Improved ${\eta }^{\mathrm{\prime }}$-Meson Distribution Amplitudes from Inclusive $\mathrm{{\rm Y}}(1S)\to {\eta }^{\mathrm{\prime }}X$ Decay

We calculate the ${\eta }^{\mathrm{\prime }}$-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 ${\eta }^{\mathrm{\prime }}{g}^{\ast }g$ effective vertex function (EVF) involving a virtual and a real gluon from the available data on the hard part of the ${\eta }^{\mathrm{\prime }}$-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 $B_2^{(q)}$ and $B_2^{(g)}$ of the quark-antiquark and gluonic distribution amplitudes of the ${\eta }^{\mathrm{\prime }}$-meson. The resulting constraints are combined with the existing ones on these coefficients from an analysis of the ${\eta }^{\mathrm{\prime }}-\gamma$ transition form factor and the requirement of positivity of the EVF, yielding $B_2^{(q)}({\mu }_0^2)=-0.008\pm 0.054$ 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.