Comprehensive analysis of the wave function of a hadronic resonance and its compositeness

Abstract

We develop a theoretical framework to investigate the two-body composite structure of a resonance as well as a bound state from its wave function. For this purpose, we introduce both one-body bare states and two-body scattering states, and define the compositeness as a fraction of the contribution of the two-body wave function to the normalization of the total wave function. Writing down explicitly the wave function for a resonance state obtained with a general separable interaction, we formulate the compositeness in terms of the position of the resonance pole, the residue of the scattering amplitude at the pole and the derivative of the Green function of the free two-body scattering system. At the same time, our formulation provides the elementariness expressed with the resonance properties and the two-body effective interaction, and confirms the sum rule showing that the summation of the compositeness and elementariness gives unity. In this formulation the Weinberg's relation for the scattering length and effective range can be derived in the weak binding limit. The extension to the resonance states is performed with the Gamow vector, and a relativistic formulation is also established. As its applications, we study the compositeness of the $\Lambda (1405)$ resonance and the light scalar and vector mesons described with refined amplitudes in coupled-channel models with interactions up to the next to leading order in chiral perturbation theory. We find that $\Lambda (1405)$ and $f_0(980)$ are dominated by the $\bar{K}N$ and $K\bar{K}$ composite states, respectively, while the vector mesons $\rho (770)$ and $K^{\ast }(892)$ are elementary. We also briefly discuss the compositeness of $N(1535)$ and $\Lambda (1670)$ obtained in a leading-order chiral unitary approach.

34 pages, 1 figure, version accepted for publication in PTEP

References in corpus (21)

• Big-Bang Nucleosynthesis
• Heavy quarkonium: progress, puzzles, and opportunities
• Effective Kbar N interaction based on chiral SU(3) dynamics
• Origin of resonances in the chiral unitary approach
• Multi-quark hadrons from Heavy Ion Collisions
• Measurement of the Sigma pi photoproduction line shapes near the Lambda(1405)
• Photoproduction of Lambda(1405) and Sigma^{0}(1385) on the proton at E_γ= 1.5-2.4 GeV
• Evidence of the Θ^+ in the γd \to K^+K^-pn reaction
• Couplings in coupled channels versus wave functions in the case of resonances: application to the two $\Lambda (1405)$ states
• Remarks on pole trajectories for resonances
• Electromagnetic mean squared radii of Lambda(1405) in chiral dynamics
• Hadron mass scaling near the s-wave threshold
• On the nature of the $\Lambda (1405)$ and $\Lambda (1670)$ from their $N_c$ behavior in chiral dynamics
• Structure of the Lambda(1405) baryon resonance from its large Nc behavior
• Mixing of pseudoscalar-baryon and vector-baryon in the $J^P=1/2^{-}$ sector and the $N^{\ast }$(1535) and $N^{\ast }$(1650) resonances
• Nature of the sigma meson as revealed by its softening process
• Elementarity of composite systems
• The composite nature of the $\Lambda (1520)$ resonance
• Study of the $\gamma d\to K^{+}{K}^{-}np$ reaction and an alternative explanation for the "${\Theta }^{+}(1540)$ pentaquark" peak
• How and when can one identify hadronic molecules in the baryon spectrum
• Double-pole nature of $\Lambda (1405)$ studied with coupled-channel complex scaling method using complex-range Gaussian basis

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