Symmetry-Breaking and Charge Operator in Su(9) Grand-Unification Models

We present a thorough analysis of U(1) generators in all possible symmetry-breaking modes that reduce SU(9) down to ${\mathrm{SU}}_{\mathrm{c}}$(3)\ifmmode\times\else\texttimes\fi{}${\mathrm{SU}}_{\mathrm{W}}$(2)\ifmmode\times\else\texttimes\fi{}U(1). Two types of representations that satisfy a set of grand-unification guidelines are considered. The first type embeds SU(5) trivially and the second type has nontrivial embedding of SU(5). Acceptable representations of both types are found in SU(9). Projection matrices as well as the contribution of U(1) eigenvalues to the weak hypercharge are given in all physically interesting modes of symmetry breaking. An example is explicitly worked out for the evolution of coupling strengths to determine the intermediate mass scales responsible for invisible axions and the delayed decay of the proton while maintaining a satisfactory value of ${\mathrm{sin}}^{2}$${\mathrm{theta}}_{\mathrm{W}}$ at low energies. Fermion contents are also discussed.