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

Abstract

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 $a(Q^2)\equiv \hat{I}{\pm }_s(Q^2)/\ddot{I}€$ is the running QCD coupling. Now we generalise the resummation to spacelike quantities $D(Q^2)=a(Q^2{)}^{\hat{I}{½}_0}+O(a^{\hat{I}{½}_0+1})$ and timelike quantities $F(\ddot{I}ƒ)=a(\ddot{I}ƒ{)}^{\hat{I}{½}_0}+O(a^{\hat{I}{½}_0+1})$, where $\hat{I}{½}_0$ is in general a noninteger number ($0<\hat{I}{½}_0\le 1$). 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.

v2: 39 pp, 7 figures; final version as published in JPG; added references; improved text; new appendix (D) added