A unification of left-right SU(3)L × SU(3)R, colour SU(3)C and family SU(3)F symmetries in a maximal rank-8 subgroup of E8 is proposed as a landmark for future explorations beyond the Standard Model (SM). We discuss the implications of this scheme in a supersymmetric (SUSY) model based on the trinification gauge [SU(3)] 3 and global SU(3)F family symmetries. Among the key properties of this model are the unification of SM Higgs and lepton sectors, a common Yukawa coupling for chiral fermions, the absence of the µ-problem, gauge couplings unification and proton stability to all orders in perturbation theory. The minimal field content consistent with a SM-like effective theory at low energies is composed of one E6 27-plet per generation as well as three gauge and one family SU(3) octets inspired by the fundamental sector of E8. The details of the corresponding (SUSY and gauge) symmetry breaking scheme, multi-scale gauge couplings' evolution, and resulting effective low-energy scenarios are discussed.

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In this work, we present a trinification-based grand unified theory incorporating a global SU(3) family symmetry that after a spontaneous breaking leads to a left-right symmetric model. Already at the classical level, this model can accommodate the matter content and the quark Cabbibo mixing in the Standard Model (SM) with only one Yukawa coupling at the unification scale. Considering the minimal low-energy scenario with the least amount of light states, we show that the resulting effective theory enables dynamical breaking of its gauge group down to that of the SM by means of radiative corrections accounted for by the renormalisation group evolution at one loop. This result paves the way for a consistent explanation of the SM breaking scale and fermion mass hierarchies.

Using the one-loop Coleman-Weinberg effective potential, we derive a general analytic expression for all the derivatives of the effective potential with respect to any number of classical scalar fields. The result is valid for a renormalisable theory in four dimensions with any number of scalars, fermions or gauge bosons. This result corresponds to the zero-external momentum contribution to a general one-loop diagram with N scalar external legs. We illustrate the use of the general result in two simple scalar singlet extensions of the Standard Model, to obtain the dominant contributions to the triple couplings of light scalar particles under the zero external momentum approximation.

We present a supersymmetric (SUSY) model based on trinification ½SUð3Þ 3 and family SUð3Þ F symmetries embedded into a maximal subgroup of E 8 , where the sectors of light Higgs bosons and leptons are unified into a single chiral supermultiplet. The common origin of gauge trinification and of the family symmetry from E 8 separates the model from other trinification-based GUTs, as it protects, in particular, the Standard Model fermions from gaining mass until the electroweak symmetry is broken. Furthermore, it allows us to break the trinification symmetry via vacuum expectation values in SU(3)-adjoint scalars down to a left-right symmetric theory. Simultaneously, it ensures the unification of the gauge and Yukawa couplings as well as proton stability. Although the low-energy regime (e.g., mass hierarchies in the scalar sector determined by a soft SUSY-breaking mechanism) is yet to be established, these features are one key to revive the once very popular trinification-based GUTs.