Two-Pole Structure of $\Lambda(1405)$ with Temporal Evolution and Spatial Distribution
The $\Lambda(1405)$ is a special hadron resonance associated with two poles of the scattering amplitudes, and its nature remains under debate since its discovery before the birth of the quark model. In this work we study the structures of these two poles and their temporal evolution, including their difference, interference, and synergy. Each pole can usually be represented by the Gamow vector $|\psi^{\rm Gamow}\rangle$ in the complex momentum space $|\vec p e^{-i\theta}\rangle$. We construct its representation $|\psi^{\rm phys}\rangle$ in the real momentum space $|\vec p\rangle$ through the analytic continuation of the Gamow wavefunction, which also satisfies the Hamiltonian eigenvalue equation with the assistance of a virtual state vector. Both the decreasing behavior of the resonance and the production of the decayed scattering states can be simultaneously described by the temporal evolution $|\psi^{\rm phys},t\rangle=\exp(-iH t) \, |\psi^{\rm phys}\rangle$. The state $|\psi^{\rm phys},t=0\rangle$ gives the finite-range confinement of the resonance while $|\psi^{\rm phys},t\to \infty\rangle$ provides a Breit-Wigner-like distribution of the final scattering states whose appearance probability is nonzero as $r\to \infty$. In the two-channel system $\pi\Sigma$-$\bar{K}N$, we first dynamically generate the two poles of $\Lambda(1405)$ and then discuss their temporal evolutions and spatial distributions which can produce results consistent with experimental measurements such as the $\pi\Sigma$ invariant-mass spectrum and provide a new path to study hadron resonances.