Renormalization-Group Evolution of the None-local Matrix Element of $B$-meson In QCD and $B\to \pi$ transition form factor

We consider the renormalization-group evolution of the matrix element of $\langle 0| \bar{q}(z)_\beta [z,0]b(0)_\alpha| \bar{B}\rangle$, which can be used to define the distribution amplitudes for $B$ meson and widely applied in studies of $B$ meson decays. The contribution to the renormalization constant of the non-local operator $\bar{q}(z)_\beta [z,0]b(0)_\alpha$ is considered up to one-loop order in QCD. Since the quark fields in this operator are not directly coupled fields, momentum can not flow freely through this non-local operator. Momentum involved in this operator can be treated stringently in coordinate space. We find that the ultraviolet divergences regulated by dimensional parameter $\epsilon$ cancel with each other, and the evolution effect vanishes. The matrix element $\langle 0| \bar{q}(z)_\beta [z,0]b(0)_\alpha| \bar{B}\rangle$ escapes from the renormalization-group evolution. We then apply the matrix element in calculating $B\to\pi$ transition form factor, where the matrix element is obtained by using the $B$ meson wave function calculated in QCD-inspired potential model. By comparing with experimental data for the semileptonic decay of $B\to \pi \ell\nu$ and light-cone sum rule calculation, we analyse the perturbative and non-perturbative contributions to $B\to\pi$ transition form factor in the frame work of perturbative QCD approach. We find that the effectiveness of the suppression of Sudakov factor to soft contribution depends on the end-point behavior of $B$ meson wave function, and with the $B$-meson wave function used in this work, soft contribution can not be well suppressed. The hard contribution to the $B\to\pi$ transition form factor is about 59\%, and soft contribution can be as large as 41\% in the naive calculation. To make the perturbative calculation reliable, a soft momentum cutoff in the calculation and soft form factor have to be introduced.