Improved $η^\prime$-Meson Distribution Amplitudes from Inclusive $Υ(1S) \to η^\prime X$ Decay

We calculate the $η^\prime$-meson energy spectrum in the $Υ(1S) \to η^\prime g g g \to η^\prime X$ decay in the leading-order perturbative QCD in the static-quark limit for the orthoquarkonium.Our principal result is the extraction of parameters of the $η^\prime g^* g$ effective vertex function (EVF) involving a virtual and a real gluon from the available data on the hard part of the $η^\prime$-meson energy spectrum. The perturbative-QCD based framework provides a good description of the available CLEO data, allowing one to constrain the lowest Gegenbauer coefficients $B^{(q)}_2$ and $B^{(g)}_2$ of the quark-antiquark and gluonic distribution amplitudes of the $η^\prime$-meson. The resulting constraints are combined with the existing ones on these coefficients from an analysis of the $η^\prime - γ$ transition form factor and the requirement of positivity of the EVF, yielding $B^{(q)}_2 (μ_0^2) = -0.008 \pm 0.054$ and $B^{(g)}_2 (μ_0^2) = 4.6 \pm 2.5$ for $μ_0^2 = 2 GeV^2$. This reduces significantly the current uncertainty on these coefficients.

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