  In the paper, we present QCD predictions for $\eta_{c} + \gamma$ production
at an electron-position collider up to next-to-next-to-leading order (NNLO)
accuracy without renormalization scale ambiguities. The NNLO total
cross-section for $e^{+}+e^{-}\to\gamma+\eta_{c}$ using the conventional
scale-setting approach has large renormalization scale ambiguities, usually
estimated by choosing the renormalization scale to be the $e^+ e^-$
center-of-mass collision energy $\sqrt{s}$. The Principle of Maximum
Conformality (PMC) provides a systematic way to eliminate such renormalization
scale ambiguities by summing the nonconformal $\beta$ contributions into the
QCD coupling $\alpha_s(Q^2)$. The renormalization group equation then sets the
value of $\alpha_s$ for the process. The PMC renormalization scale reflects the
virtuality of the underlying process, and the resulting predictions satisfy all
of the requirements of renormalization group invariance, including
renormalization scheme invariance. After applying the PMC, we obtain a
scale-and-scheme independent prediction, $\sigma|_{\rm NNLO, PMC}\simeq 41.18$
fb for $\sqrt{s}$=10.6 GeV. The resulting pQCD series matches the series for
conformal theory and thus has no divergent renormalon contributions. The large
$K$ factor which contributes to this process reinforces the importance of
uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme
independent conformal series and the $\rm Pad\acute{e}$ approximation approach,
we predict $\sigma|_{\rm NNNLO, PMC+Pade} \simeq 21.36$ fb, which is consistent
with the recent BELLE measurement $\sigma^{\rm obs}$=$16.58^{+10.51}_{-9.93}$
fb at $\sqrt{s} \simeq 10.6$ GeV. This procedure also provides a first estimate
of the NNNLO contribution.
