Electromagnetic Interactions and the Subgroups ofSU(6)

Various sets of relations concerning the electromagnetic properties of the constituents of the $\mathrm{SU}\left(6\right)\mathrm{}$ supermultiplets were recently derived by various authors. A systematic study of these relations is presented here, using subgroups of $\mathrm{SU}\left(6\right)\mathrm{}$ which are generalizations of the $U$-spin subgroup of $\mathrm{SU}\left(3\right)\mathrm{}$. It is shown that two different assumptions on the transformation properties of the electromagnetic contributions to the particle masses are not in contradiction with the present data: (a) Invariance to all orders under $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_U\otimes \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R\otimes \mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_P$, where $R$ and $P$ are the total quark spins of the $Q=-\frac{1}{3}$ and $Q=\frac{2}{3}$ quarks, respectively. (b) Second-order contributions from an $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_U$-invariant interaction which transforms like a 35. Moreover, the relation $p-n=N^{*+}-N^{*0\mathrm{}}$ is obtained from all the various assumptions which are stronger than the usual $U$-spin invariance. This relation may serve as a criterion for the usefulness of any discussion of electromagnetic mass differences according to $\mathrm{SU}\left(6\right)\mathrm{}$.