Study of $B\to \pi \rho$, $\pi \omega$ decays in the modified perturbative QCD approach

We investigate the $B\to\pi\rho$ and $\pi\omega$ decay processes using the modified perturbative QCD (PQCD) approach. Sudakov factors arising from the resummation of the double logarithms for the contributions of the transverse momenta of partons can increase the applicability of perturbative calculations. In this work we find that infrared contributions cannot be fully suppressed by Sudakov factors alone when the $B$ meson wave function obtained by solving the wave equation in the QCD-inspired relativistic potential model is used in the calculation of $B$ decays. Therefore, it is necessary to introduce a soft cutoff scale $\mu_c$ to separate the hard and soft contributions. Color-octet contributions for the intermediate quark-antiquark pairs are considered. Such contributions are crucial to deminish the discrepancies between the theoretical calculation and experiemntal data. Using the isospin symmetry, we do find the optimal parameter space for the nonperturbative inputs, which can result in excellent agreement between the theoretical calculation and experimental data of $B\to\pi\rho$ and $B\to\pi\omega$ decays.