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Trinification

## Summary

In physics, the trinification model is a Grand Unified Theory proposed by Alvaro De Rújula, Howard Georgi and Sheldon Glashow in 1984.[1][2]

## Details

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.

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

## References

1. ^ De Rujula, A.; Georgi, H.; Glashow, S. L. (1984). "Trinification of all elementary particle forces". In Kang, K.; Fried, H.; Frampton, F. (eds.). Fifth Workshop on Grand Unification. Singapore: World Scientific.
2. ^ Hetzel, Jamil; Stech, Berthold (2015-03-25). "Low-energy phenomenology of trinification: An effective left-right-symmetric model". Physical Review D. 91 (5): 055026. arXiv:1502.00919. doi:10.1103/PhysRevD.91.055026. ISSN 1550-7998.