Axial anomaly and the triality symmetry of leptons and hadrons

Abstract

We apply the supersymmetric model of É. Cartan to the pseudoscalar meson decay into two photons, ${\pi }_0\to \gamma \gamma$, $\eta \to \gamma \gamma$ and ${\eta }^{\prime }\to \gamma \gamma$. In the book of É. Cartan published in 1966, Dirac spinors ${}^t(A,B)$ and ${}^t(C,D)$ and vector fields $E$ and $E^{\prime }$ were introduced and five supersymmetric transformations $G_{23},G_{12},G_{13},G_{123}$ and $G_{132}$ were considered. The Pauli spinor is treated as a quaternion and the Dirac spinor is treated as an octonion. In the pseudoscalar meson decay, when the two final vector fields belong to the same group ($EE$ or $E^{\prime }{E}^{\prime }$), we call the diagram rescattering diagram. When they belong to different groups ($E{E}^{\prime }$), the diagram is called twisted diagram. Assuming the triality selection rules of octonions, dark matter is interpreted as matter emitting photons in a different triality sector than that of electromagnetic probes in our world.

Comments: 24 pages, 24 figures, References and comments added, to be published in Few-Body Systems(Springer)