Axial Anomaly and the Triality Symmetry of Leptons and Hadrons

Abstract

We apply the supersymmetric model of \'E. Cartan to the pseudoscalar meson decay into two photons, $\pi_0\to\gamma\gamma$, $\eta\to\gamma\gamma$ and $\eta'\to\gamma\gamma$. In the book of \'E. Cartan published in 1966, Dirac spinors ${^t(}A,B)$ and ${^t(}C,D)$ and vector fields $E$ and $E'$ 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 $E'E'$), we call the diagram rescattering diagram. When they belong to different groups ($EE'$), 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.