renormalon ambiguities. In this section we clarify the fact that renormalon ambiguities in the short-distance coefficients and operator matrix elements are artifacts of dimensional-regularization factorization schemes and are absent in cutoff schemes. In Sec. \[sec:method\], we describe our method for computing the renormalon ambiguities in the operator matrix elements. We use this method, in Sec. \[sec:computation\], to calculate the renormalon ambiguities in the NRQCD matrix elements for decays of S-wave quarkonia. In Sec. \[sec:gremm-kapustin\], we examine the renormalon content of the various quantities that appear in the Gremm-Kapustin [@gremm-kapustin] relation. We also discuss the ambiguities in various definitions of the heavy-quark mass. Finally, we summarize out results and discuss their implications in Sec \[sec:conclusion\].

Renormalons in the Bubble-Chain Model {#sec:renormalons}
=====================================

As we have mentioned, the bubble-chain model, gives the exact large-$N_f$ behavior of QCD. In it, 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. That is, one replaces the factor $\alpha_s(\mu^2)$ in the one-loop diagram with

$$\begin{aligned}
&&\alpha_s(\mu^2)\sum_{n=0}^\infty
\left[-\beta_0\alpha_s(\mu^2)\ln 
(l^2e^C/\mu^2)\right]^n\label{bubbles}\\
&&={\alpha_s(\mu^2)\over 1+\beta_0\alpha_s(\mu^2)\ln(l^2e^C/\mu^2)}
\label{bubble-sum}\\
&&\equiv \alpha_s^{(1)}(l^2),\end{aligned}$$

\[bubble-all\]

where $l$ is the Euclidean momentum of the gluon in the one-loop expression, $C$ is a renormalization-scheme-dependent constant ($C=-5/3$ 