Dynamics of chemical equilibrium of hadronic matter close to $T_c$

Quick chemical equilibration times of hadrons (specifically, $p\mathrm{p̄}$, $K\mathrm{K̄}$, $\Lambda \mathrm{\Lambda ̄}$, and $\Omega \mathrm{\Omega ̄}$ pairs) within a hadron gas are explained dynamically using Hagedorn states, which drive particles into equilibrium close to the critical temperature. Within this scheme, we use master equations and derive various analytical estimates for the chemical equilibration times. We compare our model to recent lattice results and find that for both $T_c=176$ MeV and $T_c=196$ MeV, the hadrons can reach chemical equilibrium almost immediately, well before the chemical freeze-out temperatures found in thermal fits for a hadron gas without Hagedorn states. Furthermore, the ratios $p/\pi$, $K/\pi$, $\Lambda /\pi$, and $\Omega /\pi$ match experimental values well in our dynamical scenario.

Figure 1

Comparison of trace anomaly to lattice QCD results from Refs. [18, 19], where $T_c=196$ MeV. HS is in reference to our model including Hagedorn states.

Figure 2

Comparison of entropy density to lattice QCD results from Refs. [18, 19], where $T_c=196$ MeV. HS is in reference to our model including Hagedorn states.

Figure 3

Comparison of energy density to lattice QCD results from Ref. [17], where $T_c=176$ MeV. HS is in reference to our model including Hagedorn states.

Figure 4

Comparison of entropy density to lattice QCD results from Ref. [17], where $T_c=176$ MeV. HS is in reference to our model including Hagedorn states.

Figure 5

Hagedorn states decay into multiple pions and an $X\mathrm{X̄}$ pair.

Figure 6

Average number of $\Lambda$’s and $\Omega$’s. The $\Omega$’s are calculated within our canonical ensemble and the $\Lambda$’s are calculated in both our canonical ensemble and a microcanonical ensemble.

Figure 7

Comparison of the chemical equilibrium times for p’s, K’s, $\Lambda$’s, and $\Omega$’s when $\alpha =1$ and ${\beta }_i=1$, where (a) $T_H=176$ MeV and (b) $T_H=196$ MeV. The band at the bottom of the panel is the range of chemical equilibrium times for the Hagedorn states (see Table ).

Figure 8

Graph of the number of proton-antiproton pairs, kaon-antikaon pairs, and $\Lambda$-anti-$\Lambda$ pairs when both the resonances and the pions are held in equilibrium for $T_H=176$ MeV.

Figure 9

Graph of the number of proton-antiproton pairs, kaon-antikaon pairs, and $\Lambda$-anti-$\Lambda$ when both the resonances and the pions are held in equilibrium for $T_H=196$ MeV.

Figure 10

Comparison of the chemical equilibration times of the pions to the total chemical equilibration time for (a) $T_H=176$ MeV and (b) $T_H=196$ MeV.

Figure 11

Entropy per pion for a hadron gas in chemical equilibrium within the fireball ansatz.

Figure 12

The temperature-time relationship is directly linked to the average transversel velocity chosen in Eq. (18) within the fireball model ansatz.

Figure 13

Ratio of the entropy of the Hagedorn states to the total entropy.

Figure 14

Comparison of the effective pion numbers when (a) $T_H=176$ MeV or (b) $T_H=196$ MeV.

Figure 15

Comparison of the total number of $X\mathrm{X̄}$ and their effective numbers when (a) $T_H=176$ MeV or (b) $T_H=196$ MeV.

Figure 16

Results for the (a) $p\mathrm{p̄}$, (b) $K\mathrm{K̄}$, and (c) $\Lambda \mathrm{\Lambda ̄}$ when the pions and Hagedorn resonances are held in equilibrium for $T_H=176$ MeV.

Figure 17

Results for the (a) $p\mathrm{p̄}$, (b) $K\mathrm{K̄}$, and (c) $\Lambda \mathrm{\Lambda ̄}$ when the pions and Hagedorn resonances are held in equilibrium for $T_H=196$MeV.

Figure 18

Results for the (a) $p$’s and (b) pions with various initial conditions for $T_H=176$ MeV.

Figure 19

Results for the (a) $p$’s and (b) pions with various initial conditions for $T_H=196$ MeV.

Figure 20

Results for the ratio of $p$’s with various initial conditions for (a) $T_H=176$ MeV or (b) $T_H=196$ MeV. Note that for STAR $p/{\pi }^{-}=0.11$ and $\mathrm{p̄}/{\pi }^{-}=0.082$.

Figure 21

Results for the (a) $K$’s and (b) pions with various initial conditions for $T_H=176$ MeV.

Figure 22

Results for the (a) $K$’s and (b) pions with various initial conditions for $T_H=196$ MeV.

Figure 23

Results for the ratio of $K$’s with various initial conditions for (a) $T_H=176$ MeV or (b) $T_H=196$ MeV. Note that for STAR $K^{+}/{\pi }^{-}=0.16$ and $K^{-}/{\pi }^{-}=0.15$.

Figure 24

Results for the (a) $\Lambda$’s and (b) pions with various initial conditions for $T_H=176$ MeV.

Figure 25

Results for the (a) $\Lambda$’s and (b) pions with various initial conditions for $T_H=196$ MeV.

Figure 26

Results for the ratio of $\Lambda /\pi$’s with various initial conditions for (a) $T_H=176$ MeV or (b) $T_H=196$ MeV. Note that for STAR $\Lambda /{\pi }^{-}=0.54$ and $\mathrm{\Lambda ̄}/{\pi }^{-}=0.41$.

Figure 27

Plot of the various ratios including all initial conditions defined in Table . The points show the ratios at $T=110$ MeV for the various initial conditions (circles are for $T_H=176$ MeV and diamonds are for $T_H=196$ MeV). The experimental results for STAR and PHENIX are shown by the black error bars.

Figure 28

Results for the (a) $\Omega$’s and (b) pions with various initial conditions for $T_H=176$ MeV.

Figure 29

Results for the (a) $\Omega$’s and (b) pions with various initial conditions for $T_H=196$ MeV.

Figure 30

Results for the ratio of $\Omega /\pi$’s with various initial conditions for (a) $T_H=176$ MeV or (b) $T_H=196$ MeV. Note that for STAR $\Omega /{\pi }^{-}=9.5\times {10}^{-4}$ and $\mathrm{\Omega ̄}/{\pi }^{-}=9.6\times {10}^{-4}$.

Figure 31

(a) Numerical and analytical results for the pions and Hagedorn states, where $\eta =0.9$ at $T=175$ MeV for $T_H=176$ MeV when ${\beta }_i=1.1$ and $\alpha =0.9.$ (b) Numerical results for the same initial conditions including $K\mathrm{K̄}$ pairs with $\varphi =0$.