Reconsideration of the QCD corrections to the ${\eta }_c$ decays into light hadrons using the principle of maximum conformality

Abstract

In the paper, we analyze the ${\eta }_c$ decays into light hadrons at the next-to-leading order QCD corrections by applying the principle of maximum conformality (PMC). The relativistic correction at the $\mathcal{O}({\alpha }_s{v}^2)$-order level has been included in the discussion, which gives about $10\%$ contribution to the ratio $R$. The PMC, which satisfies the renormalization group invariance, is designed to obtain a scale-fixed and scheme-independent prediction at any fixed order. To avoid the confusion of treating $n_f$-terms, we transform the usual $\overline{\mathrm{MS}}$ pQCD series into the one under the minimal momentum space subtraction scheme. To compare with the prediction under conventional scale setting, $R_{\mathrm{Conv},\mathrm{mMOM}-\mathrm{r}}=\left(4.12_{-0.28}^{+0.30}\right)\times 10^3$, after applying the PMC, we obtain $R_{\mathrm{PMC},\mathrm{mMOM}-\mathrm{r}}=\left(6.09_{-0.55}^{+0.62}\right)\times 10^3$, where the errors are squared averages of the ones caused by $m_c$ and ${\Lambda }_{\mathrm{mMOM}}$. The PMC prediction agrees with the recent PDG value within errors, i.e. $R^{\exp }=\left(6.3\pm 0.5\right)\times 10^3$. Thus we think the mismatching of the prediction under conventional scale-setting with the data is due to improper choice of scale, which however can be solved by using the PMC.

Comments: 5 pages, 2 figures