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.
Hunting resonance poles with Rational Approximants
基于数学上定义良好的Padé理论，提出了一种理论上安全的提取极点质量和共振宽度的新方法。特别地，由于Montessus de Ballore定理，我们能够展开振幅的第二黎曼单，以搜索复平面中谐振极点的位置。该方法是系统的，并提供了一个独立于模型的处理的预测和相应的近似误差。这封信部分涵盖了作者在2010年6月28日至7月3日在法国蒙彼利埃举行的第15届国际量子色动力学会议：QCD 10（25周年纪念）和2010年8月30日至9月3日在西班牙马德里举行的夸克禁闭和强子谱IX上提交的材料。
