Unitary model analysis of $f_0(500)$ pole positions by continuously varying $m_{\pi }$: comparison with discrete lattice predictions

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.