Trinification

In physics, the trinification model is a Grand Unified Theory proposed by Alvaro De Rújula, Howard Georgi and Sheldon Glashow in 1984.{{cite book|first1=A. |last1=De Rujula |first2=H. |last2=Georgi |first3=S. L. |last3=Glashow |chapter=Trinification of all elementary particle forces |title=Fifth Workshop on Grand Unification |editor-first1=K. |editor-last1=Kang |editor-first2=H. |editor-last2=Fried |editor-first3=F. |editor-last3=Frampton |publisher=World Scientific |location=Singapore |year=1984}}{{Cite journal|last=Hetzel|first=Jamil|last2=Stech|first2=Berthold|date=2015-03-25|title=Low-energy phenomenology of trinification: An effective left-right-symmetric model|url=https://link.aps.org/doi/10.1103/PhysRevD.91.055026|journal=Physical Review D|language=en|volume=91|issue=5|pages=055026|doi=10.1103/PhysRevD.91.055026|issn=1550-7998|arxiv=1502.00919}}

Details

It states that the gauge group is either

: $SU(3)_C\times SU(3)_L\times SU(3)_R$

: $[SU(3)_C\times SU(3)_L\times SU(3)_R]/\mathbb{Z}_3$ ;

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

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

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

: $(1,3,\bar{3})\rightarrow 2\,(1,2)_{-\frac{1}{2}}\oplus(1,2)_{\frac{1}{2}}\oplus2\,(1,1)_0\oplus(1,1)_1$ ,

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.

Trinification is a maximal subalgebra of E₆, whose matter representation {{math|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 $\mathbf{16}\oplus\mathbf{\overline{16}}$ .

References

Category:Grand Unified Theory