Conditions for anSUL(3)⊗U(1)gauge theory

We study the possibility of constructing an ${\mathrm{SU}}_L\mathrm{}\left(3\right)\mathrm{}\otimes \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ gauge theory of weak and electromagnetic interactions consistent with experiment and allowing room for the incorporation of renormalizable strong interactions. Greater than usual attention is paid to the stability of the Higgs potential and serious difficulties are encountered. We argue that a satisfactory $\mathrm{SU}{\mathrm{}\left(3\right)\mathrm{}}_L\otimes \mathrm{U}\mathrm{}\left(1\right)\mathrm{}$ model based on elementary Higgs fields requires a global or discrete symmetry beyond SU(3) in the Higgs sector.