New Origin of a Bilinear Mass Matrix Form

The charged lepton mass formula can be explained when the masses are propotinal to the squared vacuum expectation values (VEVs) of scalar fields. We introduce U(3) flavor symmetry and its nonet scalar field Φ, whose VEV structure plays an essential role for generating the fermion mass spectrum. We can naturally obtain bilinear form of the Yukawa coupling Yij ∝ ∑ k〈Φik〉〈Φkj〉 without the non-renormalizable interactions, when the flavor symmetry is broken only through the Yukawa coupling and tadpole terms. We also speculate the possible VEV structure of 〈Φ〉. The observed mass spectra of the quarks and leptons might provide an important clue for the underlying theory. For the charged lepton sector, we know the following empirical mass relation[1, 2], me +mμ +mτ = 2 3 ( √ me + √ mμ + √ mτ ) , (1) which can give a remarkable prediction mτ = 1776.97 MeV from the observed values of me and mμ. (The observed value is m obs τ = 1776.99 +0.29 −0.26 MeV [3].) This mass relation seems to give remarkable hints for the origin of the mass spectrum. In order to get the mass relation (1), an interesting idea was proposed in Ref.[2]: the mass spectrum originates not in the structure of the Yukawa coupling constants Yij but of the vacuum expectation values (VEVs) vis of scalars φis as v 1 + v 2 2 + v 2 3 = 2 3 (v1 + v2 + v3) . (2) Here we encounter following two questions. (i) How can we obtain the VEV relation (2) naturally? (ii) How to build a model in which mei has a bilinear form mei ∝ v i , (3) naturally? The first question seems to be related to a permutation symmetry of S3[4] or higher symmetries which contain S3. The second question can be solved by the seesaw-type mass generation mechanism for the charged fermions [5]. However, in the seesaw-type model, we must identify the scalar φ as the three Higgs doublets withO(102) GeV VEVs, which may induce the unwanted large flavor changing neutral current (FCNC) [7]. On the other hand, φs are not Higgs doublets in the Froggatt-Nielsen-type model[6] so that the FCNC problem might be avoided. However, it should be emphasized that the bilinear form is just an assumption in the Froggatt-Nielsen-type model. The purpose of this paper is to propose a new mechanism which induces the bilinear form mei ∝ v2 i in the framework of a SUSY scenario. The SUSY model which leads to the VEV relation (2) and the bilinear form has been firstly proposed by Ma [8], where four Higgs fields (ηi, ξi, ζi, ψi) were introduced 1. The bilinear structure mei ∝ v2 i has been realized via mei ∝ 〈η0 i 〉 ∝ 〈ζ0 i 〉〈σi〉 ∝ 〈σi〉, where 〈σi〉 satisfies the VEV relation (2). This model is well organized but there are too many Higgs doublets. In this paper, we will try to construct a new model which naturally induces the bilinear form of Yij ∝ ∑ k〈Φik〉〈Φkj〉 in the different way from Refs.[8] and [9]. We will introduce only one SU(2)L-singlet superfield Φ which plays a role of giving the VEV relation (2) in addition to the conventional set of Higgs doubles, Hd and Hu, which give the masses of the charged leptons (and also the down-quarks) and the neutrinos (and also the up-quarks), respectively. Under an flavor symmetry, the leptons Li and Ei are transformed as L = UXL ′, E = UXE ′, (4) where Li and Ei are the left-handed SU(2)L doublets and the SU(2)L singlets, respectively. We do not specify whether the transformation UX is continuous or discrete. In the conventional model, the Yukawa interaction of the charged lepton sector is given by