  We compute NRQCD long-distance matrix elements that appear in the inclusive
production cross sections of $P$-wave heavy quarkonia in the framework of
potential NRQCD. The formalism developed in this work applies to strongly
coupled charmonia and bottomonia. This makes possible the determination of
color-octet NRQCD long-distance matrix elements without relying on measured
cross section data, which has not been possible so far. We obtain results for
inclusive production cross sections of $\chi_{cJ}$ and $\chi_{bJ}$ at the LHC,
which are in good agreement with measurements.
