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Physical Motivations for the channels B → V 1 V 2 and their Decay Kinematics

A complete study of the processes B → ππV (V = 1−−) is performed both in the framework of the helicity formalism and the effective lagrangian approach. Emphasis is put on the factorization hypothesis and the importance of the ρ−ω mixing in enhancing the direct CP violation. New results involving some branching ratios and the ratio of the Penguin/Tree amplitude are given in details. 1 Physical Motivations for the channels B → V1V2 and their Decay Kinematics In the framework of the LHCb experiment devoted to the search for CP violation and rare B decays, special care is given to the B decays into two vector mesons, B → V1V2, Vi = 1. (i) B decays being governed by weak interactions, the vector-mesons are polarized and their final states have specific angular distributions; which allows one to cross-check the Standard Model (SM) predictions and to perform tests of models beyond the SM. (ii) In the special case of two neutral vector mesons with V̄ 0 = V 0 ; orbital angular momentum l, total spin S and CP eigenvalues are related by the following relations : l = S = 0, 1, 2 =⇒ CP = (−1) , which implies a mixing of different CP eigenstates, proving a CP non-conservation process. According to Dunietz et al [1], tests of CP violation in a model independent way can be performed and severe constraints on models beyond the SM can be set. Because the B meson has spin 0, the final two vector mesons, V1 and V2, have the same helicity λ1 = λ2 = −1, 0,+1, and their angular distribution is isotropic in the B rest frame. Let Hw be the weak Hamiltonian describing the B decays. Any transition amplitude between the initial and final states will have the following form: Hλ = 〈V1(λ)V2(λ)|Hw|B〉 (1) where the common helicity is λ = −1, 0,+1. Then, each vector meson Vi will decay into two pseudoscalar mesons, ai and bi ; ai(bi) can be either a pion or a kaon which angular distributions depend on Vi polarization. The helicity frame of a vector-meson Vi is defined in the B rest frame such that the direction of the Z-axis is given by its momentum ~pi. Schematically, the whole process gets the form: B −→ V1 + V2 −→ (a1 + b1) + (a2 + b2) . The corresponding decay amplitude, Mλ ( B → ∑2 i=1(ai + bi) ) , is factorized according to the relation,