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The non-diagonal correlators of vector and scalar currents are considered atthree-loop order in QCD. The full mass dependence is computed in the case whereone of the quarks is massless and the other one carries mass $M$. We exploitthe decoupling relations between the full theory and the heavy quark effectivetheory (HQET) in order to obtain the logarithmic parts of the leading thresholdterms. With the help of conformal mapping and Pad\'e approximation numericalestimates for the non-logarithmic terms are extracted which in turn lead to aprediction of the correlator in HQET at order $\alpha_s^2$. As applications ofthe vector and scalar correlator we consider the single-top-quark productionvia the process $q\bar{q}\to t\bar{b}$ and the decay rate of a charged Higgsboson into hadrons, respectively. In both cases the computed NLO correctionsare shown to be numerically much less important than the leading ones. On thecontrary, the NLO order QCD corrections to the HQET sum rule for the leptonicdecay rate of a heavy-light meson proves to be comparable to the leading one.Comment: 31 page

Heavy-light current correlators at order $\alpha_\mathrm{s}^2$ in QCD and HQET