The NLO contributions to the scalar pion form factors and the O(αs2) annihilation corrections to the B→ππB→ππ decays

In this paper, by employing the kTkT factorization theorem, we made the first calculation for the space-like scalar pion form factor Q2F(Q2)Q2F(Q2) at the leading order (LO) and the next-to-leading order (NLO) level, and then found the time-like scalar pion form factor Fa,I′(1) 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%10% decrease to the LO result in the considered Q2Q2 region; (b) the NLO time-like scalar pion form factor Fa,I′(1) describes the O(αs2) contribution to the factorizable annihilation diagrams of the considered B→ππB→ππ decays, i.e. the NLO annihilation correction; (c) the NLO part of the form factor Fa,I′(1) is very small in size, and is almost independent of the variation of cutoff scale μ0μ0, but this form factor has a large strong phase around −55°−55° and may play an important role in producing large CP violation for B→ππB→ππ decays; and (d) for B0→π+π−B0→π+π− and π0π0π0π0 decays, the newly known NLO annihilation correction can produce only a very small enhancement to their branching ratios, less than 3%3% in magnitude, and therefore we could not interpret the well-known ππ-puzzle by the inclusion of this NLO correction to the factorizable annihilation diagrams.