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Perturbative QCD (pQCD) running coupling a(Q^2) (=alpha_s(Q^2)/pi) isexpected to get modified at low spacelike momenta 0 < Q^2 < 1 GeV^2 so that,instead of having unphysical (Landau) singularities it remains smooth andfinite there, due to infared (IR) fixed point. This behavior is suggested by:Gribov-Zwanziger approach, Dyson-Schwinger equations (DSE) and other functionalmethods, lattice calculations, light-front holographic mapping AdS/CFT modifiedby a dilaton background, and by most of the analytic (holomorphic) QCD models.All such couplings, A(Q^2), differ from the pQCD couplings a(Q^2) at |Q| > 1GeV by nonperturbative (NP) terms, typically by some power-suppressed terms~1/(Q^2)^N. Evaluations of low-energy physical QCD quantities in terms of suchA(Q^2) couplings (with IR fixed point) at a level beyond one-loop are usuallyperformed with (truncated) power series in A(Q^2). We argue that such anevaluation is not correct, because the NP terms in general get out of controlas the number of terms in the power series increases. The series consequentlybecome increasingly unstable under the variation of the renormalization scale,and have a fast asymptotic divergent behavior compounded by the renormalonproblem. We argue that an alternative series in terms of logarithmicderivatives of A(Q^2) should be used. Further, a Pad\'e-related resummationbased on this series gives results which are renormalization scale independentand show very good convergence. Timelike low-energy observables can beevaluated analogously, using the integral transformation which relates thetimelike observable with the corresponding spacelike observable.

Techniques of evaluation of QCD low-energy physical quantities with running coupling with infrared fixed point