  In this article, we study the axial-vector mesons $Z_b(10610)$ and
$Z_b(10650)$ with the $C\gamma_\mu-C\gamma_5$ type and
$C\gamma_\mu-C\gamma_\nu$ type interpolating currents respectively by carrying
out the operator product expansion to the vacuum condensates up to
dimension-10. In calculations, we explore the energy scale dependence of the
QCD spectral densities of the hidden bottom tetraquark states in details for
the first time, and suggest a formula
$\mu=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_b)^2}$ with the effective mass
${\mathbb{M}}_b=5.13\,\rm{GeV}$ to determine the energy scales. The numerical
results favor assigning the $Z_b(10610)$ and $Z_b(10650)$ as the
$C\gamma_\mu-C\gamma_5$ type and $C\gamma_\mu-C\gamma_\nu$ type hidden bottom
tetraquark states, respectively. We obtain the mass of the $J^{PC}=1^{++}$
hidden bottom tetraquark state as a byproduct, which can be compared to the
experimental data in the futures. Furthermore, we study the strong decays
$Z_b^\pm(10610)\to\Upsilon\pi^{\pm}\, ,\,\eta_b\rho^{\pm}$ with the three-point
QCD sum rules, the decay widths also support assigning the $Z_b(10610)$ as the
$C\gamma_\mu-C\gamma_5$ type hidden bottom tetraquark state.
