Hidden Symmetry of the CKM and Neutrino Mapping Matrices

Abstract

We propose that the smallness of the light quark masses is related to the smallness of the T violation in hadronic weak interactions. Accordingly, for each of the two quark sectors ("upper" and "lower") we construct a 3\times 3 mass matrix in a bases of unobserved quark states, such that the "upper"and "lower" basis states correspond exactly via the $W^\pm$ transitions in the weak interaction. In the zeroth approximation of our formulation, we assume T conservation by making all matrix elements real. In addition, we impose a "hidden symmetry" (invariance under simultaneous translations of all three basis quark states in each sector), which ensures a zero mass eigenstate in each sector. Next, we simultaneously break the hidden symmetry and T invariance by introducing a phase factor e^{iχ} in the interaction for each sector. The Jarlskog invariant J_{CKM}, as well as the light quark masses are evaluated in terms of the parameters of the model. We find a simple relation with J_{CKM}=(m_dm_s/m_b^2)^{1/2}Aλ^3\cos(χ/2), with A and λthe Wolfenstein parameters. Setting J_{CKM}=3.08 \times 10^{-5}, m_b=4.7GeV, m_s=95MeV, A=0.818 and λ=0.227, we find m_d\cos^2(χ/2) \simeq 2.4MeV, consistent with the accepted value m_d=3-7MeV. We make a parallel proposal for the lepton sectors. With the hidden symmetry and in the approximation of T invariance, both the masses of e and ν_1 are zero. The neutrino mapping matrix V_νis shown to be of the same Harrison-Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant {\cal J}_ν.