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

TL;DR

This paper calculates the NLO corrections to scalar pion form factors using $k_T$ factorization and assesses their impact on $B o \pi\pi$ decays, finding small magnitude but significant phase effects relevant for CP violation.

Contribution

First calculation of space-like and time-like scalar pion form factors at NLO using $k_T$ factorization, including analytic continuation and implications for $B o \pi\pi$ decay corrections.

Findings

NLO correction to scalar pion form factor is small but can reduce LO result by up to 10%.

The NLO annihilation correction has a large strong phase, affecting CP violation.

NLO correction slightly increases branching ratios, insufficient to solve the $\pi\pi$ puzzle.

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 $F'^{(1)}_{\rm a,I}$ 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.