An Isospin Analysis of $CP$ Violation in $B_{d}\to Dπ$, $D^{*}π$ and $Dρ$

By use of current experimental data, we carry out an isospin analysis of the weak decays $B\rightarrow D\pi, D^{*}\pi$ and $D\rho$. It is found that only in $B\rightarrow D\rho$ the strong phase shift of two different isospin amplitudes can be approximately neglected. We derive some useful relations between the $CP$-violating measurables and the weak and strong transition phases, and illustrate the different effects of final-state interactions on $CP$ violation in $B_{d}\rightarrow D\pi, D^{*}\pi$ and $D\rho$.