Resonances and poles in the second Riemann sheet
In this work we study basic properties of unstable particles and scalar hadronic resonances, respectively, within simple quantum mechanical and quantum field theoretical (effective) models. We start with the basic ideas of quantum field theory. In particular, we introduce the Feynman propagator for unstable scalar resonances and motivate the idea that this kind of correlation function should possess complex poles which parameterize the mass and decay width of the considered particle. We also briefly discuss the problematic scalar sector in particle physics, emphasizing that hadronic loop contributions dominate its dynamics. A whole chapter is dedicated to the method of analytic continuation of complex functions through branch cuts into the so called second Riemann sheet. This method is crucial in order to describe physics of scalar resonances because the Feynman propagator of interacting quantum field theories will have branch cuts in the complex energy plane and complex poles in the second Riemann sheet. We then apply these concepts to a simple non-relativistic Lee model and demonstrate the physical implications, i.e., the motion of the propagator poles and the behaviour of the spectral function. Besides that, we investigate the time evolution of a particle described by such a model. In the last chapter, we finally concentrate on a simple quantum field theoretical model which describes the decay of a scalar state into two (pseudo)scalar ones. It is investigated how the motion of the propagator poles is influenced by loop contributions of the two (pseudo)scalar particles. We perform a numerical study for a hadronic system involving a scalar seed state (alias the σ-meson) that couples to pions. The unexpected emergence of a putative stable state below the two-pion threshold is investigated and it is clarified under which conditions such a stable state appears.