Constraint on color gauge groups

Abstract

Assuming that the colored gauge group $G_c$ is not necessarily the SU(3) but an arbitrary simple compact Lie group, we find that cases of $G_c$ being one of the groups SU(2), $\mathrm{SO}\mathrm{}\left(4l\right)\mathrm{}$, $\mathrm{SO}\mathrm{}(2l+1)\mathrm{}$, Sp ($2l$), ${\mathrm{G}}_2$, ${\mathrm{F}}_4$, ${\mathrm{E}}_7$, and ${\mathrm{E}}_8$ Will lead to a difficulty with the quark-line rule. Therefore, only the groups $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ ($n\ge 3$), $\mathrm{SO}\mathrm{}(4l+2)\mathrm{}$ ($l\ge 1$), and ${\mathrm{E}}_6$ remain to be possible candidates for $G_c$. It is argued that the same restriction also applies to any unified simple group $G$ of all strong, weak, and electromagnetic interactions.