Chemical equilibration in weakly coupled QCD

We study thermalization, hydrodynamization, and chemical equilibration in out-of-equilibrium quark-gluon plasma starting from various initial conditions using QCD effective kinetic theory, valid at weak coupling. In nonexpanding systems gauge bosons rapidly lose information of the initial state and achieve kinetic equilibrium among themselves, while fermions approach the equilibrium distribution only at a later time. In systems undergoing rapid longitudinal expansion, both gluons and quarks are kept away from equilibrium by the expansion, but the evolution is well described by fluid dynamics even before local thermal equilibrium is reached. For realistic couplings we determine the ordering between the separate hydrodynamization, chemical equilibration and thermalization time scales to be ${\tau }_{\text{hydro}}<{\tau }_{\mathrm{chem}}<{\tau }_{\text{therm}}$.

Figure 1

The rate of change of total fermion number density (blue lines) for a thermal gluon system with no initial quark density and $\lambda =0.1$. The dashed and dotted lines show the contributions to the rate from elastic $2\leftrightarrow 2$ and inelastic $1\leftrightarrow 2$ processes, while the solid lines are the sum of the two. The corresponding changes in the gluon number density (multiplied by $-1$) are shown by red lines.

Figure 2

The various effective temperatures [see Eq. (18)] as a function of time for a nonexpanding system with initial conditions of thermal gluons and no fermions. The time axis is scaled by ${\tau }_R=4\pi \eta /s/T_{\text{final}}$, where $\eta /s\approx 1900$ for $\lambda =0.1$. The red lines are the gluonic effective temperatures; the blue lines are the effective temperatures of the fermionic sector with $T_4^{\mathrm{eff}}>T_3^{\mathrm{eff}}>\cdots >T_{-1}^{\mathrm{eff}}$.

Figure 3

Chemical equilibration of fermion energy density (as a function of equilibrium energy density) for different coupling constants $\lambda =10$, 1.0, 0.1, which correspond to effective $\eta /s=1$, 35, 1900. Two cases shown: with zero initial fermion energy $e_q\approx 0$ and with zero initial gluon energy $e_g\approx 0$.

Figure 4

Chemical equilibration for an isotropic overoccupied gluon state with no initial fermions present. The effective temperature $T_{\alpha }$ of gluon (red lines) and fermion (blue lines) distribution functions are shown for $\lambda =0.1$. Note that for fermions $T_4^{\mathrm{eff}}>T_3^{\mathrm{eff}}>\cdots >T_{-1}^{\mathrm{eff}}$, but overoccupied gluons start with inverted ordering.

Figure 5

Chemical equilibration of fermion energy density (as a function of equilibrium energy density) for different coupling constants $\lambda =10$, 1.0, 0.1, which correspond to effective $\eta /s=1$, 35, 1900 for the system initialized with overoccupied gluon density (colored lines) and thermal initial conditions (gray lines).

Figure 6

The effective temperature $T_{\alpha }^{\mathrm{eff}}$ ($\alpha =-1,\dots ,4$) evolution in a longitudinally expanding system for QCD ($N_c=3$, $N_f=3$) and Yang-Mills ($N_c=3$, $N_f=0$). Initial conditions given by an anisotropic overoccupied gluon state ($\lambda =5$, $\xi =10$, $\eta /s\approx 2.75$) with no initial fermions, Eq. (29). Axes are scaled by time-dependent relaxation time and asymptotic temperatures ${\tau }_R(\tau )=(4\pi \eta /s)/T_{\mathrm{id}}(\tau )$ and $T_{\mathrm{id}}(\tau )$.

Figure 7

Fermion energy fraction in a longitudinally expanding system with different coupling constants $\lambda$. The system is initialized with an anisotropic overoccupied gluon state, Eq. (29) ($\xi =10$). The time axis is scaled by corresponding kinetic relaxation time ${\tau }_R(\tau )$, Eq. (28), for each value of $\lambda$.

Figure 8

(a) Total pressure anisotropy $P_L/P_T$ evolution in kinetic theory with overoccupied initial conditions Eq. (29). (b) Total energy density evolution relative to ideal estimate $e/e_{\mathrm{id}}=(T/T_{\mathrm{id}}{)}^4$ given by Eq. (27). Comparison with first order viscous temperature Eq. (34) shown by the black dashed curve.

Figure 9

Shear viscosity over entropy ratio as a function of the coupling constant $\lambda =N_c{g}^2$ in our leading order kinetic theory implementation with two regularization schemes of the elastic collision kernel (see the text). The dashed line corresponds to the next-to-leading-log result from Ref. [7].