The NLO contributions to the scalar pion form factors and the O (αS2) (at) annihilation corrections to the B → ππ decays

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-a,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% decrease to the LO result in the considered Q(2) region; (b) the NLO time-like scalar pion form factor F-a,I((1)) describes the O(alpha(2)(S)) contribution to the factorizable annihilation diagrams of the considered B -> pi pi decays, i.e. the NLO annihilation correction; (c) the NLO part of the form factor F-a,I((1)) is very small in size, and is almost independent of the variation of cutoff scale mu(0), but this form factor has a large strong phase around -55 degrees and may play an important role in producing large CP violation for B -> pi pi decays; and (d) for B-0 -> 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 rer-puzzle by the inclusion of this NLO correction to the factorizable annihilation. diagrams. (C) 2015 The Authors. Published by Elsevier B.V.