Dynamical generation of resonances in the P33 partial wave

TL;DR

This paper explores how resonances in the P33 partial wave can form dynamically through pion interactions and how genuine quark states influence these resonances, providing insights into the nature of the $ ext{Δ}(1232)$ and $ ext{Δ}(1600)$.

Contribution

It demonstrates the emergence of resonances without initial quark states and analyzes their evolution with interaction strength, linking them to known baryon resonances.

Findings

Two resonances emerge at ~1200 MeV and ~1400 MeV without quark states.

Genuine quark states mix with dynamical resonances, explaining the $ ext{Δ}(1232)$ and $ ext{Δ}(1600)$.

Helicity amplitudes' $Q^2$ dependence can distinguish resonance models.

Abstract

We investigate the formation of resonances in the P33 partial wave with the emphasis on possible emergence of dynamically generated quasi-bound states as a consequence of a strong $p$-wave pion attractive interaction in this partial wave, as well as their possible interaction with the genuine quark excited states. By using the Laurent-Pietarinen expansion we follow the evolution of the $S$-matrix poles in the complex energy plane as a function of the interaction strength. Already without introducing a genuine quark resonant state, two physically interesting resonances emerge with pole masses around 1200 MeV and 1400 MeV, with the dominant $\pi N$ and $\pi\Delta$ component, respectively. The added genuine resonant state in the $(1s)^3$ quark configuration mixes with the lower dynamically generated resonance forming the physical $\Delta(1232)$ resonance, and pushes the second dynamical resonance to around 1500 MeV, which allows it to be identified with the $\Delta(1600)$ resonance. Adding a second resonant state with one quark promoted to the $2s$ orbit generates another pole whose evolution remains well separated from the lower two poles. We calculate the helicity amplitudes at the pole and suggest that their $Q^2$ dependence could be a decisive test to discriminate between different models of the $\Delta(1600)$ resonance.