Next-to-leading-order corrections to $B\to \pi$ form factors in $k_T$ factorization

We calculate next-to-leading-order (NLO) corrections to the $B\to \pi$ transition form factors at leading twist in the $k_T$ factorization theorem. Light partons off shell by $k_T^2$ are considered in the quark diagrams, in the effective diagrams for the $B$-meson wave function defined with the effective heavy-quark field, and in the effective diagrams for the pion wave function. It is explicitly demonstrated that the infrared logarithms $\ln k_T^2$ cancel between the above sets of diagrams, as deriving the $k_T$-dependent NLO hard kernel from their difference. The infrared finiteness of the hard kernel confirms the application of the $k_T$ factorization theorem to $B$-meson semileptonic decays. The NLO pion wave function is identical to those constructed from the pion transition and electromagnetic form factors, consistent with its universality. Choosing the renormalization and factorization scales lower than the $B$-meson mass, the NLO corrections are under control: they amount only up to 30% of the form factors at large recoil of the pion, when varying models for the meson wave functions.

Figure 1

Leading-order quark diagrams for the $B\to \pi$ transition form factors with $\otimes$ representing the weak vertex.

Figure 2

Self-energy corrections to Fig. 1a.

Figure 3

Vertex corrections to Fig. 1a.

Figure 4

Box and pentagon diagrams.

Figure 5

$O({\alpha }_s)$ diagrams for the $B$-meson wave function.

Figure 6

$O({\alpha }_s)$ diagrams for the pion wave function.

Figure 7

Renormalization scale $t_s({\mu }_f)$, defined in Eq. (61), as a function of momentum fractions $x_1$ and $x_2$ for a typical factorization scale ${\mu }_f=1.5\mathrm{GeV}$ and the ratio ${\zeta }_1/m_B=25$.

Figure 8

Ratios of the NLO corrections over the total contributions to the $B\to \pi$ form factors for three different choices of the renormalization and factorization scales.

Figure 9

LO and NLO contributions to the $B\to \pi$ form factors with the nonasymptotic pion distribution amplitudes in Eq. (57) and the first scenario for the scale choice, ${\mu }_f=t$ and $\mu =t_s({\mu }_f)$.

Figure 10

LO and NLO contributions to the $B\to \pi$ form factors with the asymptotic pion distribution amplitudes and the first scenario for the scale choice, ${\mu }_f=t$ and $\mu =t_s({\mu }_f)$.

Figure 11

LO and NLO contributions to the $B\to \pi$ form factors with the $B$-meson distribution amplitudes in Eq. (62) and the first scenario for the scale choice: ${\mu }_f=t$ and $\mu =t_s({\mu }_f)$.

Figure 12

LO and NLO contributions to the $B\to \pi$ form factors with the $B$-meson distribution amplitudes in Eq. (59), however, varying the shape parameter ${\omega }_0$ from 0.30 GeV to 0.40 GeV and the first scenario for the scale choice: ${\mu }_f=t$ and $\mu =t_s({\mu }_f)$.

Figure 13

Theoretical uncertainties of the $B\to \pi$ form factors with the first scenario for the scale choice: ${\mu }_f=t$ and $\mu =t_s({\mu }_f)$.