  In the framework of QCD Sum Rules we investigate the $q q^\prime \bar{Q}
\bar{Q}$ tetraquark structure with quantum number $J^P = 1^+$, which are
embedded in two types of configurations, the $[8_c]_{q \bar{Q}} \otimes
[8_c]_{q^\prime \bar{Q}}$ and $[1_c]_{q \bar{Q}} \otimes [1_c]_{q^\prime
\bar{Q}}$ with $Q = b, c$, $q = u$, and $q^\prime = d, s$. Our finding confirms
the Lattice QCD result that the bottom tetraquark states could exist and their
masses are evaluated. In the calculation, the non-perturbative condensate
contributions up to dimension eight in operator product expansion are
considered, and those terms linear to the strange quark mass $m_s$ are kept. It
is found that for octet-octet configuration the masses of potential tetraquark
states are about $11.28$ GeV for the $ud\bar{b}\bar{b}$ system, and $11.31$ and
$11.34$ GeV for the $us\bar{b}\bar{b}$ system, which are above the
corresponding two-meson thresholds. For molecular configuration, the
corresponding masses are found below the thresholds, that is $10.36$ GeV and
$10.48$ GeV, respectively. The possible tetraquark decay channels are analyzed
and the strong decay rates are evaluated. The mass dependence on the radiative
correction and the condensates is estimated. Moreover, the doubly charmed
tetraquark states are also considered.
