Effective theories for real-time correlations in hot plasmas

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We discuss the sequence of effective theories needed to understand the qualitative, and quantitative, behavior of real-time correlators < A(t) A(0) > in ultra-relativistic plasmas. We analyze in detail the case where A is a gauge-invariant conserved current. This case is of interest because it includes a correlation recently measured in lattice simulations of classical, hot, SU(2)-Higgs gauge theory. We find that simple perturbation theory, free kinetic theory, linearized kinetic theory, and hydrodynamics are all needed to understand the correlation for different ranges of time. We emphasize how correlations generically have power-law decays at very large times due to non-linear couplings to long-lived hydrodynamic modes.