  Resonance, bound-state, and virtual-state pole positions of the $f_0(500)$
scalar meson are computed as a continuous function of pion mass in the
framework of a unitarized and analytic coupled-channel model for scalar mesons,
described as dynamical quark-antiquark states. The $f_0(500)$ is modeled with
both light and strange $q\bar{q}$ seeds, mixing with each other through the
common $S$-wave $\pi\pi$, $K\bar{K}$, and $\eta\eta$ meson-meson decay
channels. The few model parameters are fitted to experimental $S$-wave $\pi\pi$
phase shifts up to 1 GeV. In the case of the physical $\pi^\pm$ mass of 139.57
MeV, resonance poles at $(460-i222)$ MeV and $(978-i37.2)$ MeV are found for
the $f_0(500)$ and $f_0(980)$, respectively. Resonance, bound-state, and
virtual-state pole trajectories are computed and plotted as a function of pion
masses up to 500 MeV, both in the complex-energy and complex-momentum planes.
The results are discussed and compared to the most advanced lattice QCD
computations employing interpolators that correspond to the $q\bar{q}$ and
meson-meson channels in the present model, that is, for a few discrete values
of the unphysical pion mass in those lattice calculations.
