Axial anomaly and the triality symmetry of octonion

TL;DR

This paper explores the axial anomaly within the framework of octonions and their triality symmetry, proposing a novel connection between symmetry transformations and dark matter detection issues.

Contribution

It introduces a model combining octonion algebra and gauge theory to explain axial anomaly and suggests a link to dark matter phenomena via symmetry transformations.

Findings

Octonion triality symmetry affects fermion and gauge field interactions.

Certain symmetry operations could render electromagnetic signals undetectable from galaxies.

Axial anomaly is linked to matter field symmetry, not vacuum symmetry.

Abstract

With an assumption that in the Yang-Mills Lagrangian, a left-handed fermion and a right-handed fermion both expressed as the quaternion makes an octonion, and the gauge field can be treated as self-dual, we calculate the axial current and two vector currents triangle diagram of Bardeen, which yields the contribution of the axial anomaly. The octonion possesses the triality symmetry, and there are 5 symmetry operations $G_{ij}$ and $G_{ijk}$ ($ijk=123)$, in which mixing of spinors and vectors occur. $G_{23}$ does not mix vectors and spinors, but mismatch of the spinor and vector fields occurs. Hence, electro magnetic (EM) wave emitted from galaxies transformed by the five transformations would not be detected by EM detectors in our galaxy, and the source would be regarded as dark matter. The axial anomaly appears as a reflection of the symmetry of the matter field and not as a reflection of the symmetry of the pure vacuum, which is consistent with recent arguments on condensates and confinement.