Probing the Weizs\"acker-Williams gluon Wigner distribution in pp collisions

We show that the diffractive forward production of two quarkonia, especially the ${\chi }_{cJ}$ and ${\eta }_c$ states, in proton-proton or proton-ion collisions can access the Weizs"acker-Williams gluon Wigner distribution of the proton. We use the hybrid factorization approach where the collinear, double gluon PDF is applied for one of the protons and the $k_T$-dependent (Wigner or GTMD) distribution for the other. The production of quarkonia is treated in the NRQCD framework. A particularly concise formula is obtained for double ${\chi }_{J=1}$ production.