Heavy - light current correlators at order at $\alpha^2_s$ in QCD and HQET

The non-diagonal correlators of vector and scalar currents are considered at three-loop order in QCD. The full mass dependence is computed in the case where one of the quarks is massless and the other one carries mass $M$. We exploit the decoupling relations between the full theory and the heavy quark effective theory (HQET) in order to obtain the logarithmic parts of the leading threshold terms. With the help of conformal mapping and Pad\'e approximation numerical estimates for the non-logarithmic terms are extracted which in turn lead to a prediction of the correlator in HQET at order $\alpha_s^2$. As applications of the vector and scalar correlator we consider the single-top-quark production via the process $q\bar{q}\to t\bar{b}$ and the decay rate of a charged Higgs boson into hadrons, respectively. In both cases the computed NLO corrections are shown to be numerically much less important than the leading ones. On the contrary, the NLO order QCD corrections to the HQET sum rule for the leptonic decay rate of a heavy-light meson proves to be comparable to the leading one