Collective Flow, Kinetic Freeze-Out, and a Data-Driven Baryon Chemical Potential in Au+Au Collisions at $\sqrt{s_{NN}} = 7.7$--$39$ GeV
We study how transverse and longitudinal collective flow affect kinetic freeze-out conditions and the net baryon density in 0--5\% central Au+Au collisions at $\sqrt{s_{NN}} = 7.7$--$39$ GeV, using a covariant statistical fireball model fitted to STAR identified-hadron transverse-momentum spectra at three fixed longitudinal velocities $v_z = 0$, $0.2$, and $0.4$. \rev{Longitudinal flow shifts the extracted temperature upward through a kinematic $v_z$--$T$ degeneracy rather than by hardening the spectra; imposing $T \le T_c \approx 155$--$160$ MeV bounds the longitudinal flow to $v_z \lesssim 0.6$.} \nw{A simultaneous $\pi^+$, $K^+$, $p$ fit with shared $(T, v_T)$ sharpens the extraction \fx{by a factor of $3.7$--$5.3$ on $T$ relative to the single-species proton fit} and gives $T_{\rm kin}\simeq107$--$115$~MeV, reproducing STAR's independent blast-wave values \fx{to within $0.9\sigma$} and lying well below the chemical freeze-out temperature.} \fx{Further including the antiproton spectra, with a normalization shared between $p$ and $\bar p$, breaks the fugacity--normalization degeneracy and makes the baryon chemical potential data-driven: the $\bar p/p$ ratio fixes $\mu_B/T$, the meson spectra fix $T$, and together they give $\mu_B \simeq 262$--$63$~MeV at kinetic freeze-out, satisfying $\mu_B/T_{\rm kin} \simeq (0.85$--$0.88)\,\mu_B^{\rm ch}/T_{\rm ch}$: the $\bar p/p$ ratio is thus largely, but not exactly, set at the chemical surface, the $12$--$15\%$ deficit being significant at up to $3.2\sigma$. The resulting kinetic freeze-out trajectory in the $(\rho_B,\varepsilon)$ plane shows that decoupling occurs in a dilute system, reaching only $\rho_B \simeq 0.03\,n_0$ at $7.7$~GeV, \fx{a factor of $\simeq25$} below the chemical-surface compression maximum, while longitudinal flow raises the inferred \fx{$\rho_B$} by up to a factor of $\sim$4.}