Quark Masses and Mixing Angles

Inspired by the vector-like phenomenon of chiral gauge theories at short distances, we postulate that the W-gauge boson has a vector-like gauge coupling in the high-energy region, beside its purely right-handed gauge coupling observed in the low-energy region. It is discussed that the top quark acquires its mass via spontaneous symmetry breaking, while the u, d, c, s, b quarks acquire their masses via explicit symmetry breakings. We discover four relationships between four inter-generation mixing angles and six quark masses by examining Dyson equations for quark self-energy functions. A preliminary analysis shows that the hierarchical pattern of quark masses is closely related to the hierarchical pattern of inter-generation mixing angles. In particular, the CP-violating phase is predicted to be approximately 79. September, 1996 PACS 11.15Ha, 11.30.Rd, 11.30.Qc a) E-mail address: [email protected] 1. Motivation The vector-like phenomenon of chiral gauge theories (e.g., the Standard Model) at short distances, shown by the “No-Go” theorem of Nielsen and Ninomiya [1], is that the fermion spectrum and gauge coupling of regularized chiral gauge theories must be vector-like if one insists on preserving chiral gauge symmetries. This seems to run into a paradox that the successful Standard Model possesses, its very peculiar features of purely left-handed gauge coupling of W-boson and only left-handed neutrinos. However, this inconsistency may imply what are the Nature’s possible choices for the Standard Model at short distances [2, 3, 4]. There have been many attempts to find a resolution to this paradox. It is currently a very important issue for chiral gauge theories and the problem has not been completely resolved. We will not enter into the details of this issue, instead, inspired by this vector-like phenomenon, we postulate that the effective gauge coupling of W-bosons to quarks is vector-like in the highenergy region, beside its purely left-handed gauge coupling observed in the low-energy region: Γwμij(q) = igw(q)γμVij(PL + f(q)) (1) f(q) 6= 0, q ∼ Λ, (2) where gw(q) is a renormalized coupling constant and Vij is the CabibboKobayashi-Maskawa (CKM) [5] representation of the mixing matrix that is parameterized by the following four inter-generation mixing angles [6], θ12 , θ23 , θ13 , δ13, (3) where θ12 = θc is the Cabibbo angle and δ13 is the CP-violating phase. In eq.(1), the non-vanishing of the vertex function f(q) in the high-energy region q ∼ Λ, which may be the Planck scale, implies that the gauge coupling (1) is vector-like. Whereas, to coincide with the parity-violating gauge coupling observed in low-energy experiments, it is assumed there is an intermediate energy-threshold ǫ between the weak scale (∼ 250GeV) and the cutoff Λ, 250GeV < ǫ < Λ, (4) at which the vertex function f(q) vanishes f(q)|q→ǫ = 0. (5)