  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 \sigma-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.

High Energy Physics - Phenomenology