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We present a formulation to give renormalon-free predictions consistentlywith fixed order perturbative results. The formulation has a similarity toLee's method in that the renormalon-free part consists of two parts: one isgiven by a series expansion which does not contain renormalons and the other isthe real part of the Borel integral of a singular Borel transform. The mainnovel aspect is to evaluate the latter object using a dispersion relationtechnique, which was possible only in the large-$\beta_0$ approximation. Here,we introduce an "ambiguity function," which is defined by inverse Mellintransform of the singular Borel transform. With the ambiguity function, we canrewrite the Borel integral in an alternative formula, which allows us to obtainthe real part using analytic techniques similarly to the case of thelarge-$\beta_0$ approximation. We also present detailed studies ofrenormalization group properties of the formulation. As an example, we applyour formulation to the fixed-order result of the static QCD potential, whoseclosest renormalon is already visible.

Formulation for renormalon-free perturbative predictions beyond large-$\beta_0$ approximation