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

Abstract

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 $F_{\mathrm{a},\mathrm{I}}^{\prime (1)}$ 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 LO result in the considered $Q^2$ region; (b) the NLO time-like scalar pion form factor $F_{\mathrm{a},\mathrm{I}}^{\prime (1)}$ describes the $\mathcal{O}({\alpha }_s^2)$ 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 $F_{\mathrm{a},\mathrm{I}}^{\prime (1)}$ 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.

Comments: 26 pages, 12 figures, 1 Table; Minor corrections