on the spectrum of mesons, page 2    1 3

Scattering and resonances
     The knowledge necessary for the construction of such model of quarks and mesons, is basically scattering theory. You at first may wonder what scattering has to do with confinement. But, if you think a minute, or two, about the consequences of quark-pair creation, then you understand why there is a relation between confinement and scattering. In the figure this relation is depicted. The thing in the middle of the diagram, is the resonance we are interested in. However, because of quark-pair creation it decays in two newly formed mesons. Hence, the whole process of the formation of a resonance may be viewed as the scattering of two mesons. Hence, you need to know about the S matrix, its relation to the scattering cross section, and the singularity structure of the scattering amplitude. The latter is achieved by extending linear momenta and energy to the complex plane. Resonances are then described by complex poles in the scattering amplitude, whereas, bound states correspond to real poles. Studying how to control pole positions by varying the model parameters, is exciting, and teaches you a lot about the properties of scattering amplitudes and cross sections.

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