Current-Generated Algebra and Mass Levels of the Hadrons

Abstract

The mass splittings of the $U\left(3\right)\mathrm{}$ [or $\mathrm{SU}\left(3\right)\mathrm{}$] multiplets of the hadrons are investigated under the following assumptions: (i) The Hamiltonian decomposes into an invariant part plus an eighth component of an octet of $U\left(3\right)\mathrm{}$, and the latter is a space integral of the scalar current transforming like $\frac{1}{2}\left(q^{†}\beta {\lambda }_{8}q\right)$, where $q$ stands for the quark. (ii) An algebra of the positive parity operators generating a nonchiral $U\left(3\right)\mathrm{}\times U\left(3\right)\mathrm{}$ is a "good symmetry," where the generators consist of the scalar currents and the fourth components of the vector currents. It is shown that if a universal constant with the dimensions of a mass together with the average mass of the multiplet are given, then the splitting is calculated exactly for any multiplet under the above assumptions. A kind of competition of the kinematical group $U\left(3\right)\mathrm{}$ and the dynamical group $U\left(3\right)\mathrm{}\times U\left(3\right)\mathrm{}$ can predict the average masses of $U\left(3\right)\mathrm{}$ multiplets if they belong to the same multiplet of $U\left(3\right)\mathrm{}\times U\left(3\right)\mathrm{}$. The multiplets that concern us are the ${\mathrm{½}}^{+}$ octet, ${\frac{3}{2}}^{+}$ decuplet, ${\frac{3}{2}}^{-}$ octet, $0^{-}$ nonet, $2^{+}$ nonet, and $1^{-}$ nonet. The results show good agreement with experiment.