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 $\sqrt{s_{NN}} = 7.7$--$39$ 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 $\mu_B$, 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 $\mu_B \sim 420$ MeV at $7.7$ GeV to $\sim 200$ MeV at $39$ GeV, independently of $v_z$. Reconstructing the freeze-out trajectory in the $(\rho_B, \varepsilon)$ 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.