Direct $\boldsymbol{CP}$ Violation In $\boldsymbol{B\to \pi^{+}\pi^{-} V}$ With $\boldsymbol{\rho^{0}-\omega}$ Mixing Effects: Phenomenological Approach

We present a detailed study of direct CP violation and branching ratios in the channels $B^{0,\pm} \to \pi^{+}\pi^{-} V^{0,\pm}$, where $V$ is a vector meson ($K^{* 0,\pm}$ or $\rho^{\pm}$). Emphasis is placed upon the important role played by ${{\rho}^{0}}-{\omega}$ mixing effects in the estimation of the CP-violating asymmetry parameter, $a_{cp}$, associated with the difference of $B$ and $\bar B$ decay amplitudes. A thorough study of the helicity amplitudes is presented as a function of the pion-pion invariant mass. All of the calculations and simulations considered correspond to channels which will be analyzed at the LHCb facility.