U(3)-Flavor Nonet Scalar as an Origin of the Flavor Mass Spectra

According to an idea that the quark and lepton mass spectra originate in a VEV structure of a U(3)-flavor nonet scalar \Phi, the mass spectra of the down-quarks and charged leptons are investigated. The U(3) flavor symmetry is spontaneously and completely broken by non-zero and non-degenerated VEVs of $\Phi$, without passing any subgroup of U(3). The ratios (m_e+m_\mu+m_\tau)/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2 and \sqrt{m_e m_\mu m_\tau}/(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^3 are investigated based on a toy model.


Introduction
The observed mass spectra of the quarks and leptons might provide a promising clue for a unified understanding of the quarks and leptons. In investigating an origin of the flavor mass spectra, we may expect that an approach based on symmetries will be promising. However, when we want to introduce a flavor symmetry into our mass matrix model, we always encounter an obstacle, i.e. a no-go theorem [1] in flavor symmetries. The theorem asserts that we cannot bring any flavor symmetry into a mass matrix model as far as we consider a mass generation mechanism based on the standard model. The premises to derive the theorem are as follow: (i) the SU(2) L symmetry is unbroken; (ii) there is only one Higgs scalar in each sector (e.g. H u and H d for up-and down-quark sectors, respectively); (iii) 3 eigenvalues of Y f in each sector are non-zero and no-degenerate. Therefore, we have three options [2] to evade this no-go theorem: (A) a model with more than two Higgs scalars in each sector; (B) a model with an explicit flavor symmetry breaking term; (C) a model with a new scalar whose vacuum expectation values (VEVs) yield effective Yukawa coupling constants. The approach (B) has adopted by many authors in phenomenological studies of flavor symmetries. However, we want a model without such an explicit symmetry breaking term. The approach (A) induces a flavor-changing neutral current (FCNC) problem [3]. In order to suppress the FCNC effects, we must make those Higgs scalars heavy except for one of the linear combinations of those scalars. However, it is not so easy to build such a reasonable suppression mechanism without an explicit symmetry breaking term.
Therefore, in the present paper, we take a great interest in the option (C). For example, we consider the following superpotential terms: where Y f (f = u, d, ν, e) are not coupling constants, but U(3)-flavor nonet fields [4,5,6], and Q and L are quark and lepton SU(2) L doublet fields, respectively, and U , D, N , and E are SU(2) L singlet matter fields. The mass parameter M denotes an energy scale of the effective theory. For example, if we assume the following terms for an additional U(3)-nonet (gauge singlet) scalar Φ in the superpotential we can obtain relations Thus, if we give values of the coefficients c n (n = 3, 2, 1, 0), we can completely determine three eigenvalues of Φ , so that we can give a charged lepton mass spectrum from Eq. (1.5 We should note that the superpotential (1.2) does not include any explicit flavor symmetry breaking parameter. The most distinctive feature of the present model is that the U(3) flavor symmetry is spontaneously and completely broken by the non-zero and non-degenerate VEVs of Φ , without passing any subgroup of U(3). (For example, differently from the present model, a U(3)-nonet scalar Φ in Ref. [6] is broken, not directly, but via a discrete symmetry S 4 .) The idea mentioned above is very attractive, because the model does not include conventional Yukawa coupling constants which explicitly break the flavor symmetry. However, a straightforward application of the model (1.2) needs, at least, four different Φ fields, i.e. Φ f (f = u, d, ν, e), because we know that mass spectra in the four sectors are completely different from each other. From the economical point of view in a unification model of quarks and leptons, we will consider that the Φ fields are, at most, two, i.e. Φ u and Φ d , which couple to the up-quark and neutrino sectors and to the down-quark and charged lepton sectors, respectively.
In the present paper, at the outset, we will begin to investigate the down-quark and charged lepton sectors. In the next section, Sec.2, we give a framework of the model, and we will investigate the relations for Tr[ΦΦ]/Tr 2 [Φ] and detΦ/Tr 3 [Φ], and also discuss a relation between the down-quark masses m di and the charged lepton masses m ei . Those relations are essentially described by four parameters. In Sec.3, we will give a speculation in order to obtain explicit values of those parameters, although it is only a toy mode and it should not be seriously taken. Finally, Sec.4 will be devoted to concluding remarks. We assume the following superpotential: where Φ, Y e and Y d are U(3)-flavor nonet superfields (for convenience, we denote Φ d as Φ simply), and the m dφ -term has been added in order to give a down-quark mass formula as we state later.
In order to couple Y e and Y d with the charged lepton sector LEH d and down-quark sector QDH d , respectively, for example, we may assign additional U(1) charges (q e , −q e ) and (q d , −q d ) to the fields (Y e , E) and (Y d , D), respectively. However, such U(1) charges cannot be conserved in W Φ unless the U(1) charges are also suitably assigned to the coefficients in W Φ . For the moment, we assume such phenomenological assignments of the U(1) charges to the coefficients in W Φ . From the SUSY vacuum conditions, we obtain so that we obtain the following relations (2.6) (Hereafter, for simplicity, we will denote Φ , Y e and Y d as Φ, Y e and Y d .) Eqs.(2.5) and (2.6) mean that the charged lepton masses m ei and down-quark masses m di are given by in the diagonal basis of Φ d , so that the paremeter η is given by The values of the quark mass ratios m d /m s = 0.050 and m s /m b = 0.031 at µ = M Z [7] lead to η ≃ −0.11 and η ≃ −0.13, respectively, so that we can understand the observed ratios by taking η ≃ −0.12 within one sigma deviation. 1 (Here, in estimating the value η, we have used the values z i which are obtained from the pole mass values of the charged leptons, because the ratios are insensitive to the energy scale µ.) This seems to offer a new view of the unified understanding of the quark and lepton masses and mixings. By substituting Eqs.(2.5) and (2.6) into (2.2), we again obtain the same equation with (1.5), Here, it is convenient to define the following parameters: ms/m b is highly dependent on the energy scale µ. Therefore, in the present paper, by considering that the scenario (2.8) is applicable to the observed quark masses, we will go on investigating.
Note that those parameters α, β, γ and δ are invariant under the scale transformation of the fields The deviation from the bilinear form in the down-quark mass formula [η in the expression (2.8)] are described by In order to give explicit values of the mass spectra, we need further assumptions. In the present paper, we are interested in the ratio R which is given by (2.21), because if we can give R = 2/3, which means VEV relation we can obtain the following mass relation [10] for the charged leptons from the bilinear mass formula (2.5) on the diagonal basis of Φ . We consider that the ratio R is a fundamental quantity in the present model, so that we expect that the ratio R will be expressed by a simple form. Since the m dφ -term is an extra term from the point of view of an e ↔ d symmetry in the superpotential (2.1), we consider on trial that the ratio R will be independent of such a parameter γ =m dφ /m dd . This demands (2.27) From the requirement (2.27) for the case γ → ∞, we obtain the relation (For γ → 0, we take β → −1 from the condition (2.28), so that we again obtain the result (2.30).) Note that the ratio R is described only by one parameter α =m dd /m ee .

Speculations
In the present section, in order to speculate the values of the parameters α and β, let us put the following assumptions on trial: although there is no theoretical ground for such requirements. We consider that the ratio R is a fundamental quantity in the model, so that it is likely that the ratio is rational. Therefore, we consider that the relations (3.1) and (3.2) are also rational. Since the m dφ -term is concerned with Φ and Y d , we consider that more fundamental parameter will be λ e rather than λ φ . Therefore, we assume that the relation (3.1) will be expressed rationally in the unit of λ e , e.g. λ φ = nλ e , λ d = −(n + 1)λ e , (n = 1, 2, · · · ). Since we define Tr[Φ] > 0 and we search the solutions with η < 0, the signs of m dd and m φφ must be opposite each other. By considering the relation m dd = −(n + 1)m φφ /n from (3.4), we must take n as n > 0. Then, the parameters α and β are given by and the ratio R and parameter η are given by R = 5n + 7 9(n + 1) , .
We assume that the ratio R should be as large as possible. This demands n = 1 in Eq.(3.6) with n = 1, 2, · · · . (On the other hand, the case n = 1 gives a minimum of |η|.) Then, we can obtain α = −1/4 which gives the desirable relation Regrettably, the magnitude of the predicted value (3.9) is somewhat larger than the desirable value η ≃ −0.12 which is estimated from the formula (2.8) with the observed quark mass ratios [7]. However, we do not consider that this discrepancy (3.9) in the present toy model denies the basic idea suggested in Sec.2. 2 Rather, we consider that the order of the value (3.9) is reasonable, so that our direction is basically right. Finally, let us speculate the value of the ratio r 123 . From Eq.(2.23), we obtain Since the value r 123 should be zero in the limit m e → 0, we expect that the value is realized as small as possible. This again demands n = 1 in (3.10), and we obtain r 123 = 1 2 · 3 4 δ. (3.11) Previously, we have assumed the constraints (3.1) and (3.2) for the quadratic and cubic terms in the superpotential (2.1), while, for the tadpole term, since the tadpole term is the µ 2 φ -term alone, we cannot put such a speculative relation, so that we cannot speculate a value of δ. The value δ is completely free, although we consider that the value is also rational. Therefore, we give up the prediction of the value r 123 , and instead, we estimate of a value of δ from the observed charged lepton mass rations. The observed charged lepton masses [11] give z 1 = 0.01647, z 2 = 0.23687, z 3 = 0.97140, (3.12) as the values of z i = m ei /(m e + m µ + m τ ). Although we know that above values are excellently satisfy the relation (2.26) [i.e. (3.8)], the "masses" in the present model mean not "pole" masses, but the "running" masses. For example, if we adopt the mass values [9] at µ = 2 × 10 16 GeV which are estimated from a SUSY scenario with tan β = 10, we obtain In the present paper, we ignore such a small difference. Since we consider the parameter δ will be also expressed with a concise rational value, we take it on trial as δ = 1 3 . (3.14) Then, we obtain (3.15) so that we obtain the predicted values of z i , z 1 = 0.01642, z 2 = 0.2369 and z 3 = 0.97139, which are in good agreement with (3.12) [and also (3.13)]. The rational value δ = 1/3 is plausible, although we have no theoretical ground for δ = 1/3.

Concluding remarks
In conclusion, we have investigated the charged lepton and down-quark mass spectra on the basis of a model in which the quark and lepton mass spectra originate not in structures of Yukawa coupling constants, but in structures of VEVs of U(3)-flavor nonet (gauge singlet) fields Φ u and Φ d . We have proposed a mechanism which gives a bilinear form M e ∝ Φ d Φ d for the charged lepton mass matrix M e , and which gives a form M d ∝ Φ d Φ d + c Φ d for the down-quark mass matrix M d . The U(3)-flavor symmetry is spontaneously and completely broken without passing any subgroup of U(3), i.e. directly. The VEV spectrum Φ d = v diag(z 1 , z 2 , z 3 ) is completely determined by the coefficients in the superpotential W Φ , (2.1). The superpotential (2.1) does not include any symmetry breaking term. As shown in Sec.2, the VEV spectrum of Φ d (we have denoted Φ d as Φ simply) is essentially described by four parameters which have been defined in (2.19). Thus, the VEV spectrum is closely related to the both parameters in the charged lepton and down-quark sectors. The rations In order to reduce the number of the parameters, we have assumed that the ratio R is independent of the parameter γ =m dφ /m dd and we have gotten the constraint γ = β + 1. Then, we have obtained a simple expression of R, R = 1 − 4(1 + α)/9, so that the ratio R is given only by the parameter α =m dd /m ee .
In order to predict the charged lepton and down-quark masses, we need explicit values of the parameters α, β, γ and δ. In Sec.3, we have demonstrated a trial scenario to obtain those parameters. By assuming that these parameters have rational relations, we have obtained  3 , a concise rational value δ = 1/3 is required. Such a value δ = 1/3 seems to be plausible. However, the scenario given in Sec.3 is highly speculative, so that it should not be taken seriously. How to get those parameter values more naturally is a future task to us. However, as we have demonstrated in Sec.3, the idea that those parameter values are described by simple and rational numbers seems to be promising.
By the way, in the present model, we have assumed that the nonet field Φ is Hermitian. Therefore, the parameters in the superpotential have been taken real. This does not mean that all components of Φ are real, although three eigenvalues are real. Therefore, the model can include a CP -violating phase. However, in order to give such a phase explicitly, we will need further modification to the superpotential form (2.1).
In the present paper, we have not investigated the up-quark and neutrino mass matrices. We suppose that the up-quark and neutrino mass spectra are given by a similar mechanism for another nonet field Φ u . (There is a possibility that the observed up-quark and neutrino (Dirac) mass spectra are also described by the forms m ui ∝ (z ui ) 2 + ηz ui and m Dirac νi ∝ (z ui ) 2 if we assume a VEV spectrum z ui δ ij = (Φ u ) ij / Tr[Φ u Φ u ] different from Φ d .) However, the present formulation is applicable only to mass spectra. In order to give non-trivial flavor mixings, the diagonal bases of Φ u and Φ d must be different from each other. If we have a superpotential W which consists of two sets W u (Φ u , Y u , Y ν ) and W d (Φ d , Y d , Y e ) and in which there are no cross terms between (Φ u , Y u , Y ν ) and (Φ d , Y d , Y e ), we can take two different bases, Φ u -diagonal basis and Φ d -diagonal basis, separately. However, since, in such a model, there are no parameters which describe relations between Φ u and Φ d , we cannot predict the mixing matrices. As we stressed in Sec.1, the most distinctive feature of the present model is that the scenario does not include any explicit flavor symmetry breaking parameter. However, in order to give an explicit relation between Φ u and Φ d , we will be obliged to introduce some symmetry breaking term as a flavor-basis fixing term (for example, see Ref. [12]).
The idea that the flavor mass spectra originate in a VEV structure of a U(3)-nonet scalar seems to be promising for unified understanding of quark and lepton masses and mixings.