| SO(10) | |
|---|---|
| Name | SO(10) |
| Type | Lie group |
| Dimension | 45 |
| Algebra | so(10) |
| Classification | Special orthogonal group |
SO(10)
SO(10) is the special orthogonal group in ten dimensions, a compact simple Lie group of dimension 45 that preserves a nondegenerate symmetric bilinear form on a ten-dimensional real vector space. In quantum physics and high-energy theory it is prominent as a candidate gauge group for grand unification and as a rich structure for embedding the Standard Model fermion content, offering unified descriptions of electroweak and strong forces as well as insights into neutrino mass generation and baryogenesis.
SO(10) serves as both a mathematical object in the study of continuous symmetries and a physical gauge group in quantum field theory. Within the framework of gauge theory and quantum field theory, SO(10) provides a unifying symmetry that can accommodate the fifteen known chiral fermions of one Standard Model family plus a right-handed neutrino in a single irreducible representation. This property makes SO(10) central to model building in GUTs, linking it to experimental programs at facilities such as the Large Hadron Collider and to theoretical developments in supersymmetry and string theory.
SO(10) is defined as the group of 10×10 real orthogonal matrices with determinant +1, preserving a quadratic form of signature (10,0) (or other signatures for noncompact forms like SO(5,5)). Its Lie algebra so(10) consists of 10×10 real antisymmetric matrices and is a simple algebra of type B or D? — specifically of type D5 in the Cartan classification. The group has rank 5 and 45 generators; its root system and Dynkin diagram are those of D5. Important mathematical constructions include the spin groups: the double cover Spin(10) is the universal covering group relevant for defining spinor representations and for coupling to fermions in quantum theories. The topology, center, and fundamental group of SO(10) influence possible global forms used in physics, e.g., Spin(10) versus SO(10)/Z2.
Critical representations are the vector 10, the adjoint 45, and the two inequivalent chiral spinor representations 16 and \overline{16} of Spin(10). The 16-dimensional spinor representation is notable because a single family of Standard Model fermions plus a right-handed neutrino fits into a single 16 of Spin(10). Tensor product decompositions under embedded subgroups give the branching rules used in model building; for example, decompositions under SU(5), Pati–Salam (SU(4)×SU(2)×SU(2)) and SU(3)×SU(2)×U(1) are widely used. The Dynkin labels and weights organize multiplets; Cartan generators provide conserved quantum numbers that map to baryon and lepton number combinations relevant for proton decay estimates.
SO(10) GUTs were proposed to extend earlier unification schemes such as SU(5) and the Pati–Salam model. Advantages include automatic incorporation of right-handed neutrino states enabling see-saw mechanism explanations of small neutrino masses, left–right symmetry embeddings, and potential explanations for charge quantization. SO(10) naturally accommodates Yukawa coupling structures and can be combined with supersymmetry (as in SUSY SO(10) models) to address the hierarchy problem and gauge coupling unification observed in renormalization group evolution, which is tested against precision results from experiments at the CERN laboratories and flavor factories like Belle II.
Breaking SO(10) to the Standard Model gauge group requires intermediate steps realized by Higgs fields in various representations (e.g., 45, 126, 210, 16, 10). Common symmetry-breaking chains include SO(10) → SU(5) × U(1), SO(10) → SU(4)×SU(2)×SU(2), SO(10) → SU(3)×SU(2)×SU(2)×U(1), and variants with intermediate U(1) factors. Model builders use specific Higgs sectors to achieve desired doublet–triplet splitting and to suppress rapid proton decay via mechanisms such as the Dimopoulos–Wilczek mechanism. Global and discrete symmetries, including R-parity in supersymmetric extensions, are often imposed to control unwanted operators and to provide candidates for dark matter such as the lightest supersymmetric particle.
SO(10)-based models predict several experimental signatures: proton decay channels with lifetimes dependent on intermediate scales and GUT-scale physics probed by experiments like Super-Kamiokande and planned detectors such as Hyper-Kamiokande; neutrino mass and mixing patterns testable in oscillation experiments like NOvA and DUNE; gauge coupling unification predictions constrained by precision measurements at LEP and the LHC; and potential new gauge bosons or scalar states accessible at colliders or in low-energy observables. SO(10) also links to cosmological phenomena: leptogenesis scenarios for baryon asymmetry, implications for inflationary model building, and effects on relic abundances. The interplay between theoretical constraints from renormalization group analyses and experimental bounds continues to guide viable SO(10) constructions and motivates searches across particle physics facilities and neutrino observatories.
Category:Lie groups Category:Grand Unified Theory