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In this paper, by employing the $k_{T}$ factorization theorem, we made thefirst calculation for the space-like scalar pion form factor $Q^2 F(Q^2)$ atthe leading order (LO) and the next-to-leading order (NLO) level, and thenfound the time-like scalar pion form factor $F'^{(1)}_{\rm a,I}$ by analyticcontinuation from the space-like one. From the analytical evaluations and thenumerical results, we found the following points: (a) the NLO correction to thespace-like scalar pion form factor has an opposite sign with the LO one but isvery small in magnitude, can produce at most $10\%$ decrease to LO result inthe 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 thefactorizable 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 thevariation of cutoff scale $\mu_0$, but this form factor has a large strongphase around $-55^\circ$ and may play an important role in producing large CPviolation 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 produceonly a very small enhancement to their branching ratios, less than $3\%$ inmagnitude, and therefore we could not interpret the well-known $\pi\pi$-puzzleby the inclusion of this NLO correction to the factorizable annihilationdiagrams.

The NLO contributions to the scalar pion form factors and the ${\cal O}(\alpha_s^2)$ annihilation corrections to the $B\to \pi\pi$ decays