SO(10)

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.

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, Georgi found the SO(10) theory just a few hours before finding SU(5) at the end of 1973.

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.

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.

breaks the gauge group down to the Georgi–Glashow SU(5).

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 Dimopoulos-Wilczek mechanism, or the choice of diag(1,1,1,0,0,-1,-1,-1,0,0) of <45>.

One may either include three copies of singlet representations φ and a Yukawa coupling

(the "double seesaw mechanism"); or else, add the Yukawa interaction

(The standard electroweak U(1)Y is a linear combination of the (1,1)0 bosons.)

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.