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We argue that the topological charge of nonabelian gauge theory isunphysical. To show this statement, we use the Adler-Bardeen theorem and theBecchi-Rouet-Stora-Tyutin symmetry which are warranted by the perturbativefiniteness of the chiral anomaly, thus being free of Gribov ambiguity. Inaddition to the original argument using the unphysical gauge field componentcollinear to the spatial derivative of the gauge function, we show theunobservability of the topological charge using the Ward-Takahashi identity. Wethen present the consequences of this finding and show the consistency withmany physical pictures and ideas that have been developed around the topologyof nonabelian gauge theory. The most important ones are the resolution of theStrong CP problem, the unobservability of topological instantons, the physicalrelevance of the axial $U(1)$ symmetry, the independence of the vacuum energyon the vacuum angle, and the impossibility to realize the sphaleron inducedbaryogenesis and chiral magnetic effects. The axial $U(1)$ symmetry and theunphysical $\theta$-term imply that the physical complex phase of theCabibbo-Kobayashi-Maskawa matrix is the sole source of CP violation in thestandard model. The unphysical sphaleron also means that the lepton number isphenomenologically free from the baryon number, and their violations may bemodeled separately. We also comment on the consistency with the results oflattice calculations.

Unobservability of the topological charge in nonabelian gauge theory: Ward-Takahashi identity and phenomenological aspects