Infrared fixed point of QCD, critical flavor number and triality automorphism of octonions

Abstract

We show that the discrepancy on the critical flavor number of fermions $N_f^c$ for the appearance of the infrared fixed point based on the t'Hooft anomaly matching condition and derived from the Schr\"odinger functional method ($N_f^c\sim 9)$ and the experimental analysis of the JLab group using Bjorken sum rule and GDH sum rule, and our lattice simulation($N_f^c\sim 3$) could be resolved by assuming the topological structure of the infrared fixed point is not that of $U(1)^3$ but that of $G_2$ with triality automorphism of octonions which appear in the product of quaternions. The agreement of the infrared fixed point of the running coupling measured in lattice simulations with use of the quaternion real condition and the prediction of the BLM renormalization theory might be due to the boundary condition on $S^3\times R$ manifold of quaternion. The form factor of a proton measured via Ward identity through the difference of the inverse propagator at momentum p+q/2 and at p-q/2 agrees with the phenomenological dipole fit.