Resonating group method for baryon-baryon interactions with unequal oscillator frequencies and its application to the $NΔ$ system in a chiral quark model

TL;DR

The paper addresses a foundational inconsistency in quark‑model BB studies that assume a common oscillator frequency for all baryons. It develops a quark‑level RGM formalism for two‑baryon systems with unequal frequencies, including bound‑state and scattering treatments with CM motion removed, using generator coordinates and a Gaussian relative basis with b_{AB}^2 = 1/(M_A ω_A) + 1/(M_B ω_B). Applied to the NΔ system in a chiral SU(3) quark model, the approach reveals significant differences from traditional equal‑ω results, including a nonzero confinement contribution to the adiabatic interaction and no bound NΔ state, with phase shifts across several partial waves notably altered. This framework enables a self‑consistent determination of qq interactions and provides a robust foundation for exploring BB interactions and exotic multiquark states such as dibaryons in a unified manner.

Abstract

The resonating group method (RGM) is widely used to investigate baryon-baryon interactions at the quark level, typically under the assumption that the two baryons involved share an identical harmonic-oscillator frequency. In reality, however, when a specific interaction Hamiltonian is given, different baryons should have unequal oscillator frequencies due to distinct interaction potentials induced by their different quantum numbers. In this work, we develop a new quark-level RGM formalism for baryon-baryon systems with unequal oscillator frequencies, with the aim of describing both single baryons and baryon-baryon interactions in a consistent framework. We present the formalism for solving bound-state and scattering problems, with particular emphasis on constructing the wave functions of two-baryon systems with unequal oscillator frequencies. The proposed formalism is then applied to the $NΔ$ system within a chiral SU(3) quark model, where the quark-quark interaction includes, in addition to the one-gluon exchange (OGE) and a phenomenological confinement potential, the nonet scalar and pseudoscalar meson exchanges arising from the spontaneous breaking of chiral SU(3) symmetry. The distinctive features of the newly developed formalism are elucidated by comparing the results from the new formulation with those from traditional calculations.

Figures (6)

• Figure 1: Kinetic-energy contribution to the adiabatic interaction in the $S$-wave $N\Delta$ system (isospin $I=1$) as a function of the generator coordinate $S_i$. Solid lines: results from the present work. Dashed lines: results from traditional RGM calculations where the oscillator frequency for $\Delta$ is set equal to that for $N$.
• Figure 2: Confinement-potential contribution to the adiabatic interaction in the $S$-wave $N\Delta$ system ($I=1$). Notations are the same as in Fig. \ref{['fig:kine']}.
• Figure 3: Contributions from the OGE potential and from $\sigma$- and $\pi$-meson exchanges to the adiabatic interaction in the $S$-wave $N\Delta$ system ($I=1$). Notations are the same as in Fig. \ref{['fig:kine']}.
• Figure 4: Total Hamiltonian-induced adiabatic interactions for the $S$-wave $N\Delta$ systems with $I=1$ (upper panel) and $I=2$ (lower panel). Notations are the same as in Fig. \ref{['fig:kine']}.
• Figure 5: $N\Delta$ ($I=1$) phase shifts for the $^3S_1$ and $^3D_1$ partial waves. Notations are the same as in Fig. \ref{['fig:kine']}.