Factorial growth at low orders in perturbative QCD: Control over truncation uncertainties

A method, known as ``minimal renormalon subtraction'' [Phys. Rev. D 97 (2018) 034503, JHEP 2017 (2017) 62], relates the factorial growth of a perturbative series (in QCD) to the power~$p$ of a power correction $\Lambda^p/Q^p$. ($\Lambda$ is the QCD scale, $Q$ some hard scale.) Here, the derivation is simplified and generalized to any~$p$, more than one such correction, and cases with anomalous dimensions. Strikingly, the well-known factorial growth is seen to emerge already at low or medium orders, as a consequence of constraints on the $Q$ dependence from the renormalization group. The effectiveness of the method is studied with the gluonic energy between a static quark and static antiquark (the ``static energy''). Truncation uncertainties are found to be under control after next-to-leading order, despite the small exponent of the power correction ($p=1$) and associated rapid growth seen in the first four coefficients of the perturbative series.