Highly occupied gauge theories in $2+1$ dimensions: A self-similar attractor

Motivated by the boost-invariant Glasma state in the initial stages in heavy-ion collisions, we perform classical-statistical simulations of SU(2) gauge theory in $2+1$ dimensional space-time both with and without a scalar field in the adjoint representation. We show that irrespective of the details of the initial condition, the far-from-equilibrium evolution of these highly occupied systems approaches a unique universal attractor at high momenta that is the same for the gauge and scalar sectors. We extract the scaling exponents and the form of the distribution function close to this nonthermal fixed point. We find that the dynamics are governed by an energy cascade to higher momenta with scaling exponents $\alpha =3\beta$ and $\beta =-1/5$. We argue that these values can be obtained from parametric estimates within kinetic theory indicating the dominance of small momentum transfer in the scattering processes. We also extract the Debye mass nonperturbatively from a longitudinally polarized correlator and observe an IR enhancement of the scalar correlation function for low momenta below the Debye mass.

Figure 1

The gauge distributions $f_{\mathrm{E}}$ at the initial time $Qt=0$ for three different initial conditions are shown by dashed lines; note that these same initial conditions are used for both 2D and $2\mathrm{D}+\mathrm{sc}$ theories. At a later time $Qt=4000$, full lines show the distributions from these initial conditions in the 2D theory and dashed-dotted lines in the $2\mathrm{D}+\mathrm{sc}$ theory. These six curves overlap so well that they are indistinguishable in this plot, demonstrating the attractor property of the common nonthermal fixed point for both theories.

Figure 2

Spectra prior to rescaling (top) and rescaled occupation numbers and momenta (bottom) are shown for the 2D theory at different times. For the rescaling, we used the reference time $Q{t}_r=500$ and the scaling exponents $\beta =-1/5$ and $\alpha =3\beta$. The gray dashed line corresponds to a power law $p^{-1}$. The blue arrow indicates the maximal value of the Debye mass $m_D$ for the times displayed.

Figure 3

Self-similar evolution of the $2\mathrm{D}+\mathrm{sc}$ theory for different times with rescaled occupation numbers and momenta for the gauge distribution $f_{\mathrm{E}}$ (top) and the scalar distribution $f_{\pi }$ (bottom). The same values for $\alpha$ and $\beta$ as in Fig. 2 are used. For comparison, we show the corresponding $f_{\mathrm{E}}$ of the 2D theory for $Qt=2000$ as a dark-blue line.

Figure 4

Different hard scales ${\mathrm{\Lambda }}^2(t)$ for the 2D and $2\mathrm{D}+\mathrm{sc}$ theories. The gauge-invariant definitions (17) and (18) are shown with dashed lines compared to the perturbative integral expressions ${\mathrm{\Lambda }}_{\mathrm{pert}}^2(t)$ in (19) as points. The curves are rescaled with $t^{2\beta }$ with $\beta =-1/5$.

Figure 5

Hard scales ${\mathrm{\Lambda }}_E^2(t)$ for different initial conditions in 2D theory. The gauge-invariant definition (17) is shown as dashed lines and the perturbative integral expressions (19) as points. The curves are rescaled with $t^{2\beta }$ with $\beta =-1/5$.

Figure 6

Correlation functions at $Qt=2000$ for 2D (top) and $2\mathrm{D}+\mathrm{sc}$ theories (bottom) shown for different discretization parameters. Black dashed-dotted lines correspond to fitting $⟨E_L{E}_L^{*}⟩$ to the functional form (20).

Figure 7

Time evolution of $m_D^2$ extracted from fits of the form (20) to the longitudinal correlator in Fig. 6 and shown for 2D and $2\mathrm{D}+\mathrm{sc}$ theories. The black dashed line corresponds to a power law $t^{2\beta }$ with $\beta =-1/5$, leading to Eq. (21).