Inparticle physics,SO(10) refers to agrand unified theory (GUT) based on thespin group Spin(10). The shortened name SO(10) is conventional^[1] among physicists, and derives from theLie algebra or less precisely theLie group of SO(10), which is aspecial orthogonal group that isdouble covered by Spin(10).

SO(10) subsumes theGeorgi–Glashow andPati–Salam models, and unifies allfermions in ageneration into a single field. This requires 12 newgauge bosons, in addition to the 12 ofSU(5) and 9 ofSU(4)×SU(2)×SU(2).

Before theSU(5) theory behind theGeorgi–Glashow model,^[2]Harald Fritzsch andPeter Minkowski, and independentlyHoward Georgi, found that all the matter contents are incorporated into a single representation,spinorial 16 of SO(10).^[3] However, Georgi found the SO(10) theory just a few hours before finding SU(5) at the end of 1973.^[4]

It has thebranching rules to [SU(5)×U(1)_χ]/Z₅.

${\displaystyle 45\to {24}_0\oplus {10}_{-4}\oplus {\overline{10}}_4\oplus 1_0}$

${\displaystyle 16\to {10}_1\oplus {\overline{5}}_{-3}\oplus 1_5}$

${\displaystyle 10\to 5_{-2}\oplus {\overline{5}}_2.}$

If thehypercharge is contained within SU(5), this is the conventionalGeorgi–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. Thesuperpotential may then includerenormalizable terms of the formTr(45 ⋅ 45);Tr(45 ⋅ 45 ⋅ 45); 10 ⋅ 45 ⋅ 10, 10 ⋅ 16* ⋅ 16 and 16* ⋅ 16. The first three are responsible to thegauge symmetry breaking at low energies and give theHiggs mass, and the latter two give the matter particles masses and theirYukawa 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 asflipped SU(5).

Another important subgroup is either [SU(4) × SU(2)_L × SU(2)_R]/Z₂ orZ₂ ⋊ [SU(4) × SU(2)_L × SU(2)_R]/Z₂ depending upon whether or not theleft-right symmetry is broken, yielding thePati–Salam model, whose branching rule is

${\displaystyle 45\to (15,1,1)\oplus (6,2,2)\oplus (1,3,1)\oplus (1,1,3)}$

${\displaystyle 16\to (4,2,1)\oplus (\overline{4},1,2).}$

The symmetry breaking of SO(10) is usually done with a combination of (( a 45_H OR a 54_H) AND ((a 16_H AND a${\displaystyle {\overline{16}}_H}$) OR (a 126_H AND a${\displaystyle {\overline{126}}_H}$)) ).

Let's say we choose a 54_H. When this Higgs field acquires a GUT scaleVEV, we have a symmetry breaking toZ₂ ⋊ [SU(4) × SU(2)_L × SU(2)_R]/Z₂, i.e. thePati–Salam model with aZ₂left-right symmetry.

If we have a 45_H 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), theGeorgi–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 minimalleft-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 theDimopoulos-Wilczek mechanism aka the "missing VEV mechanism" and it is proportional toB−L.

The choice of a 16_H and a${\displaystyle {\overline{16}}_H}$ breaks the gauge group down to the Georgi–Glashow SU(5). The same comment applies to the choice of a 126_H and a${\displaystyle {\overline{126}}_H}$.

It is the combination of BOTH a 45/54 and a 16/${\displaystyle \overline{16}}$ or 126/${\displaystyle \overline{126}}$ which breaks SO(10) down to theStandard Model.

The electroweak Higgs doublets come from an SO(10) 10_H. 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-mediatedproton decays. Seedoublet-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/${\displaystyle \overline{16}}$ or 126/${\displaystyle \overline{126}}$ sector interacts with the 45 sector.^[5]

The matter representations come in three copies (generations) of the 16 representation. TheYukawa coupling is 10_H 16_f 16_f. This includes a right-handed neutrino. One may either include three copies ofsinglet representationsφ and a Yukawa coupling (the "double seesaw mechanism"); or else, add the Yukawa interaction or add thenonrenormalizable coupling. Seeseesaw mechanism.

The 16_f field branches to [SU(5)×U(1)_χ]/Z₅ and SU(4) × SU(2)_L × SU(2)_R as

${\displaystyle 16\to {10}_1\oplus {\overline{5}}_{-3}\oplus 1_5}$

${\displaystyle 16\to (4,2,1)\oplus (\overline{4},1,2).}$

The 45 field branches to [SU(5)×U(1)_χ]/Z₅ and SU(4) × SU(2)_L × SU(2)_R as

${\displaystyle 45\to {24}_0\oplus {10}_{-4}\oplus {\overline{10}}_4\oplus 1_0}$

${\displaystyle 45\to (15,1,1)\oplus (6,2,2)\oplus (1,3,1)\oplus (1,1,3)}$

and to the standard model [SU(3)_C × SU(2)_L × U(1)_Y]/Z₆ as

The four lines are the SU(3)_C, SU(2)_L, and U(1)_B−L bosons; theSU(5) leptoquarks which don't mutateX charge; thePati-Salam leptoquarks and SU(2)_R bosons; and the new SO(10) leptoquarks. (The standardelectroweak U(1)_Y is a linear combination of the(1,1)₀ bosons.)

• These graphics refer to theX bosons andHiggs bosons.
• 
  Dimension 6 proton decay mediated by theX boson${\displaystyle (3,2{)}_{-\frac{5}{6}}}$ in SU(5) GUT
• 
  Dimension 6 proton decay mediated by theX boson${\displaystyle (3,2{)}_{\frac{1}{6}}}$ in flipped SU(5) GUT

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 allnonperturbative 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 onnon-spin manifolds.^[6][7]

• Flipped SO(10)

• Grand Unified Theory