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

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 $({\rho }_B,\epsilon )$ 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.