Even-wave harmonic oscillator theory of baryonic states: A new classification

An even-wave harmonic oscillator (h.o.) model for the quark-quark interaction proposed recently for the baryon spectrum is described with a detailed mathematical formulation. The mechanism, which formally admits of a relativistic extension of the Feynman-Kislinger-Ravndal type, leaves unchanged the usual h.o. predictions for 56 states (symmetric) for all $L$ values even and odd, but totally keeps out the 20 states (antisymmetric). It changes the structure of the 70 states considerably, while retaining the principal feature of linear rise of ${\mathrm{}\left(\mathrm{mass}\right)\mathrm{}}^2$ with $J$ through the interplay of two reduced slopes of magnitudes $\frac{1}{2}\alpha$ and $\frac{1}{2}\sqrt{3}\alpha$, compared to $\alpha$ for the 56 spectrum. The new features of the 70 states are (i) a dual spectrum leading to considerable mass splitting compared to the usual h.o. model without SU(6)-breaking effects, (ii) prediction of a unique (70, $0^{+}$) supermultiplet lower than the (70, $1^{-}$), and (iii) the prediction of low radial excitations because of the reduced slopes. The immediate experimental successes are (i) an understanding of $P_{11}\mathrm{}\left(1470\right)\mathrm{}$ together with possible $\Delta$, $\Sigma$, $\Lambda$ counterparts, (ii) two distinct mass groupings manifest in (70, $1^{-}$) states, and (iii) plausible explanation of $P_{11}^{\prime }\mathrm{}\left(1750\right)\mathrm{}$ as a radial excitation of $P_{11}\mathrm{}\left(1470\right)\mathrm{}$. The mass splittings of $\Delta$, $\Sigma$, $\Lambda$ from their $N$ counterparts, compared for 56 and 70 states, conform extremely well to the ratio of the average slope $\delta =\frac{1}{4}\alpha (1+\sqrt{3})\approx 0.68 \alpha$ for 70 states to that ($\alpha$) for the 56, thus facilitating the prediction of $\Delta$, $\Sigma$, $\Lambda$ positions from those of $N$ states for different quantum numbers. Extra predictions of states are discussed in terms of an extended classification scheme given by an ordered set of four quantum numbers ($n_x{l}_x{n}_y{l}_y$) defined in the text.