The QCD beta function extracted from polarized electron proton scattering data obtained at JLab and the lattice simulation in the MOM scheme suggest that the critical flavor number for the presence of IR fixed point is about three. In analyses of Schrödinger functional scheme, however, critical flavor number for the presence of IR fixed point and the conformality is larger than nine.
In the QCD analysis, when quarks are expressed in quaternion basis, the product of quaternions are expressed by octonions and the octonion posesses the triality symmetry. Since the triality has the effect of multiplying the falvor number, it could explain the apparent large critical flavor number in the Schrödinger functinal scheme. In this scheme, larger degrees of freedom in adjusting data of different scales on the boundary are necessary than in the MOM scheme.
In weak interaction, there is no clear lepton‐flavor violation except in the neutrino oscillation. If the triality is assigned to the lepton flavors(e,μ and τ) and they are assumed to be exact symmetry, or the electro‐magnetic interaction preserves tiality, but the strong interaction is triality blind, there is a possibility of explaining the neutrino oscillation through triality mixing of the matter field.
The self energy of gluons, ghost and gauge bosons due to self‐dual gauge fields and leptonic decays of B,D and mesons are discussed.