Trinification

In physics, the trinification model is a Grand Unified Theory proposed by Alvaro De Rújula, Howard Georgi and Sheldon Glashow in 1984.

Details

It states that the gauge group is either

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

or

: U(3)_C\times SU(3)_L\times SU(3)_R\mathbb_3;

and that the fermions form three families, each consisting of the representations: $\mathbf Q=(3,\bar,1)$, $\mathbf Q^c=(\bar,1,3)$, and $\mathbf L=(1,3,\bar)$. 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 $(1,3,\bar)$ and maybe also a $(1,\bar,3)$ 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 U(2)\times U(1)\mathbb_2.

The fermions branch (see restricted representation) as

:$(3,\bar,1)\rightarrow(3,2)_\oplus(3,1)_$,

:$(\bar,1,3)\rightarrow 2\,(\bar,1)_\oplus(\bar,1)_$,

:$(1,3,\bar)\rightarrow 2\,(1,2)_\oplus(1,2)_\oplus2\,(1,1)_0\oplus(1,1)_1$,

and the gauge bosons as

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

:$(1,8,1)\rightarrow(1,3)_0\oplus(1,2)_\oplus(1,2)_\oplus(1,1)_0$,

:$(1,1,8)\rightarrow 4\,(1,1)_0\oplus 2\,(1,1)_1\oplus 2\,(1,1)_$.

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

:$(1,3,\bar)_H(3,\bar,1)(\bar,1,3)$

and

:$(1,3,\bar)_H(1,3,\bar)(1,3,\bar)$.

Note that calling representations things like $(3,\bar,1)$ 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

:$\pi_2\left(\frac\right)=\mathbb$,

this model predicts 't Hooft–Polyakov magnetic monopoles.

Trinification is a maximal subalgebra of E₆, whose matter representation has exactly the same representation and unifies the $(3,3,1)\oplus(\bar,\bar,1)\oplus(1,\bar,3)$ fields. E₆ adds 54 gauge bosons, 30 it shares with SO(10), the other 24 to complete its $\mathbf\oplus\mathbf$.

References

{{Reflist

Grand Unified Theory