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We discuss exclusive elastic double diffractive axial-vector $\chi_c(1^+)$and tensor $\chi_c(2^+)$ mesons production for different meson polarisations inproton-(anti)proton collisions at the Tevatron energy. The amplitude for theprocess is derived within the $k_t$-factorisation approach using unintegratedgluon distributions (UGDFs). Differential cross sections for different $\chi_c$polarisations are calculated. Angular distributions of $J/\psi$ meson from theradiative $\chi_c(1^+,2^+)$ decays are derived. Prospects for experimentalselection of different spin states of $\chi_c$ mesons are discussed.

Polarisation effects in the central exclusive $\chi_c$ production and the $J/\psi$ angular distributions