in the $\overline{MS}$ scheme), $\beta_0$ is the one-loop contribution to the QCD beta function, and we identify the expression (\[bubble-all\]) as the one-loop running coupling constant $\alpha_s^{(1)}(l^2)$. At this stage, $\beta_0$ contains only the fermion-loop contribution. However, in the bubble-chain model, one includes the effects of gluons on the running of the coupling by promoting $\beta_0$ to the complete QCD expression $\beta_0=(33/2-N_f)/(6\pi)$.

Now, suppose that a part of the original one-loop integral has the IR behavior $$\int^\Lambda dl\, l^m.
\label{one-loop}$$ Here we have introduced a cutoff $\Lambda$ on the magnitude of $l$, in order to focus on the low-momentum region. We assume that the integral is IR finite ($m\ge -1$), since any infrared divergences are absorbed into the operator matrix elements in the factorization procedure. If one inserts the expression (\[bubbles\]) into the one-loop integral (\[one-loop\]) and integrates term by term, then it is easy to see, using $$\int_0^\mu dl\, l^m \ln^n (l^2/\mu^2)=2^n
\mu^{m+1}(-1)^n n!/(m+1)^{n+1},$$ that terms in the resulting series grow as $n!$. Hence, the series is only asymptotic.

One way to associate a well-defined function with an asymptotic series is through the Borel transform. If a function $f(\alpha_s)$ has a power-series expansion $$f(\alpha_s) \;=\;  \sum_{n=0}^{\infty}\, a_n \alpha_s^{n} \;,
