Trinification

In physics, the trinification model is a Grand Unified Theory proposed by Alvaro De Rújula, Howard Georgi and Sheldon Glashow in 1984.¹²

It states that the gauge group is either

${\displaystyle SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}}$

or

${\displaystyle [SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}]/\mathbb {Z} _{3}}$;

and that the fermions form three families, each consisting of the representations: ${\displaystyle \mathbf {Q} =(3,{\bar {3}},1)}$, ${\displaystyle \mathbf {Q} ^{c}=({\bar {3}},1,3)}$, and ${\displaystyle \mathbf {L} =(1,3,{\bar {3}})}$. 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 ${\displaystyle (1,3,{\bar {3}})}$ and maybe also a ${\displaystyle (1,{\bar {3}},3)}$ scalar field called the Higgs field which acquires a vacuum expectation value. This results in a spontaneous symmetry breaking from

${\displaystyle SU(3)_{L}\times SU(3)_{R}}$ to ${\displaystyle [SU(2)\times U(1)]/\mathbb {Z} _{2}}$.

The fermions branch (see restricted representation) as

${\displaystyle (3,{\bar {3}},1)\rightarrow (3,2)_{\frac {1}{6}}\oplus (3,1)_{-{\frac {1}{3}}}}$,

${\displaystyle ({\bar {3}},1,3)\rightarrow 2\,({\bar {3}},1)_{\frac {1}{3}}\oplus ({\bar {3}},1)_{-{\frac {2}{3}}}}$,

${\displaystyle (1,3,{\bar {3}})\rightarrow 2\,(1,2)_{-{\frac {1}{2}}}\oplus (1,2)_{\frac {1}{2}}\oplus 2\,(1,1)_{0}\oplus (1,1)_{1}}$,

and the gauge bosons as

${\displaystyle (8,1,1)\rightarrow (8,1)_{0}}$,

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

${\displaystyle (1,1,8)\rightarrow 4\,(1,1)_{0}\oplus 2\,(1,1)_{1}\oplus 2\,(1,1)_{-1}}$.

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

${\displaystyle (1,3,{\bar {3}})_{H}(3,{\bar {3}},1)({\bar {3}},1,3)}$

and

${\displaystyle (1,3,{\bar {3}})_{H}(1,3,{\bar {3}})(1,3,{\bar {3}})}$.

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

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

this model predicts 't Hooft–Polyakov magnetic monopoles.

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