UV cascade in classical Yang-Mills theory via kinetic theory

We show that classical Yang-Mills theory with statistically homogeneous and isotropic initial conditions has a kinetic description and approaches a scaling solution at late times. We find the scaling solution by explicitly solving the Boltzmann equations, including all dominant processes (elastic and number-changing). Above a scale ${\tilde{p}}_{\max }\propto t^{\frac{1}{7}}$ the occupancy falls exponentially in $p$. For asymptotically late times and sufficiently small momenta the occupancy scales as $f(p)\propto 1/p$, but this behavior sets in only at very late time scales. We find quantitative agreement of our results with lattice simulations, for times and momenta within the range of validity of kinetic theory.

Figure 1

Scaling solution for the occupancy, when ${\tilde{m}}_{\mathrm{D}}=0.08$. Kinetic theory results use the momentum discretization method (dashed black line) and via the Ansatz method (green dash-dotted curve). Lattice results are shown using a large volume (red circles, used in the IR for $\tilde{p}\lesssim 1.2$) and a smaller volume but with careful extrapolation to the continuum limit (blue circles, used in the UV for $\tilde{p}\gtrsim 0.4$). The beginning of the arrow marks the scale ${\tilde{m}}_{\mathrm{D}}$ above which the lattice and kinetic theory results should agree.

Figure 2

Components of the Boltzmann equation, expressed as relative contributions to the particle number evolution $(t/f)df/dt$, computed with the discrete momentum method with ${\tilde{m}}_{\mathrm{D}}=0.08$. $1\leftrightarrow 2$ processes (red dash-dot), $2\leftrightarrow 2$ processes (blue dashes), the sum (black solid).

Figure 3

(a) Evolution of the scaling solution as a function of the screening scale ${\tilde{m}}_{\mathrm{D}}$ computed using the discrete momentum method. The solutions coincide with each other for $\tilde{p}>1$, and exhibit a bump feature at the scale ${\tilde{m}}_{\mathrm{D}}$. The dashed red line is a fit to the UV part of the spectrum. (b) The same as (a), but the $x$-axis is scaled so that the screening scale stays at fixed $x=1$ as a function of time. While at early times when ${\tilde{m}}_{\mathrm{D}}\lesssim 1$, a power-law $\tilde{f}\propto {\tilde{p}}^{\alpha }$ with $\alpha <-1$ may be seen in the data (in particular $\alpha \sim -4/3$ for ${\tilde{m}}_{\mathrm{D}}\sim 0.1$), at later times when a proper scaling window has emerged, the solution for ${\tilde{m}}_{\mathrm{D}}<\tilde{p}<1$ approaches $\tilde{f}\propto {\tilde{p}}^{-1}$.

Figure 4

$2\to 3$ processes in the limit where $\theta \sim m_{\mathrm{D}}/p_{\max }$ and $q\sim m_{\mathrm{D}}$ give rise to effective $1\leftrightarrow 2$ splittings.