Octonion Internal Space Algebra for the Standard Model

Abstract

The paper surveys recent progress in the search for an appropriate internal space algebra for the Standard Model (SM) of particle physics. As a starting point serve Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure which implements the splitting of the octonions $\mathbb{O}=\mathbb{C}\oplus {\mathbb{C}}^3$ reflecting the lepton-quark symmetry. Such a complex structure in $C{\ell }_{10}$ is generated by the $C{\ell }_6(\subset C{\ell }_8\subset C{\ell }_{10})$ volume form, ${\omega }_6={\gamma }_1\cdots {\gamma }_6$, left invariant by the Pati-Salam subgroup of $Spin(10)$, $G_{\mathrm{PS}}=Spin(4)\times Spin(6)/{\mathbb{Z}}_2$. While the $Spin(10)$ invariant volume form ${\omega }_{10}={\gamma }_1...{\gamma }_{10}$ is known to split the Dirac spinors of $C{\ell }_{10}$ into left and right chiral (semi)spinors, $\mathcal{P}=\frac{1}{2}(1-i{\omega }_6)$ is interpreted as the projector on the 16-dimensional \textit{particle subspace} (annihilating the antiparticles). The standard model gauge group appears as the subgroup of $G_{PS}$ that preserves the sterile neutrino (identified with the Fock vacuum). The ${\mathbb{Z}}_2$-graded internal space algebra $\mathcal{A}$ is then included in the projected tensor product: $\mathcal{A}\subset \mathcal{P}C{\ell }_{10}\mathcal{P}=C{\ell }_4\otimes \mathcal{P}C{\ell }_6^0\mathcal{P}$. The Higgs field appears as the scalar term of a superconnection, an element of the odd part, $C{\ell }_4^1$, of the first factor. The fact that the projection of $C{\ell }_{10}$ only involves the even part $C{\ell }_6^0$ of the second factor guarantees that the colour symmetry remains unbroken. As an application we express the ratio $\frac{m_H}{m_W}$ of the Higgs to the $W$-boson masses in terms of the cosine of the {\it theoretical} Weinberg angle.

arXiv commentExtended version of a lecture presented at the Workshop Octonions and the Standard Model, Perimeter Institute, Waterloo, Canada, February-May 2021, and at the 14th International Workshop Lie Theory and Its Applications to Physics (LT 14), Sofia, June 2021. v3: exposition improved, references added, 34 pages

References in corpus (26)

• Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I
• Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra
• Exceptional quantum geometry and particle physics
• The standard model in noncommutative geometry and fermion doubling
• Generations: Three Prints, in Colour
• Implications of Higgs Discovery for the Strong CP Problem and Unification
• Grand Unification in the Spectral Pati-Salam Model
• Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra
• $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y\left(\times U(1{)}_X\right)$ as a symmetry of division algebraic ladder operators
• Exceptional quantum geometry and particle physics II
• Charge quantization from a number operator
• Octonions, exceptional Jordan algebra and the role of the group F_4 in particle physics
• The standard model, the Pati-Salam model, and "Jordan geometry"
• Division algebraic symmetry breaking
• Euclidean Jordan Algebras, Hidden Actions, and $J$-Kepler Problems
• Superconnections and the Higgs Field
• Octions: An $E_8$ description of the Standard Model
• Spin(11,3), particles and octonions
• Sterile Neutrino Dark Matter and Leptogenesis in Left-Right Higgs Parity
• One generation of standard model Weyl representations as a single copy of $\mathbb{R}\otimes \mathbb{C}\otimes \mathbb{H}\otimes \mathbb{O}$
• SO(9) characterisation of the Standard Model gauge group
• Superconnection in the spin factor approach to particle physics
• Superselection of the weak hypercharge and the algebra of the Standard Model
• One-loop corrections to the spectral action
• Chirality, a new key for the definition of the connection and curvature of a Lie-Kac super-algebra
• Spectral action in matrix form

Cited by in corpus (5)

• Three generations of colored fermions with $S_3$ family symmetry from Cayley-Dickson sedenions
• Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Collineation groups of octonionic and split-octonionic planes
• Weyl Neutrinos In Plane Symmetric Spacetimes