In particle physics, SO(10) refers to a grand
unified theory (GUT) based on the spin group
Spin(10).
The name SO(10) is conventional among physicists,
and derives from the Lie group SO(10), which
is a special orthogonal group that is double
covered by Spin(10).
== History ==
Before the SU(5) theory behind the Georgi–Glashow
model, Harald Fritzsch and Peter Minkowski
and independently Howard Georgi found that
all the matter contents are incorporated into
a single representation, spinorial 16 of SO(10).
(Historical note: the before in the previous
sentence is misleading: Georgi found the SO(10)
theory a few hours before finding SU(5) at
the end of 1973.)
== Important subgroups ==
It has the branching rules 
to [SU(5)×U(1)χ]/Z5.
45
→
24
0
⊕
10
−
4
⊕
10
¯
4
⊕
1
0
{\displaystyle 45\rightarrow 24_{0}\oplus
10_{-4}\oplus {\overline {10}}_{4}\oplus 1_{0}}
16
→
10
1
⊕
5
¯
−
3
⊕
1
5
.
{\displaystyle 16\rightarrow 10_{1}\oplus
{\bar {5}}_{-3}\oplus 1_{5}.}
10
→
5
−
2
⊕
5
¯
2
.
{\displaystyle 10\rightarrow 5_{-2}\oplus
{\bar {5}}_{2}.}
If the hypercharge is contained within SU(5),
this is the conventional Georgi–Glashow
model, with the 16 as the matter fields, the
10 as the electroweak Higgs field and the
24 within the 45 as the GUT Higgs field.
The superpotential may then include renormalizable
terms of the form Tr(45 ⋅ 45); Tr(45 ⋅ 45
⋅ 45); 10 ⋅ 45 ⋅ 10, 10 ⋅ 16* ⋅ 16
and 16* ⋅ 16.
The first three are responsible to the gauge
symmetry breaking at low energies and give
the Higgs mass, and the latter two give the
matter particles masses and their Yukawa couplings
to the Higgs.
There is another possible branching, under
which the hypercharge is a linear combination
of an SU(5) generator and χ.
This is known as flipped SU(5).
Another important subgroup is either [SU(4)
× SU(2)L × SU(2)R]/Z2 or Z2 ⋊ [SU(4) × SU(2)L
× SU(2)R]/Z2 depending upon whether or not
the left-right symmetry is broken, yielding
the Pati–Salam model, whose branching rule
is
16
→
(
4
,
2
,
1
)
⊕
(
4
¯
,
1
,
2
)
.
{\displaystyle 16\rightarrow (4,2,1)\oplus
({\bar {4}},1,2).}
== Spontaneous symmetry breaking ==
The symmetry breaking of SO(10) is usually
done with a combination of (( a 45H OR a 54H)
AND ((a 16H AND a
16
¯
H
{\displaystyle {\overline {16}}_{H}}
) OR (a 126H AND a
126
¯
H
{\displaystyle {\overline {126}}_{H}}
)) ).
Let's say we choose a 54H.
When this Higgs field acquires a GUT scale
VEV, we have a symmetry breaking to Z2 ⋊ [SU(4)
× SU(2)L × SU(2)R]/Z2, i.e. the Pati–Salam
model with a Z2 left-right symmetry.
If we have a 45H instead, this Higgs field
can acquire any VEV in a two dimensional subspace
without breaking the standard model.
Depending on the direction of this linear
combination, we can break the symmetry to
SU(5)×U(1), the Georgi–Glashow model with
a U(1) (diag(1,1,1,1,1,-1,-1,-1,-1,-1)), flipped
SU(5) (diag(1,1,1,-1,-1,-1,-1,-1,1,1)), SU(4)×SU(2)×U(1)
(diag(0,0,0,1,1,0,0,0,-1,-1)), the minimal
left-right 
model (diag(1,1,1,0,0,-1,-1,-1,0,0)) or SU(3)×SU(2)×U(1)×U(1)
for any other nonzero VEV.
The choice diag(1,1,1,0,0,-1,-1,-1,0,0) is
called the Dimopoulos-Wilczek mechanism aka
the missing VEV mechanism and it is proportional
to B−L.
The choice of a 16H and a
16
¯
H
{\displaystyle {\overline {16}}_{H}}
breaks the gauge group down to the Georgi–Glashow
SU(5).
The same comment applies to the choice of
a 126H and a
126
¯
H
{\displaystyle {\overline {126}}_{H}}
.
It is the combination of BOTH a 45/54 and
a 16/
16
¯
{\displaystyle {\overline {16}}}
or 126/
126
¯
{\displaystyle {\overline {126}}}
which breaks SO(10) down to the Standard Model.
== The electroweak Higgs and the doublet-triplet
splitting problem ==
The electroweak Higgs doublets come from an
SO(10) 10H.
Unfortunately, this same 10 also contains
triplets.
The masses of the doublets have to be stabilized
at the electroweak scale, which is many orders
of magnitude smaller than the GUT scale whereas
the triplets have to be really heavy in order
to prevent triplet-mediated proton decays.
See doublet-triplet splitting problem.
Among the solutions for it is the Dimopoulos-Wilczek
mechanism, or the choice of diag(0,0,0,1,1,0,0,0,-1,-1)
of .
Unfortunately, this is not stable once the
16/
16
¯
{\displaystyle {\overline {16}}}
or 126/
126
¯
{\displaystyle {\overline {126}}}
sector interacts with the 45 sector.
== Matter ==
The matter representations come in three copies
(generations) of the 16 representation.
The Yukawa coupling is 10H 16f 16f.
This includes a right-handed neutrino.
We can either include three copies of singlet
representations φ and a Yukawa coupling
<
16
¯
H
>
16
f
ϕ
{\displaystyle 16_{f}\phi
}
(see double seesaw mechanism) or add the Yukawa
interaction
<
126
¯
H
>
16
f
16
f
{\displaystyle 16_{f}16_{f}}
or add the nonrenormalizable coupling
<
16
¯
H
><
16
¯
H
>
16
f
16
f
{\displaystyle 16_{f}16_{f}}
. See seesaw mechanism.
== Proton decay ==
These graphics refer to the X bosons and Higgs
bosons.
Note that SO(10) contains both the Georgi–Glashow
SU(5) and flipped SU(5).
== See also ==
Flipped SO(10)
== Notes ==
