Abstract
We discuss in general terms pole trajectories of resonances coupling to a continuum channel as some strength parameter is varied. It is demonstrated that, regardless of the underlying dynamics, the trajectories of poles that couple to the continuum in a partial wave higher than s-wave are qualitatively the same, while in case of s-waves the pole trajectory can reveal important information on the internal structure of the resonance. In addition we show that only molecular (or extraordinary) states appear near thresholds naturally, while more compact structures need a significant fine tuning in the parameters.This study is of current relevance especially in strong interaction physics, since lattice QCD may be employed to deduce the pole trajectories for hadronic resonances as a function of the quark mass thus providing additional, new access to the structure of s-wave resonances.
Highlights
If all mesons were qq states there would be no natural reason for poles in scattering amplitudes to occur very close to thresholds
Weinberg derived relations between the scattering length, a, the effective range, r, and the wave function renormalization constant for the particle described by the S-matrix pole, Z, a=2
In order to illustrate with a realistic example what was described in the previous sections, we show the results for the pole trajectories of the ρ-meson and the σ -meson calculated within the inverse amplitude method (IAM) [26]
Summary
If all mesons were qq states there would be no natural reason for poles in scattering amplitudes to occur very close to thresholds. In this paper we look carefully at the way that the manifestations of poles in scattering amplitudes change as the poles approach thresholds as some strength parameter is varied — here one may think of varying the quark masses in lattice QCD calculations. Generic trajectories of s-wave resonances do lead to poles whose real part of the position is below threshold, but whose imaginary part of the position does not vanish, before giving rise to virtual bound states, and bound states, as some strength parameter is varied. While this observation is in line with the findings of Refs.
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