  The investigation of the scalar gluonium correlator is interesting because it
carries the quantum numbers of the vacuum and the relevant hadronic current is
related to the anomalous trace of the QCD energy-momentum tensor in the chiral
limit. After reviewing the purely perturbative corrections known up to
next-next-to-leading order, the behaviour of the correlator is studied to all
orders by means of the large-beta_0 approximation. Similar to the QCD Adler
function, the large-order behaviour is governed by the leading ultraviolet
renormalon pole. The structure of infrared renormalon poles, being related to
the operator product expansion are also discussed, as well as a low-energy
theorem for the correlator that provides a relation to the renormalisation
group invariant gluon condensate, and the vacuum matrix element of the trace of
the QCD energy-momentum tensor.
