Violation of non-Abelian Bianchi identity and QCD topology

Abstract If a gauge field in QCD has a singularity of the Dirac type, the non-Abelian Bianchi identity is violated. The violation of the non-Abelian Bianchi identity (VNABI) Jμ (x) is equal to 8 degenerate Abelian magnetic monopole currents kᵃ (x) \ \ (a=1 8). Abelian monopoles could exist without any artificial Abelian projection. When Jμ condenses in the vacuum as suggested in latttice QCD simulations, color confinement of QCD is realized by the Abelian dual Meissner effect (ADME). It is important to study the effect of VNABI besides the role of color confinement mechansim. It is found that VNABI affects topological features of QCD. Firstly, the topological charge density ₜ (x) = Tr G (x) G^* (x) is not expressed by a total derivative of the Chern-Simons density Kμ (x), but has an additional term L (x) = 2Tr (Jμ (x) Aμ (x) ), that is, ∂μKμ (x) = (g2/16π2) (ρt (x) − L (x) ). Secondly, the axial U (1) anomaly is similarly modified as j ⁵ (x) =2m (x) ₅ (x) + (g²/8 ²) (ₜ (x) -L (x) ) while the Atiyah-Singer index theorem is formally unchanged. However, the integrated term Λ = (g2/16π2) ∫d4xL (x) is seemingly not integer nor gauge-invariant. Hence if Λ remains non-zero, VNABI is not allowed in QCD. Abelian monopoles as VNABI are regarded as a generalization of the Dirac monopole to a U (1) subgroup in non-Abelian gauge theories as discussed by Wu-Yang. When we make use of similar arguments, the additional term Λ is proved to vanish theoretically. We also evaluate the term Λ in the framework of Monte-Carlo simulations on 244 lattices at the lattice spacing between 0. 05 ∼ 0. 17fm adopting an tadpole-improved SU (2) gluonic action. To reduce ultraviolet fluctuations, we introduce a partial gauge fixing like the Maximal Center gauge (MCG). The term Λ is largely fluctuating around zero and |Λ| is decreasing as β becomes larger. Then the gradient flow method is applied. The term Λ tends to vanish rapidly after small gradient flow time tflow. This suggests that the term Λ becomes zero in the continuum limit. Hence VNABI could exist in QCD. The biggest effect of VNABI on QCD topology seems to be that self-dual instantons can not be a classical solution of QCD at space-time points where VNABI occurs. One has to find an alternative mechanism explaining integer topological charge, etc. A new interesting relation is derived with respect to the topological charge. When Λ = 0, an Abelian counter term Qₐ (g²/16 ²) d⁴x Tr (f f ^*) written by Abelian field strengths is proved theoretically to satisfy Qa=3Qt where Qt = (g2/16π2) ∫d4xρt (x). The fact sugggests that the Abelian magnetic and electric fields in the presence of Jμ condensation could have a role in explaining the topological charge Qt in place of instantons.