Violation of non-Abelian Bianchi Identity and QCD Topology

Abstract If a gauge field in quantum chromodynamics (QCD) has a singularity of the Dirac type, the non-Abelian Bianchi identity is violated. The violation of the non-Abelian Bianchi identity (VNABI) $J_{\mu }(x)$ is equal to 8 degenerate Abelian magnetic monopole currents $k^a_{\mu }(x)\ \ (a=1\sim 8)$. Abelian monopoles could exist without any artificial Abelian projection. When $J_{\mu }$ condenses in the vacuum as suggested in lattice QCD simulations, color confinement of QCD is realized by the Abelian dual Meissner effect. It is important to study the effect of VNABI besides the role of the color confinement mechanism. It is found that VNABI affects topological features of QCD. Firstly, the topological charge density $\rho _t(x)=\text{Tr}G_{\mu \nu }(x)G^{*}_{\mu \nu }(x)$ is not expressed by a total derivative of the Chern–Simons density $K_{\mu }(x)$, but has an additional term $L(x)= 2\text{Tr}(J_{\mu }(x)A_\mu (x))$, i.e. $\partial _\mu K_\mu (x)=(g^2/16\pi ^2)(\rho _t(x)-L(x))$. Secondly, the axial $U(1)$ anomaly is similarly modified as $\partial _{\mu }j_{\mu }^5(x)=2m\bar{\psi }(x)\gamma _5\psi (x)+(g^2/8\pi ^2)(\rho _t(x)-L(x))$ while the Atiyah–Singer index theorem is formally unchanged. However, the integrated term $\Lambda =(g^2/16\pi ^2)\int d^4xL(x)$ is seemingly neither integer nor gauge-invariant. Hence, if $\Lambda$ remains nonzero, 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 $\Lambda$ is proved to vanish theoretically. We also evaluate the term $\Lambda$ in the framework of Monte-Carlo simulations on $24^4$ lattices at the lattice spacing between $0.05\sim 0.17$ fm adopting a tadpole-improved $SU(2)$ gluonic action. To reduce ultraviolet fluctuations, we introduce a partial gauge fixing like the Maximal Center gauge. The term $\Lambda$ is largely fluctuating around zero and $|\Lambda |$ decreases as $\beta$ becomes larger. Then the gradient flow method is applied. The term $\Lambda$ tends to vanish rapidly after small gradient flow time $t_{\rm flow}$. This suggests that the term $\Lambda$ 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 cannot 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 $\Lambda =0$, an Abelian counter term $Q_a\equiv (g^2/16\pi ^2)\int d^4x \text{Tr}(f_{\mu \nu }f_{\mu \nu }^{*})$ written by Abelian field strengths is proved theoretically to satisfy $Q_a = 3 Q_t$ where $Q_t=(g^2/16\pi ^2)\int d^4x \rho _t(x)$. The fact suggests that the Abelian magnetic and electric fields in the presence of $J_{\mu }$ condensation could have a role in explaining the topological charge $Q_t$ in place of instantons.