Physics:Trinification

Short description: Grand Unified Theory in physics

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

Details

It states that the gauge group is either

[math]\displaystyle{ SU(3)_C\times SU(3)_L\times SU(3)_R }[/math]

or

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

and that the fermions form three families, each consisting of the representations: [math]\displaystyle{ \mathbf Q=(3,\bar{3},1) }[/math], [math]\displaystyle{ \mathbf Q^c=(\bar{3},1,3) }[/math], and [math]\displaystyle{ \mathbf L=(1,3,\bar{3}) }[/math]. 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 [math]\displaystyle{ (1,3,\bar{3}) }[/math] and maybe also a [math]\displaystyle{ (1,\bar{3},3) }[/math] scalar field called the Higgs field which acquires a vacuum expectation value. This results in a spontaneous symmetry breaking from

[math]\displaystyle{ SU(3)_L\times SU(3)_R }[/math] to [math]\displaystyle{ [SU(2)\times U(1)]/\mathbb{Z}_2 }[/math].

The fermions branch (see restricted representation) as

[math]\displaystyle{ (3,\bar{3},1)\rightarrow(3,2)_{\frac{1}{6}}\oplus(3,1)_{-\frac{1}{3}} }[/math],

[math]\displaystyle{ (\bar{3},1,3)\rightarrow 2\,(\bar{3},1)_{\frac{1}{3}}\oplus(\bar{3},1)_{-\frac{2}{3}} }[/math],

[math]\displaystyle{ (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 }[/math],

and the gauge bosons as

[math]\displaystyle{ (8,1,1)\rightarrow(8,1)_0 }[/math],

[math]\displaystyle{ (1,8,1)\rightarrow(1,3)_0\oplus(1,2)_{\frac{1}{2}}\oplus(1,2)_{-\frac{1}{2}}\oplus(1,1)_0 }[/math],

[math]\displaystyle{ (1,1,8)\rightarrow 4\,(1,1)_0\oplus 2\,(1,1)_1\oplus 2\,(1,1)_{-1} }[/math].

Note that there are two Majorana neutrinos per generation (which is consistent with neutrino oscillations). Also, each generation has a pair of triplets [math]\displaystyle{ (3,1)_{-\frac{1}{3}} }[/math] and [math]\displaystyle{ (\bar{3},1)_{\frac{1}{3}} }[/math], and doublets [math]\displaystyle{ (1,2)_{\frac{1}{2}} }[/math] and [math]\displaystyle{ (1,2)_{-\frac{1}{2}} }[/math], which decouple at the GUT breaking scale due to the couplings

[math]\displaystyle{ (1,3,\bar{3})_H(3,\bar{3},1)(\bar{3},1,3) }[/math]

and

[math]\displaystyle{ (1,3,\bar{3})_H(1,3,\bar{3})(1,3,\bar{3}) }[/math].

Note that calling representations things like [math]\displaystyle{ (3,\bar{3},1) }[/math] 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

[math]\displaystyle{ \pi_2\left(\frac{SU(3)\times SU(3)}{[SU(2)\times U(1)]/\mathbb{Z}_2}\right)=\mathbb{Z} }[/math],

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 [math]\displaystyle{ (3,3,1)\oplus(\bar{3},\bar{3},1)\oplus(1,\bar{3},3) }[/math] fields. E₆ adds 54 gauge bosons, 30 it shares with SO(10), the other 24 to complete its [math]\displaystyle{ \mathbf{16}\oplus\mathbf{\overline{16}} }[/math].