pp to ppη' reaction at high energies

We discuss double-diffractive (double-elastic) production of the $\eta'$-meson in the $pp \to p\eta'p$ reaction within the formalism of unintegrated gluon distribution functions (UGDF). We estimate also the contribution of $\gamma^* \gamma^* \to \eta'$ fusion. The distributions in the Feynman $x_F$ (or rapidity), transferred four-momenta squared between initial and final protons ($t_1$, $t_2$) and azimuthal angle difference between outgoing protons ($\Phi$) are calculated. The deviations from the $\sin^2(\Phi)$ dependence predicted by one-step vector-vector-pseudoscalar coupling are quantified and discussed. The results are compared with the results of the WA102 collaboration at CERN. Most of the models of UGDF from the literature give too small cross section as compared to the WA102 data and predict angular distribution in relative azimuthal angle strongly asymmetric with respect to $\pi/2$ in disagreement with the WA102 data. This points to a different mechanism at the WA102 energy. Predictions for RHIC, Tevatron and LHC are given. We find that the normalization, $t_{1,2}$ dependences as well as deviations from $\sin^2(\Phi)$ of double-diffractive double-elastic cross section are extremely sensitive to the choice of UGDF. Possible implications for UGDFs are discussed.