Constraining the Phase-Transition EoS using the Energy Dependence of Directed Flow

Abstract

We propose a hybrid equation of state (VDF+MIT EoS) to describe the hadron-quark phase transition in dense nuclear matter. By coupling this EoS with the AMPT-HC transport model and comparing to recent experimental data on proton and $Λ$ directed flow $v_1$, we constrain the transition to likely occur near $5ρ_0$--$6ρ_0$, ruling out transitions below $3ρ_0$. Furthermore, we introduce the energy derivative of the mid-rapidity $v_1$ slope, $d(dv_1/dy)/d(\sqrt{s_{NN}})$, as a weakly model-dependent observable. Its zero crossing provides a direct signature of the phase transition critical point, offering a new tool for mapping the QCD phase diagram in future experiments.

Figures (7)

• Figure 1: Pressure $P$ as a function of baryon number density $\rho$ for the three hybrid VDF+MIT EoSs. The calculated results for the three distinct EoSs are represented by a red solid line (VDF1+MIT), a blue dashed line (VDF2+MIT), and a yellow dotted-dashed line (VDF3+MIT). The hollow pentagrams indicate the transition points from the hadronic to the quark phase. The calculated EoSs are compared with constraints from astrophysical observations and experimental measurements (represented by the different color shaded band).
• Figure 2: Mass-radius relations derived from the VDF+MIT EoS. Shaded regions represent observational constraints from PSR J0030+0451 (green) PSRJ0030p0451, PSR J0740+6620 (purple) PSRJ0740p6620APSRJ0740p6620B, and the GW190425 event (blue) NS1. The grey band denotes the predicted theoretical maximum mass limit $M_{\max}$maxM1maxM2.
• Figure 3: Time evolution of the maximum central compressional baryon number density ($\rho_{\text{max}}/\rho_0$) achieved in Au+Au collisions at various beam energies. The simulations were performed using the AMPT-HC transport model coupled with the VDF+MIT hybrid EoS.
• Figure 4: Time evolution of the proton directed flow $v_1$ slope at mid-rapidity in Au+Au collisions at $10-40\%$ centrality and center-of-mass energies of $\sqrt{s_{NN}} = 3, 4.5, \text{ and } 7.7 \, \text{GeV}$. The simulations were performed using the AMPT-HC transport model. Results using the three distinct VDF+MIT hybrid EoSs are shown by the red solid line (VDF1+MIT), the blue dashed line (VDF2+MIT), and the yellow dotted-dashed line (VDF3+MIT). The green dotted-dashed line represents the result from the pure hadronic cascade mode (no mean field).
• Figure 5: Directed flow ($v_1$) distribution as a function of rapidity ($y$) for protons and $\Lambda$ hyperons produced in Au+Au collisions. Model results corresponding to the three VDF+MIT hybrid EoSs are shown by the red solid line (VDF1+MIT), the blue dashed line (VDF2+MIT), and the yellow dotted-dashed line (VDF3+MIT), respectively. Experimental data for protons are represented by yellow circles and data for $\Lambda$ hyperons by yellow open stars. The experimental data points are sourced from the RHIC-STAR experiment in Au+Au collisions at center-of-mass energies of $\sqrt{s_{NN}} = 3.0, 4.5, \text{ and } 7.7 \, \text{GeV}$starplb2022star45star77star77p.