  Roy-equation analyses on lattice data of $\pi\pi$ scattering phase shifts at
$m_\pi=391$MeV reveals that the lowest $f_0$ meson becomes a bound state under
this condition. In addition, there is a pair of complex poles below threshold
generated by crossing symmetry [X.-H. Cao et al., Phys. Rev. D 108, 034009
(2023)]. We use the $N/D$ method to partially recover crossing symmetry of the
$O(N)$ $\sigma$ model amplitude at leading order of $1/N$ expansion, and
qualitatively reproduce the pole structure and pole trajectories with varying
pion masses as revealed by Roy-equation analyses. The $\sigma$ pole trajectory
with varying temperature is also discussed and found to be similar to its
properties when varying $m_\pi$. As the temperature increases, the complex
$\sigma$ poles firstly move from the second Riemann sheet to the real axis
becoming two virtual state poles, and then one virtual state pole moves to the
first sheet turning into a bound state pole and finally tends to the pion pole
position at high temperature which is as expected from the chiral symmetry
restoration. Our results provide further evidences that the lowest $f_0$ state
extracted from experiments and lattice data plays the role of $\sigma$ meson in
the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss
the problems of the effective potential in the situation when $m_\pi$ and
temperature get large.
