Direct CP violation in B → ρ 0 ( ω ) PS

We calculate the direct CP violating asymmetry parameter, a CP , for B → π + π − π and B → π + π − K decays, in the case where ρ 0 − ω mixing effects are taken into account. We find that the direct CP asymmetry for B − → π + π − π − , ¯ B 0 → π + π − π 0 , B − → π + π − K − and ¯ B 0 → π + π − ¯ K 0 , reaches its maximum when the invariant mass π + π − is in the vicinity of the ω meson mass. The inclusion of ρ 0 − ω mixing provides an opportunity to erase, without ambiguity, the phase uncertainty mod(π) in the determination of the CKM angles α in case of b → u and γ in case of b → s. Direct CP violating asymmetries in B decays occur through the interference of, at least, two amplitudes with different weak phase, φ, and strong phase, δ. The extraction of the weak phase φ (which is determined by a combination of CKM matrix elements) is made through the measurement of a CP violating asymmetry. However, one must know the strong phase δ which is not still well determined in any theoretical framework. In this regard, the isospin symmetry violating mixing between ρ 0 and ω can be extremely important, since it can lead to a large CP violation in B decays such as B → ρ 0 (ω)Y → π + π − Y (Y represents a meson) because the strong phase passes through 90 o at the ω resonance. In any phenomenological treatment of the weak decays of hadrons, the starting point is the weak effective Hamiltonian at low energy. It is obtained by integrating out the heavy fields from the Standard Model Lagrangian. The Operator Product Expansion is used to separate the calculation of the amplitude, A(B → F) ∝ C i (µ)F |O i |B(µ), into two distinct physical regimes. One is called hard, represented by C i (µ) and calculated by a perturbative approach. The other is called soft, described by O i (µ) and derived by using a non-perturbative approach. The operators, O i (µ), can be understood as local operators which govern effectively a given decay, reproducing the weak interaction of quarks …