  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.
