The heavy quarkonium inclusive decays using the principle of maximum conformality

Abstract

AbstractThe 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.