Investigation of Resonances in the $Σ({1/2}^{-})$ System Based on the Chiral Quark Model

TL;DR

This study addresses the Σ(1/2^−) resonance problem by examining both three-quark and five-quark contributions within a chiral quark model. It employs the Gaussian Expansion Method to construct precise multiquark orbital wave functions and the complex-scaling method to identify resonances, testing two parameter sets to gauge stability. The three-quark sector yields two states around $1.75$–$1.82$ GeV, consistent with Σ(1750) and Σ(1900)$, while the five-quark sector produces bound states in $N\bar{K}$, $N\bar{K}^*$, and $Σπ$, which become resonances $R(1302)$, $R(1395)$, and $R(1830)$ under channel coupling, revealing a two-pole structure analogous to the Λ(1380)–Λ(1405) system. However, the $R(1830)$ width is too small to fully account for the experimental Σ(1750)/Σ(1900) widths unless mixing with three-quark components is included, suggesting a mixed-state nature for these resonances with decays dominated by $\Lambda\pi$; the results motivate further unquenched studies and experimental searches in the $\Lambda\pi$ spectrum.

Abstract

In this work, we investigate the resonance structures in the $Σ(1/2^-)$ system from both three-quark and five-quark perspectives within the framework of the chiral quark model. An accurate few-body computational approach, the Gaussian Expansion Method, is employed to construct the orbital wave functions of multiquark states. To reduce the model dependence on parameters, we fit two sets of parameters to check the stability of the results. The calculations show that our results remain stable despite changes in the parameters. In the three-quark calculations, two $Σ(1/2^-)$ states are obtained with energies around 1.8~GeV, which are good candidates for the experimentally observed $Σ(1750)$ and $Σ(1900)$. In the five-quark configuration, several stable resonance states are identified, including $Σπ$, $N \bar{K}$, and $N \bar{K}^{*}$. These resonance states survive the channel-coupling calculations under the complex-scaling framework and manifest as stable structures. Our results support the existence of a two-pole structure for the $Σ(1/2^-)$ system, predominantly composed of $Σπ$ and $N \bar{K}$ configurations, analogous to the well-known $Λ(1380)$-$Λ(1405)$ ($Σπ$-$N \bar{K}$) system. On the other hand, although the energy of the $N \bar{K}^{*}$ configuration is close to that of $Σ(1750)$ and $Σ(1900)$, the obtained width is not consistent with the experimental values. This suggests that the $N \bar{K}^{*}$ state needs to mix with three-quark components to better explain the experimental $Σ(1750)$ and $Σ(1900)$ states. According to our decay width calculations, the predicted two resonance states are primarily composed of $Σπ$ and $N \bar{K}$, with their main decay channel being $Λπ$.

Figures (6)

• Figure 1: Comparison of energy levels between $\Sigma$ and $\Lambda$ baryons.
• Figure 2: Schematic complex energy distribution
• Figure 3: Complex-scaling results for the $\Sigma(1/2^{-})$ five-quark system in the 1200-2200 MeV range for the first set of parameters.
• Figure 4: Complex-scaling results for the $\Sigma(1/2^{-})$ five-quark system in the 1200-2200 MeV rangefor the second set of parameters.
• Figure 5: Calculated decay widths of resonance states in the $\Sigma(1/2^{-})$ five-quark system.(a)$\Sigma \pi \rightarrow \Lambda \pi$. (b)$N \bar{K} \rightarrow \Lambda \pi$. (c)$N \bar{K} \rightarrow \Sigma \pi$. (d)$N \bar{K}^{*} \rightarrow \Lambda \pi$. (e)$N \bar{K}^{*} \rightarrow \Sigma \pi$. (f)$N \bar{K}^{*} \rightarrow N \bar{K}$. (g)$N \bar{K}^{*} \rightarrow \Sigma \eta$.