  We study the prompt production of the $\chi_c(1^+)$ and $\chi_b(1^+)$ mesons
at high energies. Unlike $\chi(0^+,2^+)$ production, $\chi (1^+)$ mesons cannot
be created at LO via the fusion of two on-mass-shell gluons, that is
$gg\to\chi_{c,b}(1^+)$ are not allowed. However, the available experimental
data show that the cross sections for $\chi_c(1^+)$ and $\chi_c(2^+)$ are
comparable. We therefore investigate four other $\chi(1^+)$ production
mechanisms: namely, (i) the standard NLO process $gg\to\chi_{c,b}(1^+)+g$, (ii)
via gluon virtuality, (iii) via gluon reggeization and, finally, (iv) the
possibility to form $\chi_{c,b}(1^+)$ by the fusion of three gluons, where one
extra gluon comes from another parton cascade, as in the Double Parton
Scattering processes.
