  In this paper, by employing the $k_{T}$ factorization theorem, we made the
first calculation for the space-like scalar pion form factor $Q^2 F(Q^2)$ at
the leading order (LO) and the next-to-leading order (NLO) level, and then
found the time-like scalar pion form factor $F'^{(1)}_{\rm a,I}$ by analytic
continuation from the space-like one. From the analytical evaluations and the
numerical results, we found the following points: (a) the NLO correction to the
space-like scalar pion form factor has an opposite sign with the LO one but is
very small in magnitude, can produce at most $10\%$ decrease to LO result in
the considered $Q^2$ region; (b) the NLO time-like scalar pion form factor
$F'^{(1)}_{\rm a,I}$ describes the ${\cal O}(\alpha_s^2)$ contribution to the
factorizable annihilation diagrams of the considered $B \to \pi\pi$ decays,
i.e. the NLO annihilation correction; (c) the NLO part of the form factor
$F'^{(1)}_{\rm a,I}$ is very small in size, and is almost independent with the
variation of cutoff scale $\mu_0$, but this form factor has a large strong
phase around $-55^\circ$ and may play an important role in producing large CP
violation for $B\to \pi\pi$ decays; and (d) for $B^0 \to \pi^+\pi^-$ and $
\pi^0\pi^0$ decays, the newly known NLO annihilation correction can produce
only a very small enhancement to their branching ratios, less than $3\%$ in
magnitude, and therefore we could not interpret the well-known $\pi\pi$-puzzle
by the inclusion of this NLO correction to the factorizable annihilation
diagrams.
