  The $B \to \kappa \bar \kappa$ decays are investigated for the first time in
the perturbative QCD formalism based on the $k_T$ factorization theorem, where
the light scalar $\kappa$ is assumed as a two-quark state. Our numerical
results and phenomenological analyses on the CP-averaged branching ratios and
CP-violating asymmetries show that: (a) the $B_s^0 \to \kappa^+ \kappa^-$ and
$B_s^0 \to \kappa^0 \bar \kappa^0$ decays have large decay rates around ${\cal
O}(10^{-5})$, which could be examined by the upgraded Large Hadron Collider
beauty and/or Belle-II experiments in the near future; (b) a large decay rate
about $3 \times 10^{-6}$ appears in the pure annihilation $B_d^0 \to \kappa^+
\kappa^-$ channel, which could provide more evidences to help distinguish
different QCD-inspired factorization approaches, even understand the
annihilation decay mechanism; (c) the pure penguin modes $B_d^0 \to \kappa^0
\bar \kappa^0$ and $B_s^0 \to \kappa^0 \bar \kappa^0$ would provide a promising
ground to search for the possible new physics because of their zero direct and
mixing-induced CP violations in the standard model. The examinations with good
precision from the future experiments will help to further study the
perturbative and/or nonperturbative QCD dynamics involved in these considered
decay modes.
