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 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 1974 GeorgiâÂÂGlashow and PatiâÂÂSalam 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) (GeorgiâÂÂGlashow model) and 9 of (PatiâÂÂSalam model).
Before the SU(5) theory behind the GeorgiâÂÂGlashow 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). However, Georgi found the SO(10) theory just a few hours before finding SU(5) at the end of 1973.
It has the branching rules to .
If the hypercharge is contained within SU(5), this is the conventional GeorgiâÂÂGlashow model, with the 16 as the matter fields, the 10 as the electroweak Higgs field and the 24 within the 45 as the Grand Unified Theory (GUT) Higgs field. The superpotential may then include renormalizable terms of the form , , and . 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 . This is known as flipped SU(5).
Another important subgroup is either or , depending upon whether or not the leftâÂÂright symmetry is broken, yielding the PatiâÂÂSalam model, whose branching rule is
The symmetry breaking of SO(10) is usually done with a combination of ( (a 45<sub>H</sub> a 54<sub>H</sub>) ((a 16<sub>H</sub> a <sub>H</sub>) (a 126<sub>H</sub> a <sub>H</sub>)) ).
Choose a 54<sub>H</sub>. When this Higgs field acquires a GUT scale vacuum expectation value (VEV), we have a symmetry breaking to , i.e. the PatiâÂÂSalam model with a Z<sub>2</sub> leftâÂÂright symmetry.
If we have a 45<sub>H</sub> 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 , the GeorgiâÂÂGlashow model with a U(1) (diag(1,1,1,1,1,âÂÂ1,âÂÂ1,âÂÂ1,âÂÂ1,âÂÂ1)), flipped SU(5) (diag(1,1,1,âÂÂ1,âÂÂ1,âÂÂ1,âÂÂ1,âÂÂ1,1,1)), (diag(0,0,0,1,1,0,0,0,âÂÂ1,âÂÂ1)), the minimal leftâÂÂright model (diag(1,1,1,0,0,âÂÂ1,âÂÂ1,âÂÂ1,0,0)) or for any other nonzero VEV.
The choice diag(1,1,1,0,0,âÂÂ1,âÂÂ1,âÂÂ1,0,0) is called the DimopoulosâÂÂWilczek mechanism aka the "missing VEV mechanism" and it is proportional to BâÂÂL.
The choice of a 16<sub>H</sub> and a <sub>H</sub> breaks the gauge group down to the GeorgiâÂÂGlashow SU(5). The same comment applies to the choice of a 126<sub>H</sub> and a <sub>H</sub>.
It is the combination of a 45/54 and a 16/ or 126/ that breaks SO(10) down to the Standard Model.
The electroweak Higgs doublets come from an . 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.
Among the solutions for it is the DimopoulosâÂÂWilczek mechanism, or the choice of diag(1,1,1,0,0,âÂÂ1,âÂÂ1,âÂÂ1,0,0) of . Unfortunately, this is not stable once the 16/ or 126/ sector interacts with the 45 sector.
The matter representations come in three copies (generations) of the 16 representation. The Yukawa coupling is . This includes a right-handed neutrino. One may either include three copies of singlet representations and a Yukawa coupling (the "double seesaw mechanism"); or else, add the Yukawa interaction or add the nonrenormalizable coupling .
The 16<sub>f</sub> field branches to and as
The 45 field branches to and as
and to the standard model as
The four lines are the SU(3)<sub>C</sub>, SU(2)<sub>L</sub>, and U(1)<sub>BâÂÂL</sub> bosons; the SU(5) leptoquarks that do not mutate X charge; the PatiâÂÂSalam leptoquarks and SU(2)<sub>R</sub> bosons; and the new SO(10) leptoquarks. (The standard electroweak U(1)<sub>Y</sub> is a linear combination of the bosons.)
Note that SO(10) contains both the GeorgiâÂÂGlashow 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 â 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.