Spin—unitary-spin splitting of SU(8) supermultiplets

The surprising narrowness of the $J$ or $\psi \mathrm{}\left(3.1\right)\mathrm{}$ is interpreted as indication of a pure $c\overline{c}$ state, and hence as evidence for the $\mathrm{SU}\mathrm{}\left(8\right)\mathrm{}\to \mathrm{SU}\mathrm{}\left(6\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_{S_c}\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{Y_c}$ symmetry-breaking chain (${\overrightarrow{\mathrm{S}}}_c=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{m}\mathrm{e}\mathrm{d}-\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{k}-\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}$, $Y_c=\mathrm{hypercharm}\mathrm{generator}$) instead of an approximate $\mathrm{SU}\mathrm{}\left(8\right)\mathrm{}\to \mathrm{SU}\mathrm{}\left(4\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_S$ chain ($\overrightarrow{\mathrm{S}}=\mathrm{quark}\mathrm{spin}\mathrm{generators}$) which would imply strong mixings. Decompositions under both chains of the $s$-wave $q\overline{q}$ meson states of the 64 = 1 + 63 of SU(8) and of the $3q$ baryon states of the three-particle symmetric 120 representation are given. The most general mass-splitting operators with breaking in the $Y$ and $Y_c$ directions for these two multiplets are derived, which commute with the Casimir operators of the $\mathrm{SU}\mathrm{}\left(6\right)\mathrm{}\times \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_{S_c}\times \mathrm{U}{\mathrm{}\left(1\right)\mathrm{}}_{Y_c}$ chain, which contain only one- and two-body operators, and which are invariant under rotations. Two independent mass relations follow for mesons containing charmed quarks; six, for baryons containing charmed quarks. In an appendix, for reference relative to previous SU(6)-symmetric quark-model mass analyses, the reduced numerical coefficients as determined by the meson 36 of SU(6) are listed.