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Kinetic Freeze-Out Conditions and Net Baryon Density in Au+Au Collisions at $\sqrt{s_{NN}} = 7.7$--$39$ GeV within a Collective Flow Fireball Model

Sk Noor Alam, Victor Roy

Abstract

We investigate the effects of transverse and longitudinal collective flow on kinetic freeze-out conditions and net baryon density in 0--5\% central Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--$39$ GeV within the RHIC Beam Energy Scan program. Using a covariant statistical fireball model, we fit the transverse momentum spectra of protons and positive pions from STAR data to simultaneously extract the kinetic freeze-out temperature $T$, baryon chemical potential $μ_B$, and transverse flow velocity $v_T$, for three fixed longitudinal velocities $v_z = 0.0$, $0.2$, and $0.4$. Longitudinal flow induces a systematic upward shift in $T$, spanning $143$--$171$ MeV, $150$--$180$ MeV, and $175$--$209$ MeV for $v_z = 0$, $0.2$, and $0.4$, respectively, arising from a kinematic degeneracy between $v_z$ and $T$ in the Lorentz-invariant distribution function rather than from any hardening of the $p_T$ spectra. While temperatures extracted at $v_z = 0$ and $0.2$ are broadly consistent with the QCD crossover temperature $T_c \approx 155$--$160$ MeV expected from lattice QCD, the values obtained at $v_z = 0.4$ significantly exceed $T_c$, suggesting that the system would no longer be described by a hadron resonance gas at these conditions and indicating that large longitudinal velocities may be physically disfavored at these collision energies. The baryon chemical potential decreases monotonically from $μ_B \sim 420$ MeV at $7.7$ GeV to $\sim 200$ MeV at $39$ GeV, independently of $v_z$. Reconstructing the freeze-out trajectory in the $(ρ_B, \varepsilon)$ plane, we identify a maximum in net baryon density at $\sqrt{s_{NN}} \lesssim 11.5$ GeV, with dynamical flow enhancing the inferred baryon compression by up to $\sim$20\% relative to the static limit. These results provide refined freeze-out benchmarks for future hydrodynamic modeling and for experiments at FAIR and low-energy RHIC runs.

Kinetic Freeze-Out Conditions and Net Baryon Density in Au+Au Collisions at $\sqrt{s_{NN}} = 7.7$--$39$ GeV within a Collective Flow Fireball Model

Abstract

We investigate the effects of transverse and longitudinal collective flow on kinetic freeze-out conditions and net baryon density in 0--5\% central Au+Au collisions at -- GeV within the RHIC Beam Energy Scan program. Using a covariant statistical fireball model, we fit the transverse momentum spectra of protons and positive pions from STAR data to simultaneously extract the kinetic freeze-out temperature , baryon chemical potential , and transverse flow velocity , for three fixed longitudinal velocities , , and . Longitudinal flow induces a systematic upward shift in , spanning -- MeV, -- MeV, and -- MeV for , , and , respectively, arising from a kinematic degeneracy between and in the Lorentz-invariant distribution function rather than from any hardening of the spectra. While temperatures extracted at and are broadly consistent with the QCD crossover temperature -- MeV expected from lattice QCD, the values obtained at significantly exceed , suggesting that the system would no longer be described by a hadron resonance gas at these conditions and indicating that large longitudinal velocities may be physically disfavored at these collision energies. The baryon chemical potential decreases monotonically from MeV at GeV to MeV at GeV, independently of . Reconstructing the freeze-out trajectory in the plane, we identify a maximum in net baryon density at GeV, with dynamical flow enhancing the inferred baryon compression by up to 20\% relative to the static limit. These results provide refined freeze-out benchmarks for future hydrodynamic modeling and for experiments at FAIR and low-energy RHIC runs.
Paper Structure (7 sections, 27 equations, 4 figures, 10 tables)

This paper contains 7 sections, 27 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Model of a Dynamic Fireball
  3. Calculation of Energy Density and Net Baryon Density
  4. Results and Discussion
  5. Fitting the Transverse Momentum Spectra
  6. Conclusions and Outlook
  7. ACKNOWLEDGMENTS

Figures (4)

  • Figure 1: (Color online) Midrapidity ($|y| <0.1$) transverse momentum spectra for positive protons in Au+Au collisions at $\sqrt{s_{NN}} =$ 7.7 to 39 GeV for 0-5$\%$ centrality with longitudinal velocity $V_z$ = 0.0 of the dynamic fireball. The red lines are fits using Eq.\ref{['Eq:eq15']}PhysRevC.96.044904.
  • Figure 2: (Color online) Midrapidity ($|y| <0.1$) transverse momentum spectra for positive pions in Au+Au collisions at $\sqrt{s_{NN}} =$ 7.7 to 39 GeV for 0-5$\%$ centrality with longitudinal velocity $V_z$ = 0.0 of the dynamic fireball. The red lines are fits using Eq.\ref{['Eq:eq15']}PhysRevC.96.044904.
  • Figure 3: (Color online) Baryon chemical potential vs temperature for three longitudinal velocities $v_z =0$ (black circles), $v_z =0.2$ (red squares), and $v_z =0.4$ (green triangles).
  • Figure 4: (Color online) The Hadronic freeze-out points in the $\rho_{B}$-$\varepsilon$ phase plane as obtained in the statistical model with the values of $\mu_B$ and T obtained from the fitting of the transverse momentum of proton at the different STAR energies with different dynamic fireball's longitudinal velocities.