- Source: Trinification
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
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}} }
.
