Periodic Theta Vacuum Model and the QCD Instanton Vacuum

Abstract

We formulate a toy model for the $θ$ vacuum that embodies the n-point moments of the even and odd topological charge densities. The partition function is $2π$ periodic only after resumming over all winding modes. We apply the same analysis to the quenched instanton vacuum at finite $θ$ and derive saddle-point results for the vacuum energy density that are manifestly $2π$ periodic whatever $θ$. We briefly discuss the consequences of unquenching the fermions on our results.