Abstract
The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium eta _Q (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) for eta _c and left( ^{+0.0}_{-9%}right) for eta _b, by varying the scale within the range of [m_Q, 4m_Q]. The principle of maximum conformality (PMC) provides a systematic way to fix the alpha _s-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 inclusive decay width can be greatly improved. Taking the relativistic correction {mathcal {O}}(alpha _{s}v^2) into consideration, the ratios of the eta _{Q} decays to light hadrons or gamma gamma are: R^mathrm{NNLO}_{eta _c}|_{mathrm{PMC}}=(3.93^{+0.26}_{-0.24})times 10^3 and R^mathrm{NNLO}_{eta _b}|_{mathrm{PMC}}=(22.85^{+0.90}_{-0.87})times 10^3, respectively. Here the errors are for Delta alpha _s(M_Z) = pm 0.0011. As a step forward, by applying the Padacute{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 and R^{mathrm{NNNLO}}_{eta _b}|_{mathrm{PAA+PMC}}=(26.02^{+1.24}_{-1.17})times 10^3. The R^{mathrm{NNNLO}}_{eta _c}|_{mathrm{PAA+PMC}} ratio agrees with the latest PDG value R_{eta _c}^mathrm{{exp}}=(5.3_{-1.4}^{+2.4})times 10^3, indicating the necessity of a strict calculation of NNNLO terms.
Highlights
The heavy quarkonium, being a common bound state of Quantum Chromodynamics (QCD) which consists of a pair of heavy quark and antiquark, has been continuously studied either experimentally or theoretically
Using the principle of maximum conformality (PMC) single-scale method, we show that the resultant PMC conformal series is renormalization scale independent, and the precision of the ηQ inclusive decay width can be greatly improved
The nonrelativistic QCD (NRQCD) factorization theory provides us an effective framework to deal with heavy quarkonium processes [3], which factorizes the pQCD approximant into the non-perturbative but universal longdistance matrix elements (LDMEs) and the perturbatively calculable short-distance coefficients
Summary
The heavy quarkonium, being a common bound state of Quantum Chromodynamics (QCD) which consists of a pair of heavy quark and antiquark, has been continuously studied either experimentally or theoretically. People adopts the “guessed” typical momentum flow of the process such as m Q as the renormalization scale with the purpose of eliminating the large logarithmic terms or minimizing the contributions of the higher-order loop diagrams [12,13] Such a naive treatment, though conventional, directly violates the renormalization group invariance [14] and does not satisfy the self-consistency requirements of the renormalization group [15], leading to a scheme-dependent and scaledependent less reliable pQCD prediction in lower orders. By using the PMC, the effective coupling constant is fixed by using the β-terms of the pQCD series, which are arranged by the general degeneracy relations in QCD [21] among different perturbative orders.
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