Generalized isospin, generalized mass groups, and generalized Gell-Mann--Okubo formalism
The current concepts of isospin and baryon mass groups are only well-adapted to deal with baryon multiplets involving both the u and d quarks, and some other quark k. In this paper, we generalize isospin and mass groups to accommodate baryon multiplets involving quarks of any flavor, and the Gell-Mann--Okubo (GMO) formalism is generalized accordingly. Generalized isospin proves to be a simple and valuable framework when working in non-udk baryon multiplets, and provides new quantum numbers that allows us to distinguish $\Lambda$-like baryons from $\Sigma$-like baryons in the non-udk multiplets. The generalized GMO formalism allows us to quantify the quality of flavor symmetries seen in baryon multiplets, and also allows us to predict the masses of all observable $J^P = \frac{1}{2}^+$ and $\frac{3}{2}^+$ baryons with an estimated accuracy on the order of 50 MeV in the worst cases, on mass scales that span anywhere from 1000 MeV to 15000 MeV.