Vacuum Nodes in QCD at θ=π: Exact Results

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We show that the vacuum functional of 3+1 dimensional non-abelian gauge theories vanishes for some classical field configurations ψ₀(A)=0 when the coefficient of the CP violating θ--term becomes θ=π (mod. 2π). Some of these classical configurations are explicitly identified and include sphalerons. The results shed new light into the non-perturbative behavior of non-abelian gauge theories and suggest a relevant role for these classical configurations in the confinement mechanism at θ=0.