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

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

and that the fermions form three families, each consisting of the representations: $\mathbf{Q}=(3,\overline{3},1)$, ${\mathbf{Q}}^c=(\overline{3},1,3)$, and $\mathbf{L}=(1,3,\overline{3})$. 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 $(1,3,\overline{3})$ and maybe also a $(1,\overline{3},3)$ 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 $[SU(2)\times U(1)]/{\mathbb{\mathbb{Z}}}_2$.

The fermions branch (see restricted representation) as

$(3,\overline{3},1)\to (3,2{)}_{\frac{1}{6}}\oplus (3,1{)}_{-\frac{1}{3}}$,

$(\overline{3},1,3)\to 2(\overline{3},1{)}_{\frac{1}{3}}\oplus (\overline{3},1{)}_{-\frac{2}{3}}$,

$(1,3,\overline{3})\to 2(1,2{)}_{-\frac{1}{2}}\oplus (1,2{)}_{\frac{1}{2}}\oplus 2(1,1{)}_0\oplus (1,1{)}_1$,

and the gauge bosons as

$(8,1,1)\to (8,1{)}_0$,

$(1,8,1)\to (1,3{)}_0\oplus (1,2{)}_{\frac{1}{2}}\oplus (1,2{)}_{-\frac{1}{2}}\oplus (1,1{)}_0$,

$(1,1,8)\to 4(1,1{)}_0\oplus 2(1,1{)}_1\oplus 2(1,1{)}_{-1}$.

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 $(\overline{3},1{)}_{\frac{1}{3}}$, and doublets $(1,2{)}_{\frac{1}{2}}$ and $(1,2{)}_{-\frac{1}{2}}$, which decouple at the GUT breaking scale due to the couplings

$(1,3,\overline{3}{)}_H(3,\overline{3},1)(\overline{3},1,3)$

and

$(1,3,\overline{3}{)}_H(1,3,\overline{3})(1,3,\overline{3})$.

Note that calling representations things like $(3,\overline{3},1)$ 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

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

this model predicts 't Hooft–Polyakov magnetic monopoles. Trinification is a maximal subalgebra of E₆, whose matter representation 27 has exactly the same representation and unifies the $(3,3,1)\oplus (\overline{3},\overline{3},1)\oplus (1,\overline{3},3)$ fields. E₆ adds 54 gauge bosons, 30 it shares with SO(10), the other 24 to complete its $\mathbf{1}\mathbf{6}\oplus \overline{\mathbf{1}\mathbf{6}}$.