Borel resummation with low order perturbations in QCD

The perturbative expansions in weak coupling constant in quantum field theories are in general divergent, with the coefficients growing in factorial. One way to make a rigorous sense out of the divergent series is to resum the series using Borel transform. When the large order behavior of the expansion is sign-alternating Borel resummation may yield the full amplitude. But in QCD the series are in general of same-sign and not Borel resummable, and must be augmented by nonperturbative effects. However, Borel resummation can still be useful in these cases when a precise definition of the nonperturbative effects are required, or the coupling is large enough that the usual perturbative amplitude from the low order perturbations is not sufficiently accurate for the purpose. The same-sign large order behavior of weak coupling expansion gives a singularity (renormalon) on the contour of the Borel integral. When the coupling is small the Borel integral receives most of its contribution from the immediate neighborhood of the origin, where the Borel transform can be well-described by the low order perturbations, and the singular behavior of the Borel transform is not of much concern, but as the coupling increases the contribution from the region near the singularity becomes important, and one needs to have an accurate description of the Borel transform in the region that contains the origin as well as the nearest singularity to the