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The inconsistency between the fixed-order (FO) and contour-improved (CI)representation of the QCD corrections to the inclusive hadronic tau decay widthlimits the precision to which the strong coupling can be determined from thisprocess. It has recently been shown that subtracting the infrared renormalondivergence related to the gluon condensate resolves the discrepancy. Here wesuggest to employ the gradient flow to define gauge-invariant regularizedoperators and to use the corresponding condensates in the operator productexpansion. The associated rearrangement of the perturbative series results inautomatic renormalon subtraction without the need to determine explicitly theStokes constants that normalize the divergent asymptotic series. Applying thismethod to the gluon condensate, we find that the CI series is modified and nowagrees with the (unmodified) FO series. This conclusively demonstrates thepreference for the fixed-order approach, as has been advocated long ago.

Gradient-flow renormalon subtraction and the hadronic tau decay series