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

We calculate the $\eta^\prime$-meson energy spectrum in the decay $\Upsilon (1S) \to \eta^\prime g g g \to \eta^\prime X$ 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 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 $\eta-\gamma$ and $\eta^\prime-\gamma$ transition form factors and the requirement of positivity of the EVF, yielding $B^{(q)}_2(\mu_0^2) = 0.010 \pm 0.068$ and $B^{(g)}_2(\mu_0^2) = 5.6 \pm 3.4$ for $\mu_0^2=2$ GeV$^2$. This reduces significantly the current uncertainty on these coefficients. The resulting EFV $F_{\eta^\prime g^* g} (q^2, 0, m_{\eta^\prime}^2)$, including the $\eta^\prime$-meson mass effects, is presented.