We compute the next-to-leading order O(g) correction to the thermal photon production rate in a QCD plasma. The NLO contributions can be expressed in terms of gauge invariant condensates on the light cone, which are amenable to novel sum rules and Euclidean techniques. We expect these technologies to be generalizable to other NLO calculations. For the phenomenologically interesting value of α s = 0.3, the NLO correction represents a 20% increase and has a functional form similar to the LO result.

We follow the time evolution of non-Abelian gauge bosons from far-from-equilibrium initial conditions to thermal equilibrium by numerically solving an effective kinetic equation that becomes accurate in the weak coupling limit. We consider isotropic initial conditions that are either highly overoccupied or underoccupied. We find that overoccupied systems thermalize through a self-similar cascade reaching equilibrium in multiples of a thermalization time t eq ≈ 72:=ð1 þ 0.12 log λ −1 Þ × 1=λ 2 T, whereas underoccupied systems undergo a "bottom-up" thermalization in a time t eq ≈ ½34: þ 21: logðQ=TÞ= ð1 þ 0.037 log λ −1 Þ × ðQ=TÞ 1=2 =λ 2 T, where Q is the characteristic momentum scale of the initial condition. We apply this result to model initial stages of heavy-ion collisions and find rapid thermalization roughly in a time Qt eq ≲ 10 or t eq ≲ 1 fm=c. Non-Abelian far-from-equilibrium plasmas occur in many cosmological pre-or reheating scenarios [1] or due to possible cosmological phase transitions [2], as well as in the early stages of heavy-ion collisions. These far-from-equilibrium systems may be overoccupied, such that the energy is spread out in longer wavelength modes than in thermal equilibrium but with stronger fields. This is the case, e.g., for fields generated through parametric resonance, and in heavy-ion collisions, at least in the limit of asymptotically large center of mass energies, where the initial condition may be described by using the colorglass-condensate framework [3]. Alternately, far-fromequilibrium systems may be underoccupied, such that the system consists of fewer, but more energetic, quasiparticles than the corresponding thermal system. This is the case in, e.g., Planck-suppressed decay of inflatons [4]. Also, even though the initial condition of heavy-ion collisions is overoccupied, it has been demonstrated by Baier, Mueller, Schiff, and Son [5] (see also [6]) that the longitudinal expansion renders the prethermal fireball underoccupied before it reaches local thermal equilibrium.This has motivated several numerical [7][8][9][10][11][12] and analytical [13,14] works to study simple far-from-equilibrium model systems, in particular, that of a single species of gauge bosons in a (nonexpanding) flat space-time with statistically isotropic initial conditions at weak coupling, which we investigate in this Letter with both over-and underoccupied initial conditions. In both cases, we follow the time evolution of the system from the initial far-fromequilibrium state to thermal equilibrium and extract the thermalization time, which we define as the exponent governing relaxation of the deviation of the first moment of the distribution function, hjpji ¼ R p jpjfðpÞ= R p fðpÞ, from its equilibrium value hjpji T at late times:In the overoccupied case, early dynamics fall onto a nonthermal attractor solution: if the initial momentum scale characterized by Q 2 ≡ hp 2 ðt ¼ 0Þi is much smaller than the momentum scale of the target thermal system, then the scattering time of the initial system τ ini...

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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 scalep max ∝ t 1 7 the occupancy falls exponentially in p. For asymptotically late times and sufficiently small momenta the occupancy scales as f (p) ∝ 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.