  The branching ratios and CP violations of the $B\to D\pi$ decays, including
both the color-allowed and the color-suppressed modes, are investigated in
detail within QCD framework by considering all diagrams which lead to three
effective currents of two quarks. An intrinsic mass scale as a dynamical gluon
mass is introduced to treat the infrared divergence caused by the soft
collinear approximation in the endpoint regions, and the Cutkosky rule is
adopted to deal with a physical-region singularity of the on mass-shell quark
propagators. When the dynamical gluon mass $\mu_g$ is regarded as a universal
scale, it is extracted to be around $\mu_g = 440$ MeV from one of the
well-measured $B\to D\pi$ decay modes. The resulting predictions for all
branching ratios are in agreement with the current experimental measurements.
As these decays have no penguin contributions, there are no direct $CP$
asymmetries. Due to interference between the Cabibbo-suppressed and the
Cabibbo-favored amplitudes, mixing-induced CP violations are predicted in the
$B\to D^{\pm}\pi^{\mp}$ decays to be consistent with the experimental data at
1-$\sigma$ level. More precise measurements will be helpful to extract weak
angle $2\beta+\gamma$.
