Triplicated Trinification

Gauge-coupling unification is just as successful in the standard model with six Higgs doublets as it is in the minimal supersymmetric standard model. However, the gauge couplings unify at 10 GeV, which yields rapid proton decay in the SU(5) model. I propose that the grand-unified gauge group is instead SU(3)c × SU(3)L × SU(3)R, in which baryon number is conserved by the gauge interactions. Nature appears to come in triplicate. The elementary fermions of nature come in three identical generations of quarks and leptons, distinguished only by their couplings to the Higgs field. We have no understanding of why nature chooses to triplicate itself. In contrast, there is only a single Higgs field in the standard model, which is responsible for breaking the electroweak symmetry and generating the masses of all the particles. This is the simplest model of electroweak symmetry breaking, and it is consistent with all data. However, it seems odd that there should be only one Higgs field, when the fermion fields come in triplicate. The gauge sector of the standard model, based on the symmetry SU(3)c×SU(2)L×U(1)Y , does not come in triplicate. It was observed long ago that this gauge symmetry can be unified into an SU(5) gauge group with a single gauge coupling [1]. When the SU(5) gauge symmetry is broken at a high energy scale, the SU(3)c×SU(2)L×U(1)Y gauge couplings evolve to their low-energy values [2]. However, precision measurements reveal that the low-energy values of the gauge couplings are not consistent with SU(5) grand unification. As is well known, the minimal supersymmetric standard model nudges the relative evolution of the gauge couplings just enough to bring them into accord with SU(5) grand unification [3, 4, 5]. The reason for this is three-fold. First, the relative evolution of the gauge couplings is unaffected when one adds a complete SU(5) representation [2]. Since the fermions are in complete SU(5) representations, the addition of their superpartners does not affect the relative evolution of the gauge couplings [6]. Second, the superpartners of the gauge bosons (which are not in complete SU(5) representations) change only the unification scale, since they have the same gauge structure as the gauge bosons [6]. Third, the minimal supersymmetric standard model requires two Higgs doublets in order to generate masses for all the fermions. Since the Higgs field is not in a complete SU(5) representation, the addition of a second Higgs doublet, as well as the superpartners of these two Higgs doublets, modifies the relative evolution of the gauge couplings. Thus it is the extension of the Higgs sector that is behind the successful SU(5) unification of the gauge couplings in the minimal supersymmetric standard model [7, 8]. In the renormalization-group equations responsible for the evolution of the gauge couplings, a (chiral) fermion field counts twice as much as a (complex) scalar field with the same gauge quantum numbers, at least at leading order. Thus the successful SU(5) unification of the gauge couplings in the minimal supersymmetric standard model, with its two Higgs doublets and their fermionic superpartners, can be mimicked by the standard model with six Higgs doublets. Since six is a multiple of three, this implies a triplication of the Higgs sector, in keeping with the triplication of the fermion fields. Thus, with respect to the unification of gauge couplings, the six-Higgs-doublet standard model is on the same footing as the minimal supersymmetric standard model. However, the unification of the gauge couplings occurs at a lower scale in the six-Higgs-doublet model. The evolution of the gauge couplings is given at leading order by 1 αi(μ) − 1 αi(μ′) = bi 2π ln (