The $N^*(1710)$ as a resonance in the $\pi\pi N$ system

TL;DR

This study models the $ star(1710)$ resonance as a $\pi\pi N$ system using Faddeev equations and chiral Lagrangians, identifying a resonance consistent with experimental data and suggesting the complexity of the Roper resonance.

Contribution

It introduces a novel approach to identify the $ star(1710)$ as a $\pi\pi N$ resonance through Faddeev equations and chiral potentials, without relying on traditional quark models.

Findings

Identifies a $ star(1710)$ resonance at 1704 MeV with quantum numbers $I=1/2$, $J^ ext{pi}=1/2^+$.

Finds the resonance where the $\pi\pi$ subsystem matches the $\sigma$ resonance region.

Does not find evidence for the Roper resonance or higher isospin states.

Abstract

We study the $\pi \pi N$ system by solving the Faddeev equations, for which the input two-body $t$-matrices are obtained by solving the Bethe-Salpeter equation in the coupled channel formalism. The potentials for the $\pi \pi$, $\pi N$ sub-systems and their coupled channels are obtained from chiral Lagrangians, which have been earlier used to study resonances in these systems successfully. In this work, we find a resonance in the $\pi\pi N$ system with a mass of $1704 - i 375/2$ MeV and with quantum numbers $I=1/2$, $J^\pi =1/2^+$. We identify this state with the $N^*(1710)$. This peak is found where the energies of the $\pi \pi$ sub-system fall in the region of the $\sigma$ resonance. We do not find evidence for the Roper resonance in our study indicating a more complex structure for this resonance, nor for any state with total isospin $I=3/2$ or $5/2$.