The $η^\prime g^* g^*$ vertex with arbitrary gluon virtualities in the perturbative QCD hard scattering approach

Abstract

We study the $η^\prime g^* g^*$ vertex for arbitrary gluon virtualities in the time-like and space-like regions, using the perturbative QCD hard scattering approach and an input wave-function of the $η^\prime$-meson consistent with the measured $η^\prime γ^* γ$ transition form factor. The contribution of the gluonic content of the $η^\prime$-meson is taken into account, enhancing the form factor over the entire virtuality considered. However, data on the electromagnetic transition form factor of the $η^\prime$-meson is not sufficient to quantify the gluonic enhancement. We also study the effect of the transverse momenta of the partons in the $η^\prime$-meson on the $η^\prime g^* g^*$ vertex, using the modified hard scattering approach based on Sudakov formalism. Analytic expressions for the $η^\prime g^* g^*$ vertex are presented in limiting kinematic regions and parametrizations are given satisfying the QCD anomaly, for real gluons, and perturbative QCD behavior for large gluon virtualities, in both the time-like and space-like regions. Our results have implications for the inclusive decay $B \to η^\prime X$ and exclusive decays, such as $B \to η^\prime (K,K^*)$, and in hadronic production processes $N + N (\bar N) \to η^\prime X$.