Hadron production in heavy ion collisions: Fragmentation and recombination from a dense parton phase

We discuss hadron production in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC). We argue that hadrons at transverse momenta $P_T<5\text{GeV}$ are formed by recombination of partons from the dense parton phase created in central collisions at RHIC. We provide a theoretical description of the recombination process for $P_T>2\text{GeV}$. Below $P_T=2\text{GeV}$ our results smoothly match a purely statistical description. At high transverse momentum hadron production is well described in the language of perturbative QCD by the fragmentation of partons. We give numerical results for a variety of hadron spectra, ratios, and nuclear suppression factors. We also discuss the anisotropic flow $v_2$ and give results based on a flow in the parton phase. Our results are consistent with the existence of a parton phase at RHIC hadronizing at a temperature of $175\text{MeV}$ and a radial flow velocity of $0.55c$.

This article appears in the following collection:

Figure 1

Sketch of trajectories of hadrons and their precursors moving close to the speed of light in the $t–r$ plane where $t$ is the time in the lab frame and $r$ the radial coordinate. The hypersurface $\Sigma$ (solid line) and a generalization with finite width (the shell given by dashed lines) are shown. The emission time $\Delta {\tau }_E$ during which recombination occurs is given by the conditions in the surrounding medium and the trajectory. We approximate this short time by taking the hypersurface $\Sigma$ infinitely thin. A radially expanding hypersurface does not alter the argument.

Figure 2

(Color online) Spectrum of $u$ and $s$ quarks at hadronization in a central $\text{Au}+\text{Au}$ collision at RHIC. Perturbative partons from hard QCD processes with subsequent energy loss (dashed lines) and the thermal phase with $T=175\text{MeV}$ and radial flow $v_T=0.55c$ (solid lines) are shown.

Figure 3

(Color online) Hadron spectra at midrapidity as a function of transverse momentum $P_T$ for central $\text{Au}+\text{Au}$ collisions at $\sqrt{S}=200\text{GeV}$. We show fragmentation (dotted line), recombination (dashed line) and the sum of both contributions (solid line) at $b=3\text{fm}$ vs data from PHENIX and STAR (for $\Phi$, ${\Xi }^{+}+{\Xi }^{-}$, and $\Omega +\overline{\Omega }$ recombinations only). $\Omega +\overline{\Omega }$ data are minimum bias, therefore the result of a calculation with impact parameter $b=10\text{fm}$ is also shown. All data are preliminary except ${\pi }^0$, $p$, and $\overline{p}$ data from PHENIX. All error bars give statistical errors only, except for ${\pi }^0$ data from PHENIX which give the total error. See Sec. 5 for more details.

Figure 4

(Color online) Relative deviation $r_{\delta }$ of the $\delta$-function approximation from calculations using wide wave functions for neutral pions and protons.

Figure 5

(Color online) The ratio $r\left(P_T\right)=R∕(R+F)$ of recombined hadrons to the sum of recombination and fragmentation for ${\pi }^0$ (solid line), $K_s^0$ (dashed line), and $p$ (dotted lines). For protons and pions different impact parameters $b=0$, 7.5, and $12\text{fm}$ (from top to bottom) are shown. $K_s^0$ is for $b=0\text{fm}$ only.

Figure 6

(Color online) The spectrum of charged hadrons $(h^{+}=h^{-})∕2$ for four different impact parameters 3, 7.5 (divided by 25), 10 (/200), and $12\text{fm}\left(\mathrm{/1000}\right)$ (from top to bottom) in comparison with data from STAR and PHENIX in different centrality bins. Contributions from fragmentation only (dotted) and the sum of recombination and fragmentation ($R+F$, solid lines) are shown.

Figure 7

(Color online) Hadron ratios $p∕{\pi }^0$, $\overline{p}∕{\pi }^0$, $\overline{p}∕p$, $K^{+}∕{\pi }^{+}$, $K^{-}∕{\pi }^{-}$, $K^{-}∕K^{+}$, $(\Lambda +\overline{\Lambda })∕4K_s^0$, $2({\Xi }^{-}+{\Xi }^{+})∕(\Lambda +\overline{\Lambda })$, and $2{\pi }^0∕(h^{+}+h^{-})$ as functions of transverse momentum $P_T$. The calculation for $\Xi$ baryons only takes into account recombination. We show data from STAR, PHENIX, and BRAHMS and results from the statistical model (dash-dotted lines). Where several curves for the statistical model are shown these are for different strangeness fugacities 1.0, 0.9, and 0.8 (from top to bottom).

Figure 8

(Color online) The spectrum of neutral pions for impact parameters 3, 5.5 (divided by 5), 7.5 (∕25), 9 (∕100), 10 (∕200), 11 (∕500), 12 (∕1000), 13 (∕2000), and $13.9\text{fm}$ (∕5000) (from top to bottom) in comparison with data from PHENIX in different centrality bins. Contributions from fragmentation only (dotted lines) and the sum of recombination and fragmentation ($R+F$, solid lines) are shown. $13.9\text{fm}$ calculation is fragmentation $\left(F\right)$ only.

Figure 9

(Color online) The spectrum of neutral kaons for impact parameters 3, 10 (∕25), and $12\text{fm}$ (∕200) (from top to bottom) in comparison with data from STAR in different centrality bins. Contributions from fragmentation only (dotted lines) and the sum of recombination and fragmentation ($R+F$, solid lines) are shown.

Figure 10

(Color online) The ratio $p∕{\pi }^0$ for three different impact parameters 3, 7.5, and $13\text{fm}$ (solid lines, top to bottom) compared to the statistical model (dash-dotted line) and PHENIX data.

Figure 11

(Color online) Nuclear modification factor $R_{AA}$ for ${\pi }^0$ at impact parameters 3 (bottom) and $10\text{fm}$ (top). Normalization by $p+p$ is via our own calculation (solid lines) or via PHENIX $p+p$ results (dashed lines). Data on $R_{AA}$ are from PHENIX Collaboration with point-to-point errors only.

Figure 12

(Color online) $R_{\text{CP}}$ for neutral pions (bottom) and protons (top) given by the ratio of particle yields at impact parameters 3 and $12\text{fm}$ compared to data from PHENIX.

Figure 13

(Color online) $R_{\text{CP}}$ for charged hadrons given by the ratio of particle yields at impact parameters 3 and $12\text{fm}$ compared to data from STAR and PHENIX.

Figure 14

(Color online) $R_{\text{CP}}$ for $K_s^0$ (bottom) and $\Lambda +\overline{\Lambda }$ (top) given by the ratio of particle yields at impact parameters 3 and $12\text{fm}$ compared to data from STAR.

Figure 15

(Color online) Elliptic flow $v_2$ in the parton phase as a function of transverse momentum $p_T$. The flow in the thermal phase (dashed line) and a pQCD calculation (dotted line) are shown. The solid line interpolates between the two domains. The shaded region shows the region where the interpolation takes place.

Figure 16

(Color online) $v_2$ for positively charged pions. We show recombination only (dashed line), fragmentation only (dotted line), and the full calculation (solid line). The result of a calculation in the $\delta$-function approximation (NW) for the wave function is also shown (dash-dotted line). Data are taken from PHENIX Collaboration [57].

Figure 17

(Color online) Upper panel: anisotropic flow for $p$ and ${\pi }^{+}$ compared to PHENIX data [57]. Lower panel: $v_2$ for $K^{+}$ and $\Lambda +\overline{\Lambda }$ compared to preliminary STAR data ($\Lambda +\overline{\Lambda }$,$K_s^0$) [70] and PHENIX data ($K^{+}$,$K^{-}$) [57].

Figure 18

(Color online) $v_2$ for charged hadrons. Again we show the contributions from different mechanisms as in Fig. 16. Data are preliminary and taken from STAR Collaboration.

Figure 19

(Color online) The anisotropy $v_2∕n$ for pions (bottom) and protons (top) as a function of transverse momentum $p_T∕n$ using the scaling law (89) with $n=2$ for pions and $n=3$ for protons. Data points are pions and protons from PHENIX using the same scaling law.