The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium $$\eta _Q$$ηQ (Q being c or b) has been done in the literature within the framework of nonrelativistic QCD. One may observe that the NNLO decay width still has large conventional renormalization scale dependence due to its weaker pQCD convergence, e.g. about $$\left( ^{+4\%}_{-34\%}\right) $$-34%+4% for $$\eta _c$$ηc and $$\left( ^{+0.0}_{-9\%}\right) $$-9%+0.0 for $$\eta _b$$ηb, by varying the scale within the range of $$[m_Q, 4m_Q]$$[mQ,4mQ]. The principle of maximum conformality (PMC) provides a systematic way to fix the $$\alpha _s$$αs-running behavior of the process, which satisfies the requirements of renormalization group invariance and eliminates the conventional renormalization scheme and scale ambiguities. Using the PMC single-scale method, we show that the resultant PMC conformal series is renormalization scale independent, and the precision of the $$\eta _Q$$ηQ inclusive decay width can be greatly improved. Taking the relativistic correction $${\mathcal {O}}(\alpha _{s}v^2)$$O(αsv2) into consideration, the ratios of the $$\eta _{Q}$$ηQ decays to light hadrons or $$\gamma \gamma $$γγ are: $$R^\mathrm{NNLO}_{\eta _c}|_{\mathrm{PMC}}=(3.93^{+0.26}_{-0.24})\times 10^3$$RηcNNLO|PMC=(3.93-0.24+0.26)×103 and $$R^\mathrm{NNLO}_{\eta _b}|_{\mathrm{PMC}}=(22.85^{+0.90}_{-0.87})\times 10^3$$RηbNNLO|PMC=(22.85-0.87+0.90)×103, respectively. Here the errors are for $$\Delta \alpha _s(M_Z) = \pm 0.0011$$Δαs(MZ)=±0.0011. As a step forward, by applying the Pad$$\acute{e}$$e´ approximation approach (PAA) over the PMC conformal series, we obtain approximate NNNLO predictions for those two ratios, e.g. $$R^{\mathrm{NNNLO}}_{\eta _c}|_{\mathrm{PAA+PMC}} =(5.66^{+0.65}_{-0.55})\times 10^3$$RηcNNNLO|PAA+PMC=(5.66-0.55+0.65)×103 and $$R^{\mathrm{NNNLO}}_{\eta _b}|_{\mathrm{PAA+PMC}}=(26.02^{+1.24}_{-1.17})\times 10^3$$RηbNNNLO|PAA+PMC=(26.02-1.17+1.24)×103. The $$R^{\mathrm{NNNLO}}_{\eta _c}|_{\mathrm{PAA+PMC}}$$RηcNNNLO|PAA+PMC ratio agrees with the latest PDG value $$R_{\eta _c}^\mathrm{{exp}}=(5.3_{-1.4}^{+2.4})\times 10^3$$Rηcexp=(5.3-1.4+2.4)×103, indicating the necessity of a strict calculation of NNNLO terms.

In the paper, we present QCD predictions for γ + ηc production at an electron-positron collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for e+ + e− → γ + ηc 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 $$ \sqrt{s} $$ s . The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal β contributions into the QCD coupling αs(Q2). The renormalization group equation then sets the value of αs 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 renormalization scale-and-scheme independent prediction, σ|NNLO,PMC ≃ 41.18 fb for $$ \sqrt{s} $$ 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 Padé approximation approach, we predict σ|NNNLO,PMC+Pade ≃ 18.99 fb, which is consistent with the recent BELLE measurement $$ {\sigma}^{\mathrm{obs}}={16.58}_{-9.93}^{+10.51} $$ σ obs = 16.58 − 9.93 + 10.51 fb at $$ \sqrt{s} $$ s ≃ 10.6 GeV. This procedure also provides a first estimate of the NNNLO contribution.