The flavor symmetry in the standard model and the triality symmetry

TL;DR

This paper explores the triality symmetry of octonions in fermion and boson fields within the standard model, proposing a connection to dark matter and flavor degrees of freedom.

Contribution

It introduces the concept of triality symmetry in fermion and boson fields and suggests its potential role in dark matter and flavor structure in the standard model.

Findings

Triality symmetry relates fermion and boson spinors and vectors.

Electromagnetic interactions may be sensitive to triality, affecting dark matter detection.

Quarks are likely insensitive to triality, implying larger flavor degrees of freedom.

Abstract

A Dirac fermion is expressed by a 4 component spinor which is a combination of two quaternions and which can be treated as an octonion. The octonion possesses the triality symmetry, which defines symmetry of fermion spinors and bosonic vector fields. The triality symmetry relates three sets of spinors and two sets of vectors, which are transformed among themselves via transformations $G_{23}, G_{12}, G_{13}$, $G_{123}$ and $G_{132}$. If the electromagnetic (EM) interaction is sensitive to the triality symmetry, i.e. EM probe selects one triality sector, EM signals from the 5 transformed world would not be detected, and be treated as the dark matter. According to an astrophysical measurement, the ratio of the dark to ordinary matter in the universe as a whole is almost exactly 5. We expect quarks are insensitive to the triality, and triality will appear as three times larger flavor degrees of freedom in the lattice simulation.