  The polarizations of promptly produced $\chi_\mathrm{c1}$ and
$\chi_\mathrm{c2}$ mesons are studied using data collected by the CMS
experiment at the LHC, in proton-proton collisions at $\sqrt{s} = $ 8 TeV. The
$\chi_\mathrm{c}$ states are reconstructed via their radiative decays
$\chi_\mathrm{c}$ $\to$ $\mathrm{J}/\psi\, \gamma$, with the photons being
measured through conversions to e$^+$e$^-$, which allows the two states to be
well resolved. The polarizations are measured in the helicity frame, through
the analysis of the $\chi_\mathrm{c2}$ to $\chi_\mathrm{c1}$ yield ratio as a
function of the polar or azimuthal angle of the positive muon emitted in the
$\mathrm{J}/\psi$ $\to$ $\mu^+\mu^-$ decay, in three ranges of
$\mathrm{J}/\psi$ transverse momentum. While no differences are seen between
the two states in terms of azimuthal decay angle distributions, they are
observed to have significantly different polar anisotropies. The measurement
favors a scenario where at least one of the two states is strongly polarized
along the helicity quantization axis, in agreement with nonrelativistic quantum
chromodynamics predictions. This is the first measurement of significantly
polarized quarkonia produced at high transverse momentum.
