Tri-unification: a separate SU(5) for each fermion family

In this paper we discuss SU(5)³ with cyclic symmetry as a possible grand unified theory (GUT). The basic idea of such a tri-unification is that there is a separate SU(5) for each fermion family, with the light Higgs doublet(s) arising from the third family SU(5), providing a basis for charged fermion mass hierarchies. SU(5)³ tri-unification reconciles the idea of gauge non-universality with the idea of gauge coupling unification, opening the possibility to build consistent non-universal descriptions of Nature that are valid all the way up to the scale of grand unification. As a concrete example, we propose a grand unified embedding of the tri-hypercharge model \({\text{U}}{\left(1\right)}_{Y}^{3}\) based on an SU(5)³ framework with cyclic symmetry. We discuss a minimal tri-hypercharge example which can account for all the quark and lepton (including neutrino) masses and mixing parameters. We show that it is possible to unify the many gauge couplings into a single gauge coupling associated with the cyclic SU(5)³ gauge group, by assuming minimal multiplet splitting, together with a set of relatively light colour octet scalars. We also study proton decay in this example, and present the predictions for the proton lifetime in the dominant e⁺π⁰ channel.