In numerical simulations of nonabelian plasma instabilities in the hard-loop approximation, a turbulent spectrum has been observed that is characterized by a phase-space density of particles n(p)∼p−ν with exponent ν≃2, which is larger than expected from relativistic 2↔2 scatterings. Using the approach of Zakharov, L’vov and Falkovich, we analyze possible Kolmogorov coefficients for relativistic (m≥4)-particle processes, which give at most ν=5/3 perturbatively for an energy cascade. We discuss non-perturbative scenarios which lead to larger values. As an extreme limit we find the result ν=5 generically in an inherently non-perturbative effective field theory situation, which coincides with results obtained by Berges et al. in large-N scalar field theory. If we instead assume that scaling behavior is determined by Schwinger–Dyson resummations such that the different scaling of bare and dressed vertices matters, we find that intermediate values are possible. We present one simple scenario, which would single out ν=2.
High Energy Phys Vertex Function Loop Order Energy Cascade Scaling Exponent
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V. Zakharov, V. L’vov, G. Falkovich, Kolmogorov Spectra of Turbulence (Springer, Berlin, 1992)