Extended Friedberg-Lee hidden symmetries, quark masses,and CP violation with four generations

Motivated in part by the several observed anomalies involving CP asymmetries of B and Bs decays, we consider the standard model with a 4th sequential family (SM4) which seems to offer a rather simple resolution. We initially assume T-invariance by taking the up and down-quark 4 x 4 mass matrix to be real. Following Friedberg and Lee (FL), we then impose a hidden symmetry on the unobserved (hidden) up and down-quark SU(2) states. The hidden symmetry for four generations ensures the existence of two zero-mass eigenstates, which we take to be the (u,c) and (d,s) states in the up and down-quark sectors, respectively. Then, we simultaneously break T-invariance and the hidden symmetry by introducing two phase factors in each sector. This breaking mechanism generates the small quark masses mu, mc and md, ms, which, along with the orientation of the hidden symmetry, determine the size of CP-violation in the SM4. For illustration we choose a specific physical picture for the hidden symmetry and the breaking mechanism that reproduces the observed quark masses, mixing angles and CP-violation, and at the same time allows us to further obtain very interesting relations/predictions for the mixing angles of t and t'. For example, with this choice we get Vtd ∼ (Vcb/Vcd-Vts/Vus) + O(λ2) and Vt'b∼Vt'd#smbullet#(Vcb/Vcd), Vtb'Vt'd#smbullet#(Vts/Vus), implying that Vt'd > Vt'b, Vtb'. We furthermore find that the Cabibbo angle is related to the orientation of the hidden symmetry and that the key CP-violating quantity of our model at high energies, JSM4 (triplebond) Im(VtbVt'b*Vt'b#prime#Vtb'*), which is the high-energy analogue of the Jarlskog invariant of the SM, is proportional to the light-quark masses and the measured Cabibbo-Kobayashi-Maskawa quark-mixing matrix angles: |JSM4|A3λ5 x (√(mu/mt) + √mc/mt'-√(md/mb) + √ms/mb') ∼ 10-5, where A ∼ 0.81 and λ = 0.2257 are the Wolfenstein parameters. Other choices for the orientation of the hidden symmetry and/or the breaking mechanism may lead to different physical outcomes. A general solution, obtained numerically, will be presented in a forthcoming paper.