Hidden symmetry of the CKM and neutrino-mapping matrices

摘要

We propose that the smallness of the light quark masses is related to the smallness of the T (i.e. CP) violation in hadronic weak interactions. Accordingly, for each of the two quark sectors ("upper" and "lower") we construct a 3 x 3 mass matrix in a bases of unobserved quark states, such that the "upper" and "lower" basis states correspond exactly via the W-+/- 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(iz) 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. Comparing formulas, we find that most unknown factors drop out, resulting in a simple relation with J(CKM) = (m(d)m(s)/m(b)(2))(1/2) A lambda(3) cos1/2 chi, to leading order in chi and m(s)/m(b), with A, lambda the Wolfenstein parameters. (Because of the large top quark mass, the contribution from upper quark sector can be neglected.) Setting J(CKM) = 3.08 x 10(-5), m(b) = 4.7 GeV (1s mass), m(s) = 95 MeV, A = 0.818, and lambda = 0.227, we find m(d)cos(2) 1/2 chi similar or equal to 2.4 MeV, consistent with the accepted value m(d) = 3 - 7 MeV. 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 nu(1) are zero. The neutrino-mapping matrix V-nu is shown to be of the same Harrison-Perkins-Scott form which is in agreement with experiments. We also examine the correction due to T violation, and evaluate the corresponding Jarlskog invariant J(nu). (c) 2007 Elsevier Inc. All rights reserved.