  In this article, we construct the diquark-antidiquark type current operators
to study the axialvector $B_c$-like tetraquark states with the QCD sum rules.
In calculations, we take the energy scale formula as a powerful constraint to
choose the ideal energy scales of the QCD spectral densities and add detailed
discussions to illustrate why we take the energy scale formula to improve the
QCD sum rules for the doubly heavy tetraquark states. The predicted masses
$M_{Z_{\bar{b}c}(1^{+-})}=7.30\pm0.08\,\rm{GeV}$ and
$M_{Z_{\bar{b}c}(1^{++})}=7.31\pm0.08\,\rm{GeV}$ lie between the conventional
$2\rm{P}$ and $3\rm{P}$ $B_c$ states with $J^P=1^+$ from the potential quark
models, which is a typical feature of the diquark-antidiquark type $B_c$-like
tetraquark states.
