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We calculate spectral functions of the relativistic $O(4)$ model fromreal-time lattice simulations in classical-statistical field theory. While inthe low and high temperature phase of the model, the spectral functions oflongitudinal $(\sigma)$ and transverse $(\pi)$ modes are well described byrelativistic quasi-particle peaks, we find a highly non-trivial behavior of thespectral functions in the cross over region, where additional structuresappear. Similarly, we observe a significant broadening of the quasi-particlepeaks, when the amount explicit $O(4)$ symmetry breaking is reduced. We furtherdemonstrate that in the vicinity of the $O(4)$ critical point, the spectralfunctions develop an infrared power law associated with the critical dynamics,and comment on the extraction of the dynamical critical exponent $z$ from oursimulations.

Spectral functions and critical dynamics of the $O(4)$ model from classical-statistical lattice simulations