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

Abstract

Motivated in part by the several observed anomalies involving CP asymmetries of B and B_s 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 4x4 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 m_u, m_c and m_d, m_s 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 V_{td} ~ (V_{cb}/V_{cd} - V_{ts}/V_{us}) + O(\lambda^2) and V_{t'b} ~ V_{t'd}x(V_{cb}/V_{cd}), V_{tb'} ~ V_{t'd}x(V_{ts}/V_{us}), implying that V_{t'd} > V_{t'b},V_{tb'}. 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, J_{SM4} = Im[V_{tb} V_{t'b}^* V_{t'b'} V_{tb'}^*], which is the high-energy analogue of the Jarlskog invariant of the SM, is proportional to the light-quark masses and the measured CKM angles.