Renormalon-inspired resummations for vector and scalar correlators -- estimating the uncertainty in αs(M2τ) and α(M2Z)

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients containing the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known next-to-leading order (NLO), and next-NLO (NNLO) results, we employ the Complete Renormalization Group Improvement (CORGI) approach in which all RG-predictable ultraviolet logarithms are resummed to all-orders, removing all dependence on the renormalization scale. We can also obtain fixed-order CORGI results. Including suitable weight-functions we can numerically integrate these results for the D-function in the complex energy plane to obtain so-called 'contour-improved' results for the ratio R and its tau decay analogue Rτ. We use the difference between the all-orders and fixed-order (NNLO) results to estimate the uncertainty in αs(M2Z) extracted from Rτ measurements, and find αs(M2Z)=0.120±0.002. We also estimate the corresponding uncertainty in α(M2Z) arising from hadronic corrections by considering the uncertainty in R(s), in the low-energy region, and compare with other estimates. Analogous resummations are also given for the scalar correlator. As an adjunct to these studies we show how fixed-order contour-improved results can be obtained analytically in closed form at the two-loop level in terms of the Lambert W-function and hypergeometric functions