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

Abstract

We calculate the $η^\prime$-meson energy spectrum in the decay $Υ(1S) \to η^\prime g g g \to η^\prime X$ in the leading-order perturbative QCD in the static quark limit for the Orthoquarkonium. Our principal result is the extraction of parameters of the $η^\prime g^* g$ 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 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 $η-γ$ and $η^\prime-γ$ transition form factors 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. The resulting EFV $F_{η^\prime g^* g} (p^2, 0, m_{η^\prime}^2)$, including the $η^\prime$-meson mass effects, is presented.