Natural parity states of SU(6)×O(3) in baryon spectra: Evidence for (56,${\mathrm{odd}}^{\mathrm{-}}$) via Δ states

The possibility of some natural odd parity Δ states in the latest PDG tables belonging to (56, ${\mathrm{odd}}^{\mathrm{-}}$) members of SU(6)×O(3) which start with (N=3) is explored dynamically, by employing a new complex harmonic oscillator (HO) representation which provides a compact ${\mathit{S}}_3$-symmetric description and is also realizable as a solution of a 3D dynamical equation (BSE) based on a vectorlike harmonic confinement. The complex representation permits the identification of an additional quantum number (${\mathit{N}}_{\mathit{a}}$) over and above those already realizable in the standard (real) HO basis, and also facilitates the solution of the above BSE in terms of the Casimir terms of several distinct SO(2,1) algebras deducible from the former. The model is precalibrated to a fairly representative cross section of the known (N,Δ) spectra so as to provide the needed mandate for extrapolation. Fair agreement with expected mass locations for several new ${\mathrm{\Delta }}^{\mathrm{-}}$ resonances is found in terms of appropriate (56, ${\mathrm{odd}}^{\mathrm{-}}$) quantum numbers which are not only unique in cases such as ${\mathit{G}}_{39}$ and ${\mathit{I}}_{3,13}$, but also compete very favorably with parallel (${70}^{\mathrm{-}}$) assignments in several others. More vigorous searches are therefore recommended for such ${\mathrm{\Delta }}^{\mathrm{-}}$ states which are much less contaminated by mixing than are the corresponding ${\mathit{N}}^{\mathrm{-}}$ states.