Predictions of baryon directed flow in heavy-ion collisions at high baryon density

TL;DR

Problem addressed: constraining the high-baryon-density QCD equation of state using the proton directed flow $v_1$ in heavy-ion collisions. Method: predictions are made with the three-fluid dynamics model ($3$FD) employing hadronic, first-order phase transition (1PT), and crossover EoS, with the THESEUS UrQMD afterburner for final-state effects, across $\sqrt{s_{NN}} \approx 3$–$11.5$ GeV and comparison to STAR data. Key results: the crossover EoS reproduces $v_1$ data below $4.5$ GeV and above $7.7$ GeV; in the $4.5$–$7.7$ GeV window, $v_1$ shows non-monotonic behavior with a weak antiflow at $7.2$ GeV and a smaller wiggle than in the strong 1PT scenario; data disfavor a strong first-order transition, favoring a weak or crossover transition. The study highlights that the proton $v_1$ is a robust EoS probe largely unaffected by late-stage dynamics, and provides testable predictions for future experiments.

Abstract

Predictions of the proton directed flow ($v_1$) in semicentral Au+Au collisions in the energy range between 4.5 and 7.7 GeV are done. The calculations are performed within the model of three-fluid dynamics with crossover equation of state, which well reproduces the proton $v_1$ both below 4.5 GeV and above 7.7 GeV, as well as bulk observables in the energy range of interest. It is predicted that the proton flow evolves non-monotonously. At the energy of 7.2 GeV it exhibits antiflow (i.e. negative slope of $v_1(y)$) in the midrapidity. At 7.7 GeV, the flow returns to the normal pattern in accordance with the STAR data. The midrapidity $v_1$-slope excitation functions within the first-order phase and crossover transitions to quark-gluon phase (QGP) turn out to be qualitatively similar, but the amplitude of the wiggle in the crossover scenario is much smaller than that in the strong first-order phase transition. Therefore, the change of sign followed by minimum at 7.2 GeV in the $v_1$-slope excitation function indicates onset of (weak phase or crossover) transition to QGP. The second change of the sign around 10 GeV results from interplay between incomplete baryon stopping and transverse expansion of the system.

Figures (9)

• Figure 1: Directed flow of protons as function of rapidity in semicentral Au+Au collisions at collision energies of $\sqrt{s_{NN}}=$ 3 and 4.5 GeV. Results are calculated within the 3FD model (upper row of panels) and the THESEUS Ivanov:2024gkn (lower row of panels) with hadronic, 1PT, and crossover EoSs. STAR data are from Refs. STAR:2020davSTAR:2021yiu.
• Figure 2: Dynamical trajectories of the matter in the central cell of the colliding Au+Au nuclei in semicentral collisions (impact parameter is $b=$ 6 fm) at energies $\sqrt{s_{NN}}=$ 3, 4.5, 7.2 and 11.5 GeV. The trajectories are plotted in terms of the baryon density ($n_B$, scaled by the normal nuclear density $n_0$) and temperature $T$. The trajectories are presented for the three EoSs. The wide shadowed area displays the region of the crossover EoS, where the QGP fraction $W_{QGP}$ lies between 0.1 and 0.5. The shadowed region in the 1PT panel indicates the mixed phase, where $0<W_{QGP}<1$.
• Figure 3: Evolution of the isentropic speed of sound ($c_s$) as function of the baryon density ($n_B$, scaled by the normal nuclear density $n_0$) along the dynamical trajectories displayed in Fig. \ref{['fig:T-nB_b=6fm']}. The evolution is displayed from the instants (indicated by star symbols) that are close to the trajectory turning points, when the matter is sufficiently thermalized. The trajectories are presented for the 1PT EoS.
• Figure 4: The same as in Fig. \ref{['fig:Cs-vs-nb-tph-b6']} but for the crossover EoS.
• Figure 5: Directed flow of protons as function of rapidity in semicentral ($b=$ 6 fm) Au+Au collisions at collision energies of $\sqrt{s_{NN}}=$ 3, 3.2, 3.5 and 3.9 GeV. Results are calculated within the 3FD model with hadronic, 1PT, and crossover EoSs. STAR data are from Refs. STAR:2021yiuSTAR:2025twg.