  In this paper we explore practicable ways for self-consistent calculations of
spectral functions from analytically continued functional renormalization group
(aFRG) flow equations. As a particularly straightforward one we propose to
include parametrizations of self-energies based on explicit analytic one-loop
expressions. To exemplify this scheme we calculate the spectral functions of
pion and sigma meson of the $O(4)$ model at vanishing temperature in the broken
phase. Comparing the results with those from previous aFRG calculations, we
explicitly demonstrate how self-consistency at all momenta fixes the tight
relation between particle masses and decay thresholds. In addition, the
two-point functions from our new semi-analytic FRG scheme have the desired
domain of holomorphy built in and can readily be studied in the entire
cut-complex frequency plane, on physical as well as other Riemann sheets. This
is very illustrative and allows, for example, to trace the flow of the
resonance pole of the sigma meson across an unphysical sheet. In order to
assess the limitations due to the underlying one-loop structure, we also
introduce a fully self-consistent numerical scheme based on spectral
representations with scale-dependent spectral functions. The most notable
improvement of this numerically involved calculation is that it describes the
three-particle resonance decay of an off-shell pion, $\pi^* \to
\sigma\pi\to3\pi$. Apart from this further conceptual improvement, overall
agreement with the results from the considerably simpler semi-analytic one-loop
scheme is very encouraging, however. The latter can therefore provide a sound
and practicable basis for self-consistent calculations of spectral functions in
more realistic effective theories for warm and dense matter.
