seen extensive applications in recent years in such topics as the Operator Product Expansions (OPE) for $e^+e^-$ annihilation and $\tau$ decay[@Mueller; @Beneke; @Neubert; @BBB], Heavy Quark Effective Theory (HQET)[@BB94; @BSU; @NS94; @LMS94], and the NRQCD factorization formalism[@braaten-chen]. In this model, which we call the “bubble-chain model,” one generates the “perturbation series” by inserting into the gluon propagator in a one-loop diagram all numbers of order-$\alpha_s$ fermion-loop vacuum-polarization corrections. The resulting perturbation series gives the order-$\alpha_s$ fermion-loop contribution to the running of the coupling constant. In this expression, one then replaces the fermion-loop contribution to the one-loop QCD beta function $\beta_0$ with the full one-loop QCD beta function. As we will describe later, in the bubble-chain model, the perturbation series diverges, with terms growing as the factorial of the order.

One can attempt to resum the series by carrying out a Borel transformation. In the Borel plane, the factorial growth of the series corresponds to singularities, known as renormalons. If the renormalon singularities lie on the positive real axis, then they make the inverse Borel transform ill-defined. These ambiguities in the inverse Borel transform are potentially important because they have the same nominal size as corrections in the QCD factorization formulas 