The recent observation of B 0 d → π 0 π 0 andB 0 d → π 0 π 0 decay modes allows us to make a fresh isospin analysis of B → ππ transitions. We find that current experimental data can impose some model-independent constraints on the parameter space of direct CP violation in B d → π + π − and B d → π 0 π 0 . Furthermore, we establish a direct relationship between the weak phase α and the charge-averaged branching fractions and CP-violating asymmetries of B → ππ decays.
The present neutrino oscillation data allow [Formula: see text] (or [Formula: see text]) for the neutrino mass spectrum and support [Formula: see text] and [Formula: see text] as two good approximations for the PMNS lepton flavor mixing matrix [Formula: see text]. We show that these intriguing possibilities can be a very natural consequence of the translational[Formula: see text]-[Formula: see text] reflection symmetry — the effective Majorana neutrino mass term keeps invariant under the transformations [Formula: see text], [Formula: see text] and [Formula: see text] (for [Formula: see text] or 3), where [Formula: see text] is the charge conjugation of a constant spinor field [Formula: see text]. By extending such a working flavor symmetry to the canonical seesaw mechanism at a superhigh-energy scale, we calculate its soft breaking effects at the electroweak scale by using the one-loop renormalization-group equations.
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