Scaling of elliptic flow, recombination, and sequential freeze-out of hadrons in heavy-ion collisions

The scaling properties of elliptic flow of hadrons produced in ultrarelativistic heavy-ion collisions are investigated at low transverse momenta, $p_T\lesssim 2$ GeV. Utilizing empirical parametrizations of a thermalized fireball with collective-flow fields, the resonance recombination model (RRM) is employed to describe hadronization via quark coalescence at the hadronization transition. We reconfirm that RRM converts equilibrium quark distribution functions into equilibrated hadron spectra including the effects of space-momentum correlations on elliptic flow. This provides the basis for a controlled extraction of quark distributions of the bulk matter at hadronization from spectra of multistrange hadrons which are beligeved to decouple close to the critical temperature. The resulting elliptic flow from empirical fits at the BNL Relativistic Heavy Ion Collider exhibits transverse kinetic-energy and valence-quark scaling. Utilizing the well-established concept of sequential freeze-out, the scaling at low momenta extends to bulk hadrons ($\pi$, $K$, $p$) at thermal freeze-out, albeit with different source parameters compared to chemical freeze-out. Elliptic-flow scaling is thus compatible with both equilibrium hydrodynamics and quark recombination.

Figure 1

Comparison of $\varphi$-meson $p_T$ spectra computed from (i) resonance recombination of strange quarks using blast-wave flow fields for strange quarks (rectangles), and (ii) a direct application of an equilibrium blast-wave distribution (solid line) with the same flow field as for quarks (solid line). The hadronization temperature has been chosen at $T_c=180$ MeV and the radial flow at the surface as $v_0=0.55$.

Figure 2

Comparison of elliptic flow for $\varphi$, $D$, and $J/\psi$ mesons (top to bottom) computed from (i) resonance recombination of strange quarks (symbols) and (ii) a direct application of the blast-wave expression at the meson level (lines). Both quark and meson distributions are based on the same anisotropic flow field constructed from the RL parametrization with $T_c=180$ MeV, $R_x=7.5$ fm, $R_y=8.5$ fm, ${\rho }_0=0.66$, and ${\rho }_2=0.07$.

Figure 3

Blast-wave fits to STAR data [13, 29, 30, 31] in semicentral Au-Au($\sqrt{s_{\mathit{NN}}}=200$) collisions for transverse-momentum spectra (a) and elliptic flow (b) of $\varphi$, ${\Xi }^{+}$, and $(\Omega +\mathrm{\Omega ̄})/2$ using the RL ansatz with parameters specified in Table . Panel (c) shows the resulting $\Omega /\varphi$ ratio on a linear scale, compared with STAR data [29].

Figure 4

(a) Equilibrated transverse-momentum distributions of quarks on a hypersurface with temperature just above $T_c$ for a midcentral fireball at RHIC energies as predicted by the RRM model from group-I data. (b) The same for the $v_2$ of the equilibrated quarks. The quark masses used (in GeV) are $m_{u/d}=0.3$, $m_s=0.45$, $m_c=1.5$, and $m_b=4.5$.

Figure 5

Blast-wave fits using the RL fireball parametrization to proton and kaon $p_T$ spectra (a) and elliptic flow (b) in mid-central Au-Au collisions at RHIC. The flow-field source parameters are listed in Table . Experimental data are taken from Refs. [3, 37, 38].

Figure 6

Parameter contours for ${\rho }_2$ and $R_y/R_x$ enclosing the regions of consistency with RL blast-wave fits to $p_T$ spectra and $v_2$ in the thermal regime for both freeze-out groups. The temperature $T$ and average radial-flow rapidity ${\rho }_0$ have been fixed at $(T,{\rho }_0)=(180\hspace{0.5em}\mathrm{MeV},0.75)$ and (110 MeV, 0.93) for groups I and II, respectively.

Figure 7

${\mathrm{KE}}_T$ and $n_q$ scaling of $v_2$ for different hadrons obtained within the sequential freeze-out scenario with blast-wave parameters given in Table . The result of a slightly later freeze-out for pions at temperature 100 MeV is also shown.

Figure 8

Comparison of the generic ${\mathrm{KE}}_T$-scaling behavior of hadron-$v_2$ extracted from Fig. 7 (thick solid line) to results for interchanging the freeze-out configurations for group-I and group-II particles. Specifically, the $\Omega$ baryon (dash-dotted line) is frozen out from the late source at $T_{\mathrm{fo}}=110$ MeV, while pions (dotted line) and kaons (dashed line) are frozen out from the early source at $T_c=180$ MeV (source parameters as quoted in Table ).