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We discuss exclusive elastic double diffractive axial-vector \chi_c(1^{+})meson production in proton-antiproton collisions at the Tevatron. The amplitudefor the process is derived within the k_t-factorisation approach withunintegrated gluon distribution functions (UGDFs). We show that the famousLandau-Yang theorem is not applicable in the case of off-shell gluons.Differential cross sections for different UGDFs are calculated. We compareexclusive production of \chi_c(1^+) and \chi_c(0^+). The contribution of\chi_c(1^+) to the J/\Psi + \gamma channel is smaller than that of the\chi_c(0^+) decay, but not negligible and can be measured. The numerical valueof the ratio of the both contributions is almost independent of UGDFs modeling.

Elastic double diffractive production of axial-vector \chi_c(1^{++}) mesons and the Landau-Yang theorem