}$  &  $0$  &  $0$      &  $-21.00$  &  $-18.53$  &  $-29.43$  \\
				$z$    &  $6.309\times10^{-4}$  &  $1.781\times10^{-2}$  &  $0$  &  $17.85$  &  $-41.99$  &  $-27.79$  &  $-39.24$  \\
				$z^2$  &  $5.047\times10^{-4}$  &  $1.336\times10^{-2}$  &  $0$  &  $0$      &  $\boxed{6.999}$   &  $-55.58$  &  $-58.86$  \\
				$z^3$  &  $4.206\times10^{-4}$  &  $1.069\times10^{-2}$  &  $0$  &  $0$      &  $41.99$   &  $\boxed{-32.42}$  &  $-117.7$  \\
				$z^4$  &  $3.605\times10^{-4}$  &  $8.907\times10^{-3}$  &  $0$  &  $0$      &  $21.00$   &  $55.58$   &  $\boxed{-104.0}$  \\
				$z^5$  &  $3.154\times10^{-4}$  &  $7.634\times10^{-3}$  &  $0$  &  $0$      &  $14.00$   &  $27.79$   &  $117.7$   \\
				\bottomrule
			\end{tabular}
			\label{tab:w_monomial}
		}
	\end{center}
\end{table}

In Tab.~\ref{tab:w_monomial}, we show the residues of the dominant UV and IR poles for the first six
monomials, which are the building blocks for most of the moments used in the literature. Residues of the double poles are shown as boxed numbers. The results of this table can be used to understand
a few features of specific cases. For example, in the Borel transform of the kinematic moment,
Eq.~\eqref{wtau}, a partial cancellation of the leading UV renormalon is manifest: its residue is reduced
by a factor of 3.3. The perturbative series associated with this moment is expected to display a more
tamed behaviour, with the asymptotic nature setting in later.





We  are now in a position to reassess some of the findings of Ref.~\cite{Beneke:2012vb} in the light of these
results.
One of the main observations of Ref.~\cite{Beneke:2012vb}  is that the perturbative series for moments of weight
functions that
contain the monomial $x$ tend to be badly behaved, in the sense that the series never stabilise around
the true value of
the function, it displays what was called a ``run-away behaviour". This can be directly linked to the fact
that the Borel
transform of these moments are the only ones that have the singularity at $u=2$. The contribution of this
renormalon
to the coefficients of the series is fixed sign and it is large at higher orders.





In order to establish the correspondence between the leading renormalons and the behaviour of the
perturbative series, we will make use of an even simpler model.
Since the series is dominated by the leading renormalons, we can construct an approximation to
the result in large-$\beta_0$ using only the leading UV pole and the first two IR poles, which
corresponds to truncating the sum in Eq.~\eqref{eq:BorelAdlerLb0} at its first term. We know from the
works of Ref.~\cite{Beneke:2008ad,Beneke:2012vb,Boito:2018rwt} that such a minimalistic model should be largely sufficient to capture the
main features of the full result in large-$\beta_0$. In Fig.~\ref{fig:Adler_Lb} we confirm this
expectation by plotting the results for the Adler function in large-$\beta_0$ and in its truncated
version, normalized to the value of the Borel integral in each case, which removes an overall
normalization effect that is immaterial here (throughout this paper we use $\alpha_s(m_\tau^2)=0.316(10)$~\cite{Patrignani:2016xqp}). In Fig.~\ref{fig:Adler_Lb}, one sees that the results are essentially identical for our purposes. However, the simplicity of this model prevents the study of moments that are maximally sensitive to
condensates with dimension $D \geq 8$, because the corresponding renormalon poles  are not included.



\begin{figure}[!t] 
\begin{center}
{\includegraphics[width=.7\columnwidth,angle=0]{Figs/LB0-complete-Adler}}
\caption{Adler function order by order in $\alpha_s$ in large-$\beta_0$ and in the truncated version that includes only the leading UV and the first two IR renormalon poles. Both results are normalised to the respective Borel integrals. The horizontal band gives the ambiguity arising from the IR poles, in the prescription of Ref.~\cite{Beneke:2008ad}. Here and elsewhere we use $\alpha_s(m_\tau^2)=0.316(10)$~\cite{Patrignani:2016xqp}.}
\label{fig:Adler_Lb}
\end{center}
\end{figure}


\begin{figure}[!t]
	\begin{center}
		\subfigure[$\delta^{(0)}$, $w(x)=x$, truncated large-$\beta_0$]{\includegraphics[width=.49\columnwidth,angle=0]{Figs/LB0_delta_FO_CI_w_1_T}\label{fig:TruncatedLbweqx}}
		\subfigure[$\delta^{(0)}$, $w(x)=1-x^2$, truncated large-$\beta_0$] {\includegraphics[width=.49\columnwidth,angle=0]{Figs/LB0_delta_FO_CI_w_4_T}\label{fig:TruncatedLbweq1mx2}}
		\subfigure[Contributions to $\delta^{(0)}_x$ (FOPT).]{\includegraphics[width=.49\columnwidth,angle=0]{Figs/LB0_delta_FO_CI_w_1_T_poles}\label{fig:TruncatedLbPolesweqx}}
		\subfigure[Contributions to $\delta^{(0)}_{1-x^2}$ (FOPT).] {\includegraphics[width=.49\columnwidth,angle=0]{Figs/LB0_delta_FO_CI_w_4_T_poles}\label{fig:TruncatedLbPolesweq1mx2}}
		\caption{Perturbative series for the moments $w(x)=x$ and $w(x)=1-x^2$ order by order in $\alpha_s$ in the truncated version of the large-$\beta_0$ limit (see text for details) with $s_0=m_\tau^2$ (upper panels).  Breakdown of the contributions from the leading  singularities for the same moments in FOPT (lower panels). Results are normalized to the respective Borel integrals. The horizontal band represents the ambiguity arising from IR renormalon poles, calculated as in Ref.~\cite{Beneke:2008ad}.}
		\label{fig:TruncatedLbw}
	\end{center}
\end{figure}

We start by considering the moment of $w(x)=x$. The perturbative expansion of $\delta^{(0)}_x$ in the
truncated model for FOPT and CIPT are displayed in Fig.~\ref{fig:TruncatedLbweqx}. The FOPT series shows
the ``run-away behaviour'' identified in Ref.~\cite{Beneke:2012vb}. It overshoots the true value, at first, and
later crosses it and runs into the asymptotic regime with almost no stable region. The CIPT series is
better behaved but also overshoots the true value and then runs into the asymptotic behaviour, with sign
alternating coefficients, much earlier than FOPT. To understand this pattern we can use the Borel
transform of $\delta^{(0)}_{x}$ which is rather simple in the truncated model,
\begin{equation}
B[\delta^{(0)}_x]_T (u) = \frac{4 \text{e}^{5u/3}\sin(\pi u)}{27\pi^2}\Big{[} \frac{8}{u} + \frac{8(5u-4)}{3(2-u)^2} -
\frac{(13u+19)}{3(1+u)^2} - \frac{(17