Hierarchy spectrum of SM fermions: from top quark to electron neutrino

In the SM gauge symmetries and fermion content of neutrinos, charged leptons and quarks, we study the effective four-fermion operators of Einstein-Cartan type and their contributions to the Schwinger-Dyson equations of fermion self-energy functions. The study is motivated by the speculation that these four-fermion operators are probably originated due to the quantum gravity that provides the natural regularization for chiral-symmetric gauge field theories. In the chiral-gauge symmetry breaking phase, as to achieve the energetically favorable ground state, only the top-quark mass is generated via the spontaneous symmetry breaking, and other fermion masses are generated via the explicit symmetry breaking induced by the top-quark mass, four-fermion interactions and fermion-flavor mixing matrices. A phase transition from the symmetry breaking phase to the chiral-gauge symmetric phase at TeV scale occurs and the drastically fine-tuning problem can be resolved. In the infrared fixed-point domain of the four-fermion coupling for the SM at low energies, we qualitatively obtain the hierarchy patterns of the SM fermion Dirac masses, Yukawa couplings and family-flavor mixing matrices with three additional right-handed neutrinos $\nu^f_R$. Large Majorana masses and lepton-symmetry breaking are originated by the four-fermion interactions among $\nu^f_R$ and their left-handed conjugated fields $\nu^{fc}_R$. Light masses of gauged Majorana neutrinos in the normal hierarchy ($10^{-5}-10^{-2}$ eV) are obtained consistently with neutrino oscillations. We present some discussions on the composite Higgs phenomenology and forward-backward asymmetry of $t\bar t$-production, as well as remarks on the candidates of light and heavy dark matter particles (fermions, scalar and pseudoscalar bosons).

In the SM gauge symmetries and fermion content of neutrinos, charged leptons and quarks, we study the effective four-fermion operators of Einstein-Cartan type and their contributions to the Schwinger-Dyson equations of fermion self-energy functions. The study is motivated by the speculation that these four-fermion operators are probably originated due to the quantum gravity that provides the natural regularization for chiral-symmetric gauge field theories. In the chiral-gauge symmetry breaking phase, as to achieve the energetically favorable ground state, only the top-quark mass is generated via the spontaneous symmetry breaking, and other fermion masses are generated via the explicit symmetry breaking induced by the top-quark mass, four-fermion interactions and fermion-flavor mixing matrices.
A phase transition from the symmetry breaking phase to the chiral-gauge symmetric phase at TeV scale occurs and the drastically fine-tuning problem can be resolved. In the infrared fixed-point domain of the four-fermion coupling for the SM at low energies, we qualitatively obtain the hierarchy patterns of the SM fermion Dirac masses, Yukawa couplings and family-flavor mixing matrices with three additional right-handed neutrinos ν f R . Large Majorana masses and lepton-symmetry breaking are originated by the four-fermion interactions among ν f R and their left-handed conjugated fields ν f c R . Light masses of gauged Majorana neutrinos in the normal hierarchy (10 −5 − 10 −2 eV) are obtained consistently with neutrino oscillations. We present some discussions on the composite Higgs phenomenology and forward-backward asymmetry of tt-production, as well as remarks on the candidates of light and heavy dark matter particles (fermions, scalar and pseudoscalar bosons).

I. INTRODUCTION
The parity-violating (chiral) gauge symmetries and spontaneous/explicit breaking of these symmetries for the hierarchy pattern of fermion masses have been at the center of a conceptual elaboration that has played a major role in donating to mankind the beauty of the Standard Model (SM) for fundamental particle physics. On the one hand the composite Higgs-boson model or the Nambu-Jona-Lasinio (NJL) [1] with effective four-fermion operators, and on the other the phenomenological model [2] of the elementary Higgs boson, they are effectively equivalent for the SM at low energies and provide an elegant and simple description for the chiral electroweak symmetry breaking and intermediate gauge boson masses. The experimental measurements of Higgs-boson mass 126 GeV [3] and top-quark mass 173 GeV [4], as well as the other SM fermion masses and family-mixing angles, in particular neutrino oscillations, begin to shed light on this most elusive and fascinating arena of fundamental particle physics.
The patterns of the SM fermion masses and family-mixing matrices are equally fundamental, and closely related. Since Gatto et. al [5] tried to find the relation between the Cabibbo mixing angle and light-quark masses, the tremendous effort and many models have been made to study the relation of the SM fermion masses and family-mixing matrices from the phenomenological and/or theoretical view points [5]- [67], where the references are too many to be completely listed. In literature the most of effort based on phenomenological models assuming a particular texture in the original fermion-mass matrices in quark and/or lepton sectors to find the fermion-family mixing matrices as functions of observed fermion masses, i.e., the eigenvalues of the original fermion-mass matrices. Whereas some other models try to find the relations of fermion masses and familymixing matrices on the basis of theoretically model-building approaches, for example, the left-right symmetric scenario [5]- [9] and [16,42], string theory phenomenology [43,44] or the scenario of effective vector-like W ± -coupling at high energies [25,26]. In the model-independent approach, the fermion-mass matrices with different null matrix elements (texture zeros) are considered to find the relations of fermion mass and mixing patterns [45]- [50]. The gauge symmetries of grand unification theories, like SO(10)-theory, and/or the fermion-flavor symmetries, like horizontal or family discrete symmetry, are adopted to find non-trivial relations of fermion mass and mixing patterns [15]- [18], [50]- [55] and [61]- [67]. As the precision measurements for neutrino oscillations are progressing [40,56,57], the study of neutrino mass pattern and lepton-flavor mixing becomes vigorously crucial [58][59][60].
In this article, we approach to this long-standing problem by considering effective four-fermion operators in the framework of the SM gauge symmetries and fermion content: neutrinos, charged leptons and quarks. In order to accommodate high-dimensional operators of fermion fields in the SM-framework of a well-defined quantum field theory at the high-energy scale Λ, it is essential and necessary to study: (i) what physics beyond the SM at the scale Λ explains the origin of these operators; (ii) which dynamics of these operators undergoes in terms of their dimensional couplings (e.g., G) and energy scale µ; (iii) associating to these dynamics, where infrared (IR) and ultraviolet (UV) stable fixed points of these couplings locate and what characteristic energy scales are; (iv) in the IR-domain and UV-domain (scaling regions) of these stable IR and UV fixed points, which operators become physically relevant (effectively dimension-4) and renormalizable following renormalization group (RG) equations (scaling laws), and other irrelevant operators are suppressed by the cutoff at least O(Λ −2 ).
We briefly recall that the strong technicolor dynamics of extended gauge theories at the TeV scale was invoked [68,69] to have a natural scheme incorporating the four-fermion operator of Bardeen, Hill and Lindner (BHL) t t -condensate model [70] in the context of a well-defined quantum field theory at the high-energy scale Λ. The four-fermion operator (1) [69,70,72]. On the other hand, the relevant operator (1) can be constructed on the basis of the SM phenomenology at low-energies. It was suggested [70][71][72] that the symmetry breakdown of the SM could be a dynamical mechanism of the NJL type that intimately involves the top quark at the high-energy scale Λ, since then, many models based on this idea have been studied [73].
Nowadays, the known top-quark and Higgs boson masses completely determine the boundary conditions of the RG equations for the top-quark Yukawa couplingḡ t (µ) and Higgs-boson quartic couplingλ(µ) in the composite Higgs boson model (1). Using the experimental values of top-quark and Higgs boson masses, we obtained [74,75] the unique solutionsḡ t (µ) andλ(µ)to these RG equations, provided the appropriate non-vanishing form-factorZ H (µ) = 1/ḡ 2 t (µ) of the composite Higgs boson at the energy-scale E ∼ TeV, where the effective quartic couplingλ(µ) of composite In Ref. [79], after a short review that recalls and explains the quantum-gravity origin of fourfermion operators at the cutoff Λ, the BHL t t -condensate model and the SSB, we show that due to four-fermion operators (i) there are the SM gauge symmetric vertexes of quark-lepton interactions; (ii) the one-particle-irreducible (1PI) vertex-function of W ± -boson coupling becomes approximately vector-like at TeV scale. Both interacting vertexes contribute the explicit symmetry breaking (ESB) terms to the Schwinger-Dyson (SD) equations of fermion self-energy functions. As a result, once the top-quark mass is generated via the SSB, the masses of third fermion family (ν τ , τ, b) are generated by the ESB via quark-lepton interactions and W ± -boson vector-like coupling. Within the third fermion family, we qualitatively study the hierarchy of fermion masses and effective Yukawa couplings in terms of the top-quark mass and Yukawa coupling [79].
In this article, we generalize this study into three fermion families of the SM by taking into account the flavor mixing of three fermion families. behaves an effective and renormalizable field theory in low energies. To achieve these SM relevant operators, a finite field theory of chiral-gauge interactions should be well-defined by including the quantum gravity that naturally provides a space-time regularization (UV cutoff). As an example, the finite superstring theory is proposed by postulating that instead of a simple space-time point, the fundamental space-time "constituents" is a space-time "string". The Planck scale is a plausible cut-off, at which all principle and symmetries are fully respect by gauge fields and particle spectra, fermions and bosons.
In this article, we do not discuss how a fundamental theory at the Planck scale induces highdimensional operators. Instead, as a postulation or motivation, we argue the presence of at least four-fermion operators beyond the SM from the following point view. A well-defined quantum field theory for the SM Lagrangian requires a natural regularization (UV cutoff Λ) fully preserving the SM chiral-gauge symmetry. The quantum gravity naturally provides a such regularization of discrete space-time with the minimal lengthã ≈ 1.2 a pl [80], where the Planck length a pl ∼ 10 −33 cm and scale Λ pl = π/a pl ∼ 10 19 GeV. However, the no-go theorem [81] tells us that there is no any consistent way to regularize the SM bilinear fermion Lagrangian to exactly preserve the SM chiralgauge symmetries, which must be explicitly broken at the scale of fundamental space-time cutof a. This implies that the natural quantum-gravity regularization for the SM should lead us to consider at least dimension-6 four-fermion operators originated from quantum gravity effects at short distances [103]. As a model, we adopt the four-fermion operators of the torsion-free Einstein-Cartan Lagrangian within the framework of the SM fermion content and gauge symmetries. We stress that a fundamental theory at the UV cutoff is still unknown.

B. Einstein-Cartan theory with the SM gauge symmetries and fermion content
The Lagrangian of torsion-free Einstein-Cartan (EC) theory reads, where the gravitational Lagrangian L EC = L EC (e, ω), tetrad field e µ (x) = e a µ (x)γ a , spinconnection field ω µ (x) = ω ab µ (x)σ ab , the covariant derivative D µ = ∂ µ − igω µ and the axial current J d =ψγ d γ 5 ψ of massless fermion fields. The four-fermion coupling G relates to the gravitationfermion gauge coupling g and fundamental space-time cutoffã.
Within the SM fermion content, we consider massless left-and right-handed Weyl fermions ψ f L and ψ f R carrying quantum numbers of the SM symmetries, as well as three right-handed Weyl sterile neutrinos ν f R and their left-handed conjugated fields ν f c R = iγ 2 (ν R ) * , where "f " is the fermion-family index. Analogously to the EC theory (2), we obtain a torsion-free, diffeomorphism and local gauge-invariant Lagrangian where we omit the gauge interactions in D µ and axial currents read The four-fermion coupling G is unique for all four-fermion operators and high-dimensional fermion operators (d > 6) are neglected.
By using the Fierz theorem [82], the dimension-6 four-fermion operators in Eq. (3) can be written as [83] which preserve the SM gauge symmetries. Equations (5) and (6) where the two component Weyl fermions ψ f L and ψ f R respectively are the SU L (2) × U Y (1) gauged doublets and singlets of the SM. For the sake of compact notations, ψ f R are also used to represent ν f R , which have no any SM quantum numbers. All fermions are massless, they are four-component Dirac In Eq. (7), f and f ′ (f, f ′ = 1, 2, 3) are fermion-family indexes summed over respectively for three lepton families (charge q = 0, −1) and three quark families (q = 2/3, −1/3). Eq. (7) preserves not only the SM gauge symmetries and global fermion-family symmetries, but also the global symmetries for fermion-numbers conservations. We adopt the effective four-fermion operators (7) in the context of a well-defined quantum field theory at the high-energy scale Λ.

C. SM gauge-symmetric four-fermion operators
Neglecting the flavor-mixing of three fermion families (f = f ′ ) to simply notations, we explicitly show SM gauge symmetric four-fermion operators in Eq. (7). In the quark sector, the four-fermion operators are where a, b and i, j are the color and flavor indexes of the top and bottom quarks, the quark SU L (2) doublet ψ ia L = (t a L , b a L ) and singlet ψ a R = t a R , b a R are the eigenstates of electroweak interaction. The first and second terms in Eq. (9) are respectively the four-fermion operators of top-quark channel [70] and bottom-quark channel, whereas "terms" stands for the first and second quark families that can be obtained by substituting t → u, c and b → d, s [74,75,86].
In the lepton sector with three right-handed sterile neutrinos ν ℓ R (ℓ = e, µ, τ ), the four-fermion operators in terms of gauge eigenstates are, preserving all SM gauge symmetries, where the lepton SU L (2) doublets ℓ i L = (ν ℓ L , ℓ L ), singlets ℓ R and the conjugate fields of sterile neutrinos ν ℓc R = iγ 2 (ν ℓ R ) * . Coming from the second term in Eq. (7), the last term in Eq. (10) preserves the symmetry U lepton (1) for the lepton-number conservation, although (ν ℓ R ν ℓc R ) violates the lepton number of family "ℓ" by two units. Similarly, from the second term in Eq. (7) there are following four-fermion operators where quark fields u ℓ a,R = (u, c, t) a,R and d ℓ a,R = (d, s, b) a,R .

D. Four-fermion operators of quark-lepton interactions
Although the four-fermion operators in Eq. (7) do not have quark-lepton interactions, we consider the following SM gauge-symmetric four-fermion operators that contain quark-lepton interactions [26], where ℓ i L = (ν e L , e L ) and ψ Lia = (u La , d La ) for the first family. The (· · ·) represents for the second and third families with substitutions: e → µ, τ , ν e → ν µ , ν τ , and u → c, t and d → s, b. The four-fermion operators (12) of quark-lepton interactions are not included in Eq. (7), since leptons and quarks are in separated representations of SM gauge groups. They should be expected in the framework of Einstein-Cartan theory and SO(10) unification theory [87].
In order to study the mass generation of three fermion families by the mixing of three fermion families we generalize the quark-lepton interacting operators (12) to analogously to the four-fermion operators in Eq. (7).

III. GAUGE VS MASS EIGENSTATES IN FERMION-FAMILY SPACE
Due to the unique four-fermion coupling G and the global fermion-family U L (3) × U R (3) symmetry of Eq. (7), one is allowed to perform chiral transformations U L ∈ U L (3) and U R ∈ U R (3) so that f = f ′ , the four-fermion operators (7) are only for each fermion family without the familyflavor-mixing and all fermion fields are Dirac mass eigenstates. In this section, neglecting gauge interactions we discuss the unitary chiral transformations from gauge eigenstates to mass eigenstates in quark and lepton sectors, so as to diagonalize in the fermion-family space the four-fermion operators (7) and two-fermion operators (ψψ), the latter is relating to fermion mass matrices.

A. Quark sector
For the quark sector, the four-fermion operators (7) are where the SU L (2) × U Y (1) doublets ψ f L and singlets ψ f R are the SM gauge eigenstates, SU (3)color index "a"is summed overψ af L ψ f ′ aR →ψ f L ψ f ′ R , f and f ′ are family indexes of three fermion families. The first term is for the (2/3)-charged sector, ψ f ′ R ⇒ u f ′ R represented by the u-quark sector u f ′ ⇒ (u, c, t), the second term is for the (−1/3)-charged sector, Due to the unique four-fermion coupling G and the global fermion-family U u and so that in Eq. (14) the fermion-family indexes f = f ′ , i.e., δ f f ′ respectively for the u-quark sector and the d-quark sector. As a result, all quark fields are mass eigenstates, the four-fermion operators (14) are "diagonal" only for each quark family without family-mixing, In this representation, the vacuum expectation values of two-fermion operators ψ f Rψ f L + h.c., i.e., quark-mass matrices are diagonalized in the fermion-family space by the biunitary transformations where all quark masses (eigenvalues) are positive, U L and U R are related by and V u,d is an unitary matrix, see for example [88,89].
Using unitary matrices U u L,R (15) and U d L,R (16), up to a diagonal phase matrix we define the unitary quark-family mixing matrices, where the first element is the CKM matrix The experimental values [90] of CKM matrix are adopted to calculate the fermion spectrum in this article.

B. Lepton sector
For the lepton sector, the four-fermion operators (7) are where Dirac lepton fields ℓ f L and ℓ f R are the SM SU L (2)-doublets and singlets respectively, ν f R are three sterile (Dirac) neutrinos and ν f c R = iγ 2 (ν f R ) * are their the conjugate fields. Analogously to the quark sector (14), we perform four unitary chiral transformations from gauge eigenstates to mass eigenstates and so that in Eq. (22) the fermion-family indexes f = f ′ , i.e., δ f f ′ respectively for the Dirac ν-neutrino sector f → ν ⇒ (ν e , ν µ , ν τ ) and the charged ℓ-lepton sector f → ℓ ⇒ (e, µ, τ ). As a result, all lepton fields are mass eigenstates, the four-fermion operators (22) are "diagonal" only for each lepton family without family-mixing, and the vacuum expectation values of two-lepton operators ℓ f c. and ν f c Rν f R + h.c., i.e., lepton-mass matrices are diagonalized in the fermion-family space by the biunitary transformations and where all lepton masses (eigenvalues) are positive. The Dirac neutrino mass matrix can be expressed and V ν is an unitary matrix. This also applies for charged lepton sector (ν → ℓ), see [88,89]. In the following sections, we adopt the bases of mass-eigenstates and drop the subscriptions 1, 2, 3 for simplifying the notations in M u,d diag (18,19), M ν,ℓ diag (27,26) and M diag (28).
Using unitary matrices U ν L,R (23) and U ℓ L,R (24), up to a phase we define the unitary leptonfamily mixing matrices, where the first element is the PMNS matrix We adopt the most recent updated range [91] of PMNS matrix elements to calculate the fermion spectrum in this article. We can also define the notation for the last element that will be used later. Note that each of the unitary matrices U ν,ℓ,u,d L in Eqs. (15,16) and (23,24) is unique up to a diagonal phase matrix P ν,ℓ,u,d where in the last equality we assume the CP-conservation for Majorana fields ν f c R and ν f R so that their matrix M = M * and transformation (U ν R ) * = U ν R are real. Comparing Eq. (27) to Eq. (32), we find that the Dirac neutrino mass matrix M ν = H ν V ν (27)

C. Quark-lepton interaction sector
Using the same chiral transformations (15), (16), (23) and (24) in quark and lepton sectors, we obtain that in the fermion-family space the four-fermion operators (13) are "diagonal" (f = f ′ ), and we rewrite these operators in terms of Dirac mass eigenstates = where f = f ′ and four unitary mixing matrices between lepton and quark families are defined by analogously to the mixing matrices (21) in the quark sector and (30) in the lepton sector. Relating is expected to have a hierarchy structure, namely, in the fermion-family space the diagonal elements are the order of unit, while the off-diagonal elements are much smaller than the order of unit.
Equations (21), (30), and (35) give the mixing matrices of mass and gauge eigenstates of three fermion families, due to the W ± -boson interaction and four-fermion interactions (3). The elements of these unitary matrices are not completely independent each other, as we have already known from the CKM and PMNS matrices. As will be shown, these mixing matrices and mass spectra of the SM fermions are fundamental, and closely related.
Henceforth, all fermion fields are mass eigenstates, two-fermion mass operators and four-fermion operators are "diagonal" in the fermion-family space.

IV. SPONTANEOUS SYMMETRY BREAKING
In this section, we briefly recall and discuss that in the IR-domain of the IR-stable fixed point G c , the relevant four-fermion operator (9) undergoes the SSB and becomes an effectively bilinear and renormalizable Lagrangian that follows the RG-equations to approach the SM physics in the low-energy. This is necessary and fundamental for studying the origin of SM fermion masses in this article.

A. The IR fixed-point domain and only top-quark mass generated via the SSB
Apart from what is possible new physics at the scale Λ explaining the origin of these effective four-fermion operators (7), it is essential and necessary to study: (i) which dynamics of these operators undergo in terms of their couplings as functions of running energy scale µ; (ii) associating to these dynamics where the infrared (IR) or ultraviolet (UV) stable fixed point of physical couplings locates; (iii) in the domains (scaling regions) of these stable fixed points, which physically relevant operators that become effectively dimensional-4 renormalizable operators following RG equations (scaling laws), while other irrelevant operators are suppressed by the cutoff at least O(Λ −2 ).
In the IR-domain of the IR-stable fixed point G c , the four-fermion operator (1) was shown [70] to become physically relevant and renormalizable operators of effective dimension-4, due to the SSB dynamics of NJL-type. Namely, the Lagrangian (1) [74,75].
It seems that via the SSB dynamics the four-fermion operator (17) leads to the quark- We turn to the lepton sector. The first and second four-fermion operators in Eq. (10) or (25) relate to the lepton Dirac mass matrix. At first glance, it seems that the four-lepton operators undergo the SSB leading to the lepton-condensation M ℓ of the lepton sector (q = 0 and q = −1) satisfying 3 + 3 mass-gap equations of NJL type. Actually, the first and second four-fermion operators in Eq. (10) or (25) do not undergo the SSB and two lepton Dirac mass matrices (q = 0 and q = −1) are zero matrices, i.e., M ν diag = (0, 0, 0) and M ℓ diag = (0, 0, 0). The reason is that the effective four-lepton coupling (GN c )/N c is N c -smaller than the critical value (GN c ) of the effective four-quark coupling for the SSB in the quark sector, in addition to the reason of energetically favorable solution for the SSB ground state discussed above.
Therefore, in the IR-domain where the SSB occurs, except the top quark, all quarks and leptons are massless and their four-fermion operators (17) and (25), as well as repulsive four-fermion operators (5), are irrelevant dimension-6 operators. Their tree-level amplitudes of four-fermion scatterings are suppressed O(Λ −2 ), thus such deviations from the SM are experimentally inaccessible today [83].
The heaviest quark which acquires its mass via the SSB is identified and named as the top quark. The heaviest fermion family is named as the third fermion family of fermions ν τ , τ, t, b, where the top quark is. We study their mass spectra in Ref. [79]. As will be discussed, these third-family quarks and leptons are grouped together for their heavy masses, due to the fermions ν τ , τ, b have the largest mixing with the top quark.

B. The t t -condensate model
We briefly recall the BHL t t -condensate model [70] for the full effective Lagrangian of the low-energy SM in the IR-domain, and the analysis [74,75] of RG equations based on experimental boundary conditions, as well as experimental indications of the composite Higgs boson.

The scaling region of the IR-stable fixed point and BHL analysis
Using the approach of large N c -expansion with a fixed value GN c , it is shown [70] that the top-quark channel of operators (9) undergoes the SSB dynamics in the IR-domain of IR-stable fixed point G c , leading to the generation of top-quark mass where Z HY and Z 4H are proper renormalization constants of the Yukawa coupling and quartic coupling in Eq. (37). The SSB-generated top-quark mass m t (µ) =ḡ 2 Higgs-boson is described by its pole-mass m 2 , and effective quartic couplingλ(µ), provided thatZ H (µ) > 0 andλ(µ) > 0 are obeyed. After the proper wave-function renormalizationZ H (µ), the Higgs boson behaves as an elementary particle, as long asZ H (µ) = 0 is finite.
In the IR-domain where the SM of particle physics is realized, the full one-loop RG equations for running couplingsḡ t (µ 2 ) andλ(µ 2 ) read where one can find A, B and RG equations for running gauge couplingsḡ 2 1,2,3 in Eqs. (4.7), (4.8) of Ref. [70]. The solutions to these ordinary differential equations are uniquely determined, once the boundary conditions are fixed.
As a result, we obtained the unique solution (see Fig. 1) for the composite Higgs-boson model (1) or (37) as well as at the energy scale E More detailed discussions can be found in Ref. [79]. The interested readers are referred to Ref. [74] for the resolution to drastically fine-tuning problem.

Experimental indications of composite Higgs boson ?
To end this section, we discuss the experimental indications of composite Higgs boson. In the IR-domain, the dynamical symmetry breaking of four-fermion operator G(ψ ia L t Ra )(t b R ψ Lib ) of the top-quark channel (9) accounts for the masses of top quark, W and Z bosons as well as a Higgs boson composed by a top-quark pair (tt) [70]. It is shown [74,75] that this mechanism consistently gives rise to the top-quark and Higgs masses, provided the appropriate value of non-vanishing form-factor of composite Higgs boson at the high-energy scale E 5 TeV.
Due to its finite form factor (42), the composite Higgs boson behaves as if an elementary Higgs particle, the deviation from the SM is too small to be identified by the low-energy collider signatures at the present level [75]. More detailed analysis of the composite Higgs boson phenomenology is indeed needed. It deserves another lengthy article for this issue, nevertheless we present a brief discussion on this aspect. The non-vanishing form-factorZ H (µ) means that after conventional wave-function and vertex renormalizations Z  Fig. 3). This means that the composite Higgs boson becomes more tightly bound as the the energy scale µ increases.
On the other hand, that the effective Yukawa couplingḡ t (µ) and quartic couplingλ(µ) decrease as the energy scale µ increases in the range m H < µ < E implies some effects on the rates or crosssections of the following three dominate processes of Higgs-boson production and decay [84,85] or other relevant processes. Two-gluon fusion produces a Higgs boson via a top-quark loop, which is proportional to the effective Yukawa couplingḡ t (µ). Then, the produced Higgs boson decays into the two-photon state by coupling to a top-quark loop, and into the four-lepton state by coupling to two massive W -bosons or two massive Z-bosons. Due to thet t-composite nature of Higgs boson, and their corresponding non-diagonal mass matrices are Eqs. (18), (19), (27) and (26). The unitary quark-lepton mixing matrices (35) make the transformations from lepton diagonal mass-matrices to quark diagonal mass-matrices, vice versa.
Apart from the SSB-generated top-quark mass m sb t , all other fermion masses m eb f are ESBgenerated and related to the top-quark mass m sb t by the mixing matrices (34) or (35). Analogously to Eq. (36) for the t t , in terms of two-fermion operators in mass eigenstates, we define Dirac quark, lepton and neutrino bare masses at the energy scale E, as well as Majorana mass M m eb where the color index a is summed over in Eq. (46) and the lepton-family index ℓ is summed over in Eq. (49), whereas in Eqs. (47) and (48) ℓ = e, µ, τ respectively indicates each of three fermion families (mass eigenstates). In Eqs. (46)(47)(48)(49), the notation · · · does not represent new SSBcondensates, but the 1PI functions of fermion mass operatorψ L ψ R , i.e., the self-energy functions Σ f that satisfy the self-consistent SD equations or mass-gap equations.
We use the quark-lepton interaction of the third family as an example to show the quark-lepton interactions contribute to the SD-equations of fermion self-energy functions [79]. The quark-lepton interaction (12) of the third family reads where ℓ i L = (ν τ L , τ L ) and ψ Lia = (t La , b La ). Once the top quark mass m sb t is generated by the SSB, the quark-lepton interactions (50) introduce the ESB terms to the SD equations (mass-gap equations) for other fermions.
In order to show these ESB terms, we first approximate the SD equations to be self-consistent mass gap-equations by neglecting perturbative gauge interactions and using the large N c -expansion to the leading order, as indicated by Fig. 2. The quark-lepton interactions (50), via the tadpole diagrams in Fig. 2, contribute to the tau lepton mass m eb τ and tau neutrino mass m eb τν , provided the bottom quark mass m eb b and top quark mass m sb t are not zero. The latter m sb t is generated by the SSB, see Sec. IV. The former m eb b is generated by the ESB due to the W ± -boson vector-like coupling and top-quark mass m sb t , see next Sec. V B. Corresponding to the tadpole diagrams in Fig. 2, the mass-gap equations of tau lepton and tau neutrino are given by Here we use the self-consistent mass-gap equations of the bottom and top quarks (see Eq. 2.1 and 2.2 in Ref. [70]) On the other hand, if the tau-neutrino mass m eb ντ and tau-lepton mass m eb τ are not zero, they also contribute to the self-consistent mass-gap equations for m sb t and m eb b . These discussions can be generalized to the three-family case by replacing t → t, c, u and ν τ → ν τ , ν µ , ν e in Eqs. (51) and (54); b → b, s, d and τ → τ, µ, e in Eqs. (52) and (53), and summing all contributions. All these self-consistent mass-gap equations are coupled together.

B. W ± -boson coupling to right-handed fermions
In addition to the ESB terms due to quark-lepton interactions, the effective vertex of W ± -boson coupling to right-handed fermions [79], at the energy scale E, also introduces the ESB terms to the Schwinger-Dyson equations. This is the main reason for the nontrivial bottom-quark mass m b , once the top-quark mass m t is generated by the SSB [79]. This will be generalized to other fermions in Sec. VI.
Before leaving this Section, we would like to mention that the vector-like feature of W ± -boson coupling at high energy E is expected to have some collider signatures (asymmetry) on the decay channels of W ± -boson into both left-and right-handed helicity states of two high-energy leptons or quarks [74,77]. The collider signatures should be more evident in high energies, where heavier fermions are produced. In fact, at the Fermilab Tevatron pp collisions the CDF [92] and D0 [93] experiments measured the forward-backward asymmetry in top-quark pair production where the number N t (cos θ) of outgoing top quarks in the direction θ w.r.t. the incoming proton beam. This is larger than the asymmetry within the SM. In addition to the s-channel of one gauge boson (γ, g, Z 0 ) exchange, the process d(p 1 )d(p 2 ) → t(k 1 )t(k 2 ), i.e., down-quark pair to top-quark pair, has the t-channel of one SM W-boson exchange. Its contributions to the asymmetry (56) and total tt-production rate were studied [78] by assuming a new massive boson W ′ with leftand right-handed couplings (g L , g R ) to the top and down quarks. Performing the same analysis as that in Ref. [78], we can explain the asymmetry (56) by using the SM boson masses (≈ M z ) and renormalized SU L (2)-couplingḡ 2 2 (M z ) ≈ 0.45 with (g L = 1, g R = Γ W ≈ γ w ≈ 0.57). The detailed analysis will be presented somewhere else. However, we want to point out that the analogous asymmetry should be also present in the bb channel, since the vector-like coupling (55) is approximately universal for all fermions [79].

VI. SCHWINGER-DYSON EQUATIONS FOR FERMION SELF-ENERGY FUNCTIONS
In order to understand how fermion masses are generated by the ESB and obey their RG equations, we are bound to study the Schwinger-Dyson (SD) equations for fermion self-energy functions Σ f . The SD equations are generalized from the third family [79] to the three families.

A. Chiral symmetry-breaking terms in SD equations
In a vector-like gauge theory, SD equations for fermion self-energy functions were intensively studied in Ref. [94]. In the Landau gauge, SD equations for quarks are given by where the integration p ′ ≡ d 4 p ′ /(2π) 4 is up to the cutoff E. In Eqs. (57) and (58), only for the top quark the SSB-generated mass term m sb t = 0, see the simplest mass-gap equation (54) where α c = π/3, the W -contributions are approximately boundary terms in the integral SD equations (57) and (58), see Fig. 4 in Ref. [79] for the third family.
We recall that in the SM the W ± boson does not contribute to the SD equations for fermion self-energy functions Σ f . However, due to the nontrivial vertex function (55), the W ± gauge boson has the vector-like contributions to SD equations [25,26]. These contributions not only introduce additional ESB terms, but also mix up SD equations for self-energy functions of different fermion fields via the CKM mixing matrix

B. Twelve coupled SD equations for SM quark and lepton masses
Following the approach of Ref. [94], we convert integral equations (57) and (58) to the following boundary value problems (x = p 2 , α = e 2 /4π): and where the fine structure constant α f (α f ′ ) corresponds to the quark sector 2/3(−1/3) and the QCD contributions are not explicitly shown.
Analogously, we obtain the following boundary value problems in the lepton sector: and where U ℓ ℓℓ ′ ν is the PMNS mixing matrix U ℓ = U νe † L U e L of CKM-type in the lepton sector. The boundary conditions (61), (63), (65) and (67)  These twelve inhomogeneous SD equations admit massive solutions [94,95] where γ ≪ 1 is the anomalous dimension of fermion mass operators, and running fermion masses at an infrared scale µ and mass-shell conditions read whereḡ f (µ) is the corresponding Yukawa coupling, see more discussions in Ref. [79].

C. Realistic massive solutions
Once the top-quark mass m sb t in Eq. (61) is generated by the SSB, the SD equations (63) for d, s, b quarks acquire inhomogeneous α w -terms via flavor mixing. Vice versa, once d, s, b quarks are massive, the SD equations (61) for u, c, t quarks acquire inhomogeneous α w -terms via flavor mixing as well. In the same way for the lepton sector, α w -terms due to the lepton-flavor mixing in Eqs. (65) and (67)

VII. THE HIERARCHY SPECTRUM OF SM FERMION MASSES
In this section, we focus on approximately finding the qualitative fermion masses first for the third family (ν τ , τ, t, b), then for the second family (ν µ , µ, c, s) and the first family (ν e , e, u, d), in order to understand what is the dominate contribution to each fermion mass and how the hierarchy of fermion masses is built in by the fermion-family mixing.

A. The third fermion family
This family is much more massive than the first and second fermion families in coupled SD equations. Therefore, we treat the massive solution (m ντ , m τ , m b , m t ) of the third fermion family as a leading term and those for the first and second fermion families as perturbations in SD equations.

Approximate fermion mass-gap equations for the third family
In Eqs. (61), (63), (65) and (67), we use Eq. (68) to calculate the term where |U tb | ≈ 1.03 [90], |U ℓ τ ντ | ≈ (0.590 → 0.776) [91]. The dominate contributions in the RHS of these equations can be figured out. We obtain the approximate solution to Eqs. (70) and (72), as well as the approximate solution to Eqs. (71) and (73), which are given in the last step with Equations (70)(71)(72)(73) show that at the energy scale E, the bare masses m 0 ντ , m 0 τ and m 0 b are related to the bare mass m 0 t from the SSB. The dominate contributions follow the following way: (i) the τ -neutrino acquires its mass m 0 ντ from the top-quark mass m 0 t via the quark-lepton mixing (12) and Fig. 2 (right), (ii) the bottom-quark acquires its mass m 0 b from the top-quark mass m 0 t via the CKM mixing, (iii) the τ -lepton acquires its mass m 0 τ from the bottom-quark mass m 0 b via the quark-lepton mixing and τ -neutrino mass m 0 ντ via the PMNS mixing. These fermion bare masses m 0 f due to the ESB at the scale E are in terms of the top-quark mass m 0 t due to the SSB.

Fermion masses and running Yukawa couplings
These fermion bare masses m 0 f evolve to their infrared masses m f (µ), mainly follow the topquark one m t (µ), apart from the energy-scale evolutions of the SM gauge interactions. In order to qualitatively calculate the infrared scale m f = m f (µ) as functions of the running scale µ in Eq. (69) for each fermion "f ", we neglect the corrections from perturbative gauge interactions and define the effective Yukawa couplings analogously to the top-quark mass m t (µ) =ḡ t (µ)v/ √ 2. This means that effective fermion Yukawa couplingsḡ f (µ) are functions of the top-quark oneḡ t (µ). Equations (70)(71)(72)(73) become Equation (59) gives γ w ≈ 0.85 (2/N c ) ∼ O(1), where the valueḡ 2 2 (E) ≈ 0.42. In this way, we approximately determine the finite part of vertex function Γ W (p, p ′ ) (55) or (59).
Using the Yukawa couplingḡ τ (µ) (75) andḡ t (µ) (Fig. 1), we numerically solve Eq. (77) and which is qualitatively consistent with the experimental value. Some contributions from the first and second fermion families should be expected. Analogously, using the Yukawa couplingḡ ντ (µ) (75) andḡ t (µ) (Fig. 1), we numerically calculate Eq. (76) at µ = 2 GeV and obtain the neutrino Dirac mass  Equations (39) and (40) show thatḡ t (µ) has received the contributions from gauge interactions g 1,2,3 (µ) of the SM. This means that the RG-equations of these Yukawa couplings calculated are only valid in the high-energy region where theḡ 3 (µ)-andḡ 2 (µ)-perturbative contributions toḡ t (µ) are taken into account. This is the reason that we adopt the point µ = 2 GeV to calculate m ντ (81), instead of using the mass-shell condition. The same reason will be for calculating at µ = 2 GeV the light fermion masses of the second and first families.

B. The second fermion family
In this section, we examine how the masses m ντ ,τ,t,b of the third fermion family introduce ESB terms into the SD equations of the second fermion family via SM gauge interactions and four-fermion interactions, leading to the mass generation of the second fermion family.
It is worthwhile to mention that at the lowest order (tree-level), SM neutral gauge-bosons (γ and Z 0 ) interactions and four-fermion interactions (14) and (22) do not give rise to a 1PI vertex function of the interactions among three fermion families with the same electric charge q = 0, −1, 2/3, −1/3, as an example, the black blob in Fig. 4. This indicates the separate conservations of u-quark, c-quark and t-quark numbers for the q = 2/3 sector, and the same for other charged sectors q = 0, −1, −1/3. As a result, the contributions of the 1PI self-energy functions, as shown in Fig. 4, to SD equations for fermion self-energy functions are negligible. . 4: We adopt quarks (t, c) as an example to illustrate a neutral gauge-boson γ contribution to the fermion self-energy function Σ c (p) in terms of Σ t (p), the same diagrams for other quarks (u, c, t) of q = 2/3 charged sector, (d, s, b) of q = −1/3 charged sector, as well as for other leptons (e, µ, τ ) of q = −1 charged sector, (ν e , ν µ , ν τ ) of q = 0 neutral sector.

Approximate mass-gap equations of the first fermion family
Analogously to Eqs. (70)(71)(72)(73) and Eqs. (82)(83)(84)(85) respectively for the third and second fermion family, Equations (61), (63), (65) and (67) for the first fermion family read,  [91]. The dominate contributions in the RHS of these equations can be figured out. We obtain the approximate solution to Eqs. (92) and (94), as well as the approximate solution to Eqs. (93) and (95), which are given in the last step with The dominate contributions are: (i) the ν e -neutrino acquires its mass m 0 νµ from the t-quark mass m 0 of light lepton masses and PMNS mixing angles; (iv) the d-quark dominantly acquires its mass m 0 d from quark masses m u , m c and m t via the CKM mixing, as well as a small contribution from the quark-lepton interaction M 9 , which implies the approximate relations of light quark masses and CKM mixing angles.

Running fermion masses and Yukawa couplings
Analogously to the discussion for the third fermion family from Eqs. (70)(71)(72)(73) to Eqs. (76)(77)(78), neglecting the perturbative corrections from the SM gauge interactions, and defining running fermion masses and Yukawa couplings and the gap-equations at the scale µ are obtained by replacing m 0 f → m f (µ) in Eqs. (92)(93)(94)(95). On the basis of Eqs. (92,94) and (97)  of the third, second and the first family of the SM. With the knowledge of the CKM and PMNS matrices, as well as the fermion mass spectra, we try to identify the dominate ESB contributions to the SD gap-equations, and approximately find their masses, consistently with the fermion-family mixing parameters M i . We have checked that the contributions from perturbative gauge interactions are negligible, compared with the essential contributions due to the fermion-family mixing.
As qualitative and preliminary results, without any drastic fine-tuning we approximately obtain the hierarchy pattern of 12 SM-fermion masses, see Table I,  are introduced by the top-quark mass and fermion-family mixing matrices in the two ways: (i) the fermion-family-mixing matrices (21) and (30) including the CKM and PMNS matrices introduce the ESB terms, due to the vector-like coupling α w (55) and (59) of the W ± -boson at high energies (E) (see preliminary study [25,26]); (ii) the quark-lepton-family mixing matrices (35) introduce the ESB terms, due to the quark-lepton interactions (34) at high energies (E). It is expected that the ESB terms perturbatively re-arrange the SSB generated vacuum alignment, because of the small coupling α w and fermion-family-mixing matrix elements. The Table ( In conclusion, the spectrum of fermion masses, i.e., the structure of eigenvalues of fermion mass matrices mainly depends on the ESB terms that relats to the unitary matrices or mixing matrices between three fermion-flavor families and four families of fermions with different charges. We cannot theoretically determine these matrices, except for adopting those CKM-and PMNS-matrix elements already experimentally measured. If these fermion-family mixing-matrix elements are small deviations from triviality, namely the hierarchy pattern likes the observed CKM matrix, the pattern of fermion masses is hierarchy, and vice versa. In this article, the hierarchy pattern of It should be emphasized that we have at the infrared scale 12 SD equations for 12 SM fermion masses coupled together via the fermion-family-mixing matrices (21), (30) and (35) are preliminarily qualitative, and far from being quantitatively compared with the SM fermion masses and precision tests of e.g., Yukawa couplings. Due to the fact that 12 coupled SD mass-gap equations depend on not only poorly known and totally unknown family-mixing parameters, but also running gauge couplings, the quantitative study of solving these SD equations is a difficult and challenging task. These results could be quantitatively improved, if one would be able to solve coupled SD equations by using a numerical approach in future. Our goal in this article is to present an insight into a possible scenario and understanding of the origins and hierarchy spectrum of fermion masses in the SM without drastic fine-tuning.
In the next section, we will relabel neutrino Dirac mass m ν by m D ν , discuss three heavy sterile Majorana neutrinos (ν f R + ν f c R ) and three light gauged Majorana neutrinos (ν f L + ν f c L ) in terms of their Dirac masses m D ν and Majorana masses m M ν .

VIII. NEUTRINO SECTOR
On the basis of Dirac neutrino mass eigenstates and masses calculated (see In the four-fermion operators (22) of the lepton sector, the last term reads where the conjugate fields of sterile Wely neutrinos ν f R are given by ν f c R = iγ 2 (ν f R ) * . This fourfermion operator preserves the global U lepton (1)-symmetry for the lepton-number conservation.
Similarly to the discussions of the SSB mechanism for the generation of top-quark mass in Sec. IV, the four-fermion operator (103) can generate a mass term of Majorana type, since the family index "f " is summed over as the color index "a" and the family number N f = 3 plays the similar role as the color number N c in the t t -condensate (36). We notice that the lepton-number is conserved in the ground state (vacuum state) realized by the SSB of the SM chiral gauge symmetries, whereas the lepton-number is not conserved in the ground state realized by the spontaneous symmetry breaking of the global U lepton (1)-symmetry of the Lagrangian (103).
On the basis of the mass eigenstates, the spontaneous symmetry breaking of the U lepton (1)symmetry generates the masses of Majorana type together with a sterile massless Goldstone boson, i.e. the pseudoscalar bound state and a sterile massive scalar particle, i.e. the scalar bound state both of them carry two units of the lepton number. The sterile neutrino mass m M and sterile scalar particle mass m M H satisfy the mass-shell conditions, whereḡ sterile (µ 2 ) andλ sterile (µ 2 ) obey the same RG equations (absence of gauge interactions) of Eqs. (38), (39) and (40), as well as the boundary conditions (107). However, we cannot determine the solutionsḡ sterile (µ 2 ) andλ sterile (µ 2 ), since the energy scale v sterile of boundary conditions (107) are unknown. The electroweak scale v is determined by the gauge-boson masses M W and M Z experimentally measured, the scale v sterile needs to be determined by the sterile neutrino masses m M i and the sterile scalar particle mass m M H . In fact, the scale v sterile represents the energy scale of the lepton-number violation.

B. Gauged and sterile Majorana neutrino masses
The SSB and ESB of the SM chiral gauge symmetries, as well as the spontaneous symmetry breaking of the U lepton (1)-symmetry result in the following bilinear Dirac and Majorana mass terms in terms of neutrino mass eigenstates ν f L and ν f R in the f -th fermion family, see Eqs. (27) and (28). Following the usual approach [88,89], diagonalizing the 2 × 2 mixing matrix (108) in terms of the neutrino and sterile neutrino mass eigenstates of the family "f = 1, 2, 3", we obtains two mass This corresponds to two mass eigenstates: three light gauged Majorana neutrinos (four components) and three heavy sterile Majorana neutrinos (four components) where p stands for neutrino momentum, corresponding velocity v p . The mixing angles between gauged and sterile Majorana neutrinos are The previously obtained Dirac masses m f ≡ m D f have the structure of hierarchy (see Table I). The discussions after Eq. (32) show that the Majorana masses m M f are expected to have a hierarchy structure relating to the one of Dirac masses m D f [104]. This indicates the normal hierarchy structure of neutrino mass spectrum: Dirac neutrino masses m D 1 < m D 2 < m D 3 , sterile Majorana neutrino masses m M 1 < m M 2 < m M 3 (112) and gauged Majorana neutrino masses M g 1 < M g 2 < M g 3 (111), i.e., Moreover, due to the absence of observed lepton-violating processes up to the electroweak scale and the smallness of gauged neutrino masses, it is nature to assume that the neutrino Majorana masses are much larger than their Dirac masses m M f ≫ m D f , i.e., the energy scale v sterile of the lepton-number violation is much larger than the electroweak scale v.

C. Flavor oscillations of gauged Majorana neutrinos
We first discuss the family-flavor oscillations of three light gauged Majorana neutrinos (111) in the usual framework. They are described by the PMNS mixing matrix U ℓ L = U ν † L U ℓ L , the mass and mass-squared differences of gauged Majorana neutrino mass-eigenstates (f, f ′ = 1, 2, 3), which are calculated by using Eq. (109) Equation (115) is up to the order O{(m D f ) 2 /4m M f }, and Eq. (116) is up to the order O{[(m D f ) 2 /4m M f ] 2 }. The oscillating probability from the flavor ν g α to the flavor ν g β reads The large oscillating lengths of relativistic and non-relativistic gauged neutrinos are given by (118) for non-relativstic and relativistic cases. The oscillating lengths read The large values of Majorana mass m M f and mass-squared (m M f ) 2 [see Table II] show the small oscillating lengths. The small mixing angle (113) indicates the small oscillating probabilities (130) between gauged and sterile Majorana neutrinos.
The oscillating probability between the sterile flavor ν s α and the gauged flavor ν g β reads Apart from mixing matrices, via the oscillatory factor exp[−i(E f s −E f ′ g )t] the oscillation probability depends on the sum over the mass differences ∆m f f ′ or mass-squared differences ∆m 2 f f ′ of masseigenstates (f = f ′ ) of two flavor neutrinos ν s α and ν g β . Given neutrino energies, and their masses or mass-squared differences, one can select an oscillating length L f f ′ that is relevant for a possible observation or effect. The mass spectra (Table II) where e iϕ 1 is a relative phase between U ν † L and U ℓ R , and e iϕ 2 is another relative phase between U ν † R and U ℓ L . The maxing matrix (136) is unitary, if ϕ 2 − ϕ 1 = nπ, n = 1, 2, 3, · · ·. The diagonal parts U ℓ L (PMNS) and U ℓ R respectively represent the mixing matrices for the gauged flavor oscillations (117) and sterile flavor oscillations (127), and the off-diagonal parts represent the mixing matrices for the gauged-sterile flavor oscillations (135).

IX. A SUMMARY AND SOME REMARKS
We end this lengthy article by making some relevant remarks and preliminary discussions on possible consequences of SM gauged particle, Majorana sterile and gauged neutrino spectra, Tables I and II qualitatively obtained in this article.

A. SM fermion Dirac masses and Yukawa couplings
Due to the ground-state (vacuum) alignment of the effective theory of relevant four-fermion operators, the top-quark mass is generated by the SSB, and other fermion masses are originated from the ESB terms, which are induced by the top-quark mass via the fermion-family mixing, quark-lepton interactions and vector-like W ± -boson coupling at high energies. As a consequence, the fermion masses are functions of the top-quark mass and the fermion Yukawa couplings are functions of the top-quark Yukawa coupling. Based on the approach adopted and the results obtained in Ref. [79], we study the inhomogeneous SD-equations for all SM fermion masses with the ESB terms and obtain the hierarchy patter of fermion masses and Yukawa couplings, consistently with the hierarchy patter of the fermion-family mixing matrix elements. However, we do not discuss the detailed properties of the quark-flavor mixing matrices (21), the lepton-flavor mixing matrices