Renormalon-based resummation for spacelike and timelike QCD quantities whose perturbation expansion has general form

We present a generalisation of our previous approach of a renormalon-motivated resummation of the QCD observables. Previously it was applied to the spacelike observables whose perturbation expansion was $D(Q^2) = a(Q^2) + O(a^2)$, where $a(Q^2) \equiv \alpha_s(Q^2)/\pi$ is the running QCD coupling. Now we generalise the resummation to spacelike quantities $D(Q^2) = a(Q^2)^{\nu_0} + O(a^{\nu_0+1})$ and timelike quantities $F(\sigma) = a(\sigma)^{\nu_0} + O(a^{\nu_0+1})$, where $\nu_0$ is in general a noninteger number ($0<\nu_0 \leq 1$). We evaluate with this approach a timelike quantity, namely the scheme-invariant factor of the Wilson coefficient of the chromomagnetic operator in the heavy-quark effective Lagrangian, and related quantities.