The Effective $\eta^\prime g^* g^*$ Vertex at Arbitrary Gluon Virtualities

The effective $\eta^\prime g^* g^*$ vertex is studied in the standard (Brodsky-Lepage) and modified hard scattering approaches for arbitrary gluon virtualities in the time-like and space-like regions. The contribution of the gluons in the $\eta^\prime$-meson is taken into account, and the wave-function is constrained using data on the electromagnetic transition form factor of the $\eta^\prime$ meson. Our results have implications for the inclusive decay $B \to \eta^\prime X$ and exclusive decays, such as $B \to \eta^\prime (K, K^*)$, and in hadronic production processes $N + N (\bar N) \to \eta^\prime X$.