Editing 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.<ref>{{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}}</ref><ref>{{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}}</ref>

==Details==
It states that the [[gauge group]] is either

:<math>SU(3)_C\times SU(3)_L\times SU(3)_R</math>

or

:<math>[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 of Lie groups/algebras|representations]]: <math>\mathbf Q=(3,\bar{3},1)</math>, <math>\mathbf Q^c=(\bar{3},1,3)</math>, and <math>\mathbf L=(1,3,\bar{3})</math>.  The '''L''' includes a hypothetical [[right-handed neutrino]], which may account for observed [[neutrino mass]]es (see [[neutrino oscillation]]s), and a similar sterile "flavon."

There is also a <math>(1,3,\bar{3})</math> and maybe also a <math>(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>SU(3)_L\times SU(3)_R</math> to <math>[SU(2)\times U(1)]/\mathbb{Z}_2</math>.

The fermions branch (see [[restricted representation]]) as

:<math>(3,\bar{3},1)\rightarrow(3,2)_{\frac{1}{6}}\oplus(3,1)_{-\frac{1}{3}}</math>,

:<math>(\bar{3},1,3)\rightarrow 2\,(\bar{3},1)_{\frac{1}{3}}\oplus(\bar{3},1)_{-\frac{2}{3}}</math>,

:<math>(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>(8,1,1)\rightarrow(8,1)_0</math>,

:<math>(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>(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 spinor|Majorana]] neutrinos per [[Generation (particle physics)|generation]] (which is consistent with [[neutrino oscillation]]s). Also, each generation has a pair of [[Doublet–triplet splitting problem|triplet]]s <math>(3,1)_{-\frac{1}{3}}</math> and <math>(\bar{3},1)_{\frac{1}{3}}</math>, and doublets <math>(1,2)_{\frac{1}{2}}</math> and <math>(1,2)_{-\frac{1}{2}}</math>, which decouple at the GUT breaking scale due to the couplings

:<math>(1,3,\bar{3})_H(3,\bar{3},1)(\bar{3},1,3)</math>

and

:<math>(1,3,\bar{3})_H(1,3,\bar{3})(1,3,\bar{3})</math>.

Note that calling [[Representations of Lie groups/algebras|representations]] things like <math>(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 tableau]]x or [[Dynkin diagram]]s with numbers on their vertices, but it is standard among GUT theorists.

Since the [[homotopy group]]

:<math>\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 monopole|'t Hooft–Polyakov]] [[magnetic monopole]]s.

Trinification is a [[maximal subalgebra]] of [[E6 (mathematics)#Physics|E<sub>6</sub>]], whose matter representation {{math|'''27'''}} has exactly the same representation and unifies the <math>(3,3,1)\oplus(\bar{3},\bar{3},1)\oplus(1,\bar{3},3)</math> fields. E<sub>6</sub> adds 54 [[gauge boson]]s, 30 it shares with [[SO(10)]], the other 24 to complete its <math>\mathbf{16}\oplus\mathbf{\overline{16}}</math>.

==References==
{{Reflist}}

[[Category:Grand Unified Theory]]