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 $U(1{)}_Y^3$ 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 $e^{+}{\pi }^0$ channel.