  We analyze the renormalon diagram of gauge theories on $\mathbb R^3\times
\mathbb S^1$. In particular, we perform exact one loop calculations for the
vacuum polarization in QCD with adjoint matter and observe that all infrared
logarithms, as functions of the external momentum, cancel between the vacuum
part and finite volume part, which eliminates the IR renormalon problem. We
argue that the singularities in the Borel plane, arising from the topological
neutral bions, are not associated with renormalons, but with the proliferation
of the Feynman diagrams. As a byproduct, we obtain, for the first time, an
exact one-loop result of the vacuum polarization which can be adapted to the
case of thermal compactification of QCD.
