Non-Abelian plasma instabilities: SU(3) versus SU(2)

We present the first $3+1$ dimensional simulations of non-Abelian plasma instabilities in gauge-covariant Boltzmann-Vlasov equations for the QCD gauge group SU(3) as well as for SU(4) and SU(5). The real-time evolution of instabilities for a plasma with stationary momentum-space anisotropy is studied using a hard-loop effective theory that is discretized in the velocities of hard particles. We find that the numerically less expensive calculations using the group SU(2) essentially reproduce the nonperturbative dynamics of non-Abelian plasma instabilities with higher rank gauge groups provided the mass parameters of the corresponding hard-loop effective theories are the same. In particular, we find very similar spectra for the turbulent cascade that forms in the strong-field regime, which is associated with an approximately linear growth of energy in collective fields. The magnitude of the linear growth however turns out to increase with the number of colors.

Figure 1

Comparison of average energy densities $\mathcal{E}$ for SU(2) (light) and SU(3) (dark) on logarithmic (upper panel) and linear (lower panel) scale in $3+1$ dimensional simulations for anisotropy parameter $\xi =10$ on a ${64}^3$ lattice with $a{m}_{\infty }=0.5656$ (i.e., $a=0.16({\lambda }_{*}/2$), $L=64a=13.8{\lambda }_{\mu }$) and ${\mathcal{N}}_{\mathcal{W}}=320$ ($={\mathcal{N}}_Z\times {\mathcal{N}}_{\phi }=20\times 16$). The initialization of the $\mathcal{W}$ fields is chosen with parameter $\sigma =0.1$ resulting in equal initial field amplitudes per gluon degree of freedom for both gauge groups. Shown are the total field energy densities as well as the contributions from transverse and longitudinal chromo-electric and chromo-magnetic fields. In the lower panel, the individual runs are shown by light curves.

Figure 2

Same as Fig. 1 for the energy density divided by the number of gluons $N_g$.

Figure 3

Comparison of average total field energy densities $\mathcal{E}$ for SU(2) through SU(5) on logarithmic (upper panel) and linear (lower panel) scale in $3+1$ dimensional simulations for anisotropy parameter $\xi =10$ on a ${64}^3$ lattice with ${\mathcal{N}}_{\mathcal{W}}=320$ ($=N_Z\times N_{\varphi }=20\times 16$). The initialization of the $\mathcal{W}$ fields is chosen with parameter ${\sigma }_{\mathrm{SU}(2)}={\sigma }_{\mathrm{SU}(3)}={\sigma }_{\mathrm{SU}(4)}={\sigma }_{\mathrm{SU}(5)}=0.1$.

Figure 4

The ratio of the time derivative of the energy densities of Fig. 3 for gauge groups SU(3), SU(4), and SU(5), divided by that of SU(2). For large times, this is seen to scale approximately with $N_c$. Horizontal lines at $3/2$, $4/2$, and $5/2$ mark the ratios for scaling by the number of colors.

Figure 5

The ratio of the time derivative of the energy densities for various lattice parameters. (a) ${32}^3$, $a{m}_{\infty }=0.2828$, $\sigma =0.0707$; (b) ${32}^3$, $a{m}_{\infty }=0.2828$, $\sigma =3.46$; (c) ${64}^3$, $a{m}_{\infty }=0.1414$, $\sigma =0.1$. For the $\mathrm{SU}(5)/\mathrm{SU}(2)$ slope ratio, the same parameters as for (b) have been used.

Figure 6

Spectra corresponding to Fig. 1 for times $0\le m_{\infty }t\lesssim 150$. The distance between the lines is $m_{\infty }\Delta t\approx 6$. The upper panel shows the SU(2) spectra for the magnetic ($f_A(k)$) and electric ($f_E(k)$) distribution functions. The curves at time $m_{\infty }t\approx 23$ are plotted with thick lines. The lower panel shows the electric ($f_E(k)$) distribution functions for SU(2) vs SU(3).

Figure 7

The power spectrum for SU(2) of the Coulomb gauge distribution $f(k)$ at late times $80\lesssim m_{\infty }t\lesssim 150$. The distance between the lines is $m_{\infty }\Delta t\approx 11$. The upper panel shows a comparison for SU(2) for $f_E(k)$ and $f_A(k)$. The straight central horizontal red line indicates a power-law spectrum $f\sim k^{-\nu }$ with $\nu =2$, while the dotted lines correspond to $\nu =1.8$ and 2.2. The lower panel compares the spectra $f_E(k)$ for SU(2), SU(3), SU(4), and SU(5).