SO(10)

SO(10). In theoretical physics and mathematics, SO(10) refers to the special orthogonal group of rank 5, a 45-dimensional Lie group of central importance in Grand Unified Theories (GUTs). Proposed independently by Howard Georgi and H. M. Srivastava in the 1970s, it provides a compelling framework for unifying the Standard Model gauge group into a single, simple Lie algebra. Its structure naturally accommodates all known fermions of a single generation within a single irreducible representation, the 16-dimensional spinor.

Definition and basic properties

Mathematically, SO(10) is defined as the group of 10×10 orthogonal matrices with a determinant of +1, acting on a 10-dimensional real vector space. Its corresponding Lie algebra, denoted $\backslash mathfrak\{so\}(10)$, is of type D₅ and has 45 generators. A key property is its center, which is isomorphic to Z₂, and its universal cover is the spin group Spin(10). The group contains several important maximal subgroups, including SU(5) × U(1), which connects it directly to the earlier Georgi–Glashow model. The representation theory of SO(10) is rich, with fundamental representations including the 10, the 16 and \(\overline{16}\), and the 45.

Grand Unified Theory (GUT) framework

In particle physics, SO(10) serves as the gauge group for a prominent Grand Unified Theory. This framework embeds the Standard Model gauge group SU(3) × SU(2) × U(1) via intermediate subgroups like the Pati–Salam group SU(4) × SU(2) × SU(2) or SU(5). A major theoretical virtue is its automatic inclusion of a right-handed neutrino for each generation, facilitating the see-saw mechanism to explain the small masses of active neutrinos. The unification of coupling constants in the Minimally Supersymmetric version of SO(10) is notably successful, with the predicted GUT scale near 10¹⁶ GeV. Pioneering work on these models was advanced by R. N. Mohapatra, Goran Senjanović, and Jogesh C. Pati.

Symmetry breaking and particle content

The breaking of the SO(10) symmetry to the Standard Model can proceed through various Higgs sectors and intermediate symmetries. Common breaking chains utilize Higgs fields in representations like the 45, 54, 126, or 210 of SO(10). A chain like SO(10) → SU(4)_C × SU(2)_L × SU(2)_R → Standard Model introduces right-handed gauge bosons and predicts left-right symmetry. The fermion content of a single generation is unified into the 16_F representation, which contains the 15 known quarks and leptons plus the right-handed neutrino. The Yukawa couplings for generating fermion masses often involve Higgs fields in the 10 and 126 representations.

Predictions and experimental status

SO(10) GUTs make several testable predictions despite the inaccessibility of the unification scale. The existence of right-handed neutrinos and the see-saw mechanism predicts neutrino oscillations and neutrinoless double beta decay, which are active areas of research at facilities like Super-Kamiokande and KamLAND. The models typically predict proton decay via exchange of superheavy gauge bosons like the X and Y bosons, with modes such as p → e⁺π⁰; current non-observation by the Super-Kamiokande experiment constrains model parameters. They also predict relationships between fermion masses and mixing angles at the GUT scale, and the possible existence of magnetic monopoles as predicted by Paul Dirac.

Mathematical structure and subgroups

The group-theoretic structure of SO(10) is deeply connected to Clifford algebras and spin representations in 10 dimensions. Its Dynkin diagram of type D₅ indicates it is a member of the classical Lie group family. Important subgroups include SU(5) × U(1)_χ, where the U(1) charge is related to baryon number minus lepton number (B-L), and the Pati–Salam group SU(4) × SU(2) × SU(2). The maximal subgroup SO(6) × SO(4) is isomorphic to SU(4) × SU(2) × SU(2), showcasing the unification of color and lepton number. The 16 decomposes under SU(5) as 1 ⊕ 5̄ ⊕ 10, which are precisely the representations used in the Georgi–Glashow model.

Category:Lie groups Category:Grand Unified Theory Category:Theoretical physics