum window $0.4<p_{\mathrm{T}}<2.0$ GeV/$c$.

The centrality is determined using charged particle multiplicity within $|\eta|<1.0$ excluding protons and antiprotons to avoid auto-correlation effect Luo_2013 . The centrality bin width correction Luo_2013 is applied to suppress initial volume fluctuation effect. Cumulants are calculated at each multiplicity bin and then their weighted averages are taken for each centrality bin. The weight is the number of events at the corresponding multiplicity bin. Detector efficiency correction PhysRevC.91.034907 in cumulant calculations are done by assuming binomial detector efficiency. Statistical uncertainties of cumulants are estimated by Bootstrap bootstrap and Delta methods Luo_2012 .

Figure 1 shows energy dependence of $S\sigma$ and $\kappa\sigma^{2}$ of net-proton distributions from 0-5% and 70-80% centrality bins within $|y|<0.5$ and $0.4<p_{T}<2.0$ GeV/$c$ in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 7.7 - 200 GeV PhysRevLett.126.092301 ; starcollaboration2021cumulants . The $S\sigma$ (left panel) shows a decreasing trend with the increase of collision energy both in central and peripheral collisions. The decreasing trend can be qualitatively described by HRG hrg and UrQMD urqmd_model models. The $\kappa\sigma^{2}$ (right panel) shows a non-monotonic energy dependence in central collisions while the results for peripheral collisions show monotonic energy dependence. The non-monotonic trend in central collisions can not be described by different conditions (GCE, EV, and CE) of HRG and UrQMD models.

Figure 2 shows energy dependence of $\mathrm{C_{5}/C_{1}}$ and $\mathrm{C_{6}/C_{2}}$ of net-proton distributions from 0-40% and 70-80% centrality bins within $|y|<0.5$ and $0.4<p_{T}<2.0$ GeV/$c$ in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 7.7 - 200 GeV. It is suggested from Lattice QCD and FRG calculations that $\mathrm{5^{th}}$- and $\mathrm{6^{th}}$-order cumulants show negative signs while calculations from UrQMD and HRG models are consistent with either zero or unity. In UrQMD and HRG models, no phase transition physics is implemented. The measurements of BES-I data are shown as blue circles for 0-40% and red diamonds for 70-80%. The cumulant ratio $\mathrm{C_{5}/C_{1}}$ (left panel) deviates from zero with less than 2$\sigma$ significance. In peripheral collisions $\mathrm{C_{5}/C_{1}}$ shows positive sign for all energies. The ratio $\mathrm{C_{6}/C_{2}}$ (right panel) for 0-40% decreases with negative sign with less than 2$\sigma$ significance when decreasing energy while it shows positive sign in peripheral collisions (70-80%) for all energies.

Figure 3 shows multiplicity dependence of net-proton results of $\mathrm{C_{5}/C_{1}}$ and $\mathrm{C_{6}/C_{2}}$ within $|y|<0.5$ and $0.4<p_{T}<2.0$ GeV/$c$ in $p+p$ collisions at $\sqrt{\mathrm{s}}$ = 200 GeV. We see that the cumulant ratios ($\mathrm{C_{4}/C_{2}}$, $\mathrm{C_{5}/C_{1}}$, and $\mathrm{C_{6}/C_{2}}$) from $p+p$ collisions fit into the multiplicity dependence of results from Au+Au collisions which are shown with triangles. The cumulant ratios $\mathrm{C_{5}/C_{1}}$ and $\mathrm{C_{6}/C_{2}}$ are negative for 0-40% in Au+Au collisions and positive at peripheral Au+Au collisions and $p+p$ collisions. Pythia pythia calculation using version 8.2 of $p+p$ collisions at $\sqrt{\mathrm{s}}$ = 200 GeV is positive as shown with yellow bands. The Lattice QCD calculation PhysRevD.101.074502 at $\sqrt{\mathrm{s_{NN}}}$ = 200 GeV is negative as shown with red bands. Compared with calculations from various models, it is suggested that the negative sign for central Au+Au collisions at $\sqrt{\mathrm{s_{NN}}}$ = 200 GeV is due to a smooth crossover transition between partonic and hadronic phases.

In this proceedings, we report the measurements of net-proton cumulant ratios up to $6^{th}$-order in Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 7.7 - 200 GeV and $p+p$ collisions at $\sqrt{s}$ = 200 GeV from STAR. With results from 200 GeV $p+p$ collisions and the energy dependence of $\mathrm{C_{4}/C_{2}}$, $\mathrm{C_{5}/C_{1}}$, and $\mathrm{C_{6}/C_{2}}$ from the BES-I data sets, and the comparison with LQCD calculations, we conclude: 1) QCD matter is indeed created in the 200 GeV central (0-5%) Au+Au collisions at RHIC; 2) non-monotonic energy dependence of $\mathrm{C_{4}/C_{2}}$ is observed from the most central (0-5%) Au+Au collisions PhysRevLett.126.092301 ; starcollaboration2021cumulants . Future results from BES-II and STAR fixed-target experiment $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 3 GeV data sets will allow to answer if QCD critical point exists in the covered energy region. In 2023 to 2025, STAR plans to collect 15 to 20 billion events of Au+Au collisions at $\sqrt{\mathrm{s}_{{}_{\mathrm{NN}}}}$ = 200 GeV. This will allow us to perform more precise measurements of higher-order cumulants, maybe even up to $8^{th}$-order.

This work was supported by the National Key Research and Development Program of China (Grant No. 2020YFE0202002 and 2018YFE0205201), the National Natural Science Foundation of China (Grant No. 11828501, 11890711 and 11861131009) and China scholarship council (No. 201906770055).