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In this paper we explore practicable ways for self-consistent calculations ofspectral functions from analytically continued functional renormalization group(aFRG) flow equations. As a particularly straightforward one we propose toinclude parametrizations of self-energies based on explicit analytic one-loopexpressions. To exemplify this scheme we calculate the spectral functions ofpion and sigma meson of the $O(4)$ model at vanishing temperature in the brokenphase. Comparing the results with those from previous aFRG calculations, weexplicitly demonstrate how self-consistency at all momenta fixes the tightrelation between particle masses and decay thresholds. In addition, thetwo-point functions from our new semi-analytic FRG scheme have the desireddomain of holomorphy built in and can readily be studied in the entirecut-complex frequency plane, on physical as well as other Riemann sheets. Thisis very illustrative and allows, for example, to trace the flow of theresonance pole of the sigma meson across an unphysical sheet. In order toassess the limitations due to the underlying one-loop structure, we alsointroduce a fully self-consistent numerical scheme based on spectralrepresentations with scale-dependent spectral functions. The most notableimprovement of this numerically involved calculation is that it describes thethree-particle resonance decay of an off-shell pion, $\pi^* \to\sigma\pi\to3\pi$. Apart from this further conceptual improvement, overallagreement with the results from the considerably simpler semi-analytic one-loopscheme is very encouraging, however. The latter can therefore provide a soundand practicable basis for self-consistent calculations of spectral functions inmore realistic effective theories for warm and dense matter.

Self-consistent $O(4)$ model spectral functions from analytically continued FRG flows