Trinification

Details

It states that the gauge group is either

:SU(3)_C\times SU(3)_L\times SU(3)_R

or

:[SU(3)_C\times SU(3)_L\times SU(3)_R]/\mathbb{Z}_3;

and that the fermions form three families, each consisting of the representations: \mathbf Q=(3,\bar{3},1), \mathbf Q^c=(\bar{3},1,3), and \mathbf L=(1,3,\bar{3}). The L includes a hypothetical right-handed neutrino, which may account for observed neutrino masses (see neutrino oscillations), and a similar sterile "flavon."

There is also a (1,3,\bar{3}) and maybe also a (1,\bar{3},3) scalar field called the Higgs field which acquires a vacuum expectation value. This results in a spontaneous symmetry breaking from

:SU(3)_L\times SU(3)_R to [SU(2)\times U(1)]/\mathbb{Z}_2.

The fermions branch (see restricted representation) as

:(3,\bar{3},1)\rightarrow(3,2){\frac{1}{6}}\oplus(3,1){-\frac{1}{3}},

:(\bar{3},1,3)\rightarrow 2,(\bar{3},1){\frac{1}{3}}\oplus(\bar{3},1){-\frac{2}{3}},

:(1,3,\bar{3})\rightarrow 2,(1,2){-\frac{1}{2}}\oplus(1,2){\frac{1}{2}}\oplus2,(1,1)_0\oplus(1,1)_1,

and the gauge bosons as

:(8,1,1)\rightarrow(8,1)_0,

:(1,8,1)\rightarrow(1,3)0\oplus(1,2){\frac{1}{2}}\oplus(1,2)_{-\frac{1}{2}}\oplus(1,1)_0,

:(1,1,8)\rightarrow 4,(1,1)_0\oplus 2,(1,1)1\oplus 2,(1,1){-1}.

Note that there are two Majorana neutrinos per generation (which is consistent with neutrino oscillations). Also, each generation has a pair of triplets (3,1){-\frac{1}{3}} and (\bar{3},1){\frac{1}{3}}, and doublets (1,2){\frac{1}{2}} and (1,2){-\frac{1}{2}}, which decouple at the GUT breaking scale due to the couplings

:(1,3,\bar{3})_H(3,\bar{3},1)(\bar{3},1,3)

and

:(1,3,\bar{3})_H(1,3,\bar{3})(1,3,\bar{3}).

Note that calling representations things like (3,\bar{3},1) and (8,1,1) is purely a physicist's convention, not a mathematician's, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but it is standard among GUT theorists.

Since the homotopy group

:\pi_2\left(\frac{SU(3)\times SU(3)}{[SU(2)\times U(1)]/\mathbb{Z}_2}\right)=\mathbb{Z},

this model predicts 't Hooft–Polyakov magnetic monopoles.

The trinification symmetry Lie algebra \mathfrak{su}(3)_C \oplus \mathfrak{su}(3)_L \oplus \mathfrak{su}(3)_R is a maximal subalgebra of E6, whose matter representation 27 has exactly the same representation and unifies the (3,3,1)\oplus(\bar{3},\bar{3},1)\oplus(1,\bar{3},3) fields. E6 adds 54 gauge bosons, 30 it shares with SO(10), the other 24 to complete its \mathbf{16}\oplus\mathbf{\overline{16}}.

References

References

1. (1984). "Fifth Workshop on Grand Unification". World Scientific.
2. Hetzel, Jamil. (2015-03-25). "Low-energy phenomenology of trinification: An effective left-right-symmetric model". Physical Review D.