Bilocal expansion of the Borel amplitude and the hadronic tau decay width

The singular part of Borel transform of a QCD amplitude near the infrared renormalon can be expanded in terms of higher order Wilson coefficients of the operators associated with the renormalon. In this paper we observe that this expansion gives nontrivial constraints on the Borel amplitude that can be used to improve the accuracy of the ordinary perturbative expansion of the Borel amplitude. In particular, we consider the Borel transform of the Adler function and its expansion around the first infrared renormalon due to the gluon condensate. Using the next-to-leading order Wilson coefficient of the gluon condensate operator, we obtain an exact constraint on the Borel amplitude at the first IR renormalon. We then extrapolate, using judiciously chosen conformal transformations and Pade approximants, the ordinary perturbative expansion of the Borel amplitude in such a way that this constraint is satisfied. This procedure allows us to predict the $O(\alpha_s^4)$ coefficient of the Adler function, which gives a result consistent with the estimate by Kataev and Starshenko using a completely different method. We then apply this improved Borel amplitude to the tau decay width, and obtain the strong coupling constant $\alpha_s(M_Z) =0.1193 \pm 0.0007_{exp.} \pm 0.0010_{EW+CKM} \pm 0.0009_{meth.} \pm 0.0003_{evol.}$. We then compare this result with those of other resummation methods.