  We present calculations of the $S$-wave isospin-zero and isospin-two
pion-pion scattering lengths within a nucleon-meson model with parity doubling.
Both scattering lengths are computed in various approximations, ranging from a
mean-field (MF) calculation towards the inclusion of loop corrections by means
of the functional renormalization group (FRG). The bosonic part of the
investigated nucleon-meson model is formulated in terms of stereographic
projections as a "natural" set of coordinates on the respective vacuum
manifold. We thereby elucidate subtleties concerning the truncation of the
effective action w.r.t. higher-derivative pion interactions and the
"successful" computation of the scattering lengths. As the main result, we find
simultaneous agreement for the isospin-zero and isospin-two scattering lengths
with experimental data within the $\mathrm{LPA}^{\prime}$-truncation of the
FRG, together with chiral symmetry breaking (roughly) occurring at the
characteristic scale of $4\pi f_{\pi}$. The isoscalar $\sigma$-mass is
dynamically generated by the FRG integration of momentum modes, and is a
prediction of the model. It ends being of the order of $500\ \mathrm{MeV}$,
i.e., much lower than the value ($> 1\ \mathrm{GeV}$) found in MF or one-loop
treatment of this or related models. Finally, the convergence of the
corresponding low-energy expansion of the quantum effective action in terms of
pion momenta is discussed.
