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 $hatalpha_s$, whose running is explicitly renormalisation scheme independent. The scheme dependence of $hatalpha_s$ 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 ${rm Im}Psi(s)$.
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