An Analysis of the Inclusive $\Upsilon (1S) \to \eta^\prime X$ Decay and Constraints on the $\eta^\prime$-Meson Distribution Amplitudes

We calculate the $\eta^\prime$-meson energy spectrum in the $\Upsilon (1S) \to \eta^\prime g g g \to \eta^\prime X$ 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 $\eta^\prime g^* g$ effective vertex function (EVF) involving a virtual and a real gluon from the available data on the hard part of the $\eta^\prime$-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 $\eta^\prime$-meson. The resulting constraints are combined with the existing ones on these coefficients from an analysis of the $\eta^\prime - \gamma$ transition form factor and the requirement of positivity of the EVF, yielding $B^{(q)}_2 (\mu_0^2) = -0.008 \pm 0.054$ and $B^{(g)}_2 (\mu_0^2) = 4.6 \pm 2.5$ for $\mu_0^2 = 2 GeV^2$. This reduces significantly the current uncertainty on these coefficients.