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.

Extended 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

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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