In particle physics, SO(10) refers to a grand unified theory (GUT) based on the spin group Spin(10). The shortened name SO(10) is conventional[1] among physicists, and derives from the Lie algebra or less precisely the Lie group of SO(10), which is a special orthogonal group that is double covered by Spin(10).
SO(10) subsumes the Georgi–Glashow and Pati–Salam models, and unifies all fermions in a generation into a single field. This requires 12 new gauge bosons, in addition to the 12 of SU(5) and 9 of SU(4)×SU(2)×SU(2).
Before the SU(5) theory behind the Georgi–Glashow model,[2] 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).[3] However, it is worth noting that Georgi found the SO(10) theory just a few hours before finding SU(5) at the end of 1973.[4]
It has the branching rules to [SU(5)×U(1)<sub>χ</sub>]/Z5.
45 → 240 ⊕ 10-4 ⊕ \overline{10}4 ⊕ 10
16 → 101 ⊕ \bar{5}-3 ⊕ 15
10 → 5-2 ⊕ \bar{5}2.
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)<sub>L</sub> × SU(2)<sub>R</sub>]/Z2 or Z2 ⋊ [SU(4) × SU(2)<sub>L</sub> × SU(2)<sub>R</sub>]/Z2 depending upon whether or not the left-right symmetry is broken, yielding the Pati–Salam model, whose branching rule is
45 → (15,1,1) ⊕ (6,2,2) ⊕ (1,3,1) ⊕ (1,1,3)
16 → (4,2,1) ⊕ (\bar4,1,2).
The symmetry breaking of SO(10) is usually done with a combination of ((a 45H OR a 54H) AND ((a 16H AND a
\overline{16}H
\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)<sub>L</sub> × SU(2)<sub>R</sub>]/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
\overline{16}H
\overline{126}H
It is the combination of BOTH a 45/54 and a 16/
\overline{16}
\overline{126}
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(1,1,1,0,0,-1,-1,-1,0,0) of <45>. Unfortunately, this is not stable once the 16/
\overline{16}
\overline{126}
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. One may either include three copies of singlet representations and a Yukawa coupling
<\overline{16}H>16f\phi
<\overline{126}H>16f16f
<\overline{16}H><\overline{16}H>16f16f
The 16f field branches to [SU(5)×U(1)<sub>χ</sub>]/Z5 and SU(4) × SU(2)L × SU(2)R as
16 → 101 ⊕ \bar{5}-3 ⊕ 15
16 → (4,2,1) ⊕ (\bar4,1,2).
The 45 field branches to [SU(5)×U(1)<sub>χ</sub>]/Z5 and SU(4) × SU(2)L × SU(2)R as
45 → 240 ⊕ 10-4 ⊕ \overline{10}4 ⊕ 10
45 → (15,1,1) ⊕ (6,2,2) ⊕ (1,3,1) ⊕ (1,1,3)
and to the standard model [SU(3)<sub>C</sub> × SU(2)<sub>L</sub> × U(1)<sub>Y</sub>]/Z6 as
\begin{align}45 → &(8,1)0 ⊕ (1,3)0 ⊕ (1,1)0
| ⊕ \\ &(3,2) | ||||
|
⊕
| (\bar{3},2) | ||||
|
| ⊕ \\ &(3,1) | ||||
|
⊕
| (\bar{3},1) | ||||
|
⊕ (1,1)1 ⊕ (1,1)-1 ⊕ (1,1)0 ⊕ \\ &(3,2)
|
⊕
| (\bar{3},2) | ||||
|
.\\ \end{align}
The four lines are the SU(3)C, SU(2)L, and U(1)B−L bosons; the SU(5) leptoquarks which don't mutate X charge; the Pati-Salam leptoquarks and SU(2)R bosons; and the new SO(10) leptoquarks. (The standard electroweak U(1)Y is a linear combination of the bosons.)
Note that SO(10) contains both the Georgi–Glashow SU(5) and flipped SU(5).
It has been long known that the SO(10) model is free from all perturbative local anomalies, computable by Feynman diagrams. However, it only became clear in 2018 that the SO(10) model is also free from all nonperturbative global anomalies on non-spin manifolds --- an important rule for confirming the consistency of SO(10) grand unified theory, with a Spin(10) gauge group and chiral fermions in the 16-dimensional spinor representations, defined on non-spin manifolds.[6] [7]