trinification

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

/\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 is now known 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

/\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)\rightarrow2\,(\bar{3},1)_{\frac{1}{3}}\oplus(\bar{3},1)_{-\frac{2}{3}}

(1,3,\bar{3})\rightarrow2\,(1,2)_{-\frac{1}{2}}\oplus(1,2)_{\frac{1}{2}}\oplus2\,(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

/\mathbb{Z}_2}\right)=\mathbb{Z}

, this model predicts monopoles. See 't Hooft-Polyakov monopole.

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