In physics, the trinification model is a GUT theory.
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: \( (3,{\bar {3}},1) \), \( ({\bar {3}},1,3) \) and : \( (1,3,{\bar {3}}) \).
This includes the right-handed neutrino, which can account for observed neutrino masses (but has not yet been proved to exist). See neutrino oscillations.
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 VEV. This results in a spontaneous symmetry breaking from
\( SU(3)_{L}\times SU(3)_{R} \) to \( [SU(2)\times U(1)]/{\mathbb {Z}}_{2} \)
and also,
\( (3,{\bar {3}},1)\rightarrow (3,2)_{{{\frac {1}{6}}}}\oplus (3,1)_{{-{\frac {1}{3}}}}, \)
\( ({\bar {3}},1,3)\rightarrow 2\,({\bar {3}},1)_{{{\frac {1}{3}}}}\oplus ({\bar {3}},1)_{{-{\frac {2}{3}}}}, \)
\( (1,3,{\bar {3}})\rightarrow 2\,(1,2)_{{-{\frac {1}{2}}}}\oplus (1,2)_{{{\frac {1}{2}}}}\oplus 2\,(1,1)_{0}\oplus (1,1)_{1}, \)
\( (8,1,1)\rightarrow (8,1)_{0}, \)
\( (1,8,1)\rightarrow (1,3)_{0}\oplus (1,2)_{{{\frac {1}{2}}}}\oplus (1,2)_{{-{\frac {1}{2}}}}\oplus (1,1)_{0}, \)
\( (1,1,8)\rightarrow 4\,(1,1)_{0}\oplus 2\,(1,1)_{1}\oplus 2\,(1,1)_{{-1}}. \)
See restricted representation.
Note that there are two Majorana neutrinos per generation (which is consistent with neutrino oscillations). Also, a copy of
\( (3,1)_{{-{\frac {1}{3}}}} \) and \( ({\bar {3}},1)_{\frac {1}{3}} \)
as well as
\( (1,2)_{\frac {1}{2}} \) and \( (1,2)_{-{\frac {1}{2}}} \)
per generation 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 the representations things like \( (3,{\bar {3}},1) \) and (8,1,1) is purely a physicist's convention, not a mathematician's convention, where representations are either labelled by Young tableaux or Dynkin diagrams with numbers on their vertices, but still, 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 monopoles. See 't Hooft–Polyakov monopole.
This model is suggested by Sheldon Lee Glashow, Howard Georgi and Alvaro de Rujula, in 1984. This is one of the maximal subalgebra of E6, whose matter representation 27 has exactly the same representation as above.
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