  In this work, the perturbative QCD series of the scalar correlation function
$\Psi(s)$ is investigated. Besides ${\rm Im}\Psi(s)$, which is relevant for
Higgs decay into quarks, two other physical correlators, $\Psi^{"}(s)$ and
$D^L(s)$, have been employed in QCD applications like quark mass determinations
or hadronic $\tau$ decays. $D^L(s)$ suffers from large higher-order corrections
and, by resorting to the large-$\beta_0$ approximation, it is shown that this
is related to a spurious renormalon ambiguity at $u=1$. Hence, this correlator
should be avoided in phenomenological analyses. Moreover, it turns out
advantageous to express the quark mass factor, introduced to make the scalar
current renormalisation group invariant, in terms of the renormalisation
invariant quark mass $\hat m_q$. To further study the behaviour of the
perturbative expansion, we introduce a QCD coupling $\hat\alpha_s$, whose
running is explicitly renormalisation scheme independent. The scheme dependence
of $\hat\alpha_s$ is parametrised by a single parameter $C$, being related to
transformations of the QCD scale parameter $\Lambda$. It is demonstrated that
appropriate choices of $C$ lead to a substantial improvement in the behaviour
of the perturbative series for $\Psi^{"}(s)$ and ${\rm Im}\Psi(s)$.
