Renormalon Singularities of the QCD Vacuum Polarization Function to Leading Order in $1/N_{f}$

We explicitly determine the residues and orders of all the ultra-violet (UV) and infra-red (IR) renormalon poles in the Borel plane for the QCD vacuum polarization function (Adler D-function), to leading order in an expansion in the number of quark flavours, $N_{f}$. The singularity structure is precisely as anticipated on general grounds. In particular, the leading IR renormalon is absent, in agreement with operator product expansion ideas. There is a curious and unexplained symmetry between the third and higher UV and IR renormalon residues. We are able to sum up separately UV and IR contributions to obtain closed form results involving $\zeta$-functions. We argue that the leading UV renormalon should have a more complicated structure than conventionally assumed. The disappearance of IR renormalons in flavour-saturated SU($N$) QCD is shown to occur for $N=3,6$ or 9.