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

Journal of High Energy Physics, Nov 2016

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, which 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 R f . Large Majorana masses and lepton-number symmetry breaking are originated by the four-fermion interactions among ν R f and their left-handed conjugated fields ν R fc . 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 \( t\overline{t} \)-production, as well as remarks on the candidates of light and heavy dark matter particles (fermions, scalar and pseudoscalar bosons).

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

Received: October Hierarchy spectrum of SM fermions: from top quark to electron neutrino She-Sheng Xue 0 1 2 3 ICRANet 0 1 2 3 0 Piazzale Aldo Moro 5 , 00185 Roma , Italy 1 Physics Department, Sapienza University of Rome 2 Piazza della Repubblica 10 , 65122 Pescara , Italy 3 Open Access , c The Authors In the SM gauge symmetries and fermion content of neutrinos, charged leptons and quarks, we study the e ective four-fermion operators of Einstein-Cartan type The study is motivated by the speculation that these four-fermion operators are probably originated due to the quantum gravity, which provides the natural regularization for chiral-symmetric gauge eld 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- avor mixing matrices. A phase transition from the symmetry breaking phase to the chiral-gauge symmetric phase at TeV scale occurs and the drastically problem can be resolved. In the infrared pling for the SM at low energies, we qualitatively obtain the hierarchy patterns of the SM fermion Dirac masses, Yukawa couplings and family- avor mixing matrices with three additional right-handed neutrinos Rf breaking are originated by the four-fermion interactions among conjugated elds Rfc. Light masses of gauged Majorana neutrinos in the normal hierarchy 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 ArXiv ePrint: 1605.01266 to; electron; neutrino; ne-tuning; (fermions; scalar and pseudoscalar bosons) - nicolor and Composite Models 1 Introduction Four-fermion operators beyond the SM Regularization and quantum gravity Einstein-Cartan theory with the SM gauge symmetries and fermion content SM gauge-symmetric four-fermion operators Four-fermion operators of quark-lepton interactions Gauge vs mass eigenstates in fermion-family space Quark-lepton interaction sector Spontaneous symmetry breaking xed-point domain and only top-quark mass generated via the SSB The htti-condensate model The scaling region of the IR-stable xed point Experimental indications of composite Higgs boson? Origins of explicit symmetry breaking Quark-lepton interactions W -boson coupling to right-handed fermions Schwinger-Dyson equations for fermion self-energy functions Chiral symmetry-breaking terms in SD equations Twelve coupled SD equations for SM quark and lepton masses Realistic massive solutions The hierarchy spectrum of SM fermion masses The third fermion family The second fermion family The rst fermion family Summary and discussion Approximate fermion mass-gap equations for the third family Fermion masses and running Yukawa couplings Approximate fermion mass-gap equations of the second family Running fermion masses and Yukawa couplings Approximate mass-gap equations of the rst fermion family Running fermion masses and Yukawa couplings Spontaneous symmetry breaking of Ulepton(1) symmetry Gauged and sterile Majorana neutrino masses Flavor oscillations of gauged Majorana neutrinos Flavor oscillations of sterile Majorana neutrinos Oscillations between gauged and sterile Majorana neutrinos A summary and some remarks SM fermion Dirac masses and Yukawa couplings Neutrinos and dark-matter particles 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 e ective four-fermion operators, and on the other the phenomenological model [2{7] of the elementary Higgs boson, they are e ectively 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 [8, 9] and top-quark mass 173 GeV [10, 11], 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. [12] tried to nd the relation between the Cabibbo mixing angle and light-quark masses, the tremendous e ort 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 [13{74], where the references are too many to be completely listed. In literature the most of e ort based on phenomenological models assuming a particular texture in the original fermion-mass matrices in quark and/or lepton sectors to nd 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 nd the relations of fermion masses and family-mixing matrices on the basis of theoretically model-building approaches, for example, the left-right symmetric scenario [12{16] and [23, 49], string theory phenomenology [50, 51] or the scenario of e ective -coupling at high energies [32, 33]. In the model-independent approach, the fermion-mass matrices with di erent null matrix elements (texture zeros) are considered to nd the relations of fermion mass and mixing patterns [52{57]. The gauge symmetries of grand uni cation theories, like SO(10)-theory, and/or the fermion- avor symmetries, like horizontal or family discrete symmetry, are adopted to nd non-trivial relations of fermion mass and mixing patterns [22{25, 57{62] and [68{74]. As the precision measurements for neutrino oscillations are progressing [47, 63, 64], the study of neutrino mass pattern and lepton- avor mixing becomes vigorously crucial [65{67]. In this article, we approach to this long-standing problem by considering e ective 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 elds in the SM-framework of a well-de ned quantum eld theory at the high-energy scale , it is essential and necessary to study: (i) what physics beyond the SM at the scale explains the origin of these operators; (ii) which dynamics of these operators undergoes in terms of their dimensional couplings (e.g., G, see below) and energy scale ; (iii) associating to these dynamics, where infrared (IR) and ultraviolet (UV) stable xed 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 xed points, which operators become physically relevant (e ectively dimension-4) and renormalizable following renormalization group (RG) equations (scaling laws), and other irrelevant operators are suppressed by the cuto at least O( We brie y recall that the strong technicolor dynamics of extended gauge theories at the TeV scale was invoked [75{80] to have a natural scheme incorporating the four-fermion operator L = Lkinetic + G( LiatRa)(tbR Lib); of Bardeen, Hill and Lindner (BHL) htti-condensate model [81] in the context of a wellde ned quantum eld theory at the high-energy scale . The four-fermion operator (1.1) undergoes the spontaneous symmetry breaking (SSB) dynamics responsible for the generation of top-quark and Higgs-boson masses in the domain of IR-stable xed point Gc (critical value associated with the SSB) and characteristic energy scale (vev) v 239:5 GeV. The analysis of this composite Higgs boson model shows [81] that eq. (1.1) e ectively becomes a bilinear and renormalizable Lagrangian following RG equations, together with the composite Goldstone modes for the longitudinal components of massive W and Z0 gauge bosons, and the composite scalar for the Higgs boson. The low-energy SM physics, including the values of top-quark and Higgs-boson masses, was supposed to be achieved by the RG-equations in the domain of the IR-stable xed point [78{81, 85]. On the other hand, the relevant operator (1.1) can be constructed on the basis of the SM phenomenology at low-energies. It was suggested ([81{83, 85]; the SU(3)-extension of their work in chapter 26 of the textbook [84]) 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 , since then, many models based on this idea have been studied [86, 87]. Nowadays, the known top-quark and Higgs boson masses completely determine the boundary conditions of the RG equations for the top-quark Yukawa coupling gt( ) and Higgs-boson quartic coupling ~( ) in the composite Higgs boson model (1.1). Using the experimental values of top-quark and Higgs boson masses, we obtained [88, 89] the unique solutions gt( ) and ~( )to these RG equations, provided the appropriate non-vanishing where the e ective quartic coupling ~( ) of composite Higgs bosons vanishes. The form-factor of composite Higgs boson H nite and does not vanish in the SSB phase (composite Higgs phase for small G & Gc), indicating that the tightly bound composite Higges particle behaves as if an elementary particle. On the other hand, due to large four-fermion coupling G, massive composite fermions by combining a composite Higgs boson H with an elementary fermion (H ) are formed in the symmetric phase where the SM gauge symmetries are exactly preserved [90{94]. This indicates that a second-order phase transition from the SSB phase to the SM gauge symmetric phase takes place at the critical point Gcrit > Gc. In addition the e ective quartic coupling of composite Higgs bosons vanishing at E TeV scales indicates the characteristic energy scale of such phase transition. The energy scale E is much lower than the cuto that the drastically ne-tuning (hierarchy) problem that fermion masses mf pseudoscalar decay constant f or the pseudoscalar decay constant f < E [88]. can be possibly avoided by the replacements mf < E In ref. [97], after a short review that recalls and explains the quantum-gravity origin of four-fermion operators at the cuto , the BHL htti-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 -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 ( ; ; 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 e ective Yukawa couplings in terms of the top-quark mass and Yukawa coupling [97]. In this article, we generalize this study into three fermion families of the SM by taking into account the avor mixing of three fermion families. Such avor mixing inevitably introduces the 1PI vertex-functions of quark-lepton interactions and approximately vectorlike W -boson coupling among three fermion families at TeV scale. As a consequence, these 1PI vertex-functions introduce the ESB terms into the SD-equations of the fermion self-energy functions for all SM fermions in three families. Once the top-quark mass is generated via the SSB, all other SM fermions acquire their masses via the ESB terms by (i) four-fermion interactions among fermion avors via family mixing matrices; (ii) the -boson coupling among fermion avors via the CKM or PMNS mixing matrix. The latter is dominate particularly for light quarks and leptons. As a result, we quatitatively obtain the hierarchy patterns of the SM fermion masses and family-mixing matrices, and all fermion masses and Yukawa couplings are functions of the top-quark mass and Yukawa coupling. Neutrino masses will separately be studied in the last part of the article, for This lengthy article is organized as follow. In section 2, we give an argument why fourfermion operators should be present in an e ective Lagrangian at the high-energy cuto at which the quantum gravity introduces a natural regulator for chiral gauge theories. In the framework of the SM gauge symmetries and fermion content, we discuss four-fermion operators, including quark-lepton interactions. In section 3, we describe fermion- avor mixing matrices in lepton and quark sectors, as well as quark-lepton interaction sector. In section 4, we give a brief recall that the SSB is responsible only for the top-quark and Higgs boson masses, whose values determine the unique solution to the RG equations for the topquark Yukawa and composite Higgs quartic couplings. In sections 5 and 6, we discuss the ESB terms and massive solutions of SD equations of other SM fermions. In section 7, we qualitatively present the hierarchy patterns of the SM fermions and fermion- avor mixing matrices. In the last section 8, we focus on the discussions of gauged and sterile neutrinos of Dirac or Majorana type, and their masses, mixing and oscillation. A brief summary and some remarks are given at the end of the article.1 Four-fermion operators beyond the SM Regularization and quantum gravity Up to now the theoretical and experimental studies tell us the chiral gauge- eld interactions to fermions in the lepton-quark family that is replicated three times and mixed. The spontaneous breaking of these chiral gauge symmetries and generating of fermion masses are made by the Higgs eld sector. In the IR- xed-point domain of weak four-fermion coupling or equivalently weak Yukawa coupling, the SM Lagrangian with all relevant operators (parametrizations) is realized and behaves an e ective and renormalizable eld theory in low energies. To achieve these SM relevant operators, a nite eld theory of chiral-gauge interactions should be well-de ned by including the quantum gravity that naturally provides a space-time regularization (UV cuto ). As an example, the nite 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-o , at which all principle and symmetries are fully respect by gauge elds and particle spectra, fermions and bosons. In this article, we do not discuss how a fundamental theory at the Planck scale induces high-dimensional 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-de ned quantum eld theory for the SM Lagrangian requires a natural regularization (UV cuto ) fully preserving the SM chiral-gauge symmetry. The quantum gravity naturally provides a such regularization of discrete space-time with the minimal length a~ pl = =apl 1:2 apl [98{100], where the Planck length apl 10 33 cm and scale 1019 GeV. However, the no-go theorem [101{104] tells us that there is no any consistent way to regularize the SM bilinear fermion Lagrangian to exactly preserve the SM chiral-gauge symmetries, which must be explicitly broken at the scale of fundamental space-time cuto 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 1More discussions on the experimental aspects of this scenario can be found in the refs. S.-S. Xue [130] and in the last section of [97] and references therein. from quantum gravity e ects at short distances.2 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 cuto is still unknown. Einstein-Cartan theory with the SM gauge symmetries and fermion conThe Lagrangian of torsion-free Einstein-Cartan (EC) theory reads, LEC(e; !; ) = LEC(e; !) + = @ ig! and the axial current J d = of massless fermion elds. The four-fermion coupling G relates to the gravitation-fermion gauge coupling g and fundamental space-time cuto a~. Within the SM fermion content, we consider massless left- and right-handed Weyl Rf carrying quantum numbers of the SM symmetries, as well as three right-handed Weyl sterile neutrinos Rf and their left-handed conjugated f c = i 2( R ) , where \f " is the fermion-family index. Analogously to the EC theory (2.1), we obtain a torsion-free, di eomorphism and local gauge-invariant Lagrangian L = LEC(e; !) + X + G jL jL; + 2JL jL; + 2JR jL; ; where we omit the gauge interactions in D and axial currents read 5 Lf;R ; j The four-fermion coupling G is unique for all four-fermion operators and high-dimensional fermion operators (d > 6) are neglected. eq. (2.2) can be written as [107] By using the Fierz theorem [105, 106], the dimension-6 four-fermion operators in G X +(G=2) JL JL; + JR JR; + jL jL; + 2JL jL; which preserve the SM gauge symmetries. Equations (2.4) and (2.5) represent repulsive and attractive operators respectively. The former (2.4) are suppressed by the cuto 2In the regularized and quantized EC theory [98{100] with a basic space-time cuto , in addition to dimension-6 four-fermion operators, there are high-dimensional fermion operators (d > 6), e.g., @ J @ J , which are suppressed at least by O(a~4). and cannot become relevant and renormalizable operators of e ective dimension-4. Thus the torsion-free EC theory with the attractive four-fermion operators read, L = LEC + X G X where the two component Weyl fermions Lf and Rf respectively are the SUL(2) gauged doublets and singlets of the SM. For the sake of compact notations, used to represent Rf, which have no any SM quantum numbers. All fermions are massless, they are four-component Dirac fermions Weyl neutrinos Lf and four-component sterile Majorana neutrinos Mf = ( Rfc + Rf f = ( Lf + Rf), two-component right-handed kinetic terms read Eq. (2.6) preserves not only the SM gauge symmetries and global fermion-family symmetries, but also the global symmetries for fermion-numbers conservations. We adopt the e ective four-fermion operators (2.6) in the context of a well-de ned quantum eld theory at the high-energy scale . SM gauge-symmetric four-fermion operators explicitly show SM gauge symmetric four-fermion operators in eq. (2.6). In the quark sector, the four-fermion operators are G h( LiatRa)(tbR Lib) + ( LiabRa)(bbR Lib)i + \terms"; where a; b and i; j are the color and avor indexes of the top and bottom quarks, the quark SUL(2) doublet Ra = taR; baR are the eigenstates of electroweak interaction. The rst and second terms in eq. (2.8) are respectively the four-fermion operators of top-quark channel [81] and bottom-quark channel, whereas \terms" stands for the rst and second quark families that can be obtained by substituting t ! u; c and b ! d; s [88, 89, 109]. four-fermion operators in terms of gauge eigenstates are, G h(`iL`R)(`R`Li) + (`iL R`)( R``Li) + ( R`c R`)( R` R`c)i ; second term in eq. (2.6), the last term in eq. (2.9) preserves the symmetry Ulepton(1) for the lepton-number conservation, although ( R` R`c) violates the lepton number of family \`" by two units. Similarly, from the second term in eq. (2.6) there are following four-fermion operators G h( R`c`R)(`R R`c) + ( R`cu`a;R)(u`a;R R`c) + ( R`cd`a;R)(d`a;R R`c)i ; Four-fermion operators of quark-lepton interactions Although the four-fermion operators in eq. (2.6) do not have quark-lepton interactions, we consider the following SM gauge-symmetric four-fermion operators that contain quarklepton interactions [33], G (`iLeR)(daR Lia) + (`iL Re)(uaR Lia) + ( where `iL = ( Le; eL) and the second and third families with substitutions: e ! The four-fermion operators (2.11) of quark-lepton interactions are not included in eq. (2.6), since leptons and quarks are in separated representations of SM gauge groups. They should be expected in the framework of Einstein-Cartan theory and Lia = (uLa; dLa) for the rst family. The ( ) represents for SO(10) uni cation theory [110, 111]. 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 (2.11) to G X n(`iLf efR0 )(daRf0 Lfia) + (`iLf efR0)(uaRf0 Lfia) ; o analogously to the four-fermion operators in eq. (2.6). Gauge vs mass eigenstates in fermion-family space Due to the unique four-fermion coupling G and the global fermion-family UL(3) symmetry of eq. (2.6), one is allowed to perform chiral transformations UL 2 UL(3) and without the family- avor-mixing and all fermion elds 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 (2.6) and two-fermion operators ( the latter is relating to fermion mass matrices. For the quark sector, the four-fermion operators (2.6) are where the SUL(2) UY (1) doublets f and singfletfs0 , Rff aarendthfe 0SMaregafaumgeileyigienndsetxaetseso,fS Uth(r3e)ecolor index \a"is summed over af L f0 by the u-quark sector uf0 R ) dfR0 represented by the d-quark sector df0 f0 Due to the unique four-fermion coupling G and the global fermion-family ULu(3) URu(3) symmetry for the u-quark sector and ULd(3) URd (3) symmetry for the d-quark sector in eq. (3.1), we perform four unitary chiral transformations from gauge-eigenstates to mass-eigenstates: sector and the d-quark sector. As a result, all quark elds are mass eigenstates, the fourfermion operators (3.1) are \diagonal" only for each quark family without family-mixing, In this representation, the vacuum expectation values of two-fermion operators h Rf Lf i + h:c:, i.e., quark-mass matrices are diagonalized in the fermion-family space by the biunitary ) Mduiag = (m1u; mc2; mt3) = ULuyM uURu; ) Mddiag = (m1d; ms2; mb3) = ULdyM dURd ; where all quark masses (eigenvalues) are positive, UL and UR are related by u;d = V u;d is an unitary matrix, see for example [112, 113]. de ne the unitary quark-family mixing matrices, u d Using unitary matrices UL;R (3.2) and UL;R (3.3), up to a diagonal phase matrix we ULuyULd. The experimental values [114] of CKM matrix are adopted to calculate the fermion spectrum in this article. For the lepton sector, the four-fermion operators (2.6) are X h`f `f0 `fR0 `fL + (`fL Rf0 )( Rf0 `fL) + ( Rfc Rf0 )( Rf0 Rfc)i ; L R where Dirac lepton elds `fL and `fR are the SM SUL(2)-doublets and singlets respectively, Rf are three sterile (Dirac) neutrinos and ) are their the conjugate Analogously to the quark sector (3.1), we perform four unitary chiral transformations from gauge eigenstates to mass eigenstates -neutrino sector f 7! As a result, all lepton ) ( e; ; ) and the charged `-lepton sector f 7! ` ) (e; ; ). elds are mass eigenstates, the four-fermion operators (3.9) are \diagonal" only for each lepton family without family-mixing, h`fL `fR `fR `fL + (`fL Rf)( Rf`fL) + ( Rfc Rf )( Rf Rfc)i ; and the vacuum expectation values of two-lepton operators h`fR `fL i + h:c:, h Lf Rfi + h:c: and R i + h:c:, i.e., lepton-mass matrices are diagonalized in the fermion-family space by the biunitary transformations UL = V UR; ) Md`iag = (me1; m2 ; m3) = U L`yM `U R`; ) Mdiag = (m1e ; m2 ; m3 ) = ULyM UR; where all lepton masses (eigenvalues) are positive. The Dirac neutrino mass matrix can be expressed as M and Mdiag (3.15). is an unitary matrix. This also applies for charged lepton sector ( see [112, 113]. In the following sections, we adopt the bases of mass-eigenstates and drop the subscriptions 1; 2; 3 for simplifying the notations in Mdui;adg (3.5), (3.6), Mdi;a`g (3.14), (3.13) Using unitary matrices UL;R (3.10) and U L`;R (3.11), up to a phase we de ne the unitary lepton-family mixing matrices, where the rst element is the PMNS matrix U ` = U ` ULyU L`. We adopt the most recent updated range [115] of PMNS matrix elements to calculate the fermion spectrum in this article. We can also de ne the notation for the last element that will be used later. Note that each of the unitary matrices UL These phase degrees of freedom are used here to ensure all mass eigenvalues are positive, ;`;u;d in eqs. (3.2), (3.3) and we do not consider the question of CP-violation at the moment. The Majorana mass matrix M (3.15) is a symmetric matrix, relating to the vacuum expectation value of two fermion operator ( Rfc Rf ). Using (3.16), eq. (3.15) can be where in the last equality we assume the CP-conservation for Majorana elds Rfc and Rf so that their matrix M = M to eq. (3.19), we nd that the Dirac neutrino mass matrix M = H V (3.14) and the common eigen-vectors. In fact, both mass matrices are related to the Rf matrix V M (3.19) are diagonalized by the same biunitary transformation, and they have - eld condensation, i.e., the Dirac mass matrix M h L Ri and the Majorana mass matrix M Therefore, we expect that they should have a similar structure of eigenvalues, for example the normal hierarchy structure, m1e < m2 < m3 in eq. (3.14) and m1M < m2M < m3M in eq. (3.15). We will present detail discussions on the rst, second and third four-fermion operators involving Rf in eq. (3.12), as well as the Dirac mass matrix M Majorana mass matrix M (3.15) and mixing matrix (3.17) in the last section specially Quark-lepton interaction sector Using the same chiral transformations (3.2), (3.3), (3.10) and (3.11) in quark and lepton sectors, we obtain that in the fermion-family space the four-fermion operators (2.12) are Eq: (2:12) = G X n [(`iLULey)f (U ReeR)f0 ][(daRURdy)f0 (ULd Lia)f ] +[(`iLULy)f (UR eR)f0 ][(uaRURuy)f0 (ULu Lia)f ]o = G X n[`iLf (URdeeR)f0 ][daRf0 (ULed Lia)f ] de ned by URde = URdyU Re; = URuyUR; ULed = ULeyULd; ULu = ULyULu; analogously to the mixing matrices (3.8) in the quark sector and (3.17) in the lepton sector. Relating to the ULd (ULu) in the CKM matrix ULuyULd , the matrix ULed (ULu) is expected to have a hierarchy structure, namely, in the fermion-family space the diagonal elements are the order of unit, while the o -diagonal elements are much smaller than the order of unit. Equations (3.8), (3.17), and (3.22) give the mixing matrices of mass and gauge eigenstates of three fermion families, due to the W -boson interaction and four-fermion interactions (2.2). 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 Henceforth, all fermion elds are mass eigenstates, two-fermion mass operators and four-fermion operators are \diagonal" in the fermion-family space. Spontaneous symmetry breaking In this section, we brie y recall and discuss that in the IR-domain of the IR-stable xed point Gc, the relevant four-fermion operator (2.8) undergoes the SSB and becomes an e ectively 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. xed-point domain and only top-quark mass generated via the Apart from what is possible new physics at the scale explaining the origin of these e ective four-fermion operators (2.6), 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 ; (ii) associating to these dynamics where the infrared (IR) or ultraviolet (UV) stable xed point of physical couplings locates; (iii) in the domains (scaling regions) of these stable xed points, which physically relevant operators that become e ectively dimensional-4 renormalizable operators following RG equations (scaling laws), while other irrelevant operators are suppressed by the cuto at least O( In the IR-domain of the IR-stable xed point Gc, the four-fermion operator (1.1) was shown [81] to become physically relevant and renormalizable operators of e ective dimension-4, due to the SSB dynamics of NJL-type. Namely, the Lagrangian (1.1) becomes the e ective SM Lagrangian with bilinear top-quark mass term and Yukawa-coupling to the composite Higgs boson H, which obeys the RG-equations approaching to the low-energy SM physics characterized by the energy scale v 239:5 GeV. In addition, the top-quark and composite Higgs-boson masses are correctly obtained by solving RG-equations with the appropriate non-vanishing form-factor of the Higgs boson in TeV scales [88, 89]. It seems that via the SSB dynamics the four-fermion operator the quark-condensation Mfqf0 = energetically favorable solution of the SSB ground state of the SM, only top quark is massive (mtsb = to those become the longitudinal modes of massive gauge bosons W Goldstone modes have positive contributions to the ground-state energy, and thus make the top-quark channel (1.1) undergoes the SSB dynamics and becomes relevant operator and Z0. Extra following the RG equations in the IR domain. We turn to the lepton sector. rst and second four-fermion operators in eq. (2.9) or (3.12) relate to the lepton Dirac mass matrix. At rst glance, it seems that 1) satisfying 3 + 3 mass-gap equations of NJL type. Actually, the rst and second four-fermion operators in eq. (2.9) e ective 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 (3.4) and (3.12), as well as repulsive four-fermion operators (2.4), are irrelevant dimension-6 operators. Their treelevel amplitudes of four-fermion scatterings are suppressed O( from the SM are experimentally inaccessible today [107]. 2), thus such deviations The heaviest quark which acquires its mass via the SSB is identi ed and named as the top quark. The heaviest fermion family is named as the third fermion family of fermions ; ; t; b, where the top quark is. We study their mass spectra in ref. [97]. As will be discussed, these third-family quarks and leptons are grouped together for their heavy masses, due to the fermions ; ; b have the largest mixing with the top quark. The htti-condensate model We brie y recall the BHL htti-condensate model [81] for the full e ective Lagrangian of the low-energy SM in the IR-domain, and the analysis [88, 89] of RG equations based on experimental boundary conditions, as well as experimental indications of the composite The scaling region of the IR-stable Using the approach of large Nc-expansion with a xed value GNc, it is shown [81] that the top-quark channel of operators (2.8) undergoes the SSB dynamics in the IR-domain of IR-stable xed point Gc, leading to the generation of top-quark mass mt = (1=2Nc)G Xhtatai = by the htti-condensate. As a result, the 2-divergence (tadpole-diagram) is removed by the mass gap-equation, the top-quark channel of four-fermion operator (1.1) becomes physically relevant and renormalizable operators of e ective dimension-4. Namely, the e ective SM Lagrangian with the bilinear top-quark mass term and Yukawa coupling to the composite Higgs boson H at the low-energy scale is given by [81] L = Lkinetic + gt0( LtRH + h:c:) + all renormalized quantities received fermion-loop contributions are de ned with respect to the low-energy scale . The conventional renormalization Z = 1 for fundamental fermions and the unconventional wave-function renormalization (form factor) Z~H for the composite Higgs boson are adopted Z~H ( ) = ; gt( ) = ZZHH1=Y2 gt0; ~( ) = ; ( ) = where ZHY and Z4H are proper renormalization constants of the Yukawa coupling and ~( ) > 0 are obeyed. After the proper wave-function renormalization Z~H ( ), the Higgs In the IR-domain where the SM of particle physics is realized, the full one-loop RG equations for running couplings gt( 2) and ( 2) read 16 2 dgt = = 12 gt4 ; t = ln where one can nd A, B and RG equations for running gauge couplings g12;2;3 in eqs. (4.7), (4.8) of ref. [81]. The solutions to these ordinary di erential equations are uniquely determined, once the boundary conditions are xed. In ref. [88, 89], we analyzed the RG equations (4.4) and (4.5) by using the boundary conditions based on the experimental values of top-quark and Higgs-boson masses, mt 173 GeV and mH 126 GeV, i.e., the mass-shell conditions mt(mt) = gt2(mt)v= 2 mH (mH ) = [2~(mH )]1=2v in the range 1:0 GeV . values, we uniquely solve the RG equations for the composite Higgs-boson model [81], we nd [88, 89] the e ective top-quark Yukawa coupling gt( ) (left) and e ective Higgs quartic coupling ~( ) (right) . 13:5 TeV. Note that ~(E) = 0 at E 5:14 TeV and ~( ) < 0 for 239:5 GeV. As a result, we obtained the unique solution (see gure 1) for the composite Higgs-boson model (1.1) or (4.2) as well as at the energy scale E ~(E ) = 0: More detailed discussions can be found in ref. [97]. The interested readers are referred to ref. [88] for the resolution to drastically ne-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 Lib) of the top-quark channel (2.8) accounts for the masses of top quark, W and Z bosons as well as a Higgs boson composed by a top-quark pair (tt) [81]. It is shown [88, 89] 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. tions ZH1=2H ! H, ZHY gt0 ! gt0 and Z4H 0 ! Due to its nite form factor (4.7), the composite Higgs boson behaves as if an elementary Higgs particle, the deviation from the SM is too small to be identi ed by the low-energy collider signatures at the present level [89]. 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-factor Z~H ( ) means that after conventional wave-function and vertex renormaliza0 [see eqs. (4.2) and (4.3)], the composite Higgs boson behaves as an elementary particle. The non-vanishing form-factor of compostic coupling ~( ) monotonically decrease with the energy scale ite Higgs boson is in fact related to the e ective Yukawa-coupling of Higgs boson and top increasing in the range more tightly bound as the the energy scale 5 TeV (see gure 3). This means that the composite Higgs boson becomes { 14 { On the other hand, that the e ective Yukawa coupling gt( ) and quartic coupling ~( ) decrease as the energy scale increases in the range mH < < E implies some e ects on the rates or cross-sections of the following three dominate processes of Higgs-boson production and decay [8, 108] or other relevant processes. Two-gluon fusion produces a Higgs boson via a top-quark loop, which is proportional to the e ective Yukawa coupling gt( ). 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 the t t-composite nature of Higgs boson, the one-particle-irreducible (1PI) vertexes of Higgs-boson coupling to a top-quark loop, two massive W -bosons or two massive Z-bosons are proportional to the e ective Yukawa coupling gt( ). As a result, both the Higgs-boson decaying rate to each of these three channels and total decay rate 2 are proportional to gt ( ), which does not a ect on the branching ratio of each Higgsdecay channel. The energy scale is actually the Higgs-boson energy, representing the total energy of nal states, e.g., two-photon state and four-lepton states, into which the produced Higgs boson decays. These discussions imply that the resonant amplitude (number of events) of two-photon invariant mass m 126 GeV and/or four-lepton invariant mass m4l 126 GeV is exof nal two-photon and/or four-lepton states increases, when the CM energy p pected to become smaller as the produced Higgs-boson energy increases, i.e., the energy p p collisions increases with a given luminosity. Suppose that the total decay rate or each channel decay rate of the SM Higgs boson is measured at the Higgs-boson energy = mt and the SM value of Yukawa coupling gt2(mt) = 2mt2=v gure 3). In this scenario of composite Higgs boson, as the Higgs-boson energy = 2mt, the Yukawa coupling gt2(2mt) gure 3), the variation of total decay rate or each channel decay rate is expected to be 6% for 0:06. Analogously, the variation is expected to be 9% at = 3mt, gt2(3mt) 0:95 or 11% at = 4mt, gt2(4mt) gure 3). These variations are still too small to be clearly distinguished by the present LHC experiments. Nevertheless, these e ects are the nonresonant new signatures of lowenergy collider that show the deviations of this scenario from the SM. We see that the induced (1PI) Yukawa couplings gb( ) and g ( ) [97], as well as gf ( ) (the present article) of composite Higgs boson to the bottom-quark, tau-lepton and other fermions also weakly decrease with increasing Higgs-boson energy, this implies a slight decrease of number of dilepton events in the Drell-Yan process. Origins of explicit symmetry breaking We study in this section, once the top quark mass is generated by the SSB at the scale E , other quarks and leptons acquire their masses by the explicit symmetry breaking (ESB), via both quark-lepton interactions (2.12) and fermion-family mixing. We henceforth indicate the SSB-generated top-quark mass mtsb and ESB-generated masses mefb of other fermions, they represent bare masses at the cuto energy scale E of the symmetry breaking phase. Quark-lepton interactions Once quarks acquire their bare masses meub and medb, due to the ESB or the SSB for top quark only at the scale E , four-fermion operators (3.21) contribute, via the tadpole lepton interactions (2.11) that contribute quark and lepton ESB masses meb to mass-gap equations or SD-equations (6.4){(6.11). The mixing matrix element ULbU Rb or ULbyU Rb y associates to the interacting vertex G in the left diagram. The mixing matrix element U UR associates to the interacting vertex G in the right diagram. The mixing matrix elements with the rst and second fermion family are neglected. diagram in gure 2, the bare mass terms m`eb and m`eb in mass-gap equations in the lepton sector. Vice versa once leptons acquire their bare masses, via the same tadpole diagram in gure 2, four-fermion operators (3.21) contribute the bare mass terms meub and medb in mass-gap equations in the quark sector. The superscript \sb" indicates the mass generated by the SSB. The superscript \eb" indicates the mass generated by the ESB. These are bare fermion masses at the energy scale E relationships between quark and lepton diagonal mass matrices, . As a result, from eq. (3.21) we obtain the [meeb; ; ] = (1=Nc)ULed[medb;s;b]URde; = (1=Nc)ULu[meub;c;t]URu where the four diagonal matrices are diag(meeb; meb; meb); diag(medb; mesb; mbeb); diag(mebe; meb ; meb); diag(meub; mceb; mtsb); and their corresponding non-diagonal mass matrices are eqs. (3.5), (3.6), (3.14) and (3.13). The unitary quark-lepton mixing matrices (3.22) make the transformations from lepton diagonal mass-matrices to quark diagonal mass-matrices, vice versa. are ESB-generated and related to the top-quark mass mtsb by the mixing matrices (3.21) or (3.22). Analogously to eq. (4.1) for the htti, in terms of two-fermion operators in mass eigenstates, we de ne Dirac quark, lepton and neutrino bare masses at the energy scale E , as well as Majorana mass M meqb = m`eb = meb` = mM = (1=2Nc)G X G=Nc X (1=2)Gh``i = (1=2)Gh G X where the color index a is summed over in eq. (5.4) and the lepton-family index ` is each of three fermion families (mass eigenstates). In eqs. (5.4){(5.7), the notation h does not represent new SSB-condensates, but the 1PI functions of fermion mass operator L R, i.e., the self-energy functions 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 [97]. The quark-lepton interaction (2.11) of the third family reads G (`iL R)(baR Lia) + (`iL R)(taR Lia) ; where `iL = ( L; L) and the SSB, the quark-lepton interactions (5.8) introduce the ESB terms to the SD equations (mass-gap equations) for other fermions. In order to show these ESB terms, we rst approximate the SD equations to be selfconsistent mass gap-equations by neglecting perturbative gauge interactions and using the large Nc-expansion to the leading order, as indicated by gure 2. The quark-lepton interactions (5.8), via the tadpole diagrams in gure 2, contribute to the tau lepton mass meb and tau neutrino mass meb, provided the bottom quark mass mbeb and top quark mass generated by the ESB due to the W -boson vector-like coupling and top-quark mass mtsb, see next section 5.2. lepton and tau neutrino are given by Corresponding to the tadpole diagrams in gure 2, the mass-gap equations of tau meb = (UL U meb = (ULbU Rb )2Gmbeb (2 )4 (mtsb)2] 1 = (UL U t Rt )(1=Nc)mtsb; (5.9) (mbeb)2] 1 = (ULbU Rb )(1=Nc)mbeb: Here we use the self-consistent mass-gap equations of the bottom and top quarks (see eq. 2.1 and 2.2 in ref. [81]) mbeb = 2GNcmbeb (2 )4 mtsb = 2GNcmtsb (2 )4 and the de nitions of Dirac quark, lepton and neutrino bare masses in eqs. (5.4){(5.7). It is important to note the di erence that eq. (5.12) is the mass-gap equation for the top-quark mass mtsb generated by the SSB, while eq. (5.11) is just a self-consistent mass-gap equation mass meb and tau-lepton mass meb are not zero, if the top-quark mass mtsb and bottomquark mass mbeb are not zero. This is meant to the mass generation of tau neutrino and tau lepton due to the ESB terms introduced by the quark-lepton interactions (2.11), quark masses mtsb and mbeb. On the other hand, if the tau-neutrino mass meb and tau-lepton mass meb are not zero, they also contribute to the self-consistent mass-gap equations for These discussions can be generalized to the three-family case by replacing t ! t; c; u ; ; e in eqs. (5.9) and (5.12); b ! b; s; d and ; ; e in eqs. (5.10) and (5.11), and summing all contributions. All these self-consistent mass-gap equations are coupled together. -boson coupling to right-handed fermions In addition to the ESB terms due to quark-lepton interactions, the e ective vertex of W -boson coupling to right-handed fermions [97], W (p; p0) = i pg2 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 mb, once the top-quark mass mt is generated by the SSB [97]. This will be generalized to other fermions in section 6. 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 [88, 94, 95]. 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 [116] and D0 [117] experiments measured the forward-backward asymmetry in top-quark pair production AF B = > 0) + Nt(cos = 0:19 energy functions where the number Nt(cos ) of outgoing top quarks in the direction w.r.t. the incoming proton beam. This is larger than the asymmetry within the SM. In addition to the schannel of one gauge boson ( ; g; Z0) exchange, the process d(p1)d(p2) ! t(k1)t(k2), i.e., down-quark pair to top-quark pair, has the t-channel of one SM W-boson exchange. Its contributions to the asymmetry (5.14) and total tt-production rate were studied [96] by assuming a new massive boson W 0 with left- and right-handed couplings (gL; gR) to the top and down quarks. Performing the same analysis as that in ref. [96], we can explain the asymmetry (5.14) by using the SM boson masses ( coupling g22(Mz) 0:45 with (gL = 1; gR = Mz) and renormalized SUL(2)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 (5.13) is approximately universal for all fermions [97]. 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 selff . The SD equations are generalized from the third family [97] to the 1:1 (7.11) and M1 10 3 (7.12). Note that mb; ; ( ) = gb; ; ( )v=p2. . 13:5 TeV which is qualitatively consistent with the experimental value. Some contributions from the rst and second fermion families should be expected. Analogously, using the Yukawa coupling g ( ) (7.6) and gt( ) ( gure 1), we numerically calculate eq. (7.7) at = 2 GeV and obtain the neutrino Dirac mass 235:8 MeV; for M1 = U Figure 3 shows the Yukawa couplings g ( ), g ( ) and gb( ), which are functions of gt( ), see gure 1. The variations of Yukawa couplings gb; ; ( ) are very small over the energy scale . Equations (4.4) and (4.5) show that gt( ) has received the contributions from gauge interactions g1;2;3( ) of the SM. This means that the RG-equations of these Yukawa couplings calculated are only valid in the high-energy region where the g3( )- and g2( )-perturbative contributions to gt( ) are taken into account. This is the reason that we adopt the point = 2 GeV to calculate m (7.12), instead of using the mass-shell condition. The same reason will be for calculating at The second fermion family In this section, we examine how the masses m ; ;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 gaugebosons ( and Z0) interactions and four-fermion interactions (3.1) and (3.9) do not give rise to a 1PI vertex function of the interactions among three fermion families with the same the 1PI self-energy functions, as shown in gure 4, to SD equations for fermion self-energy functions are negligible. to the fermion self-energy function c(p) in terms of 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; ; ) of q = 1 charged sector, ( e; ; ) of q = 0 neutral sector. Approximate fermion mass-gap equations of the second family mefb; (f = Neglecting the contributions from the rst fermion family, we assume that fermions in the second family mainly acquire their masses by ESB terms relating to fermion masses of the third family by the following ways: (i) family-mixing diagram, gure 4 in ref. [97], via W -boson exchange at high-energy scale E ; (ii) Eq. (5.1) via tadpole diagrams gure 2 of quark-lepton interactions (2.11) or (2.12). De ning bare fermion masses ; ; s; c), mass-gap equations (6.5), (6.7), (6.9) and (6.11) for the second fermion family can be approximately written as follow, mc0=Nc + U mt0=Nc + U s Rs ms0=Nc + U b Rb mb0=Nc L U L U + (4=Nc)M6mb0 (4=Nc)M4mt0 w jUscj2mc0 + jUstj2mt0 + ULsyU Rs ym0 + ULsyU Rs ym0 w jUscj2mc0 + jUstj2mt0 + M5mb0 (0:464 ! 0:713) and jU ` (0:441 ! 0:699) [115], and we use their central value for approximate calculations. The dominate contributions in the r.h.s. of these equations can be gured out. We obtain the approximate solution to eqs. (7.13) and (7.15), as well as the approximate solution to eqs. (7.14) and (7.16), which are given in the last step with cU Rc ) = U t Rt ) = (ULctU Rtc) (ULs yURsy)(ULbU Rb ) = (ULsbU Rbs); where eq. (3.22) is used. The dominate contributions in mass-gap equations (7.13){(7.16) to the fermion masses are: (i) the -neutrino and c-quark acquire their ESB masses m0 and mc0 from the topquark mass mt0 via the quark-lepton interactions (3.21) between the third and second families, i.e., M3 and M4; (ii) the s-quark acquires it ESB mass ms0 via the CKM mixing and the quark-lepton interactions M5; (iii) the -lepton acquires its ESB mass m0 via the PMNS mixing and the quark-lepton interactions M6. Running fermion masses and Yukawa couplings Analogously to the discussion for the third fermion family from eqs. (7.1){(7.4) to eqs. (7.7){ (7.9), neglecting the perturbative corrections from the SM gauge interactions, and de ning running fermion masses and Yukawa couplings m ( ) = g ( )v=p2; mc( ) = gc( )v=p2; m ( ) = g ( )v=p2; ms( ) = gs( )v=p2; and the mass-gap equations at the scale eqs. (7.13){(7.16). On the basis of eqs. (7.15) and (7.18) at the scale and the c-quark mass-shell connumerically obtain are obtained by replacing m0 1:2 GeV; for M4 = (ULctU Rtc) Using eqs. (7.13), (7.14), (7.16) and (7.18) at the scale , we calculate the light s-quark mass and -muon mass at the scale = 2 GeV, M3 = U M5 = ULsbU Rbs As a result, the Yukawa couplings gc( ) and g ( ) are shown in gure 5, the Yukawa coupling gs( ) and g ( ) are shown in gure 6. The variations of Yukawa couplings gc;s; ; ( ) are very small over the energy scale . 10 5 (7.20). Note that mc; ( ) = gc; ( )v=p2. In summary, the preliminary study (7.19){(7.22) shows that the pattern of fermion masses in the second family can be consistently obtained by the pattern (M3;4;5;6) of quark-lepton interactions and mixing between the third and second families. The scale -evolution of masses and Yukawa couplings are functions of the top-quark one gt( ), see The rst fermion family We turn to the masses and Yukawa couplings of the rst fermion family. The coupled SD gap-equations receive ESB contributions from the third and second families, through the CKM and PMNS mixing as well as quark-lepton interactions between fermion families. As a result, the fermion masses of the rst family are generated. Analogously to the calculations of the second family case, we neglect the perturbative contributions from gauge interactions and calculate the fermion masses at the scale = 2 GeV. Approximate mass-gap equations of the rst fermion family Analogously to eqs. (7.1){(7.4) and eqs. (7.13){(7.16) respectively for the third and second fermion family, Equations (6.5), (6.7), (6.9) and (6.11) for the rst fermion family read, +ULeuURu e m0u=Nc + ULecU Rc e mc0=Nc + ULetU Rt e mt0=Nc (11=2Nc)ULetU Rt e mt0 = (11=2Nc)M7mt0 +ULedURdem0d=Nc + ULesU Rsems0=Nc + ULebU Rbemb0=Nc (19=2Nc)ULutU Rtumt0 = (19=2Nc)M8mt0; +ULedyURdeyme0 + ULdyURd ym0 + ULdyURd ym0 4:3 KeV; for 2:2 MeV; for M7 = ULetU Rt e M8 = (ULutU Rtu) where the CKM matrix elements jUudj 8:4 10 3 [114], as well as the PMNS matrix elements jUe` e j (0:225 ! 0:517), (0:441 ! 0:699) [115]. The dominate contributions in the r.h.s. of these equations can be gured out. We obtain the approximate solution to eqs. (7.23) and (7.25), as well as the approximate solution to eqs. (7.24) and (7.26), which are given in the last step with The dominate contributions are: (i) the e-neutrino acquires its mass m0 from the t-quark mass mt0 via the quark-lepton interaction M7; (ii) the u-quark acquires its mass m0u from the t-quark mass mt0 via the quark-lepton interaction M8; (iii) the e-lepton acquires its mass me0 from the neutrino masses m0e , m0 and m0 via the PMNS mixing, which implies the approximate relation of light lepton masses and PMNS mixing angles; (iv) the d-quark dominantly acquires its mass m0d from quark masses mu; mc and mt via the CKM mixing, as well as a small contribution from the quark-lepton interaction M9, 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. (7.1){(7.4) to eqs. (7.7){ (7.9), neglecting the perturbative corrections from the SM gauge interactions, and de ning running fermion masses and Yukawa couplings m e ( ) = g e ( )v=p2; mu( ) = gu( )v=p2; me( ) = ge( )v=p2; md( ) = gd( )v=p2; and the gap-equations at the scale are obtained by replacing mf0 ! mf ( ) in eqs. (7.23){ (7.26). On the basis of eqs. (7.23), (7.25) and (7.28) at the scale , we numerically calculate the e, e, u- and d-quark masses at = 2 GeV 4:1 MeV; for M9 = (ULbdURdb) and the Yukawa couplings gu( ) and gd( ), see gure 7, and ge( ) and g e ( ), see gure 8. The variations of Yukawa couplings gu;d;e; e ( ) are very small over the energy scale . Summary and discussion originated from the SSB, inevitably introduce the inhomogeneous (ESB) terms into the SD equations for other fermion masses via the fermion-family mixing due to the quarklepton interactions and the W -boson vector-like vertex (CKM and PMNS mixing) at high energies. As a consequence, this leads to the generations of other fermion masses by the top-quark Yukawa coupling gt( ), gure 1. We approximately analyze the coupled SD gap-equations for the fermion masses and Yukawa couplings of the third, second and the rst 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 nd their masses, consistently with the fermion-family mixing parameters Mi. We have checked that the contributions from perturbative gauge interactions are negligible, compared with the essential contributions due to the fermionfamily mixing. As qualitative and preliminary results, without any drastic ne-tuning we which seems to be consistent with the SM. The top-quark mass is generated by the SSB, and others by the ESB attributed to the top-quark mass and family-mixing. All masses are calculated approximately obtain the hierarchy pattern of 12 SM-fermion masses, see table 1, and their Yukawa couplings, consistently with the parameter w (6.3) and the hierarchy pattern of 9 family-mixing parameters Mi. It is energetically favorable that the SSB solely occurs for the tt-channel (1.1) generating the top-quark mass and three Goldstone modes only. The SSB realizes the approximate However, this SSB generated vacuum alignment is re-arranged to the real ground states, where the real hierarchy pattern (table 1) is realized. Such rearrangement is due to the nontrivial ESB terms in the SD gap-equations for fermion masses, so that fermions become massive mt and fermion-family mixing matrices in the two ways: (i) the fermion-family-mixing matrices (3.8) and (3.17) including the CKM and PMNS matrices introduce the ESB terms, due to the vector-like coupling w (5.13) and (6.3) of the W -boson at high energies E (see preliminary study [32, 33]); (ii) the quark-lepton-family mixing matrices (3.22) introduce the ESB terms, due to the quark-lepton interactions (3.21) at high energies E . It is expected that the ESB terms perturbatively re-arrange the SSB generated vacuum alignment, because of the small coupling w and fermion-family-mixing matrix elements. The table 1 shows that the following relations between (i) neutrino Dirac masses and charged (2/3) quark masses; (ii) charged lepton and charged (-1/3) quark masses; 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- avor families and four families of fermions with di erent electric charge. 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 fermion masses (Yukawa couplings) is obtained consistently with the hierarchy pattern of fermion-family mixingmatrix elements. It should be mentioned that both of them are equally the basic parameters of the Nature, and they are closely related each other by the symmetries and/or dynamics of the fundamental theory. Some relations between them are given in this article, however more fundamental relations are expected in the framework of uni cation theories, e.g., SO(10)-theory. For the light quarks and leptons, they acquire their masses dominantly from the ESB terms of the W -boson coupling w-terms associating with either CKM or PMNS matrix. This implies that there are the approximate relations of light quark/lepton masses and CKM/PMNS mixing angles, as intensively studied in literature. 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 (3.8), (3.17) and (3.22), which are unknown except the well (poor) known CKM (PMNS) matrix. These mixing matrices have to be understood in a UV-fundamental theory symmetrically unifying not only gauge interactions but also three fermion families. In this sense, fermion mixing matrices are even more fundamental than fermion masses. Their values, mixing matrix elements and fermion masses in unit of the top-quark mass, are related and determined upon the chiral-symmetry-breaking ground state of the UV-fundamental theory. The presented results only show that the known hierarchical masses (100 10 8) of 12 SM Dirac fermions are related to the hierarchical pattern of 9 fermion-family mixing parameters Mi 10 7) of eqs. (7.5), (7.17) and (7.27). Since we have not understood the hierarchical mixing-matrix pattern of the UV-fundamental theory, the hierarchical fermion masses are not ultimately explained. It should be also emphasized that the presented results 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 di cult 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 In the next section, we will relabel neutrino Dirac mass m by mD, discuss three heavy in terms of their Dirac masses mD and Majorana masses mM. sterile Majorana neutrinos ( Rf + Rfc) and three light gauged Majorana neutrinos ( Lf + Lfc) Neutrino sector On the basis of Dirac neutrino mass eigenstates and masses calculated (see table 1) in previous sections, as well as some experimental results of neutrino oscillations, we calculate the mass-spectra of gauged and sterile neutrinos by taking into account the Majorana masses generated by the spontaneous symmetry breaking of the global Ulepton(1) symmetry for the lepton-number conservation. Spontaneous symmetry breaking of Ulepton(1) symmetry In the four-fermion operators (3.9) of the lepton sector, the last term reads where the conjugate elds of sterile Wely neutrinos Rf are given by Rfc = i 2( Rf) . This four-fermion operator preserves the global Ulepton(1)-symmetry for the lepton-number conservation. Similarly to the discussions of the SSB mechanism for the generation of top-quark mass in section 4, the four-fermion operator (8.1) can generate a mass term of Majorana type, since the family index \f " is summed over as the color index \a" and the family 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 Ulepton(1)-symmetry of the Lagrangian (8.1). On the basis of the mass eigenstates, the spontaneous symmetry breaking of the Ulepton(1)-symmetry generates the masses of Majorana type mM = mfM = Gh Rfc Rf i; M = HM = f=1;2;3 f 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 mM and sterile scalar particle mass mHM satisfy the mass-shell conditions, mM = gsterile(mM )vsterile=p2; (mM )2=2 = ~sterile(mHM )vs2terile; H where gsterile( 2) and ~sterile( 2) obey the same RG equations (absence of gauge interactions) of eqs. (4.3), (4.4) and (4.5), as well as the boundary conditions (8.5). However, we cannot determine the solutions gsterile( 2) and ~sterile( 2), since the energy scale vsterile of boundary conditions (8.5) are unknown. The electroweak scale v is determined by the gauge-boson masses MW and MZ experimentally measured, the scale vsterile needs to be fact, the scale vsterile represents the energy scale of the lepton-number violation. 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 Ulepton(1)-symmetry result in the following bilinear Dirac and Majorana in terms of neutrino mass eigenstates Lf and Rf in the f -th fermion family, see eqs. (3.14) and (3.15). Following the usual approach [112, 113], diagonalizing the 2 2 mixing matrix (8.6) in terms of the neutrino and sterile neutrino mass eigenstates of the family Mfg = Mfs = (mfM )2 + (mfD)2 1=2 o mfM + (mfM )2 + (mfD)2 1=2 o This corresponds to two mass eigenstates: three light gauged Majorana neutrinos (four gf = Egf = [p2 + (Mfg)2]1=2; (mfD)2=4mfM ; and three heavy sterile Majorana neutrinos (four components) sf = Esf = [p2 + (Mfs)2]1=2; where p stands for neutrino momentum, corresponding velocity vp. The mixing angles between gauged and sterile Majorana neutrinos are 2 f = tan 1(mfD=mfM ) (mfD=mfM ) The previously obtained Dirac masses mf mfD have the structure of hierarchy (see neutrino masses M1g < M2g < M3g (8.9), i.e., table 1). The discussions after eq. (3.19) show that the Majorana masses mfM are expected to have a hierarchy structure relating to the one of Dirac masses mfD.3 This indicates the normal hierarchy structure of neutrino mass spectrum: Dirac neutrino masses m1D < m2D < m3D, sterile Majorana neutrino masses m1M < m2M < m3M (8.10) and gauged Majorana (m1D)2=4m1M < (m2D)2=4m2M < (m3D)2=4m3M : 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 mM mfD, i.e., the energy scale vsterile of the lepton-number violation is much larger than the electroweak scale v. Flavor oscillations of gauged Majorana neutrinos We rst discuss the family- avor oscillations of three light gauged Majorana neutrinos (8.9) 3In ref. [107], we assume that the Majorana masses mfM are approximately equal (degenerate) miM because of the non-trivial chiral transformation UR in eqs. (3.14) and (3.15). sterile neutrino mass gauged neutrino mass neutrino masses Mfs mfM and gauged Majorana neutrino masses Mfg (mfD)2=4mfM . Here we 1; 2; 3), which are calculated by using eq. (8.7) where the second line (8.17) may be used for the case of cosmic neutrino background of temperature O(10 4) eV. These oscillations between the family avors of gauged Majorana neutrinos have been important for experiments performed in ground and underground laboratories. eq. (8.14), neutrino Dirac masses mfD (table 1) and experimental values [114]: 10 3 eV, see table 2. 10 8, and the Majorana masses m3M 108 GeV and m2M 105 GeV. As a result, two sterile Majorana neutrino masses (8.10) M3s 105 GeV, two gauged Majorana neutrino masses (8.9) M3g 108 GeV and M2s ! g (t) = X(UL` ) f (UL` ) f (UL` ) f0 (UL` ) f0 exp[ i(Egf The large oscillating lengths of relativistic and non-relativistic gauged neutrinos are Among the three neutrino mass-squared di erences (8.14), only two of them are in M3g22 = M3g12. In principle we cannot determine the Majorana masses m2M and m1M (m2M =m1M ) (m3M =m2M ) on the basis of the reasons we discussed in the paragraph of eq. (3.19). This inference (8.19) leads to the ratio m1D=m1M 10 8 and the lowest lying Majorana neutrino mass (m1D)2=4m1M Thus we tabulate the values (8.19) and (8.20) in the rst column of table 2. These results satisfy the recent cosmological constrain [126] on the total mass of three light gauged (mfD)2=4mfM Needless to say, it is important that the sensitivity of experiments and observations on neutrino masses can be reached at least to the level O(10 2) eV. Flavor oscillations of sterile Majorana neutrinos We turn to discuss the family- avor oscillations of three heavy sterile Majorana neutriare calculated by using eq. (8.8) Of[(miD)2=4miM ]2g and the de nitions are the sterile avor s to the sterile avor s reads mfDf0 , see table 2. The oscillating probability from P s ! s (t) = X(U R`) f (U R`) f (U R`) f0 (U R`) f0 exp[ i(Esf The oscillating lengths of non-relativistic and relativistic sterile neutrinos are given by Table 2 shows the large mass and mass-squared di erences (8.22) and (8.23), therefore in addition to their sterility the oscillating lengths between the avors of sterile Majorana neutrinos are too small to be relevant for experiments in ground and underground laboratories. However these oscillations could be relevant in early universe evolution, depending on the Majorana masses mM or the energy scale vsterile of the lepton-number violation. Oscillations between gauged and sterile Majorana neutrinos Following eqs. (8.7){(8.11), the oscillating probability between two mass eigenstates of gauged Majorana neutrino gf and sterile Majorana neutrino sf reads g ! sf (t) = 1 where f = 1; 2; 3 and (mfM )2=(2p); (mfD)2=4mfM (mfD)2=4mfM for non-relativstic and relativistic cases. The oscillating lengths read Lsfg = Lsfg = The large values of Majorana mass mfM and mass-squared (mfM )2 [see table 2] show the small oscillating lengths. The small mixing angle (8.11) indicates the small oscillating probabilities (8.28) between gauged and sterile Majorana neutrinos. The oscillating probability between the sterile avor s and the gauged avor g reads P s ! g (t) = X(U R`) f (UL` ) f (U R`) f0 (UL` ) f0 exp[ i(Esf Apart from mixing matrices, via the oscillatory factor exp[ i(Esf Egf0 )t] the oscillation probability depends on the sum over the mass di erences mf2f0 of mass-eigenstates (f 6= f 0) of two mff0 or mass-squared di erences . Given neutrino energies, and their masses or mass-squared di erences, one can select an oscillating length Lff0 that is relevant for a possible observation or e ect. The mass spectra (table 2) of gauged and sterile neutrinos show a large di erence of their mass scales, indicating oscillations between them at short distances. For the example M1s and M1g cases, the oscillating length is at least 10 1 GeV 1(vp=c) for p 102 GeV or 10 1 GeV 1(p=102 GeV) 102 GeV, as shown in eqs. (8.31) and (8.32). The latter implies the possibility (8.33) for very high-energy electron neutrinos converting themselves into sterile neutrinos. It seems to be hard to detect the oscillations between gauged and sterile Majorana neutrinos in experiments performed in space, ground and underground laboratories. However these oscillations could be important in early universe evolution, depending on the energy scale vsterile or mM of the lepton-number violation. Actually, the probabilities of three avor oscillations (8.15), (8.25) and (8.33) are described by the following 6 6 mixing matrix, see eq. (3.17) ` between URy and UL. The maxing matrix (8.34) is unitary, if '2 where ei'1 is a relative phase between ULy and U R`, and ei'2 is another relative phase '1 = n ; n = 1; 2; 3; . The diagonal parts UL` (PMNS) and U R` respectively represent the mixing matrices for the avor oscillations (8.15) and sterile avor oscillations (8.25), and the o -diagonal parts represent the mixing matrices for the gauged-sterile avor oscillations (8.33). 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 1 and 2 qualitatively obtained in this article. SM fermion Dirac masses and Yukawa couplings Due to the ground-state (vacuum) alignment of the e ective 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. [97], 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 quarkavor mixing matrices (3.8), the lepton- avor mixing matrices (3.17) [or (8.34)], where the ing to the coupling vertex of W -bosons. Also we do not discuss the quark-lepton mixing matrices (3.22) relates to the quark-lepton interactions. These unitary matrices are composed by the eigenvectors corresponding to eigenvalues (fermion-mass spectra) of fermion-mass matrices. They code all information about mixing angles and CP-violations. Relating to the slowly varying Yukawa coupling gt( ) of the top quark, see section 4.2.2, all fermion Yukawa couplings obtained slowly vary from 1 GeV to 13:5 TeV. These features imply that it should be hard to have any detectable nonresonant signatures in the LHC ppcollisions, showing the deviations from the SM with the elementary Higgs boson. All these results are preliminarily qualitative, and they should receive the high-order corrections and some non-perturbative contributions. It should be emphasized that these qualitative results cannot be quantitatively compared with the SM precision tests. The quantitative study is a di cult and challenging task and one will probably be able to carry on it by using a numerical approach in future. Nevertheless, these qualitative results may give us some insight into the long-standing problem of fermion-mass origin and hierarchy. Neutrinos and dark-matter particles The values of three light gauged Majorana neutrino masses Mfg give some insight into the neutrino problems that directly relate to the absolute values of neutrino masses. The -decay rate depends on m2 = P -decay rate depends on 1 is the CP eigenvalue of the mass eigenstate fg (8.9). The M3g and M2g values, as well as M1g range in table 2 seem to be in agreement with the analysis of using experimental data of mass-squared di erences (8.18) and the PMNS mixing matrix UL` in the normal hierarchy case (see for example ref. [128]). In addition to the measurements of neutrino mixing angles, it is obviously important to experimentally measure neutrino masses with a sensitivity below 10 2 eV so as to determine the neutrino features. The very massive sterile neutrinos (8.10) of Majorana type, whose masses M1s 105GeV and M3s 108GeV (see table 2), could be candidates for very massive cold dark-matter (DM) particles. While, the right-handed sterile neutrinos Dirac type, whose Dirac masses M1D (see table 2), could be considered as light, weak-interacting \warm" DM particles, in particular the one R1 with a few KeV mass. Moreover, the sterile composite scalar particle (8.4) could be probably a candidate for a massive cold DM particle, though we do not know its mass mHM (8.5), i.e., the scale of lepton-number non-conservation. What is then the candidate for light, non-interacting warm DM particle? We expect that it should be the pseudoscalar boson M (8.3), which acquires a small mass m by the analogy of the PCAC (partially conserved axial-vector current) and soft pion theorems, = f m2 M ; 5 R mfM ) in eq. (8.6). The f is the pseudoscalar boson M decay constant relating to the processes soft explicit breaking scale m~sf of Ulepton(1)-symmetry. It is worthwhile to notice that both sterile Majorana neutrinos (the candidates of cold DM particles) and the sterile pseudoscalar boson (the candidate of warm DM particle) carry two units of lepton number. This implies that the relevant processes of these sterile particles interacting with the SM particles, though very weak, should violate the lepton-number conservation and lead to the asymmetry of matter and anti-matter. At the end we mention that for strong coupling G the relevant four-fermion operators (2.12) and (3.9) present the interactions of DM and SM particles, and form gauged and neutral composite particles as resonances of masses at TeV scale, then these composite particles (resonances) decay into their constitutes | SM and/or DM particles [89, 94, 97, 107, 130]. . Both mass m and decay constant f depend on the Acknowledgments The author thanks Prof. Hagen Kleinert for discussions on the IR- and UV-stable xed points of quantum eld theories, Prof. Remo Ru ni for discussions on the Einstein gravitational theory and Prof. Zhiqing Zhang for discussions on the LHC physics. The author also thanks the anonymous referee for his/her e ort of reviewing this lengthy article. Open Access. 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She-Sheng Xue. Hierarchy spectrum of SM fermions: from top quark to electron neutrino, Journal of High Energy Physics, 2016, 72, DOI: 10.1007/JHEP11(2016)072