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

Abstract

We calculate the ${\eta }^{\prime }$-meson energy spectrum in the decay ${\rm Y}(1S)\to {\eta }^{\prime }ggg\to {\eta }^{\prime }X$ in the leading-order perturbative QCD in the static quark limit for the Orthoquarkonium. Our principal result is the extraction of parameters of the ${\eta }^{\prime }{g}^{\ast }g$ 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 $B_2^{(q)}$ and $B_2^{(g)}$ 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 $B_2^{(q)}({\mu }_0^2)=-0.008\pm 0.054$ and $B_2^{(g)}({\mu }_0^2)=4.6\pm 2.5$ for ${\mu }_0^2=2$ GeV${}^2$. This reduces significantly the current uncertainty on these coefficients. The resulting EFV $F_{{\eta }^{\prime }{g}^{\ast }g}(p^2,0,m_{{\eta }^{\prime }}^2)$, including the ${\eta }^{\prime }$-meson mass effects, is presented.

Comments: 23 pages, 8 figures; use epsfig.sty; Typos corrected, numerical analysis further refined; added an equation; to appear in the European Physical Journal C