Minimal modification to the tri-bimaximal neutrino mixing

Current experimental data on neutrino oscillations are consistent with the tri-bimaximal mixing. If future experimental data will determine a non-zero Ve3 and/or find CP violations in neutrino oscillations, there is the need to modify the mixing pattern. We find that a simple neutrino mass matrix, resulting from A4 family symmetry breaking with residual Z3 and Z2 discrete symmetries respectively for the Higgs sectors generating the charged lepton and neutrino mass matrices, can satisfy the required modifications. We assume that the so-called sequestering problem could be solved. The neutrino mass matrix is minimally modified with just one additional complex parameter compared with the one producing the tri-bimaximal mixing. One interesting feature of this minimal modification is that Vα2 is still equal to 1/3, for α=e,μ,τ. The CP violating Jarlskog factor J has a simple form (|J|=|Ve1Ve3|/23 for real neutrino mass matrix). We also discuss how this mixing matrix can be tested experimentally.