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

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

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

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

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

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

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 $\mathbf {16} \oplus \mathbf {\overline {16}}$.