Pole trajectories of the $\Lambda(1405)$ helps establish its dynamical nature

The $\Lambda(1405)$ has been one of the most controversial exotic baryons. If the $\Lambda(1405)$ possesses a two-pole molecular structure, these poles are expected to evolve differently towards the SU(3) limit. From an analysis of a recent LQCD simulation on the $\pi\Sigma-\bar{K}N$ scattering for $I=0$ and the study of the quark mass dependence of the octet baryon masses, we determine for the first time the trajectories of these poles towards the symmetric point over the $\mathrm{Tr}[M]=C$ trajectory accurately. At $m_\pi\simeq 200$ MeV, our results are consistent with the lattice simulations, and the extrapolations to the physical point, based on the NLO chiral Lagrangians, agree well with existing experimental analyses. We predict qualitatively similar trajectories at LO and up to NLO, consistent with the LO interaction's dominance. At the SU(3) symmetric point of this trajectory, both poles are on the physical sheet, and the lower pole is located at $E^{(1)}=1573(6)(6)$ MeV, becoming a SU(3) singlet, while the higher pole at $E^{(8a)}=1589(7)(5)$ MeV couples to the octet representation. Moreover, we make predictions in $I=1$ for the $\Sigma^*$ resonance. We find a resonance pole that evolves into a bound state around $m_\pi=415$ MeV in this sector. The results presented here are crucial to shed light on the molecular nature of exotic strange baryon resonances and can be tested in future LQCD simulations.