  We study the {\it CP}-averaged branching fractions and the {\it CP}-violating
asymmetries in the pure annihilation decays of $B_s^0 \to a_0^+ a_0^-$ and
$B_d^0 \to K_0^{*+} K_0^{*-}$, where $a_0\;[K_0^*]$ denotes the scalar
$a_0(980)$ and $a_0(1450)$ [$K_0^*(800) ( {\rm or}\; \kappa)$ and
$K_0^*(1430)$], with the perturbative QCD factorization approach under the
assumption of two-quark structure for the $a_0$ and $K_0^*$ states. The
numerical results show that the branching ratios of the $B_d^0 \to K_0^{*+}
K_0^{*-}$ decays are in the order of $10^{-6}$, while the decay rates of the
$B_s^0 \to a_0^+ a_0^-$ modes are in the order of $10^{-5}$. In light of the
measured modes with the same quark components in the pseudoscalar sector,
namely, $B_d^0 \to K^+ K^-$ and $B_s^0 \to \pi^+ \pi^-$, the predictions for
the considered decay modes in this work are expected to be measured at the
Large Hadron Collider beauty and/or Belle-II experiments in the (near) future.
Meanwhile, it is of great interest to find that the twist-3 distribution
amplitudes $\phi^{S}$ and $\phi^{T}$ with inclusion of the Gegenbauer
polynomials for the scalar $a_0(1450)$ and $K_0^*(1430)$ states in scenario 2
contribute slightly to the branching ratios while significantly to the {\it CP}
violations in the $B_d^0 \to {K_0^*}(1430)^+ {K_0^*}(1430)^-$ and $B_s^0 \to
a_0(1450)^+ a_0(1450)^-$ decays, which indicates that, compared to the
asymptotic $\phi^S$ and $\phi^T$, these Gegenbauer polynomials could change the
strong phases evidently in these pure annihilation decay channels. These
predictions await for the future confirmation experimentally, which could
further provide useful information to help explore the inner structure of the
scalars and shed light on the annihilation decay mechanism.
