  In this article, we take the tensor currents $\bar{Q}(x)\sigma_{\mu\nu}Q(x)$
to interpolate the $P$-wave spin-singlet heavy quarkonium states $h_Q$, and
study the masses and decay constants with the Borel sum rules and moments sum
rules. The masses and decay constants from the Borel sum rules and moments sum
rules are consistent with each other, the masses are also consistent with the
experimental data. We can take the decay constants as basic input parameters
and study other phenomenological quantities with the three-point correlation
functions via the QCD sum rules. The heavy quarkonium states $h_Q$ couple
potentially to the tensor currents $\bar{Q}(x)\sigma_{\mu\nu}Q(x)$, and have
the quark structure $\epsilon^{ijk}\xi^{\dagger}\sigma^k\zeta$ besides the
quark structure $ik_2^i \xi^{\dagger}\sigma \cdot (\vec{k}_1-\vec{k}_2)\zeta $.
In calculations, we take into account the leading-order, next-to-leading-order
perturbative contributions, and the gluon condensate, four-quark condensate
contributions in the operator product expansion. The analytical expressions of
the perturbative QCD spectral densities have applications in studying the
two-body decays of a boson to two fermions with the vertexes
$\sigma_{\mu\nu}\gamma_5$ and $\sigma_{\mu\nu}$.
