The Study of Pole Trajectory within a bare state in the coupled channel model

We investigate two-particle scattering and two-particle scattering with a bare basis state using Hamiltonian Effective Field Theory (HEFT). We analyze the distribution of two-body scattering poles in the momentum and energy planes under relativistic conditions. Compared to the non-relativistic case, there are significant differences in the distribution of bound state poles and resonance poles in the relativistic case, primarily due to the square root term in the relativistic formula. By considering pure two-particle scattering, we examine the relationship between the form factor and the number of poles. Additionally, we clearly elucidate the effects of attractive and repulsive interactions on the bound state poles and resonance poles. More importantly, we extend our model by including a bare state and explore the poles originating from the bare state or coupled channels through the trajectories of pole positions, as well as the compositeness of bound states.