It states that the gauge group is either
S U ( 3 ) C × S U ( 3 ) L × S U ( 3 ) R {\displaystyle SU(3)_{C}\times SU(3)_{L}\times SU(3)_{R}} or
[ S U ( 3 ) C × S U ( 3 ) L × S U ( 3 ) R ] / Z 3 {\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 : Q = ( 3 , 3 ¯ , 1 ) {\displaystyle \mathbf {Q} =(3,{\bar {3}},1)} , Q c = ( 3 ¯ , 1 , 3 ) {\displaystyle \mathbf {Q} ^{c}=({\bar {3}},1,3)} , and L = ( 1 , 3 , 3 ¯ ) {\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 ( 1 , 3 , 3 ¯ ) {\displaystyle (1,3,{\bar {3}})} and maybe also a ( 1 , 3 ¯ , 3 ) {\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
S U ( 3 ) L × S U ( 3 ) R {\displaystyle SU(3)_{L}\times SU(3)_{R}} to [ S U ( 2 ) × U ( 1 ) ] / Z 2 {\displaystyle [SU(2)\times U(1)]/\mathbb {Z} _{2}} .The fermions branch (see restricted representation ) as
( 3 , 3 ¯ , 1 ) → ( 3 , 2 ) 1 6 ⊕ ( 3 , 1 ) − 1 3 {\displaystyle (3,{\bar {3}},1)\rightarrow (3,2)_{\frac {1}{6}}\oplus (3,1)_{-{\frac {1}{3}}}} , ( 3 ¯ , 1 , 3 ) → 2 ( 3 ¯ , 1 ) 1 3 ⊕ ( 3 ¯ , 1 ) − 2 3 {\displaystyle ({\bar {3}},1,3)\rightarrow 2\,({\bar {3}},1)_{\frac {1}{3}}\oplus ({\bar {3}},1)_{-{\frac {2}{3}}}} , ( 1 , 3 , 3 ¯ ) → 2 ( 1 , 2 ) − 1 2 ⊕ ( 1 , 2 ) 1 2 ⊕ 2 ( 1 , 1 ) 0 ⊕ ( 1 , 1 ) 1 {\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
( 8 , 1 , 1 ) → ( 8 , 1 ) 0 {\displaystyle (8,1,1)\rightarrow (8,1)_{0}} , ( 1 , 8 , 1 ) → ( 1 , 3 ) 0 ⊕ ( 1 , 2 ) 1 2 ⊕ ( 1 , 2 ) − 1 2 ⊕ ( 1 , 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}} , ( 1 , 1 , 8 ) → 4 ( 1 , 1 ) 0 ⊕ 2 ( 1 , 1 ) 1 ⊕ 2 ( 1 , 1 ) − 1 {\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 ( 3 , 1 ) − 1 3 {\displaystyle (3,1)_{-{\frac {1}{3}}}} and ( 3 ¯ , 1 ) 1 3 {\displaystyle ({\bar {3}},1)_{\frac {1}{3}}} , and doublets ( 1 , 2 ) 1 2 {\displaystyle (1,2)_{\frac {1}{2}}} and ( 1 , 2 ) − 1 2 {\displaystyle (1,2)_{-{\frac {1}{2}}}} , which decouple at the GUT breaking scale due to the couplings
( 1 , 3 , 3 ¯ ) H ( 3 , 3 ¯ , 1 ) ( 3 ¯ , 1 , 3 ) {\displaystyle (1,3,{\bar {3}})_{H}(3,{\bar {3}},1)({\bar {3}},1,3)} and
( 1 , 3 , 3 ¯ ) H ( 1 , 3 , 3 ¯ ) ( 1 , 3 , 3 ¯ ) {\displaystyle (1,3,{\bar {3}})_{H}(1,3,{\bar {3}})(1,3,{\bar {3}})} .Note that calling representations things like ( 3 , 3 ¯ , 1 ) {\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
π 2 ( S U ( 3 ) × S U ( 3 ) [ S U ( 2 ) × U ( 1 ) ] / Z 2 ) = Z {\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 ( 3 , 3 , 1 ) ⊕ ( 3 ¯ , 3 ¯ , 1 ) ⊕ ( 1 , 3 ¯ , 3 ) {\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 16 ⊕ 16 ¯ {\displaystyle \mathbf {16} \oplus \mathbf {\overline {16}} } .